Homotopy theory of higher categories

Homotop y Theory of Higher Cate gories Carlos T. Simpson CNRS—INSMI Lab oratoir e J. A. Dieudo nn´ e Univ ersit´ e de Nice-Sophia Antipolis carlos @unic e.fr Ce papier a b´ en ´ eﬁci´ e d’une aide de l’Ag e nce Nationale de la Recherc he po rtant la r´ ef ´ erence ANR-09-BLAN-0151 - 02 (HOD AG ). This is draft material from a forthcoming b o ok to b e published by Cambridge Universit y Pr ess in the N ew Mathematical Mo nographs se- ries. This publication is in copyrigh t. c  Carlos T. Simpson 2010. v Abstract This is the ﬁrst draft of a b o ok ab out higher categories appro ached by iterating Segal’s method, a s in T amsamani’s deﬁnition o f n -ner ve and Pelissier’s thesis. If M is a tractable left pro per car tesian mo del category , we construct a tr actable left pro per ca rtesian mo del structure on the ca tegory o f M -precatego ries. The pro cedure c a n then be iterated, leading to mo de l categories o f ( ∞ , n )-categor ies. Con ten ts Pr efa c e p a ge 1 A cknow le dg ements 7 P AR T I HIGHER CA TEGORIES 9 1 History and mo ti v ation 11 2 Strict n -categorie s 29 2.1 Go dement , Interc hange o r the Eckmann-Hilton argument 31 2.2 Strict n -gro upo ids 33 3 F undamental eleme n ts of n - categori e s 43 3.1 A globular theory 43 3.2 Ident ities 46 3.3 Comp osition, equiv alence a nd trunca tion 46 3.4 Enriched catego ries 49 3.5 The ( n + 1)-catego ry of n -categor ies 50 3.6 Poincar´ e n -groupo ids 52 3.7 Int erior s 53 3.8 The case n = ∞ 54 4 The nee d for w eak com p ositi o n 56 4.1 Realization functors 57 4.2 n -group oids with one ob ject 59 4.3 The case of the standard realizatio n 60 4.4 Nonexistence of strict 3-g roup oids giving ris e to the 3-type of S 2 61 5 Simplici al approac hes 69 Contents vii 5.1 Strict simplicial categ ories 69 5.2 Segal’s delo oping machine 71 5.3 Segal categor ies 74 5.4 Rezk catego r ies 78 5.5 Quasicateg ories 81 5.6 Going betw een Segal categ ories and n -c a tegories 83 5.7 T ow ards weak ∞ -catego ries 85 6 Op eradic approac hes 87 6.1 May’s delo oping machine 87 6.2 Baez-Dolan’s deﬁnition 88 6.3 Batanin’s deﬁnition 91 6.4 T rimble’s deﬁnition and Cheng’s compar ison 95 6.5 W e ak units 97 6.6 Other notions 100 7 W eak enric hment o v er a cartesian m o del category: an introductio n 103 7.1 Simplicial ob jects in M 103 7.2 Diagrams ov er ∆ X 104 7.3 Hypo theses on M 105 7.4 Precateg ories 1 06 7.5 Unitalit y 107 7.6 Rectiﬁcation of ∆ X -diagra ms 109 7.7 Enforcing the Segal condition 110 7.8 Pro ducts, interv als and the mo del structure 112 P AR T I I CA TEGOR ICAL PRELIMINARIES 115 8 Some category theory 117 8.1 Lo cally presentable categ ories 119 8.2 Monadic pro jection 123 8.3 Miscellany ab out limits and colimits 128 8.4 Diagram catego ries 129 8.5 Enriched catego ries 130 8.6 Int ernal H om 138 8.7 Cell complexes 139 8.8 The small ob ject argument 155 8.9 Injective coﬁbra tions in diag r am categ ories 157 9 Mo del categorie s 163 9.1 Quillen mo del catego ries 163 viii Content s 9.2 Coﬁbrantly gener ated mo del categ ories 166 9.3 Combinatorial and trac table mo del categ o ries 167 9.4 Homotopy liftings and extensions 168 9.5 Left prop erne s s 171 9.6 Quillen adjunctions 175 9.7 The Kan-Quillen mo del categor y of simplicial sets 175 9.8 Mo del structures on diagra m categor ies 176 9.9 Pseudo-g e nerating sets 179 10 Cartesian m o del categories 192 10.1 Int ernal H om 195 10.2 The enriched categor y asso ciated to a cartes ian mo del categor y 197 11 Direct l e ft Bous ﬁeld lo cali zation 198 11.1 Pro jection to a subca tegory of lo cal ob jects 198 11.2 W e ak monadic pro jection 205 11.3 New weak equiv alences 210 11.4 Inv ariance prop erties 213 11.5 New ﬁbrations 218 11.6 Pushouts by new trivial coﬁbra tio ns 220 11.7 The mo del categor y structure 221 11.8 T ransfer along a left Quillen functor 22 4 P AR T I I I GENERA TORS AND R ELA TIONS 227 12 Precategories 229 12.1 Enriched preca teg ories with a ﬁxed set of ob jects 230 12.2 The Segal conditions 231 12.3 V a rying the set of ob jects 232 12.4 The categor y of precatego ries 234 12.5 Basic examples 235 12.6 Limits, colimits and lo cal presentabilit y 237 12.7 Int erpretatio ns a s pr esheaf ca tegories 243 13 Algebraic theories in mo del categories 253 13.1 Diagrams ov er the catego ries ǫ ( n ) 254 13.2 Impo sing the pro duct condition 259 13.3 Algebraic diagra m theories 266 13.4 Unitalit y 268 13.5 Unital algebra ic diagra m theories 274 13.6 The generatio n op eration 275 Contents ix 13.7 Reedy structures 276 14 W eak equiv alences 277 14.1 The mo del structures on PC ( X , M ) 278 14.2 Unitalization adjunctions 281 14.3 The Reedy structure 283 14.4 Some remark s 289 14.5 Global weak equiv alences 291 14.6 Categorie s enriched ov er ho( M ) 294 14.7 Change of enrichmen t categ ory 296 15 Coﬁbrations 300 15.1 Skeleta and coskeleta 300 15.2 Some natural precateg ories 305 15.3 Pro jective coﬁbrations 308 15.4 Injective coﬁbra tions 310 15.5 A pushout expressio n for the skeleta 312 15.6 Reedy coﬁbratio ns 313 15.7 Relationship b etw een the clas s es of coﬁbr ations 326 16 Calculus of generators and rel ations 329 16.1 The Υ precategor ies 329 16.2 Some trivial coﬁbra tions 331 16.3 Pushout by isotriv ial coﬁbr a tions 335 16.4 An element ary gener a tion step Gen 343 16.5 Fixing the ﬁbrant condition lo cally 347 16.6 Combining genera tio n steps 347 16.7 F unctoriality of the genera tion pro ces s 348 16.8 Example: generato rs and rela tio ns for 1-ca tegories 350 17 Generators and relations for Seg al categorie s 353 17.1 Segal categor ies 354 17.2 The Poincar´ e-Segal g roup oid 355 17.3 The calculus 357 17.4 Computing the lo op space 370 17.5 Example: π 3 ( S 2 ) 378 P AR T IV THE MO DE L STR UCTURE 383 18 Sequen tially free precategorie s 385 18.1 Impo sing the Segal condition on Υ 385 18.2 Sequentially free preca tegories in g eneral 386 x Contents 19 Pro ducts 396 19.1 Pro ducts of sequentially free preca tegories 396 19.2 Pro ducts of genera l precateg ories 406 19.3 The r ole of unitality , deg eneracies and higher coherences 413 19.4 Wh y we can’t truncate ∆ 416 20 In terv als 418 20.1 Contractible ob jects and interv als in M 419 20.2 Int erv als for M -enriched pr ecategor ies 421 20.3 The versalit y pro pe r ty 427 20.4 Contractibilit y o f interv als for K -precategories 429 20.5 Construction of a left Quillen functor K → M 431 20.6 Contractibilit y in g eneral 432 20.7 Pushout of trivial coﬁbratio ns 434 20.8 A versalit y prop er t y 43 9 21 The m o del category of M -enri c hed precategories 441 21.1 A standard factoriza tio n 4 41 21.2 The mo del structures 442 21.3 The cartesia n prop erty 445 21.4 Prop erties of ﬁbrant ob jects 446 21.5 The mo del categor y of strict M -enriched categor ies 447 22 Iterated higher categories 448 22.1 Initialization 449 22.2 Notations 449 22.3 The case of n -nerves 450 22.4 T runcation and equiv alences 452 22.5 The ( n + 1)-catego ry nC AT 454 R efer enc es 456 Preface The theory of n -categor ies is currently under active considera tion by a nu mber of diﬀerent resear ch groups around the world. The history o f the sub ject go es back a long wa y , on separa te but interrelated tracks in algebra ic top olog y , algebr aic geo metry , a nd categor y theory . F or a long time, the crucial deﬁnition of we akly asso ciative higher c ate go ry remained elusive, but now on the con trary w e ha ve a plethor a o f diﬀerent po ssibilities av ailable. One o f the next ma jor problems in the sub ject will be to achiev e a global compar is on betw een these diﬀer en t approa ches. Some work is starting to come out in this direction, but in the cur rent state of the theor y the v arious diﬀerent appr oaches remain distinct. After the compariso n is achiev ed, they will b e seen as r epresenting diﬀerent facets of the theory , so it is imp ortant to contin ue working in all of these diﬀerent directions. The pur po se of the presen t b o o k is to c o ncentrate on o ne o f the meth- o ds of deﬁning and working with higher catego ries, very closely based on the work of Gra eme Sega l in a lgebraic top ology many years ear lier. The notion o f “ Segal categor y”, which is a kind of categor y weakly en- riched over simplicial sets, was c o nsidered b y V ogt and Dwy er, Kan and Smith. The a pplication of this metho d to n -categor ies was introduced by Zo uhair T amsamani. And then put into a s trictly iterativ e form, with a general model catego ry as input, by Regis Pelissier following a sugges - tion o f Andr ´ e Hirschowitz. Our treatmen t w ill integrate imp or tant ideas contributed by Julie Berg ner, Clar k Bar w ick, Jac ob Lur ie a nd o thers. The guiding principle is to use the ca tegory of simplices ∆ as the basis for a ll the higher coher ency conditions which co me in when we This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 2 Pr efa c e allow w eak asso cia tivit y . The ob jects of ∆ are no nempt y ﬁnite ordinals [0] = { υ 0 } [1] = { υ 0 , υ 1 } [2] = { υ 0 , υ 1 , υ 2 } . . . whereas the morphisms are nondecreasing maps b etw een them. Kan had a lready introduced this categor y into a lg ebraic to po logy , cons id- ering simplicial sets which are functor s ∆ o → Set . T he s e mo del the homotopy t yp es of CW-complexes. One o f the big pro blems in a lgebraic to po logy in the 1 9 60’s was to deﬁne notions of delo opi ng machines . Sega l’s wa y was to consider sim- plicial s p ac es , or functor s A : ∆ o → Top , such that the ﬁrs t space is just a p o int A 0 = A ( [0]) = ∗ . In ∆ there are three nonconstant maps f 01 , f 12 , f 02 : [1 ] → [2] where f ij denotes the map sending υ 0 to υ i and υ 1 to υ j . In a simplicial space A which is a contra v arian t functor on ∆, we g e t three ma ps f ∗ 01 , f ∗ 12 , f ∗ 02 : A 2 → A 1 . Organize the ﬁrst tw o as a map into a pro duct, g iv ing a diagr a m of the form A 2 σ 2 → A 1 × A 1 A 1 f ∗ 02 ↓ . If we require the se c ond Se gal map σ 2 := ( f ∗ 01 , f ∗ 12 ) to be an iso morphism betw een A 2 and A 1 × A 1 , then f ∗ 02 gives a product on the space A 1 . The basic idea of Segal’s delooping machine is that if we only req uire σ 2 to be a weak ho mo topy equiv a lence of spaces, then f ∗ 02 gives what should b e considered as a “pro duct deﬁned up to homotopy”. One sa lient asp ect of this appro a ch is tha t no map A 1 × A 1 → A 1 is s pec iﬁe d, and indeed if the space s in volv ed hav e bad pr o p e r ties there might exist no s ection of σ 2 at all. The term “delo o ping machine” refer s to any o f several k inds of further mathematical structure on the lo op spa ce Ω X , enhancing t he basic com- po sition of loops up to ho motopy , whic h should allo w one to reconstr uct a space X up to ho motopy . In tandem with Se g al’s machine in which Pr efa c e 3 Ω X = A 1 , the no tion of op er ad intro duced by Peter May underlies the bes t known and most studied family of deloo ping machines. There were also other techniques such as PROP’s which are s ta rting to receive re- newed in terest. The v arious kinds of delo oping ma chin es a re sources for the v arious diﬀeren t approaches to hig her categor ies, after a m ultiplying eﬀect whereb y eac h delooping technique leads to several diﬀerent deﬁni- tions of higher categories . Our technical work in later par ts of the b o ok will concen trate o n the par ticular direction of iterating Segal’s approach while maintaining a discrete set of ob jects, but we will survey some o f the many o ther approa ches in later chapters of Part I. The re la tionship b etw een ca tegories and simplicial ob jects w as noticed early on with the nerve c onstruction . Given a categor y C its nerve is the simplicial set N C : ∆ o → Set such that ( N C ) m is the set of compo sable sequences o f m a rrows x 0 g 1 → x 1 g 2 → · · · x m − 1 g m → x m in C . The o p e rations of functoriality for maps [ m ] → [ p ] are obta ined using the comp osition law and the identities of C . The ﬁrst piece is just the set ( N C ) 0 = Ob( C ) o f ob jects of C , and the nerve satisﬁes a relative version of the Segal condition: σ m : ( N C ) m ∼ = − → ( N C ) 1 × ( N C ) 0 ( N C ) 1 × ( N C ) 0 · · · × ( N C ) 0 ( N C ) 1 . Conv ersely , any simplicial set ∆ o → Set satisfying these conditions comes fro m a unique ca tegory and these constructions ar e inv erses. I n other words, categories may b e considered as simplicia l sets sa tisfying the Seg a l co nditions. In comparing this with Segal’s situation, reca ll that he required A 0 = ∗ , whic h is like lo oking at a category with a single ob ject. An o b vious wa y of putting a ll o f these things together is to co nsider simplicial spa ces A : ∆ o → Top such that A 0 is a disc r ete set—to b e thought of a s the “set of ob jects”—but considered as a spac e , and such that the Seg al maps σ m : A m → A 1 × A 0 A 1 × A 0 · · · × A 0 A 1 are w eak homotopy equiv alences for all m ≥ 2. F unctors of this kind are Se gal c ate gories . W e use the same termino logy when Top is replace d by the ca tegory o f simplicia l sets K := Set ∆ o = Func (∆ o , Set ). The po ssibility of making this genera lization was clearly evident at the time of Segal’s pap ers [185] [187], but was made explicit only la ter by V ogt [184] and Dwyer, K an and Smith [92]. 4 Pr efa c e Segal catego ries provide a goo d way o f considering categories enriched ov er s paces. Ho wev er, a more elemen tary approach is av ailable, by look- ing at ca tegories st rictly enriched ov er spa ces, i.e. simplicia l c a tegories. A simplicial categor y could be v iewed as a Segal catego ry wher e the Segal maps σ m are isomorphis ms . More classically it can b e considered as a categor y enriched in Top or K , using the deﬁnitions of enriched category theory . In a s implicial categor y , the pro duct op era tion is well- deﬁned a nd strictly asso ciative. Dwyer, Kan and Smith show ed that we do n’t lose any genera lit y at this level b y r equiring strict a sso ciativity: ev ery Seg al categ ory is equiv alen t to a simplicial catego r y [92]. Unfortunately , we cannot just itera te the construction by co ntin uing to lo ok at c a tegories strictly enriched over the category of simplicial categories and s o forth. Such an iteration le ads to higher ca tegories with strict asso ciativity and strict units in the middle levels. One w ay of seeing why this isn’t go o d enough 1 is t o note that the Bergner mo del structure on s tr ict simplicial categor ies, is not car tesian: pro ducts of coﬁbrant o b jects are no longe r coﬁbrant. This sugg ests the need for a diﬀerent construction whic h pr eserves the ca rtesian condition, and Se g al’s metho d works. The iteration then says: a Seg a l ( n + 1)-ca tegory is a functor from ∆ o to the categor y o f Segal n -categor ies, who se ﬁrst element is a discrete set, and such that the Sega l maps are equiv a lences. The notion o f equiv alence needs to b e deﬁned in the inductive pro cess [20 6]. This iterative p oint of v ie w towards higher categor ie s is the topic of our bo ok. W e emphasize an algebraic appro ach within the world of homotopy theory , using Q uillen’s homoto pical algebra [175], but also paying pa r- ticular a tten tion to the pro cess of crea ting a higher category f rom gener- ators and relations. F or me this go es back to Massey’s b o ok [1 63] whic h was one of my ﬁrst references b o th for for algebraic top olog y , and for the notio n of a group presented by generato r s and relatio ns . One of the ma in inspira tions fo r the recent interest in higher categor ie s came from Gr othendieck’s manuscript Pursuing s tacks . He set out a wide vision o f the p o ssible developmen ts and applications o f the theor y of n - categorie s g oing up to n = ω . Ma n y o f his remarks contin ue to provide impo rtant resear ch dir ections, and many others remain un touched. The o ther main so urce of interest stems from the Baez-Dolan c onje c- tur es . These extend, to higher ca teg ories in a ll degrees, the relatio nships 1 Pa oli has shown [ 170] that n - groupoids can be semistrictiﬁe d in an y single degree, ho w ev er one cannot get st rictness in many diﬀerent degrees as will b e seen in Chapter 4. Pr efa c e 5 explored by ma n y resea rchers b etw een v arious categor ical structures and phenomena of knot inv ar iants and quantum ﬁeld theo r y . Hopkins a nd Lurie ha v e recently prov en a ma jor part of t hese conjectures. These mo- tiv a tio ns incite us to search for a g o o d unders tanding of the alg e bra of higher categor ies, and I hop e that the present bo o k c an contribute in a small w ay . The mathematical dis cussion o f the conten ts of the main part of the bo o k will b e c o nt inued in more detail in Chapter 7 at the end of Part I. The in tervening chapters o f Part I serve to in tro duce the problem by g iving some motiv ation for why higher categor ies a re needed, by explaining why str ict n -categ o ries a ren’t enough, and by considering some of the many other a pproaches which ar e cur r ently b eing developed. In Part I I we collect our main to ols from the theory of catego ries, including lo cally pr esentable categorie s and closed mo del categ ories. A small num ber of these items, such as the discussion of e nr iched cate- gories, could be useful for r eading Part I. The last chapter of Part I I concerns “direct left Bous ﬁeld lo caliza tion”, which is a sp ecial case o f left Bousﬁeld lo caliza tion in whic h the mo del structure can b e describ ed more explicitly . This Cha pter 11, tog ether with the general discussio n of c e ll complexes and the small ob ject argument in Chapters 8 and 9, are intended to provide some “black b oxes” whic h can then b e used in the rest of the bo o k without having to go into details of ca rdinality ar- guments and the like. It is hop ed that this will make a go o d part of the bo o k accessible to readers wishing to avoid to o many technicalities of the theory of mo del categorie s, a lthough so me familiarity is obviously necessary since our main goal is to construct a car tesian mo del structure. In Part I I I sta r ts the main work of lo oking a t weakly M -enriched (pre)categor ies. This part is en titled “Gener ators a nd relations” b ecause the pro ces s o f starting with a n M -precatego r y and passing to the a s- so ciated weakly M -enr iched category by enforcing the Sega l condition, should b e viewed a s a higher o r weakly enriched analogue of the c la ssi- cal pro c e ss of describing an algebraic ob ject by g enerator s and relations. W e develop several a sp e cts of this p oint of view, including a detailed discussion of the example o f catego ries weakly enriched over the mo del category K of simplicial sets. W e see in this case how to follo w alo ng the calculus o f gener ators and relations, taking as exa mple the calculatio n of the lo o p s pa ce o f S 2 . Part IV contains the cons truction of the cartesian mo del category , 6 Pr efa c e after the t w o steps treating s p eciﬁc elemen ts of our categorical situation: pro ducts, including the pro of of the c a rtesian condition, and in terv als. Part V, no t yet pres en t in the c urrent version, will disc uss v arious directions g oing tow ards basic techniques in the theory of higher ca te- gories, using the formalism developed in Parts II- IV. The ﬁrs t few c hapters o f Part V should contain discussio ns of inv ersion of morphisms, limits and colimits, and adjunctions, bas e d to a great extent on my preprint [194]. F or the ca se of ( ∞ , 1 )-categor ies, these topics are trea ted in Lurie’s recent bo ok [1 53] ab out the analogue of Gro thendiec k’s theo ry of top oi, using quasicatego ries. Other topics from higher categ ory theo ry which will only be discussed very brie ﬂy are the theory of higher stac ks, and the Poincar´ e n -g roup oid and v an Ka mpen theorems. F or the theory o f higher stacks the reader can consult [11 7] w hich starts from the mo del catego r ies cons tructed here. F or a discussion of the Poincar´ e n -g roup oid, the r eader can consult T amsamani’s o riginal pap er [206] as well a s Paoli’s discussion of this topic in the context of sp ecial C at n -groups [170]. W e will sp end a chapter lo oking at the Br e en-Baez-Dola n stabiliza- tion hyp othesis , following the preprin t [1 96]. This is o ne o f the ﬁrst parts of the famous Baez-Dolan c onj e ctur es . These conjecture s hav e strong ly motiv a ted the development o f higher c a tegory theory . Hopkins and Lurie hav e recen tly pro ven imp orta n t pieces of the main conjectures . The sta- bilization conjecture is a pr eliminary statement ab out the b ehavior of the notion of k -connected n -ca tegory , understandable with the basic tech- niques w e hav e develop ed here. W e hop e that this will serve as an intro- duction to an exciting c ur rent area of resear ch. Ac k no wledgemen ts I w ould ﬁrst like t o thank Zouhair T amas a mani, whose original w ork on this que s tion le d to a ll the rest. His techniques for gaining access to a theory o f n -categories using Segal’s delo o ping mac hine, set out the basic contours of the theor y , and contin ue to inform a nd guide o ur under stand- ing. I would like to thank Andr´ e Hirschowitz for muc h enco uragement and many interesting conv ersations in the c o urse of our work on de- scent for n -stacks, n -stac ks of complexe s, and hig her Brill-No ether. And to thank Andr´ e’s thes is student Regis Pelissier who to ok the argument to a next stage o f abstraction, braving the m ultiple diﬃculties not the least of which were the cloudy reaso ning and several imp or tant erro rs in one o f m y pr eprints. W e ar e fo llowing quite closely the main idea of Pelissier’s thesis, which is to iterate a co nstruction whereby a go o d mo del ca tegory M ser ves as input, and we try to get out a mo del cate- gory o f M -enr iched pr ecategor ie s. Cla rk B arwick then added a further crucial insight, which w as that the argument co uld b e broken do wn in to pieces, star ting w ith a fairly cla ssical le ft Bousﬁeld lo ca lization. Here again, Clark’s idea serves as gr oundwork for our approach. J acob Lur ie contin ued with ma ny contributions, o n diﬀerent levels most o f whic h are b eyond our immediate grasp; but still including some quite under- standable innov ations such a s the idea of intro ducing the ca tegory ∆ X of ﬁnite ordered sets decorated with elements of the set X “of ob jects”. This leads to a signiﬁcant lightening o f the hypotheses needed of M . His appro ach to ca rdinality ques tio ns in the small ob ject a rgument is groundbrea king, and we give here an alternate trea tmen t which is cer- tainly less streamlined but mig ht help the re ader to situate what is go ing on. These items are o f course sub or dinated to the us e of Smith’s re c ogni- tion principle a nd Dugger’s no tion of co mbinatorial mo del catego ry , on which o ur constructions are based. J ulie Berg ner has gained imp orta n t This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 8 A cknow le dg ements information ab out a whole rang e of model str uctures starting with her consolidation o f the Dwyer-Kan structure on the catego ry of simplicial categorie s. Her characteriza tion of ﬁbrant o b jects in the mo del struc- tures for Segal categor ie s, ca rries over easily to our case and provides the bas is fo r imp or ta n t parts of the statements of our main results. Bertrand T o en was lar g ely resp onsible for teaching me a bo ut mo del categorie s. His philos ophy that they are a g o o d down-to-earth yet p ow- erful a pproach to higher categorical ques tions, is s uﬀused throughout this work. I would like to thank Jo seph T a pia and Consta ntin T eleman for their encour agement in this direction to o, and Geor ges Maltsinio tis and Alain Br ug ui` eres w ho were able to expla in the higher categ orical meaning o f the E ckmann-Hilton arg ument in an understandable wa y . I would similarly like to thank Clemens Berger, Ronnie Brown, E ug enia Cheng, Denis-Charle s Cisinski, Delphine Dup ont, Joa chim Ko ck, Peter May , and Simona Paoli, for many interesting a nd informative conv er- sations ab out v arious a sp ects of this s ub ject; and to thank my current do ctoral students for contin uing discussions in dir ections extending the present work, which motiv a te the co mpletion of this pro ject. I would also like to thank my c o-workers on related topics , things which if they are not directly present here, hav e still contributed a lot to the motiv ation for the study o f higher categories . I would specia lly like to thank Diana Gillo oly of Cambridge University Press, and B urt T ota ro, for setting this pro ject in motio n. Paul T a ylor’s dia gram pa ck age is used for the co mm utative diagr ams and e ven for the arrows in the text. F or the title, we hav e chosen something almost the same as the title of a sp ecia l semester in Barcelo na a while back; we ap ologize fo r this ov erlap. This rese a rch is pa rtially supp or ted by the Agence Natio na le de la Recherc he, gr ant ANR-0 9 -BLAN-015 1 -02 (HOD AG). I would like to thank the Institut de Math´ ematiques de Jussieu for their hospitality during the completion of this work. P A R T I HIGHER C A TEGORIES 1 History and motiv ati o n The most basic motiv atio n for introducing higher ca tegories is the o b- serv ation that Ca t U , the ca tegory o f U -small categor ie s, natura lly has a structure of 2-catego ry: the ob jects are categ o ries, the morphisms are functors, and the 2-mor phisms are natura l transfor mations b etw een functors. If we deno te this 2-ca tegory by Ca t 2cat then its truncation τ ≤ 1 Ca t 2cat to a 1-categ ory would hav e, as morphis ms , the equiv alence classes of functors up to na tural equiv alence. While it is often neces- sary to consider t wo naturally equiv alent functors as b eing “the same” , ident ifying them formally leads to a loss of information. T op o logists are confronted with a similar situation when lo ok ing at the ca tegory of s paces. In homotopy theor y one thinks o f tw o homo- topic ma ps b etw een spaces as b eing “ the sa me”; how ev er the homotopy c ate gory ho( Top ) obtained after dividing by this equiv alence rela tion, do esn’t r etain enoug h information. This loss of information is illustrated by the questio n of diag rams. Suppos e Ψ is a small categ ory . A dia- gr am of sp ac es is a functor T : Ψ → Top , that is a s pa ce T ( x ) for each ob ject x ∈ Ψ and a map T ( a ) : T ( x ) → T ( y ) for each arrow a ∈ Ψ( x, y ), sa tis fying strict co mpatibilit y with identities and comp osi- tions. The catego r y of diagrams Func (Ψ , Top ) has a na tural sub clas s of morphisms : a morphism f : S → T o f diag rams is a levelwise we ak e quiva lenc e if each f ( x ) : S ( x ) → T ( x ) is a w eak e quiv a lence. Letting W = W Func (Ψ , Top ) denote this sub class, the homotopy category of diagrams ho( Func (Ψ , Top )) is deﬁned to be the Gabriel- Zisman lo cal- ization W − 1 Func (Ψ , Top ). There is a natural functor ho( Func (Ψ , Top )) → Func (Ψ , ho( Top )) , which is not in g eneral a n equiv alence of categor ies. In o ther words ho( Top ) do esn’t r etain eno ugh infor mation to r ecov er ho ( Func (Ψ , Top )). This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 12 History and motivation Thu s, we need to consider some k ind of extra str ucture b eyond just the homotopy category . This phenomenon o ccurs in a num ber of diﬀerent plac es. Starting in the 19 50’s a nd 19 60’s, the notio n of derive d c ate gory , a n ab elia niz e d version of ho( ), b eca me c rucial to a num b e r of ar eas in mo de r n ho- mological algebra and particular ly for algebr a ic geo metry . The notion of lo calization of a category seems to hav e b een proposed in this co n text by Serre, a nd app ears in Gr othendieck’s T ohoku pape r [106]. A sy stematic treatment is the sub ject o f Gabriel-Zisman’s b o o k [100]. As the ex a mple of diagr ams illustrates , in many derived-categorica l situations one m ust ﬁrst make some intermediate constructio ns o n un- derlying categ orical data, then pass to the derived ca teg ory . A funda- men tal exa mple o f this kind of reasoning was Deligne’s appr oach to the Ho dge theory of simplicial schemes using the notion of “mixed Ho dge complex” [8 0]. In the nona b elia n or homotopical- a lgebra cas e, Quillen’s notion o f closed mode l ca tegory formulates a go o d collection of requirements t hat can b e made o n the in termediate categorical data. Quillen in [175] a sked for a general structure which would encapsulate all of the higher homo- topical data. In one wa y of looking at it, the answer lies in the notion of higher c ate go ry . Quillen ha d alr eady provided this a nswer with his deﬁni- tion of “simplicial mo del ca teg ory”, wherein the simplicia l sub catego ry of coﬁbrant and ﬁbra n t ob jects provides a ho motopy in v arian t higher categoric al s tructure. As later b ecame clear with the work o f Dwyer and Ka n, this simplicia l ca tegory contains exa c tly the rig ht informa tio n. The notion of Quillen mo del category is still one o f the b est wa ys of ap- proaching the problem o f calculation with homoto pical ob jects, so m uc h so that w e adopt it as a ba sic langua ge for de a ling with notions of higher categorie s. Bondal and K apranov introduced the idea of enhanc e d derive d c at- e gori es [135] [43], whereby the usual derived categ o ry , which is the Gabriel-Zisma n lo ca lization of the categor y of complexes , is r eplaced b y a diﬀer ential gr ade d (dg) c ate go ry c ontaining the required higher homo - topy information. The notion of dg-categ ory ac tua lly app ears near the end of Gabriel-Zisman’s bo ok [1 00] (where it is compared with the no- tion of 2-categ o ry), and it was one o f the motiv ations for K elly’s theor y of enriched ca tegories [139]. The no tion of dg-categ ory , now further de- velopped by Keller [1 3 8], T abuada [20 3], Stanculescu [199], Batanin [22], Moriya [169] and others, is one po ssible answer to the search for higher categoric al structure in the k -linear case , pretty m uch analo gous to the History and motivation 13 notion of strict simplicia l catego r y . The c orresp onding weak notion is that of A ∞ -c ate gory used for ex a mple by F uk a ya [98] and Kontsevic h [144]. This deﬁnition is based on Stasheﬀ ’s notion of A ∞ -algebra [200], which is an e x ample of the pa ssage from delo oping ma chinery to higher categoric al theories. In the far future o ne could imagine s tarting directly with a notion of higher ca tegory and b ypassing the mo del- category s tep entirely , but for now th is r aises diﬃcult questions of b o ots tr apping. Lurie has taken this kind of prog ram a long w ay in [153] [154], using the no tion of quasic ate- gory as his basic higher-categ o rical ob ject. But even there, the underly- ing mo del category theory remains imp ortant. The rea der is invited to reﬂect on this interesting problem. The orig ina l example of the 2-categ ory of ca tegories, suggests us- ing 2-catego ries and their even tual iterative g eneralizations , as higher categoric al structure s. This p oint of view o ccured as early as Gabriel- Zisman’s b o ok, wher e they introduce a 2-ca tegory enha ncing the struc- ture of ho( Top ) as well as its analog ue for the ca tegory o f complexes, and pr o ceed to use it to tr eat q uestions ab out homotopy groups. Benab ou’s monog raph [28] introduced the notion o f we ak 2 - c ate gory , as well as v arious notions of weak functor. These are a ls o rela ted to Grothendieck’s notion of ﬁb er e d c ate gory in that a ﬁb er ed ca tegory may be viewed as so me kind of weak functor fro m the base ca tegory to the 2-catego ry of categories . Starting with Benab ou’s b o o k , it has b een c lear that there would be t wo types of ge ner alization from 2-categor ies to n -categor ies. The strict n -c a te gories ar e deﬁned recurrently as categories enriched ov er the category of strict n − 1-categories . By the E ckmann-Hilton a rgument, these don’t co nt ain enough ob jects, as we shall disc uss in Cha pter 4. F or this reas on, these are not our ma in o b jects of study and we will use t he terminolog y strict n -c ate gory . The relativ e ease of deﬁning strict n -categor ies nonetheless makes them attractive for learning some o f the basic outlines of the theory , the star ting point of Chapter 2. The other generaliza tion would b e to cons ide r we ak n -c ate gories also called lax n -c ate go ries , a nd which w e usually ca ll just “ n -ca tegories” , in which the comp os ition would be a sso ciative only up to a natural equiv- alence, and similarly fo r all other op era tions. The requirement that all equalities b etw een sequences of o p e r ations b e replaced by natur al equiv- alences a t o ne level higher, leads to a combinatorial explosion be cause the natural equiv alences themselves are to b e considered as op er ations. F or this r eason, the theory of weak 3-ca tegories developp ed b y Gor- 14 History and motivation don, Po wer, Street [104] following the path set out by Benab ou fo r 2- categorie s in [2 8], was already very complicated; for n = 4 it b ecame next to impo ssible (see how ever [211]) and developmen t o f this line stopp ed there. In fact, the pr oblem o f deﬁning and s tudying the higher op eratio ns which are needed in a w eakly asso c ia tive category , had b een co nsidered rather early o n by the top olo gists who noticed that the notion of “ H - space”, that is to say a space with an op eration which provides a gr oup ob ject in the homotopy categor y , was insuﬃcient to capture the data contained in a lo op space. One needs to sp ecify , for example, a ho motopy of asso ciativity b etw een ( x, y , z ) 7→ x ( y z ) and ( x, y , z ) 7→ ( xy ) z . This “asso cia tor” should itself b e sub ject to so me k ind of higher asso ciativity laws, c a lled c oher enc e r elations , inv olving co mp os ition of four or more elements. One of the ﬁr st discussions of the resulting higher co herence struc- tures w as Stasheﬀ ’s no tio n of A ∞ -algebra [200]. This was placed in the realm of diﬀerential gr aded a lgebra, but not lo ng therea fter the notion of “delo oping machine” came out, including Ma cLane’s notion of PR OP , then May’s ope radic and Segal’s simplicial delo oping machines. In Sega l’s ca se, the higher coherence r elations come a bo ut by requir ing not o nly t hat σ 2 be a w eak equiv alence, but that all of the “Segal maps” σ m : A m → A 1 × . . . × A 1 given by σ m = ( f ∗ 01 , f ∗ 12 , . . . , f ∗ m − 1 ,m ) should b e weak homotopy equiv a- lences. This was iter ated by Dunn [85]. In the op era dic viewp oint, the coherence relations come from contractibility of the spaces of n -ar y op- erations. By the la te 19 60’s a nd ear ly 1970 ’s, the top ologis ts had their delo oping machines well in hand. A main theme of the prese nt work is that these delo oping machines can gener ally lead to deﬁnitions of higher ca teg ories, but that do esn’t seem to have be e n done ex plicitly at the time. A related notion also app ear ed in the bo ok of Boardman and V ogt [42], that of r e- stricte d Kan c omplex . These ob jects ar e now known as “ quasicatego ries” thanks to Joy al’s w ork [12 6]. At tha t time, in alg ebraic geometr y , a n elab orate theory of derived catego ries w as being develop e d, but it relied only o n 1-catego ries which were the τ ≤ 1 of the relev an t hig he r categories. This diﬃculty was work ed around at a ll places, by tech niques of work- ing with explicit resolutions a nd complexe s . Illusie gav e the deﬁnition of we ak e quivalenc e of simplicial pr eshe aves which, in retro sp ect, leads later to the idea of higher stack via the mo del categories o f Jardine and History and motivation 15 Joy al. Somewher e in these works is the idea, which seems to hav e b een communicated to Illusie by Deligne, of loo king a t the derived categor y of diagrams a s a functor of the base ca tegory; this was later taken up by Grothendieck and Cisinski under the name “deriv ator” [109] [70]. In 198 0, Dwy er and Ka n came o ut with their theo ry of simplicial lo- calization, allowing the asso ciation o f a simplicial ca tegory to any pair ( M , W ) and giving the hig her ca tegorica l version of Gabriel- Zisman’s theory . They developped an extensiv e theory o f simplicial catego ries, in- cluding several diﬀerent constructions of the simplicial lo ca lization which inv erts the morphisms of W in a homotopic a l sense. This construction provides the do or passing from the world o f ca tegories to the world of higher catego r ies, b ecause even if we sta rt w ith a regula r 1-categor y , then lo calize b y inverting a collection of morphisms, the simplicial lo- calization is in general a simplicial catego ry which is not a 1-categ ory . The s implicial lo calization maps to the usual or Gabr iel-Zisman lo c al- ization but the latter is o nly the 1-truncation. So, if we w ant to in vert a collection of morphisms in a “ho motopically correct” wa y , we are forced to introduce some kind of highe r catego rical structure, at the very least the notion of simplicial ca tegory . Unfortunately , the imp ortance of the Dwyer-Kan constructio n doesn’t seem to hav e been generally noticed at the time. During this p erio d, the category -theorists and pa rticularly the Aus- tralian school, were working on fully under standing the theory o f strictly asso ciative n -c ategories and ∞ -c a tegories. In a so mewhat diﬀerent dir ec- tion, Lo day in tro duced the notio n of cat n -group which was obtained by iterating the internal catego ry constr uc tio n in a diﬀerent way , allowing categorie s of o b jects as well a s of mor phisms. Ro nnie Br own w orked on v a rious asp ects of the pr o blem o f rela ting these structures to ho motopy theory: the stric tly a sso ciative n -categ ories don’t mo del a ll homotopy t yp es (Brown-Higgins), whereas the cat n -groups do (Brown-Loday). A ma jor tur ning po in t in the history of higher categor ie s was Alexan- dre Grothendieck’s famous ma n uscript Pursu ing Stacks , which s tarted out a s a c o llection of letters to diﬀerent colleagues with many par ts crossed out and rewritten, the whole cir culated in mimeogra phed form. I w as lucky to be able to consult a copy in the back roo m o f the P rince- ton math library , and later to obtain a copy fro m Jean Malgoir e; a pub- lished version edited b y Geo rges Ma lts inio tis should a ppe a r s o on [108]. Grothendieck introduces the pro ble m of deﬁning a notion of weakly as- so ciative n -categor y , and p oints out that many are as of mathematics could b eneﬁt from s uc h a theory , explaining in particular how a theory 16 History and motivation of hig her stacks sho uld provide the r ig ht kind of co eﬃcient s ystem fo r higher no na be lian c o homology . Grothendieck made imp or tant progre s s in inv estigating the top ology and categor y theor y behind this questio n. He introduced the notion o f n -gr oup oid , an n -catego ry in which all ar rows are inv ertible up to equiv a- lences at the next higher level. He conjectured the e xistence of a Poinc ar´ e n -gr oup oid c onstruction Π n : Top → n -Gp d ⊂ n -Cat where n -Gp d is the collection of w eakly ass o ciativ e n -group oids. He pos- tulated that this functor should provide an equiv a lence of homo topy theories b etw een n -truncated spaces 1 and n -group oids. In his search fo r algebra ic mo dels for homotopy types, Gro thendieck was inspir ed b y one of the pioneer ing w orks in this direction, the notion of C at n -gr oups of Brown and Lo day . This is what is now known a s the “cubical” approa ch wher e the set o f ob jects can itself have a structure for example of n − 1 -categor y , s o it isn’t quite the sa me a s the approa ch we are lo o king for, commonly called the “globular” case. 2 Much o f “Pur suing stacks” is devoted to the more ge neral question of mo deling homotopy types b y algebraic ob jects such as presheav es on a ﬁxed small categor y , developing a theo ry of “test ca tegories” which has no w blossomed into a distinct sub ject in its own righ t thanks to the further w ork of Maltsiniotis [160] a nd Cisinski [68]. One of the questions their theor y aims to address is , which presheaf categor ies pr ovide g o o d mo dels for homotopy theory . One could ask a similar question with r e- sp ect to Sega l’s utilisa tion of ∆, namely whether other categ ories could be used ins tea d. W e don’t currently hav e any go o d information ab out this. As a star t, throughout the b o ok we will try to p oint out in discus- sion and coun terexamples the main plac es where special prop erties of ∆ are us e d. In the parts of “Purs uing stacks” ab out n -catego ries, the following theme emerges : the notion of n -categ ory with str ictly asso ciativ e com- po sition, is not suﬃcient. This is seen from the fact that strictly asso- ciative n -categor ies sa tisfying a weak group oid condition, do not ser ve to mo del ho motopy n -types as would b e expe c ted. F undamentally due 1 A s pace T is n -trunc a te d if π i ( T , t ) = 0 for all i > n and all base p oint s t ∈ T . The n -truncat ed space s are the ob j ects whic h a pp ear in the P ostnik ov tow er of ﬁbrations, and one can deﬁne the t runcation T → τ ≤ n ( T ) for an y spac e T , b y adding c ells of dim ension ≥ n + 2 to ki ll oﬀ the higher homotop y groups. 2 Pa oli has recently deﬁned a notion of sp e cial C at n -gr oup [170] that im p oses the globularity condition w eakly . History and motivation 17 to Go dement and the Eckmann-Hilton argument, this o bserv ation was reﬁned o v er time by Br own a nd Higgins [58] and B erger [29]. W e discuss it in so me detail in Chapter 4. Since strict n -ca tegories aren’t enough, it leads to the question o f deﬁning a notion of w eak n -ca tegory , which is the main sub ject of o ur bo o k. Thanks to a careful re ading by Georg es Ma ltsiniotis, we now know that Gr othendieck’s man uscript in fact contained a deﬁnitio n of weakly asso ciative n -gro upo id [161], and that his deﬁnition is very s imilar to Batanin’s deﬁnition of n -catego ry [16 2]. Grothendieck enunciated the deceptively simple rule [108]: Intuitiv ely , it means that whenever we hav e tw o wa ys of asso ciating to a ﬁn ite family ( u i ) i ∈ I of ob jects of an ∞ -group oid, u i ∈ F n ( i ) , sub jected to a standard set of relations on th e u i ’s, an elemen t of some F n , in terms of the ∞ -group oid structure only , then w e hav e automatically a “ homotopy” b etw een t hese built in in the v ery structure of the ∞ -groupoid, provided it mak es sense to ask f or one . . . The structure of this as a deﬁnition was not immedia tely evident up on any initial reading, all the mo re so when one takes int o account the directionality o f arrows, so “Purs uing stacks” left open the pr oblem of giving a go od deﬁnition of weakly ass o ciative n -categor y . Given the idea that an equiv a lence Π n betw een homotopy n -types and n -group oids should exis t, it bec omes po ssible to think o f replac- ing the notion o f n -group oid by the notion of n -tr unca ted s pa ce. This motiv a ted Joy al to deﬁne a closed mo del structure o n the catego ry of simplicial sheaves, and Jar dine to extend this to simplicial presheaves. These theor ies give an appro ach to the notion of ∞ -stack, a nd w ere used by Thomason, V o evo dsky , Morel and o thers in K -theor y . Also explic itly mentioned in “ Pursuing sta cks” was the limiting cas e n = ω , involving i -morphisms of all de g rees 0 ≤ i < ∞ . Again, an ω - group oid should corr esp ond, via the in verse of a Poincar´ e construction Π ω , to a full homotopy t yp e up to weak equiv alence. W e can now get back to the discussio n of simplicial categ ories. These are c a tegories enric hed over spaces, and applying Π ω (whic h is supp osed to b e compatible with pro ducts) to the mor phism spa ces, we can think of simplicial ca teg ories as being categories enriched ov er ω -gr oup oids. Such a thing is itself a n ω -category A , with the pro pe r ty that the morphism ω -categ ories A ( x, y ) are group oids. In other words, the i - morphisms are inv ertible for i ≥ 2, but not necessar ily for i = 1. Jacob Lurie intro duced the ter mino logy ( ∞ , 1) -c ate gories f or these things, wher e mo re gener ally an ( ∞ , n )-categor y would b e an ω -categ ory such that the i -morphis ms 18 History and motivation are inv ertible up to equiv alence, for i > n . The point of this discussion— of notions which hav e no t yet b een deﬁned—is to say that the notio n of s implicial c a tegory is a p erfectly go o d substitute for the notion o f ( ∞ , 1)-catego r y ev en if w e don’t know what a n ω - category is in genera l. This replacement no longer works if w e want to lo ok a t n - categor ie s with noninv ertible mor phisms at levels ≥ 2, or somewhat similarly , ( ∞ , n )-categor ies for n ≥ 2. Gr othendieck do es n’t seem to hav e been aw are of Dwyer and Kan’s work, just prior to “Pursuing stacks”, on simplicial categor ies; 3 how ev er he was well aw are tha t the notions of n -categor y for small v a lues of n had b een extensively inv estigated ear- lier in Benabo u’s bo ok ab out 2-categ ories [28], a nd Gordon, Po w ers a nd Street on 3-c ategories [1 04]. The combinatorial explosion inherent in these explicit theo r ies w as why Grothendieck asked for a diﬀerent form of deﬁnitio n w hich could w ork in general. As he forsaw in a vivid pa ssage [108, First letter, p. 1 6], there ar e cur- rently many diﬀer en t deﬁnitions of n -categor y . This started with Str eet’s prop osal in [201], o f a deﬁnitio n of weak n -c ategory as a simplicial set satisfying a certa in v ar iant of the Kan condition w he r e one takes into account the directions of a r rows, and also us ing the idea o f “thinness ”. His suggestion, in retr osp ect undoubtedly somewhat similar to J oy al’s iteration of the notion of quasica tegory , wasn’t w orked out at the time, but has recieved renewed int erest, see V erity [21 2] for example. The Seg al-style appro ach to weak to p o lo gical ca tegories was intro- duced by Dwyer, Kan, Smith [92] a nd Sc h w¨ anzl, V ogt [184], but the fact that they immediately proved a re c tiﬁca tion result relating Segal cat- egories ba ck to strict simplicial categor ies, seems to hav e slow ed down their further co nsideration of this idea . Applying Segal’s idea seems to hav e b een the topic of a letter from Breen to Grothendieck in 19 7 5, see page 7 1 below. Kapranov and V o e vo dsky in [136] consider ed a notion o f “Poincar´ e ∞ -group oid” which is a strictly asso ciative ∞ - g roup oid but where the arrows ar e inv ertible only up to equiv a lence. It now app ear s lik ely that their co ns tructions should b est b e interpreted using so me kind of weak unit condition [140]. A t around the same time in the mid-1990’s , three dis tinct approaches to deﬁning weak n -categor ies came out: Baez a nd Dolan’s a pproach used op etop es [7] [9 ], T amsama ni’s appr oach used itera tion of the Sega l de- lo oping machine [205] [206], and Batanin’s approach used globular op er- 3 Pa radoxically , Grothendiec k’s unpublished man uscript is responsi bl e in large part for the regain of in terest in Dwyer and Kan’s pu blished papers! History and motivation 19 ads [20] [21]. The Baez-Dolan and Ba ta nin approa ches will b e discus s ed in Cha pter 6. The work of Baez a nd Dolan was motiv ated b y a far-reaching progr a m of conjectures on the relations hip b etw een n -categor ies a nd physics [6 ] [10], which has led to impo rtant development s mo st notably the recent pro of b y Hopkins and Lurie. In re lationship with Grothendieck’s manuscript, as we p ointed out ab ov e, Batanin’s approach is the one which mo st closely r esembles what Grothendieck was asking for, indeed Maltsiniotis generalized the deﬁni- tion of n -gr oup oid which he fo und in “Pur s uing stacks”, to a deﬁnition of n -categor y w hich is similar to Batanin’s one [162]. In the subsequent p er io d, a n umber o f other deﬁnitions hav e a ppe ared, and p eople hav e b egun working mo re ser iously on the approach which had b een s uggested b y Str eet. B atanin, in men tioning the letter from Baez and Dolan to St reet [7], also p oints o ut that Hermida, Makk ai a nd Po w er hav e w orked o n the o p e topic ideas, leading to [1 14]. M. Rosellen suggested in 199 6 to g ive a version of the Sega l-style deﬁnition, using the theory of op er a ds. He didn’t concre tize this but T rimble gav e a deﬁnition along these lines, now playing an imp o rtant ro le in work of Cheng [65]. F urther idea s include those of Penon, Leinster’s multicategories, and others. T om Leinster has collected together ten diﬀere n t deﬁnitions in the useful comp endium [148]. The somewhat m ysterious [1 43] could also be p o int ed out. In the simplicia l direction Rezk ’s complete Segal spac es [178] can b e iterated as sugg ested by Bar wick [1 79], a nd Joy al prop ose s an iteration of the metho d o f qua sicategor ies [127]. W e shall disc uss the simplicial deﬁnitions in Chapter 5 and the op- eradic deﬁnitions in Chapter 6. One of the main ta sks in the future will be to under stand the relations hips betw een all of these approa ches. Our goal here is more down-to-earth: w e would lik e to dev elop the to ols nec- essary for w orking with T amsamani’s n -categories. W e hop e that similar to ols ca n b e developped for the other appr oaches, making an even tual compariso n theo ry in to a p ow erful metho d wher e b y the par ticular a d- v a nt ages of ea ch deﬁnition could all b e put in play at the same time. T amsamani deﬁned the Poincar´ e n -gr oup oid functor fo r his notion of n -catego ry , and showed Gro thendieck’s conjectured equiv alence with the theor y o f homotopy n -types [206]. The same has also b een done for Batanin’s theo ry , by Berger in [30]. It is interesting to note that the tw o ma in ingredients in T amsamani’s approach, the multisimplicial nerve cons truction and Segal’s delo op- ing machine, are bo th mentioned in “Pur suing stacks”. In par ticular, 20 History and motivation Grothendieck re pr o duces a letter fr om himself to Bre e n dated July 19 75, in which Gr othendieck acknowledges having recieved a prop os ed deﬁ- nition of non-strict n -categor y from Breen, a deﬁnition which a c c ord- ing to lo c. cit “...has certainly the merit of existing...” . It is not clear whether this prop osed cons tr uction was ever work ed out. Quite a ppar- ent ly , Breen’s sugg estion for using Sega l’s delooping ma chine must hav e gone along the lines of what w e are doing here. Rather than taking up this directio n, Grothendieck elabo rated a g eneral ansatz whe r eby n -categor ies would have v arious diﬀerent comp ositio n o per ations, and natural equiv a lences b etw een a ny t wo natural co mp os itions with the same so urce and ta rget, an idea now fully developed in the context of Batanin’s a nd related deﬁnitions. Once one o r more p oints o f view for deﬁning n -catego ries are in hand, the main pr oblem whic h needs to be c o nsidered is to o bta in—hope fully within the s ame p oint of view—a n n + 1-c a tegory nC AT pa rametrizing the n -c a tegories of that p oint of view. This problem, a lready c le arly po sed in “Pur suing stacks”, is o ne of o ur main go als in the more technical central part of the bo o k, fo r one mo del. It turns out that Quillen’s technique of mo del ca tegories, subsequently deep e ned by s e veral genera tio ns of mathematicians, is a gre at wa y of attacking this problem. It is by now well-kno wn that close d mo de l cat- egories pr ovide an excellent environment for studying ho motopy the- ory , a s b e came apparent from the work o f Bousﬁeld, Dwyer and K an on closed mo del categor ies of diagrams , and the gener alization of thes e ideas b y Joy al, Jardine, Thomason and V o evods k y who used model cat- egories to study simplicial preshe aves under Illusie’s conditio n of weak equiv alence. In the Sega l-style paradig m of weak enrich ment, w e lo o k at functors ∆ o → ( n − 1)Cat, so we are certainly a lso studying diagrams and it is reas onable to exp ect the no tio n of model ca tegory to bring s ome of the s ame b eneﬁts a s fo r the above-men tioned theories . T o b e more precis e ab out this motiv ation, r ecall from “P ur suing stacks” that nC AT should be an n + 1-ca tegory whose ob jects are in one-to-one corres p o ndence with the n -c a tegories of a given universe. The str ucture of n + 1-c a tegory therefor e consists of sp ecifying the mor phism ob jects Hom nC A T ( A , B ) which should themselves b e n -categ ories parametriz ing “functors” (in an a ppropriate sense) from A to B . In the explicit theories for n = 2 , 3 , 4 this is o ne of the places where a combinatorial explosion takes place: the functors from A to B hav e to be taken in a weak sense , tha t is to say we need a natura l equiv alence History and motivation 21 betw een the image of a comp osition and the co mpo s ition of the images, together w ith the a ppropriated coherence data at all levels. The following simple example shows that, ev en if w e were to consider only strict n - categor ie s , the str ict morphisms are not enough. Supp ose G is a group a nd V a n a b elia n g r oup and we set A eq ual t o the category with one o b ject and gro up of automorphisms G , and B equal to the strict n -categ ory with only one i - morphism for i < n and gro up V of n -automor phis ms of the unique n − 1-morphism; then for n = 1 the equiv alence classes of strict morphisms from A to B ar e the ele ments o f H 1 ( G, V ) so we w ould exp ect to get H n ( G, V ) in genera l, but for n > 1 there a r e no no ntrivial strict morphisms from A to B . So so me k ind o f weak notion o f functor is needed. Here is where the notion o f mo del category comes in: o ne can v iew this situation as b eing similar to the problem that usual maps b etw een simplicial sets are gene r ally to o r igid and don’t r eﬂect the homotopical maps b et ween spaces. Kan’s ﬁbrancy condition a nd Quillen’s formaliza- tion of this in the notion o f mo del category , provide the so lution: w e should require the target ob ject to be ﬁbrant a nd the sour c e ob ject to be coﬁbrant in an appr opriate model category structure. Quillen’s ax- ioms serve to guar antee that the notions o f co ﬁbrancy and ﬁbra nc y go together in the righ t w ay . So, in the application to n -categor ies we would like to deﬁne a mo del structure and then say that the usual maps A → B strictly r e spe cting the s tructure, ar e the rig h t ones provided that A is coﬁbrant and B ﬁbrant. T o obtain nC AT a further pro per t y is needed, indeed we are not just lo oking to ﬁnd the rig h t maps fro m A t o B but to get a morphism ob ject Hom nC A T ( A , B ) which should itself b e an n -c a tegory . It is natural to apply the idea o f “internal Hom ”, that is to put Hom nC A T ( A , B ) := Hom ( A , B ) using an internal Ho m in our mo del categ o ry . F or o ur pur p o s es, it is suﬃcient to co nsider Hom adjoint to the direc t pro duct op era tio n, in other words a map E → Hom ( A , B ) should be the same thing as a map E × A → B . This obviously implies impo sing further axioms on the closed mo del s tructure, in pa rticular compatibility b etw een × and c oﬁbrancy since the direct pro duct is used on the so urce side o f the map. It turns out that the req uired axioms are alr eady well-known in the notion of monoidal mo del c ate gory [12 0], 22 History and motivation which is a model category pro vided w ith an additional operatio n ⊗ , and certain axioms of compa tibilit y with the co ﬁbrant ob jects. In our case, the ope r ation is already giv en as t he direc t pro duct ⊗ = × of the mo del category , a nd a mo del category whic h is monoidal for the direct pro duct op eration will be ca lled c artesian (Chapter 10). In the present b o ok , we are concentrating on T a msamani’s approa ch to n -categor ies, which in [117] w as mo diﬁed to “ Segal n -catego ries” in the course of discussions with Andr´ e Hir s cho witz. In T a ms amani’s theory a n n -categor y is viewed as a ca tegory enriched over n − 1-ca teg ories, using Segal’s machine to deal with the enrichmen t in a ho motopically weak wa y . In Regis Pelissier’s thesis, following a q uestion p osed by Hirschowitz, this idea was pushe d to a next level: to study weak Segal-style enrichmen t ov er a more general mo del ca teg ory , with the aim of making the iteratio n formal. A s mall link w as miss ing in this pro cess a t the end of [17 1], essentially be c a use of an err or in [193] which Pelissier dis cov ered. He provided the cor rection when the iterative pro c e dur e is applied to the mo del category of simplicial sets. But in fact, his patch applies muc h more generally if we just c o nsider the op eration of functoriality under change of mo del ca tegories. This is what we will b e doing here. But instead of following Pelissier’s argument to o clo sely , so me as pec ts will b e set into a more general dis cus- sion of cer ta in kinds of left B ousﬁeld lo calizations . The idea of brea king down the co nstruction into several steps including a main step o f left Bousﬁeld lo calization, is due to Clark B arwick. The Segal 1 -categor ies are, as was originally proven in [92], equiv alent to strict simplicia l categor ies. Ber gner has shown that this equiv alence takes the form of a Quillen equiv alence b etw een mo del catego r ies [34]. How ev er, the mo del categ ory of simplicial categor ies is not a ppr opriate for the c o nsiderations desc rib ed ab ov e: it is not car tesian, indeed the pro duct of t wo co ﬁbrant simplicial categories will not b e coﬁbrant. 4 It is interesting to imagine several pos s ible wa ys around this pr oblem: one could try to systematically apply the coﬁbra n t repla cement o pe r ation; this would seem to lea d to a theo r y v ery similar to the considera tion of Gray tenso r pro ducts of Leroy [15 0] and Crans [76]; or one could hop e for a general constructio n r e placing a mo del catego r y by a cartesia n one (or per haps, g iven a mo del catego ry with monoidal structure incompatible 4 This remark also a pplies to the pro jectiv e m odel structure for weakly enri c hed Segal-st yle cat egories, whereas on the othe r hand the pro jectiv e structure is muc h more practical for calculating maps. History and motivation 23 with co ﬁbrations, co nstruct a mo no idal mo del categor y in some sense equiv alen t to it). As Berg ner p ointed out, the theories of simplicial catego ries and Se g al categorie s are also eq uiv a len t to Charles Rezk’s theory of c omplete Se gal sp ac es . As we sha ll discuss further in Chapter 5, Rezk req uir es that the Seg al maps be weak equiv alences, but rather than having A 0 be a discrete simplicial set corresp onding to the set of o b jects, he asks that A 0 be a simplicial set w eakly equiv alent to the “interior” Segal groupo id of A . Barwic k p ointed out that Rezk’s theory could also b e iterated, and Rezk’s r ecent pr eprint [1 79] shows that the res ulting mo del catego ry is cartesian. So , this route also leads to a construction of nC AT and can serve as an alternative to what w e are doing here. It should be p oss ible to extend Ber gner’s compa rison res ult to obtain equiv alences b etw een the iterates of Rezk’s theor y a nd the iterates we consider here. If our current theory is p erhaps simpler in its trea tmen t of the set of ob jects, Rezk’s theory has a better b ehavior with r esp ect to homotopy limits. As more diﬀerent po ints of view on higher categorie s ar e up a nd run- ning, the comparis on problem will b e p os ed: to ﬁnd an appropria te wa y to compare diﬀerent p oints o f vie w on n -categ o ries and (one hop es) to say that the v arious po int s of view a r e equiv alen t and in particula r that the v arious n + 1-ca tegories nC AT are equiv alent via these compar isons. Grothendieck ga ve a vivid descr iptio n of this problem (with remark able foresight, it w ould seem [148]) in the ﬁrst letter o f [108]. He p ointed out that it is no t actually clear wha t type of genera l setup one s hould use fo r such a comparison theory . V arious p ossibilities would include the mo del category for ma lism, or the forma lism of ( ∞ , 1)-catego ries starting with Dwyer-Kan lo c alization and moving through Lurie’s theory . Within the domain of simplicial theories, we hav e mentioned Ber gner’s compariso n betw een three diﬀeren t appro aches to ( ∞ , 1 )-categor ies [34]. A further compariso n of these theories with quas ic a tegories is to b e found in Lur ie [153]. A r ecent result due to Cheng [65] gives a compa rison b etw een T r im- ble’s deﬁnition and Batanin’s deﬁnition (with s ome mo diﬁca tions on bo th sides due to Cheng and L einster). Batanin’s approach used op- erads more as a wa y of encoding general alg ebraic s tr uctures, and is the closest to Grothendieck’s o riginal philosophy . While als o oper adic, T rimble’s deﬁnition is muc h closer to the philosophy w e a re developing in the present b o ok, whereby one g o es from top olo gists’ delo o ping ma- chinery (in his case, op erads) to an iterative theory of n -catego r ies. It is to be hop ed that Cheng’s result can b e expanded in v arious directions to 24 History and motivation obtain co mparisons betw een a wide r ange of theorie s, maybe using T rim- ble’s deﬁnition as a bridge tow ards the s implicial theories. This should clearly b e pursued in the near future, but it would go b eyond the scop e of the pr esent work. W e now turn to the question of p otential applications. Ha ving a go o d theory of n -categor ies should op en up the po s sibility to pursue any o f the several prog r ams such a s tha t o utlined by Grothendieck [1 08], the generaliza tion to n -s ta cks and n -ger bs of the work of Br een [51], or the progra m of Baez and Dola n in top olo gical quantum ﬁeld theor y [6]. Once the theory of n -stacks is oﬀ the gr ound this will give an alge braic approach to the “geometric n -stacks” considered in [192]. As the title indicates, Gro thendieck’s manuscript was in tended to de- velop a foundational framework for the theory of higher stacks. In turn, higher stacks should be the natural co eﬃcie n ts for no nab e lian cohomol- ogy , the idea b eing to g eneralize Giraud’s [102] to n ≥ 3 . The example of diagrams o f spac e s transla tes, via the co nstruction Π n , to a no tion of diagr am of n -gr oup oi ds . This is a strict version of the notion o f n -pr estack in gr oup oids which would b e a weak functor from the base categ ory Ψ to the n + 1-categor y GP D n of n -g roup oids. Grothendieck in tro duced the notion of n - stack whic h genera lizes to n - categorie s the classica l no tion of s tack. A full dis c ussion of this theo r y would go beyond the s cop e of the pre s ent work: we are just trying to set up the n -categ orical foundations ﬁrst. The notion of n -stack, maybe with n = ∞ , ha s a pplications in ma n y areas as predicted by Grothendieck. Going backw ards along Π n , it turns out that diagrams of spaces or equiv alen tly simplicial pr esheav es, serv e a s a v ery adequate replacement [125] [12 3] [20 9] [21 3] [168]. So , the notion of n -categor ies a s a prereq - uisite for higher stacks has prov en somewhat illus o ry . And in fact, the mo del category theo ry dev elop ed for s implicial presheav es has be en use- ful for a ttacking the theory of n -categor ies as we do her e, and also for going from a theory of n -categorie s to a theory of n -sta cks, as Ho llander has done for 1- s tacks [118] and as Hirs ch owitz and I did for n -sta cks in [117]. An n -stack on a site X will b e a morphism X → nC AT . This requires a construction for the n + 1-ca tegory nC AT , together with the appropria te notion of mor phism b etw een n + 1-ca tegories. The latter is a lmost equiv alen t to knowing how to co nstruct the n + 2 - categor y ( n + 1) C AT of n + 1-categ ories. F rom this discussio n the need for a n iterative approach to the theory of n -catego ries b ecomes clear. History and motivation 25 My own fav orite a pplica tion of stacks is that they lead in turn to a notion of nonab elian c oh omolo gy . Gro thendiec k says [108]: Thus n -stac ks, relativized ov er a top os to “ n -stacks o ver X ”, are viewed pri- marily as the natural “coeﬃcients” in order to do (co)homological algebra of dimension ≤ n ov er X . The idea of using higher ca tegories for no nab elian cohomo lo gy g o es back to Gira ud [102], and had b een extended to the ca ses n = 2 , 3 by B reen somewhat more recently [4 9]. Br een’s b o ok mo tiv ated us to pro ceed to the ca se of n -catego r ies at the b eginning of T amsa mani’s thesis w ork. Another utilisation of the notion of n -categor y is to mo del homotopy t yp es. F or this to be useful one w ould like to hav e a s simple and compac t a deﬁnition as po ssible, but also one which lends itself to ca lculation. The s implicial approach developpe d here is direct, but it is pos sible that the op eradic approa ches which will be ment ioned in Chapter 6 could b e more amenable to top olo gical computations. An iteration o f the class ical Segal delo o ping machin e has been c o nsidered by Dunn [85]. The Poincar´ e n - g roup oid of a space is a generalization of the Poincar´ e groupid Π 1 ( X ), a ba sep oint-free version o f the fundament al group π 1 ( X ) po pularized b y Ronnie Brown [55]. V an Kamp en’s theorem allows for computations of fundamental gr oups, and as Br own has often p ointed out, it ta kes a particularly nice form when written in terms of the Poincar´ e group oid: it says that if a space X is written a s a pushout X = U ∪ W V then the Poincar ´ e gr oup oid is a pushout in the 2-ca tegory of g roup oids: Π 1 ( X ) = Π 1 ( U ) ∪ Π 1 ( W ) Π 1 ( V ) . This s ays that Π 1 commutes with colimits. Extending this theory to the cas e of Poincar´ e n -group oids is one of the mo tiv a tio ns for intro ducing colimits and indeed the whole mo del- categoric machinery for n -categor ies. W e will then b e able to write, in case o f a pus ho ut of s pa ces X = U ∪ W V , Π n ( X ) = Π n ( U ) ∪ Π n ( W ) Π n ( V ) . Of cour se the pushout diagra m o f s paces should sa tisfy some excision condition as in the original V an Kamp en theorem, and this may be abstracted b y refering to simplicial sets instead. The homoto p y theory and nonab elian c ohomology motiv a tions may be combined by lo o king for a higher-c a tegorica l theory o f shap e . F or a 26 History and motivation space X we can deﬁne the nonab elian c ohomol o gy n -c ate gory H ( X , A ) with co eﬃcients in an n -stack F ov er X . This a pplies in pa rticular to the co nstant stack A X asso ciated to an n -categor y A . The functor A 7→ H ( X , A X ) is c o -repres e n ted by the universal element η X ∈ H ( X , Π n ( X ) X ) , giving a wa y of c haracter izing Π n ( X ) b y universal proper ty . This essen- tially tautologica l obser v a tion pav es the way for mor e nont rivial shap e theories, consisting of an n -category C OE F F a nd a functor Shape( X ) : C OE F F → C O E F F . A particularly useful version is when C O E F F is the n -category of certa in n -stac ks o ver a site Y , a nd Shap e( X )( F ) = Hom (Π n ( X ) Y , F ) where Π n ( X ) Y denotes the consta nt stack on Y with v alues equal to Π n ( X ). This le a ds to sub jects ge ne r alizing Mal- cev completions and ra tional homotopy theory [11 0] [11 1] [112], such as the schematization of homotopy t yp es [21 0] [137] [1 74] [169], de Rham shap es and no nab elian Hodg e theor y . One of the ma in adv an tages to a theory of hig her categor ies, is that the notions of homotopy limit a nd homotopy c ol imit , b y now classical in algebraic top ology , b ecome internal no tions in a higher ca teg ory . Indeed, they beco me direct analo gues of the notio ns of limit and colimit in a usual 1-catego ry , with c o rresp onding universal pro per ties a nd so on. This has an in teresting application to the “ ab elian” case: the structure of triangulated category is automa tic once we know the ( ∞ , 1 )-categor ical structure. This was pointed out by Bondal a nd Kapranov in the dg setting [43]: their enhanc e d triangulate d c ate gories ar e just dg-c a tegories satisfying some further axioms; the structure of triangles comes from the dg structure. Historically one can trace this observ a tion bac k to th e end of Gabriel- Zisman’s b o ok [1 00], although nob o dy s eems to have noticed it un til rediscovered b y Bonda l a nd Ka pranov. I ﬁr st lea rned of the notion of “2-limit” from the pap er of Deligne and Mumford [8 1], where it app ears at the b eginning with very little ex pla- nation (their pa per should also be added to the list o f motiv ations for developing the theor y of higher stacks). Sev eral author s hav e since con- sidered 2-limits and 2- to po i, originating with Bour n [45] and contin uing recently with W eb er [214] for example. The notions o f homotopy limits a nd colimits internalized in an ( ∞ , 1 )- category ha ve now r ecieved an impor tant foundational formulation with Lurie’s work on ∞ -top oi [153]. History and motivation 27 Po w er has given an extensive discussion of the motiv atio ns for higher categorie s stemming fro m logic and computer science, in [1 73]. He p oints out the role played by weak limits. Recen tly , Gaucher, Gra ndis and o th- ers have used higher catego rical no tions to study directed and concurrent pro cesses [101] [105]. It would be interesting to s ee how these theories int eract with the notion of ∞ -top os. Recall that ger bs play ed an imp ortant role in descen t theory and non- neutral tannakian ca tegories [82]. Curr ent developmen ts wher e the no- tion of higher ca tegory is more or less essential o n a foundational level, include “ derived alge br aic geometry ” and higher tanna kian theory . It would go b eyond our present scop e to discuss these here but the r eader may search for numerous references. Stacks and par ticular gerb es of hig her group oids have found many int eresting applicatio ns in the mathematica l physics literature, star ting with explicit co nsiderations for 1- and 2-g erb es. Unfortunately it would go b eyond our scop e to list all of these. Howev er, one o f the main con- tributions from mathematical ph ysics has b e en to highlight the utilit y of higher ca tegories whic h are no t group oids , in which there can b e non-inv ertible morphisms. Explicit ﬁrst cases come ab out when we con- sider m onoidal c ate gories : they may b e considered a s 2-categor ies with a sing le ob ject. And then br aide d monoidal c ate gories may b e cons id- ered as 3-ca tegories with a single ob ject a nd a single 1-morphis m, where the braiding isomorphis m co mes from the E ckmann-Hilton arg ument . These entered in to the v a st progra m of resea r ch on combinatorial quan- tum ﬁeld theor ies and kno t inv ar iants—again the r eader is left to ﬁll in the r e ferences here. John Baez and Jim Dolan provided a ma jor impetus to the theory of higher categ o ries, by formulating a series of conjectures ab out how the known relations hips b etw een low-dimensional ﬁeld theories and n - categorie s for s mall v alues of n , sho uld genera lize in all dim ensions [6] [8] [9] [10]. On the top o logical or ﬁeld-theoretical side, they conjecture the existence of a k -fo ld monoidal n -categor y (or equiv alently , a k - connected n + k -catego ry) repr esenting k -dimensiona l manifo lds up to cob ordism, where the highe r morphisms should corresp ond to ma nifolds with cor - ners. On the n -categoric al side, they propose a no tion o f n -c ate gory with duals in which all mo r phisms should hav e in ternal a djoint s. Then, their main conjectur e rela ting these tw o sides is that the co bo rdism n + k - category sho uld b e the univ ersal n + k -ca tegory with duals genera ted by a single mor phism in deg ree k . The sp eciﬁca tion of a ﬁeld theor y is a functor from this n + k -categor y to so me other one, a nd it suﬃces to 28 History and motivation sp ecify the ima g e o f the single generating morphism. They furthermore go on to in vestigate possible candida tes for the target categorie s of such functors, lo o king at higher Hilb ert spaces a nd other such things. W e will include s o me dis cussion (based on [196]) of one of Bae z and Dolan’s pr e - liminary conjectures, the st abilization hyp othesis , in an ulterior version of the pr esent manuscript. The Baez-Dolan conjectures step outside of the realm of n -g r oup oids, so they rea lly r equire an a ppr oach whic h can take into account non- inv ertible morphisms. In their “ n -categor ies w ith duals ” , they gener- alize the fact that the no tion of adjo int functor can be expr essed in 2-catego rical terms within the 2-categ ory 1 C AT . Ma ck a ay describ es the application o f internal adjoints to 4-ma nifo ld inv ar iants in [157]. The notion of a djo int generalizes within an n -categor y to the notion of dual of any i -morphism for 0 < i < n . At the top level of n -mo rphisms, the dual op eration should either b e: ig nored; impo sed a s additional struc- ture; or pushed to ∞ by considering directly the theor y o f ∞ -ca tegories. Of course, a morphism w hich is really invertible is auto matically dualiz- able a nd its dual is the same as its inv erse, s o the interesting n -categ ories with duals hav e to b e ones which are not n -group oids. As Cheng has p ointed out [64], in the last ca se one obtains a structur e which lo oks a lgebraica lly like an ∞ -group oid, s o the distinction b etw een inv ertible and dua liz a ble morphisms should probably b e c o nsidered as an additio nal mor e ana ly tic structure in itself. W e do n’t yet hav e the to ols to fully inv estigate the theor y o f ∞ -categor ies. F urther co mments on these iss ue s will b e made in Se c tion 5 .7. In a very recent development, Hopkins and Lurie hav e announced a pro of of a ma jor part of the Bae z -Dolan conjectures, saying that the category o f manifolds with appro priate co rners, and cob or disms as i - morphisms, is the universal n -category with duals g e nerated by a single element. This universal pr op erty allows one to deﬁne a functor from the cobor dis m n - c ategory to a n y other n -category with duals, b y simply sp ecifying a single ob ject. I hop e that some of the techniques presented here can help in understanding this fas cinating sub ject. 2 Strict n -categori es Classically , the ﬁrs t and easiest notion of higher ca tegory was that of strict n -c ate gory . W e review here some basic deﬁnitio ns, as they intro- duce impo rtant notions for weak n -ca tegories. In Chapter 4 we will p oint out why the s trict theory is generally considered not to b e suﬃcient. In the current chapter only , al l n -c ate gori es ar e me ant to b e strict n -c ate gories . F or this r e a son we try to put in the adjective “strict” as m uch as p oss ible when n > 1; but in any case, the very few times that we sp e ak of weak n -categor ies, this will b e explicitly stated. W e mostly restrict o ur atten tion to n ≤ 3. In case that is n’t alrea dy clear, it sho uld b e str essed that everything we do in this s ection (as w ell as most o f the next and even the subsequent one as well) is very w ell known and classical, so muc h so that I don’t know what are the o riginal r eferences. T o s tart with, a strict 2 -c ate gory A is a colle ction o f ob jects A 0 plus, for each pair o f ob jects x, y ∈ A 0 a categor y A ( x, y ) together with a morphism A ( x, y ) × A ( y , z ) → A ( x, z ) which is strictly asso cia tive in the obvious wa y; a nd such that a unit exists, that is an element 1 x ∈ Ob( A ( x, x )) with the prop erty that mul- tiplication by 1 x acts trivia lly on ob jects of A ( x, y ) o r A ( y , x ) and mul- tiplication b y 1 1 x acts trivially on morphisms of these categories . A strict 3 -c ate gory C is the same as a bove but where C ( x, y ) ar e s up- po sed to be strict 2 - categor ie s. Ther e is an o bvious notio n of direc t pro duct of str ict 2-categ ories, so the ab ov e deﬁnition applies mutatis mutandis . F or general n , the well-kno wn deﬁnition is most easily presented by induction on n . W e assume known the deﬁnition of strict n − 1-ca tegory This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 30 Strict n -c ate gories for n − 1, a nd we ass ume known that the ca tegory of strict n − 1- categories is clos ed under direct pro duct. A st rict n -c ate go ry C is then a catego ry enriched [1 39] ov er the categ ory of strict n − 1-categories. This means that C is comp ose d of a set of obje cts Ob( C ) together with, for eac h pair x, y ∈ Ob( C ), a morphism-obje ct C ( x, y ) which is a strict n − 1 -categor y; together w ith a strictly asso ciative comp osition law C ( x, y ) × C ( y , z ) → C ( x, z ) and a morphism 1 x : ∗ → C ( x, x ) (where ∗ denotes the ﬁnal o b ject cf below) acting as the identit y for the comp osition law. The c ate gory of strict n -c ate gories denoted nS tr C at is t he category whose ob jects are as ab ov e and whose morphis ms a re the tr ansformations strictly per serving all of the structures . Note that nS tr C at a dmits a direct pro duct: if C and C ′ are t wo strict n - categor ie s then C × C ′ is the s trict n -ca tegory with Ob( C × C ′ ) := Ob( C ) × Ob( C ′ ) and fo r ( x, x ′ ) , ( y , y ′ ) ∈ Ob( C × C ′ ), ( C × C ′ )(( x, x ′ ) , ( y , y ′ )) := C ( x, y ) × C ′ ( x ′ , y ′ ) where the direct pro duct o n the right is that of ( n − 1) S trC at . Note that the ﬁnal ob ject o f nS tr C at is the stric t n -categor y ∗ with exactly one ob ject x and with ∗ ( x, x ) = ∗ b eing the ﬁnal ob ject of ( n − 1 ) S trC at . The inductio n inherent in this deﬁnition may b e w orked out explicitly to give the deﬁnition as it is presented in [1 36] for ex a mple. In doing this one ﬁnds that underly ing a strict n -categ o ry C ar e the sets Mor i ( C ) of i - m orphisms or i -arr ows , for 0 ≤ i ≤ n . The set o f 0-mor phisms is by deﬁnition the set of ob jects Mo r 0 ( C ) := O b( C ), a nd Mor i ( C ) is the disjoint union over all pairs x, y of the Mor i − 1 ( C ( x, y )). These ﬁt together in a diagram called a (reﬂexive ) globular set : · · · Mor i +1 ( C ) s → ← t → Mor i ( C ) s → ← t → Mor i − 1 ( C ) · · · Mor 1 ( C ) s → ← t → Mor 0 ( C ) where the right ward maps are the sour c e and tar get maps and the left- ward maps are the identity maps 1 . Thes e may be deﬁned inductively using the deﬁnition we hav e giv en of Mor i ( C ). The structure o f strict n - category on this underlying globula r se t is deter mined by further c om- p o sition laws at each stage: the i -morphisms may b e comp osed with 1 The ad jective “reﬂexive” refers to the inclusion of these leftw ard “identit y” maps; a non-r eﬂexive globular set w ould hav e only t he s and t . 2.1 Go de ment, Int er cha nge or t he Eckmann-Hilton ar gument 31 resp ect to the j -morphisms for any 0 ≤ j < i , op erations denoted in [136] b y ∗ j . These are partially deﬁned dep ending on iterations of the source and target maps. F or a mo r e detailed e xplanation, refer to the standard references including [58] [201] [1 36] [28] [100]. 2.1 Go demen t, In terc hange or the Eckm ann-Hilton argumen t One o f the most imp orta n t of the axioms satisﬁed by the v arious com- po sitions in a s tr ict n -categor y is v ariously known under the name of “Eckmann-Hilton argument”, “Go dement r elations”, “interc hange rules” etc. This prop erty comes from the fact that the comp osition law C ( x, y ) × C ( y , z ) → C ( x, z ) is a mor phism with doma in the dir ect pro duct of the tw o morphism n − 1 - categorie s fro m x to y and from y to z . In a direct pro duct, compositions in the t wo factor s by deﬁnition a re independent (commute). Thus, for 1-morphisms in C ( x, y ) × C ( y , z ) (wher e the comp osition ∗ 0 for these n − 1-ca tegories is a ctually the comp osition ∗ 1 for C and we adopt the latter notation), w e hav e ( a, b ) ∗ 1 ( c, d ) = ( a ∗ 1 c, b ∗ 1 d ) . This le a ds to the formula ( a ∗ 0 b ) ∗ 1 ( c ∗ 0 d ) = ( a ∗ 1 c ) ∗ 0 ( b ∗ 1 d ) . This seemingly inno cuous formula takes on a sp ecial mea ning when we start inserting identit y maps. Supp ose x = y = z and let 1 x be the ident ity of x which may b e thought of a s a n ob ject of C ( x, x ). Let e denote the 2 -morphism of C , identit y of 1 x ; which may be thought of as a 1-mor phism o f C ( x, x ). It acts as the identit y for bo th co mpo sitions ∗ 0 and ∗ 1 (the reader may check that this follows from the part of the axioms for an n -categ o ry saying that the morphism 1 x : ∗ → C ( x, x ) is an iden tit y for the comp osition). If a, b ar e also endomorphisms of 1 x , then the ab ov e rule sp ecializes to: a ∗ 1 b = ( a ∗ 0 e ) ∗ 1 ( e ∗ 0 b ) = ( a ∗ 1 e ) ∗ 0 ( e ∗ 1 b ) = a ∗ 0 b. Thu s in this ca se the comp ositio ns ∗ 0 and ∗ 1 are the s ame. A diﬀer ent 32 Strict n -c ate gories ordering g ives the formula a ∗ 1 b = ( e ∗ 0 a ) ∗ 1 ( b ∗ 0 e ) = ( e ∗ 1 b ) ∗ 0 ( a ∗ 1 e ) = b ∗ 0 a. Therefore w e hav e a ∗ 1 b = b ∗ 1 a = a ∗ 0 b = b ∗ 0 a. This argument says, then, tha t Ob( C ( x, x )(1 x , 1 x )) is a commutativ e monoid and the tw o natural multip lications are the same. The same ar gument extends to the whole mono id str ucture on the n − 2- categor y C ( x, x )(1 x , 1 x ): Lemma 2.1. 1 The two c omp osition laws on the strict n − 2 -c ate go ry C ( x, x )(1 x , 1 x ) ar e e qual, and this law is c ommutative. In other wor ds, C ( x, x )(1 x , 1 x ) is an ab elian monoid-obje ct in the c ate gory ( n − 2 ) S trC at .  There is a pa rtial co n verse to the ab ov e obser v a tion: if the only ob ject is x a nd the o nly 1 -morphism is 1 x then nothing else c a n ha ppen a nd we get the fo llowing equiv alence of catego ries. Lemma 2.1 .2 Supp ose G is an ab elian m onoid-obje ct in the c ate gory ( n − 2) S tr C at . Then ther e is a unique st rict n -c ate gory C su ch that Ob( C ) = { x } and M or 1 ( C ) = Ob( C ( x, x )) = { 1 x } and such that C ( x, x )(1 x , 1 x ) = G as an ab eli an monoid-obje ct. This c onstruction establishes an e quivalenc e b etwe en t he c ate gories of ab elia n monoid-obje cts in ( n − 2) S trC at , and the strict n - c ate gories having only one obje ct and one 1 -morphism. Pr o of: Deﬁne the s trict n − 1-ca tegory U with Ob( U ) = { u } a nd U ( u, u ) = G with its monoid structure as c o mpo sition law. The fa ct that the co mpo sition law is commutativ e a llows it to b e used to deﬁne an asso ciative and commut ative multiplication U × U → U . Now let C b e the strict n -c a tegory with Ob( C ) = { x } and C ( x, x ) = U with the above multiplication. It is clear that this construction is inv erse to the pr evious o ne.  It is clear from the construction (the fact that the multip lication on U is aga in commutativ e) that the cons truction can be iterated any num ber of times. W e obtain the following “delo oping” coro llary . 2.2 Strict n -gr oup oids 33 Corollary 2.1.3 Supp ose C is a strict n - c ate gory with only one obje ct and only one 1 -morphism. Then ther e ex ists a strict n + 1 -c ate gory B with only one obje ct b and with B ( b, b ) ∼ = C . Pr o of: By the previo us lemmas, C cor resp onds to an ab e lian monoid- ob ject G in ( n − 2) S tr C at . Constr uct U as in the pro o f of 2.1.2 , and note that U is an ab elian monoid-ob ject in ( n − 1 ) S trC at . Now apply the result of 2 .1.2 directly to U to o btain B ∈ ( n + 1) S tr C at , w hich will hav e the desired prop erty .  2.2 Strict n -group oids Recall that a gr oup oid is a category w her e all morphisms are invertible. This deﬁnition g eneralizes to strict n -ca tegories in the following w ay , a s was p ointed out by Kapr anov and V o evodsky [136]. W e g ive a theorem stating that three versions of this deﬁnition are equiv alent (one of these equiv alences was left as an exerc is e in [136]). Note that, following [1 36], we do not re quire strict inv ertibility of morphisms, thus the notio n of strict n - group oid is more genera l than the notio n employ ed by Brown and Higgins [58]. Our discuss ion is in ma ny wa ys para llel to the tre a tmen t o f the gr oup oid condition for weak n -categor ies [206] to be dis c us sed in the next chapter, and our treatment in this section co mes in large par t fro m discussio ns with T amsamani a bo ut this. W e state a few deﬁnitio ns and results whic h will then b e prov en all together by induction on n . Theorem 2.2.1 Fix an inte ger n ≥ 1 . Supp ose A is a st r ict n -c ate gory. The fol lowing thr e e c onditio ns ar e e quivale nt (and in this c ase we say that A is a s trict n -gro upo id ). (1) A is an n -gr oup oid in the sen s e of [136]; (2) for al l x, y ∈ A , A ( x, y ) is a strict n − 1 -gr oup oid, and for any 1 -morphism f : x → y in A , the t wo morphisms of c omp osition with f A ( y , z ) → A ( x, z ) , A ( w, x ) → A ( w, y ) ar e e quivale nc es of st r ict n − 1 -gr oup oid s (se e b elow); (3) for al l x, y ∈ O b( A ) , A ( x, y ) is a st rict n − 1 -gr oup oi d, and the trunc ation τ ≤ 1 A (deﬁne d in the n ex t pr op osition) is a 1 -gr oup oid . Prop ositio n 2. 2.2 If A is a s trict n -gr oup oi d, then deﬁne τ ≤ k A to 34 Strict n -c ate gories b e the strict k -c ate gory whose i -morphisms ar e those of A for i < k and whose k - morphisms ar e the e quivalenc e classes of k - m orphisms of A under the e quivalenc e r elatio n that two ar e e quivalent if t her e is a k + 1 - morphism joining t hem. The fact that t his is an e quival enc e re lation is a statement ab out n − k -gr oup oi ds. The set τ ≤ 0 A wil l also b e denote d π 0 A . The trunc ation is again a k -gr oup oid, and for n -gr oup oids A the trunc ation c oincid es with the op er ation deﬁne d in [136]. If A is a strict n -g roup oid, deﬁne π 0 ( A ) := τ ≤ 0 ( A ). F o r x ∈ Ob( A ) deﬁne π 1 ( A , x ) := ( τ ≤ 1 ( A ))( x, x ), which is a group since τ ≤ 1 ( A ) is a group oid by t he previous prop ositio n. F o r 2 ≤ i ≤ n deﬁne b y induction π i ( A , x ) := π i − 1 ( A ( x, x ) , 1 x ). The interc hange pr op erty allows to show that this is an ab elian gr oup. These cla ssical deﬁnitions are recalled in [136]. Deﬁnition 2.2.3 A m orphism f : A → B of strict n -gr oup oids is said to b e an equiv alence if t he fol lowing e quival ent c onditions ar e satisﬁe d: (a) f induc es an isomorphism π 0 A → π 0 B , and for every obje ct a ∈ Ob( A ) f induc es an isomorphism π i ( A , a ) ∼ = → π i ( B , f ( a )) ; (b) f induc es a sur je ction π 0 A → π 0 B and, for every p air of obje cts x, y ∈ Ob( A ) , an e quivalenc e of n − 1 - gr oup oids A ( x, y ) → B ( f ( x ) , f ( y )) ; (c) if u, v ar e i -morph isms in A sharing the same sour c e and t ar get, and if r is an i + 1 -morphism in B going fr om f ( u ) to f ( v ) then ther e exists an i + 1 -m orphism t in A going fr om u to v and an i + 2 - morphism in B going fr om f ( t ) t o r (this includes the limiting c ases i = − 1 wher e u and v ar e not sp e ciﬁe d, and i = n − 1 , n wher e “ n + 1 -morphisms” me an e qualities b etwe en n -morphisms and “ n + 2 - morphisms” ar e not sp e ciﬁe d). Lemma 2. 2.4 If f : A → B and g : B → C ar e morphisms of strict n -gr oup oids and if any two of f , g and g f ar e e quivalenc es, then so is the thir d. If A f → B g → C h → D ar e morphisms of strict n -gr oup oi ds and if hg and g f ar e e quival enc es, then g is an e quivale nc e. It is clear for n = 0, so we assume n ≥ 1 and pro ceed by induct ion o n n : we assume that Theorem 2.2.1 and the subsequent Prop osition 2.2.2, Deﬁnition 2.2.3, as well a s Lemma 2.2.4 , a r e k nown for strict n − 1 - categorie s. W e ﬁr st discus s the existence o f truncatio n (Prop os ition 2.2.2 ), for 2.2 Strict n -gr oup oids 35 k ≥ 1 . Note that in this case τ ≤ k A ma y b e deﬁned as the strict k - category w ith the s a me o b jects as A and with ( τ ≤ k A )( x, y ) := τ ≤ k − 1 A ( x, y ) . Thu s the fact tha t the r elation in ques tion is an equiv alence r elation, is a statement ab out n − 1-ca tegories and known by induction. Note that the truncation operatio n clearly pr eserves any o ne of the three gro upo id conditions (1), (2), (3). Thus we may aﬃrm in a str ong sense that τ ≤ k ( A ) is a k -gr oup oid without knowing the equiv alence of the conditions (1 )- (3). Note a lso that the truncation op era tion for n -gr oup oids is the same as that deﬁned in [136] (they deﬁne truncatio n for genera l strict n - categorie s but for n -categor ies which are not group oids, their deﬁnition is diﬀer ent from that of [206] and not all that useful). F or 0 ≤ k ≤ k ′ ≤ n we hav e τ ≤ k ( τ ≤ k ′ ( A )) = τ ≤ k ( A ) . T o see this note that the equiv alence relation us ed to deﬁne the k - a rrows of τ ≤ k ( A ) is the same if taken in A or in τ ≤ k +1 ( A )—the existence of a k + 1-ar row g o ing betw een t w o k -arrows is equiv alen t to the existence of an equiv a lence cla ss of k + 1- a rrows going b etw een the tw o k - arrows. Finally using the abov e remark w e obtain the existence o f the trunca- tion τ ≤ 0 ( A ): the relation is the same a s fo r the trunca tion τ ≤ 0 ( τ ≤ 1 ( A )), and τ ≤ 1 ( A ) is a strict 1-gr oup o id in the usual sense so the arr ows are inv ertible, which shows that the relation used to deﬁne the 0 -arrows (i.e. ob jects) in τ ≤ 0 ( A ) is in fact an equiv a lence relation. W e complete our discus s ion o f truncation by noting that there is a nat- ural morphism of strict n -c ategories A → τ ≤ k ( A ), where the righ t hand side ( a priori a strict k -ca tegory) is co nsidered as a strict n - category in the obvious wa y . W e tur n next to t he notion of equiv alence (Deﬁnition 2.2.3), and prov e that conditions (a) and (b) ar e equiv alen t. T his notion for n -gro up oids will not enter in to the subsequent tr e atment of Theore m 2 .2 .1—what do es enter is the notio n of equiv alence for n − 1- group oids, which is known by induction—so we may a ssume the equiv alence o f deﬁnitions (1)-(3) for o ur discussion of Deﬁnition 2.2.3. Recall ﬁrst of all the deﬁnition of the homotop y g r oups. Let 1 i a denote the i -fold iterated identit y o f an o b ject a ; it is a n i - morphism, the identit y 36 Strict n -c ate gories of 1 i − 1 a (starting with 1 0 a = a ). Then π i ( A , a ) := τ ≤ i ( A )(1 i − 1 a , 1 i − 1 a ) . This deﬁnition is completed by setting π 0 ( A ) := τ ≤ 0 ( A ). These deﬁni- tions ar e the sa me as in [136]. No te directly from the deﬁnition that for i ≤ k the trunca tion mo r phism induces isomorphisms π i ( A , a ) ∼ = → π i ( τ ≤ k ( A ) , a ) . Also for i ≥ 1 we hav e π i ( A , a ) = π i − 1 ( A ( a, a ) , 1 a ) . One shows that the π i are ab elian for i ≥ 2 . This is a consequenc e of the E ckmann-Hilton argument discussed in the previo us section. Suppo se f : A → B is a morphism o f stric t n - group oids satisfying condition (b). F r om the immedia tely preceding formula and the induc- tive statemen t for n − 1-gro upo ids, we get that f induces is omorphisms on the π i for i ≥ 1. O n the other hand, the truncation τ ≤ 1 ( f ) s atisﬁes condition (b) fo r a mor phism o f 1-gro upo ids, and this is readily seen to imply that π 0 ( f ) is an isomo rphism. Thus f satisﬁes condition (a). Suppo se on the other hand that f : A → B is a mor phism of strict n -group oids s atisfying condition (a). Then o f cour se π 0 ( f ) is surjective. Consider t wo ob jects x, y ∈ A and lo ok at the induced morphism f x,y : A ( x, y ) → B ( f ( x ) , f ( y )) . W e claim tha t f x,y satisﬁes condition (a) for a mor phis m of n − 1- group oids. F or this, consider a 1-mo rphism fro m x to y , i.e. an ob ject r ∈ A ( x, y ). B y v ersion (2) of the group oid condition for A , multiplication by r induces a n eq uiv a le nce o f n − 1- g roup oids m ( r ) : A ( x , x ) → A ( x, y ) , and furthermore m ( r )(1 x ) = r . The sa me is tr ue in B : multiplication by f ( r ) induces an equiv alence m ( f ( r )) : B ( f ( x ) , f ( x )) → B ( f ( x ) , f ( y )) . The fact that f is a morphism implies that these ﬁt in to a co mm utative 2.2 Strict n -gr oup oids 37 square A ( x, x ) → A ( x, y ) B ( f ( x ) , f ( x )) ↓ → B ( f ( x ) , f ( y )) . ↓ The equiv alence condition (a) for f implies that the left vertical mor- phism induces isomorphisms π i ( A ( x, x ) , 1 x ) ∼ = → π i ( B ( f ( x ) , f ( x )) , 1 f ( x ) ) . Therefore the right vertical morphism (i.e. f x,y ) induces isomorphis ms π i ( A ( x, y ) , r ) ∼ = → π i ( B ( f ( x ) , f ( y )) , f ( r )) , this for all i ≥ 1. W e hav e now veriﬁed these iso mo rphisms for any base-ob ject r . A similar argument implies that f x,y induces an injection on π 0 . On the other hand, the fac t that f induces an isomor phism on π 0 implies that f x,y induces a surjection on π 0 (note that these last tw o statements a re reduced to statements ab out 1-gr oup oids by a pplying τ ≤ 1 so we don’t give further details ). All of these statements tak en together imply that f x,y satisﬁes condition (a), and b y the inductive statement of the theorem for n − 1 -group oids this implies that f x,y is an equiv alence. Thu s f satisﬁe s co nditio n (b). W e now r emark tha t c o ndition (b) is equiv alent to conditio n (c) for a morphism f : A → B . Indeed, the part o f condition (c) for i = − 1 is, by the deﬁnition of π 0 , identical to the conditio n that f induces a surjection π 0 ( A ) → π 0 ( B ). And the remaining conditions for i = 0 , . . . , n + 1 are identical to the c o nditions of (c) corres po nding to j = i − 1 = − 1 , . . . , ( n − 1 ) + 1 for all the mor phisms of n − 1- g roup oids A ( x, y ) → B ( f ( x ) , f ( y )). (In terms of u and v app earing in the condition in question, take x to b e the so urce of the source of the source . . . , and take y to b e the target of the target of the target . . . ). Thus by induction on n (i.e. b y the equiv alence ( b ) ⇔ ( c ) for n − 1-g r oup oids), the conditions (c) for f for i = 0 , . . . , n + 1, are equiv a len t to the conditions that A ( x, y ) → B ( f ( x ) , f ( y )) b e equiv alences of n − 1-group oids . Thus condition (c) for f is equiv a lent to condition (b) for f , which completes the pro of of the equiv alence of the diﬀerent parts of Deﬁnition 2.2.3. W e now pr o ceed with the pro o f of Theor e m 2.2.1. Note ﬁrst of all that the implications (1) ⇒ (2) and (2) ⇒ (3 ) ar e ea sy . W e give a 38 Strict n -c ate gories short discussion of (1 ) ⇒ (3) anyw ay , a nd then we pr ov e (3) ⇒ (2) and (2) ⇒ (1). Note also that the equiv alence (1 ) ⇔ (2) is the conten t of Propo s ition 1.6 o f [136]; we give a pro of here b ecause the pro o f o f P rop osition 1.6 was “left to the r eader” in [136]. (1) ⇒ (3) : Supp ose A is a strict n -catego ry s atisfying condition (1 ). This c o ndition (from [1 36]) is compatible with truncation, so τ ≤ 1 ( A ) satisﬁes co ndition (1) for 1 -categor ies; which in turn is equiv alent to the sta nda rd condition of b eing a 1 -group oid, so we g e t that τ ≤ 1 ( A ) is a 1- g roup oid. On the other hand, the conditions (1) fro m [136] for i -arr ows, 1 ≤ i ≤ n , include the same co nditions for the i − 1- arrows of A ( x, y ) for any x, y ∈ Ob( A ) (the reader has to v erify this b y lo oking at the deﬁnition in [136]). Thus by the inductiv e s tatement of the present theorem fo r str ic t n − 1-ca tegories, A ( x, y ) is a strict n − 1-group oid. This s hows that A s atisﬁes condition (3). (3) ⇒ (2) : Supp ose A is a strict n -catego ry s atisfying condition (3 ). It already satisﬁes the ﬁrst part of condition (2), by hypothesis. Thus we have to show the second part, for e x ample that for f : x → y in Ob( A ( x, y )), comp osition with f induces an equiv alence A ( y , z ) → A ( x, z ) (the other part is dual and has the s ame pro o f which we won’t r epe a t here). In or de r to prove this, we need to make a digressio n a b out the eﬀect of co mpo sition with 2-mor phisms. Supp ose f , g ∈ Ob( A ( x, y )) and sup- po se that u is a 2-mo rphism from f to g —this last supp osition may b e rewritten u ∈ Ob( A ( x, y )( f , g )) . Claim: Suppose z is another ob ject; we claim that if comp osition with f induces an equiv alence A ( y, z ) → A ( x, z ), then compos ition with g also induces a n eq uiv a le nce A ( y , z ) → A ( x, z ). T o prove the cla im, supp ose that h, k are tw o 1-mor phisms from y to z . W e now obtain a diagra m H om A ( y ,z ) ( h, k ) → H om A ( x,z ) ( hf , k f ) H om A ( x,z ) ( hg , k g ) ↓ → H om A ( x,z ) ( hf , k g ) , ↓ 2.2 Strict n -gr oup oids 39 where the top a rrow is given by comp o sition ∗ 0 with 1 f ; the left arrow by comp osition ∗ 0 with 1 g ; the bo tto m arr ow by compo sition ∗ 1 with the 2-mor phism h ∗ 0 u ; and the rig h t morphis m is given b y co mpo s ition with k ∗ 0 u . This dia g ram commutes (that is the “ Go demen t r ule ” or “interc hange rule” cf [136] p. 32). By the inductive statement of the present theo rem (version (2) of the group oid condition) for the n − 1- group oid A ( x, z ), the morphisms on the b ottom and on the rig h t in the ab ov e diagr am a re equiv alences. The hypothesis in the claim that f is a n equiv alence means that the mo r phism along the top of the diagram is a n equiv alence; th us b y the ﬁrst part of Lemma 2.2.4 a pplied to the n − 2- group oids in the diagr am, we get that the morphism on the left of the diagram is an equiv a lence. This provides the s econd half of the criterion (b) of Deﬁnition 2.2 .3 for s howing that the morphism of comp ositio n with g , A ( y , z ) → A ( x, z ), is a n eq uiv a le nce o f n − 1- g roup oids. T o ﬁnish the pro o f of the claim, we now verify the ﬁrst half o f criterio n (b) for the morphism of compositio n with g (in this par t w e use directly the condition (3) for A and don’t use either f or u ). Note that τ ≤ 1 ( A ) is a 1 -group oid, by the condition (3 ) which w e a re assuming . Note also that (by deﬁnition) π 0 A ( y , z ) = τ ≤ 1 A ( y , z ) and π 0 A ( x, z ) = τ ≤ 1 A ( x, z ) , and the morphism in question here is just the mor phism of comp osition by the image of g in τ ≤ 1 ( A ). Inv ertibility of this morphism in τ ≤ 1 ( A ) implies tha t the c o mpo sition morphism ( τ ≤ 1 A )( y , z ) → ( τ ≤ 1 A )( x, z ) is an iso mo rphism. T his completes veriﬁcation of the ﬁr st half o f criterion (b), so we get that comp osition with g is an equiv alence. This completes the pro of of the claim. W e now return to the pro of o f the comp osition condition for (2). The fact that τ ≤ 1 ( A ) is a 1-gr oup oid implies that given f there is another morphism h from y to x suc h that the class of f h is equal to the clas s of 1 y in π 0 A ( y , y ), a nd the class of hf is equa l to the c lass of 1 x in π 0 A ( x, x ). This means that there ex is t 2 -morphisms u from 1 y to f h , and v from 1 x to hf . By the ab ov e claim (and the fact that the c o mpo sitions with 1 x and 1 y act as the identit y and in pa r ticular are eq uiv alences), we get that co mpo sition with f h is an equiv alence { f h } × A ( y , z ) → A ( y , z ) , 40 Strict n -c ate gories and tha t comp osition with hf is a n equiv alence { hf } × A ( x, z ) → A ( x, z ) . Let ψ f : A ( y , z ) → A ( x, z ) be the mor phism of c o mpo sition with f , and let ψ h : A ( x, z ) → A ( y , z ) be the morphism o f co mp os ition with h . W e hav e seen that ψ h ψ f and ψ f ψ h are e q uiv a lences. By the second statement o f Lemma 2.2.4 a pplied to n − 1- group oids, these imply that ψ f is a n equiv alence. The pro of for compo sition in the o ther direction is the same; thus we hav e obtained condition (2) for A . (2) ⇒ (1) : Look at the condition (1) by refering to [136]: in question are the conditions GR ′ i,k and GR ′′ i,k ( i < k ≤ n ) of Deﬁnition 1.1 , p. 33 of [136]. By the inductiv e version of the pr e sent equiv alence for n − 1 - group oids and by the par t of condition (2) which says that the A ( x, y ) are n − 1- g roup oids, we obtain the conditions GR ′ i,k and GR ′′ i,k for i ≥ 1. Thu s we may now restrict our attention to the condition GR ′ 0 ,k and GR ′′ 0 ,k . F or a 1-morphism a from x to y , the co nditions GR ′ 0 ,k for all k with resp ect to a , are the same as the condition that for all w , the morphism of pr e-multiplication by a A ( w, x ) × { a } → A ( w, y ) is an eq uiv a le nce according to the version (c) o f the no tion of equiv a lence (Deﬁnition 2.2.3). Th us, co ndition GR ′ 0 ,k follows from the s e c ond part of condition (2) (for pre-multiplication). Similarly conditio n GR ′′ 0 ,k follows from the se c ond part of condition (2) for p ost-multiplication by every 1-morphism a . Thus condition (2) implies conditio n (1). This completes the pro of of Part (I) of the theor em. F or pro of o f the ﬁrst part of Lemma 2.2 .4, using the fa ct that iso- morphisms of sets s a tisfy the sa me “three for tw o” prop erty , and us- ing the characteriza tion of equiv alenc e s in terms of homotopy groups (condition (a)) we immediately ge t tw o of the three statements: that if f and g are equiv alences then g f is an equiv alence; and that if g f and g ar e equiv alences then f is an equiv alence. Suppo se now that g f and f are eq uiv alences; we w ould like to show that g is an equiv alence. First of all it is clear that if x ∈ Ob( A ) then g induces an isomo rphism 2.2 Strict n -gr oup oids 41 π i ( B , f ( x )) ∼ = π 0 ( C , g f ( x )) (res p. π 0 ( B ) ∼ = π 0 ( C )). Supp ose now tha t y ∈ Ob( B ), and c ho ose a 1 -morphism u go ing fro m y to f ( x ) for some x ∈ Ob( A ) (this is pos sible beca use f is sur jectiv e on π 0 ). By condition (2) for b eing a group oid, comp osition with u induces e quiv a lences along the top r ow of the diagram B ( y , y ) → B ( y , f ( x )) ← B ( f ( x ) , f ( x )) C ( g ( y ) , g ( y )) ↓ → C ( g ( y ) , g f ( x )) ↓ ← C ( g f ( x ) , g f ( x )) . ↓ Similarly comp osition with g ( u ) induces eq uiv alences alo ng the b ottom row. The sub-lemma for n − 1- group oids applied to the sequence A ( x, x ) → B ( f ( x ) , f ( x )) → C ( g f ( x ) , g f ( x )) as well as the hypo thesis that f is an equiv alence, imply that the right- most vertical arrow in the ab ove diagr am is an equiv alence. Again ap- plying the sub- lemma to these n − 1-g roup oids y ields that the leftmost vertical arrow is an equiv a lence. In particular g induces isomor phis ms π i ( B , y ) = π i − 1 ( B ( y , y ) , 1 y ) ∼ = → π i − 1 ( C ( g ( y ) , g ( y )) , 1 g ( y ) ) = π i ( C , g ( y )) . This completes the veriﬁcation of condition (a) for the mor phism g , completing the pro of of par t (IV) of the theorem. Finally w e pro ve the second part of Lemma 2.2.4 (from whic h w e no w adopt the notations A , B , C , D , f , g , h ). Note ﬁrst of all that applying π 0 gives the same situation for maps of sets, so π 0 ( g ) is an is omorphism. Next, supp o se x ∈ Ob( A ). The n w e obtain a s e quence π i ( A , x ) → π i ( B , f ( x )) → π i ( C , g f ( x )) → π i ( D , h g f ( x )) , such that the c o mpo sition of the ﬁrst pa ir and als o of the last pair a re iso- morphisms; thus g induces an iso morphism π i ( B , f ( x )) ∼ = π i ( C , g f ( x )). Now, by the same arg umen t a s for Part (IV) ab ov e, (using the hypoth- esis that f induces a surjection π 0 ( A ) → π 0 ( B )) we get that for any ob ject y ∈ Ob( B ), g induces an isomor phis m π i ( B , y ) ∼ = π i ( C , g ( y )). By Deﬁnition 2.2.3 (a) we hav e now shown tha t g is an equiv alence. T his completes the pro of of the sta temen ts in question.  Let nS trGpd b e the ca tegory of strict n -gr oup oids. In Cha pter 4 we shall s ee that for a n y realization functor ℜ : n S trGpd → Top pres erving homotopy g roups, top olo gical space s with nontrivial Whitehead pro d- ucts cannot be weakly equiv alent to any ℜ ( A ). This was Grothendieck’s 42 Strict n -c ate gories motiv a tion for propos ing to lo ok for a deﬁnition of weak n -catego ry . Our ﬁrst lo ok at the case of strict n - categor ie s serves nev ertheless as a guide to the o utlines o f any general theory of w eak n -categ o ries. 3 F u ndamen tal elemen ts o f n -categories The observ ation that the theory of s trict n -group oids is not enough to give a go o d mo del for homo topy n -types (detailed in the next Chapter 4), led Gr othendieck to ask for a theor y of n -categories with we akly asso cia- tive c omp ositio n . This will b e t he main sub ject o f our b o ok, in par ticular we use the terminology n -c ate gory to mean some kind of o b ject in a p o s- sible theor y with weak asso ciativity , or even ill-deﬁned comp osition, or per haps s ome other t yp e of weak ening (as will b e brieﬂy discusse d in Chapter 6). There are a certain num b er of basic element s exp ected of any theor y of n -ca tegories, and which can be expla ined without refering to a full deﬁnition. It will b e useful to start by consider ing these. Our discuss io n follows T amsa mani’s pap er [206], but r eally sums up the g eneral exp ec- tations for a theory of n -ca tegories which were developp ed ov er many years starting with Benabou and c o nt inuing through the th eory of strict n -categor ies and Grothendieck’s manuscript. F or this chapter, we will use the terminology “ n -c a tegory” to mean any ob ject in a g eneric theory o f n -catego ries. W e will sometimes use the idea that our g eneric theory sho uld admit dire c t pro ducts and dis joint sums. 3.1 A globular theory W e saw that a strict n - categor y has, in particular , an underlying globula r set. This basic structure should be conserved, in s ome f orm, in any weak theory . (OB)—An n -catego ry A should have an underlying set of obj e cts denoted This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 44 F undamental elements of n -c ate gories Ob( A ). If i = 0 then the s tructure A is iden tiﬁed with just this s et Ob( A ), that is to say a 0-categor y is just a set. (MOR)—If i ≥ 1 then for any tw o e le men ts x, y ∈ Ob( A ), there should be an n − 1 -c ate gory of morphisms fr om x to y denoted Mo r A ( x, y ). F rom these t wo things, we obtain by induction a who le family o f sets c alled the sets of i -morphisms of A for 0 ≤ i ≤ n . (PS)—With resp ect to dir ect pro ducts and disjoint s ums, we s hould have Ob( A × B ) = Ob( A ) × Ob( B ) and Ob( A ⊔ B ) = Ob( A ) ⊔ Ob( B ). The set of i -morphisms of A can be deﬁned inductiv ely by the follow- ing pro cedure. Put Mor [ A ] := a x,y ∈ Ob( A ) Mor A ( x, y ); this is the n − 1 - c ate gory of morphisms of A . By inductio n we obtain the n − i -categ o ry of i -morphisms of A , denoted by Mor i [ A ] := Mor [Mor[ · · · [ A ] · · · ]] . This is deﬁned whenever 0 ≤ i ≤ n , with Mo r 0 [ A ] := A and Mor n [ A ] being a set. Deﬁne Mor i [ A ] := Ob(Mor i [ A ]) . This is a set, ca lled the set of i -morphisms of A . F rom the above deﬁnitions we can write Mor i [ A ] = a x,y ∈ Mor i − 1 [ A ] Mor Mor i − 1 [ A ] ( x, y ) , and by c o mpatibilit y of ob jects with c opro ducts, Mor i [ A ] = a x,y ∈ Mor i − 1 [ A ] Ob(Mor Mor i − 1 [ A ] ( x, y )) . In particular, w e have maps s i and t i from Mor i [ A ] to Mor i − 1 [ A ] taking an element f ∈ Mor i [ A ] lying in the piece of the copro duct index ed by ( x, y ), to s i ( f ) := x or t i ( f ) := y resp ectively . These maps are called sour c e a nd t ar ge t and if no confusion arises, the index i may b e dropped. If u, v ∈ Mor i [ A ], let Mor i +1 A ( u, v ) denote the pr eimage of the pair ( u, v ) by the map ( s i +1 , t i +1 ). It is nonempty o nly if s i ( u ) = s i ( v ) and t i ( u ) = t i ( v ) and when using the notation Mor i +1 A ( u, v ) we gener ally mean to say that these c o nditions are supp osed to hold. Similarly , we get n − i − 1-ca teg ories denoted Mo r i +1 A ( u, v ). 3.1 A globular the ory 45 In this way , star ting just from the principles (OB) a nd (MOR) tog ether with the compatibilit y with sums in (PS), w e o btain from an n -categ o ry a collection of se ts Mor 0 [ A ] = Ob( A ); Mor 1 [ A ] , . . . , Mor n [ A ] together w ith pa irs of maps s i , t i : Mo r i [ A ] → Mor i − 1 [ A ] . They satisfy s i s i +1 = s i t i +1 , t i s i +1 = t i t i +1 . These elemen ts make our theory of n -categor ies into a globular the ory . Among other things, star ting from this structure we can draw pictures in a wa y which is us ua l for the theor y o f n -catego ries. These pictur es explain wh y the theory is called “glo bular”. A 0- morphism is just a po int , and a 1-morphism is pictured as a usual arrow r r ✲ A 2- morphism is pictur ed as r r ❘ ✒ ⇓ whereas a 3 -morphism s hould b e thought of as a sort of “pillow” which might be pictur ed as s s 46 F undamental elements of n -c ate gories 3.2 Iden tities F or eac h x ∈ Ob( A ) there should be a natural elemen t 1 x ∈ Mor A ( x, x ), called the id entity of x . One can envision theor ies in which the identit y is not well-deﬁned but exists only up to homotopy , se e Ko ck and Joy al [140] [128]. How ever, the theory considered here will have canonica l ident ities. F ollowing the same inductiv e pro cedure as in the previous section, w e get morphisms for any 0 ≤ i < n , e i : Mo r i [ A ] → Mor i +1 [ A ] such that s i +1 e i ( u ) = u and t i +1 e i ( u ) = u . W e ca ll e i ( u ) the identity i + 1 -morphism of the i -morphism u . Some autho r s intro duce a c ate gory of globules G n having ob jects M i for 0 ≤ i ≤ n , w ith genera ting mo r phisms s i , t i : M i → M i − 1 and e i : M i → M i +1 sub ject to the r elations s i s i +1 = s i t i +1 , t i s i +1 = t i t i +1 , s i +1 e i = 1 M i , t i +1 e i = 1 M i . An n -globula r set is a functor G n → Set ; with the iden tities this s ho uld be ca lled “reﬂexive”. Any n - c a tegory A induces an underlying globular set co nstructed a s ab ove. Other a utho r s (such as Batanin) use a category of glo bules which do esn’t hav e the identit y ar rows e i , leading to non- reﬂexive globular sets, indeed we shall use that notation in Section 6.3. The ﬁr st a nd bas ic idea for deﬁning a t heory of n - c ategories is that a n n -categor y should consist of an underlying globular set (with or without ident ities), plus a dditional structur a l morphisms s atisfying cer tain prop- erties. Wher eas the Batanin-type theories [20] [1 48] [16 2] ar e closest to this ideal, the Sega l-type theorie s w e cons ide r in the present bo ok will add a dditional structural sets to the basic globular set of A . 3.3 Comp osition, equiv alence and truncation F or ob jects x, y , z ∈ Ob( A ) there should b e some kind of morphism of n − 1- categor ie s Mor A ( x, y ) × Mor A ( y , z ) → Mor A ( x, z ) (3.3.1 ) corres p o nding to c omp osition . In the Seg al-type theories consider e d in this b o ok, the comp osition morphism is not well deﬁned and may not even exist, rather existing only in some homotopic sense. Nonetheless, in or der b est to motiv ate the following discus s ion, assume 3.3 Comp osition, e quiva lenc e and tru nc ation 47 for the momen t that w e know wha t c ompo sition means, particula rly how to deﬁne g ◦ f ∈ Mor 1 A ( x, z ) for f ∈ Mor 1 A ( x, y ) and g ∈ Mor 1 A ( x, y ). W e c a n then inductively deﬁne a notion of e quivalenc e . T amsama ni calls this inner e quiva lenc e [20 6] to emphasize that we are sp eaking of arrows in our n -ca tegory A whic h are equiv alences internally in A . T o be more pr ecise, we will deﬁne wha t it mea ns for f ∈ Mor 1 A ( x, y ) to b e an inner e quivalenc e b etwe en x and y . If such an f e xists, we say that x and y ar e e quivalent and write x ∼ y . Inductively we supp ose known w ha t this means for n − 1 -categor ies, and in particular within the n − 1-catego ries Mor A ( x, x ) o r Mor A ( y , y ). The deﬁnition then pro ceeds b y saying that f ∈ Mor 1 A ( x, y ) is a n inner equiv alence b etw een x a nd y , if there ex ists g ∈ Mo r 1 A ( y , x ) such that g ◦ f is equiv alent to 1 x in Mor A ( x, x ) and f ◦ g is equiv alent to 1 y in Mor A ( y , y ). This notion should b e transitive in the sense that if f is a n equiv a lence from x to y and g is an equiv alence from y to z , then g ◦ f should be an equiv alence from x to z . The r elation “ x ∼ y ” is therefore a tra nsitive equiv alence relatio n on the set Ob( A ). Deﬁne the trunc ation τ ≤ 0 ( A ) to be the q uotient set Ob( A ) / ∼ . W e can g o further and de ﬁne the 1 -categor ical truncation τ ≤ 1 ( A ), a 1-catego ry , as follows: Ob( τ ≤ 1 ( A )) := Ob( A ) , Mor 1 τ ≤ 1 ( A ) ( x, y ) := τ ≤ 0 (Mor A ( x, y )) . In o ther words, the ob jects of τ ≤ 1 ( A ) a re the same as the ob jects of A , but the morphisms o f τ ≤ 1 ( A ) are the equiv alence classes of 1-morphisms of A , under the equiv alence relation on the ob jects of the n − 1 -categor y Mor A ( x, y ). Comp osition of morphisms in τ ≤ 1 ( A ) should b e deﬁned b y composing representatives o f the equiv alence classes . O ne o f the ma in requirements for our theory of n - categories is that this comp osition in τ ≤ 1 ( A ) s hould be well-deﬁned, independent of the choice of repr esentativ es and indeed independent of the c hoice o f notion o f compo sition mor phism in tro duced at the s tart of this se c tion. Denote als o by ∼ the equiv alence r e lation o btained in the s ame way on the ob jects of the n − i -categor ies Mor i [ A ]. Noting that it is compa tible with the so urce and ta rget maps, w e get an eq uiv alence relation ∼ on Mor i A ( u, v ) for any i − 1-mor phisms u and v . The ab ov e discussion presupp osed the existence of so me kind of com- 48 F undamental elements of n -c ate gories po sition op era tio n, but in the Segal-style theory we consider in this b o ok, such a comp osition morphism is not canonica lly deﬁned. T hus, we restart the discussion without assuming exis tence of a comp ositio n morphism of n − 1-catego ries. The ﬁrst fundamental structure to b e considere d is th us: (EQUIV)—on each se t Mor i [ A ] we ha v e an equiv a lence relation ∼ com- patible with the sour ce and tar get maps, giv ing the set of i -m orphisms up to e quivalenc e Mor i [ A ] / ∼ . F or i = n this equiv alence relation should be trivial. The induced r elation on Mor i A ( u, v ) is also denoted ∼ . W e can then co ns ider the structure of comp ositio n which is well- deﬁned up to equiv alence, in other words it is given by a map on quotient sets. (COMP)—for any 0 < i ≤ n a nd any three i − 1-morphisms u, v , w sharing the same sources a nd the same targets, we hav e a w ell-deﬁned comp osition map  Mor i A ( u, v ) / ∼  ×  Mor i A ( v , w ) / ∼  → Mor i A ( u, w ) / ∼ which is assoc ia tive a nd has t he classes of iden tit y mor phisms as left and right units. These t wo str uc tur es a re compatible in the s ense that comp osition is deﬁned a fter pa ssing to the quotient by ∼ . As a matter of s implifying notation, given f ∈ Mor i A ( u, v ) and g ∈ Mor i A ( v , w ) then denote by g ◦ f any representative in Mor i A ( u, w ) for the comp osition of the cla ss of g with the class o f f . This is well-deﬁned up to equiv alence and by construction independent, up to equiv alence, of the choices of r epresen- tatives f a nd g for their equiv alence classes. Equiv alence and comp osition also satisfy the following further co m- patibilit y condition, expressing the notion o f equiv alence in terms whic h closely r esemble the classica l deﬁnition of equiv alence o f ca tegories. (EQC)—for any 0 ≤ i < n and u , v ∈ Mor i A sharing the same source a nd target (i.e. s i ( u ) = s i ( v ) and t i ( u ) = t i ( v ) in case i > 0), then u ∼ v if and o nly if there exist f ∈ Mo r i +1 A ( u, v ) and g ∈ Mor i +1 A ( v , u ) such tha t g ◦ f ∼ 1 u and f ◦ g ∼ 1 v . With these s tructures, we can deﬁne the 1 -categor ies τ ≤ 1 Mor i A ( u, v ), having ob jects the elements of Mor i A ( u, v ) and as morphisms b etw een w, z ∈ Mor i A ( u, v ) the equiv alence classes Mo r i +1 A ( w, z ) / ∼ . The co mpo - sition of (COMP) gives this a structure of 1-ca tegory , and w ∼ z if and only if w and z are isomorphic ob jects of τ ≤ 1 Mor i A ( u, v ). At the bottom level we obtain a 1-catego ry denoted τ ≤ 1 ( A ) and called the 1 -t runc ation of A , whose set of ob jects is Ob( A ) and whose set of mor phisms is 3.4 Enriche d c ate gories 49 Mor 1 [ A ] / ∼ . These constr uctions are compatible with the induction in the sense that τ ≤ 1 Mor i A ( u, v ) is indeed the 1-trunca tion of the n − i - category Mo r i A ( u, v ). Suppo se x, y ∈ Mor i − 1 [ A ] and u, v ∈ Mor i A ( x, y ). An element f ∈ Mor i +1 A ( u, v ) is sa id to be an internal e quivalenc e b etw een u and v , if its class is an isomorphism in τ ≤ 1 Mor i A ( x, y ). This is equiv alent to requiring the exis tence of g ∈ Mor i +1 A ( v , u ) such that g ◦ f ∼ 1 u and f ◦ g ∼ 1 v . 3.4 Enric hed categories The natural ﬁrst approach to the notion of n -categor y is to ask for n − 1 - categorie s o f morphisms Mo r A ( x, y ), with comp ositio n op era tions (3.3.1 ) which ar e strictly asso ciative and have the 1 x as str ic t left and right units. This gives a structur e of c ate gory enriche d over n − 1 - c ate gories . In an intuitiv e sens e the reader should think of an n - c a tegory in this wa y . Howev er, if the deﬁnition is applied inductively ov er n , tha t is to say that the n − 1-catego ries Mor A ( x, y ) are themselves enriched over n − 2-c a tegories and so forth, one gets to the notion of strict n -ca tegory considered in the pr e vious chapter. But, as we shall dis cuss in the next Chapter 4 b elow, the strict n -categor ies are not suﬃcient to capture all of the ho motopical b e havior we wan t for n ≥ 3. Paoli has shown [1 70] that homotopy n -types ca n be mo delled by semistrict n -gr oup oids, in o ther words n -categor ies which ar e str ictly enriched ov er weak n − 1-catego ries. Bergner showed a co rresp onding strictiﬁcation theorem for Segal categor ies, and the ana logous strictiﬁ- cation from A ∞ -categor ies to dg categ ories has b een known to the ex- per ts for some time. Lurie’s technique [15 3] for constr ucting the mo del category structure we consider here, gives additionally the strictiﬁca tion theorem g eneralizing Berg ner ’s result. So, as we shall discuss brieﬂy in Section 21 .5 , the ob jects of our Seg al-type theor y of n -ca teg ories can alwa ys b e assumed equiv alen t to semistric t o nes, that is to categories strictly enr ich ed ov er the mo del ca tegory for n − 1 -precatego ries. T his do esn’t mean tha t we can g o inductiv ely tow ards str ict n -catego ries be- cause the strictiﬁcation operatio n is not compatible with direct product, so if applied to the enriched morphism ob jects it destr oys the str ict en- richmen t str ucture. As Paoli notes in [170], semistrictiﬁcation at one level is as far a s w e can g o. 50 F undamental elements of n -c ate gories 3.5 The ( n + 1) -category of n -categories One o f the main goa ls of a theory of n -categor ies is to pr ovide a structure of n + 1-categ o ry on the collection o f all n -catego ries. O f course , s ome discussion of universes is needed here : the collection o f all n -categ ories in a universe U should for m an n +1-ca teg ory in a bigger containing universe V ⊃ U . This precis ion w ill b e dropp ed fr om most o f our discus sions below. Recall that the notion of 2-catego ry w as originally int ro duced b eca use of the familiar obser v a tion that “ the set o f all ca tegories is actually a 2-catego ry”, a 2-categor y to b e denoted 1 C AT . Its ob jects ar e the 1 - categorie s (in the smalle r universe); the 1 -morphisms of 1 C AT a re the functors b etw een ca tegories, a nd the 2-morphisms b etw een functor s are the natur al transformations. In general, we hop e and exp ect to o bta in a n n + 1-ca teg ory denoted nC AT , who se ob jects are the n -catego ries. The 1- mo rphisms of nC AT are the “ true” functors b etw een n -c a tegories, and w e obtain all of the Mor i [ nC AT ] for 0 ≤ i ≤ n + 1 which are higher ana logues of na tural transformatio ns and so on. The no tion o f in ternal equiv alence within nC AT itself, yields the no- tion of ext ernal e quivalenc e b etw een n -catego r ies: a functor f : A → B of n -categor ies, b y whic h we mean in the most g eneral sense a n element of Mor 1 nC A T ( A , B ), is said to b e an external e quival enc e if it is an internal equiv alence consider ed as a 1-morphis m in nC AT . In practice, a theo ry of n -ca tegories will usually inv olve deﬁning some kind of mathematically s tr uctured set or c ollection of sets, which natu- rally generates a usual 1 -c ate gory of n -categ ories, which we can denote by n C at . W e ex p ect then that Ob( nC at ) = Ob( nC AT ) but that there is a natural pro jection Mor 1 [ nC at ] → Mor 1 [ nC AT ] co mpatible with comp osition, indeed it should come from a morphism o f n + 1-catego ries nC at → nC AT (which is to say , a 1-mor phism in the n + 2-catego ry ( n + 1) C AT !). How ev er, the notion of exter na l equiv alence in nC at will not gen- erally sp e a king have the same characterization as in nC AT : if f ∈ Mor 1 [ nC at ] pro jects to an equiv alence in nC AT it mea ns that there is g ∈ Mor 1 [ nC AT ] such that f g a nd g f a re equiv alent to identities; how- ever the es sential inv erse g will not neces sarily come from a mo r phism in nC at . F or precisely this reason, one o f the m ain tasks needed to get a theory of n -catego ries oﬀ the gro und, is to give a diﬀerent deﬁnition of when a usual mor phism f ∈ Mo r 1 nC at ( A , B ) is a n ex ternal equiv alence. 3.5 The ( n + 1) - c ate gory of n -c ate gories 51 Of c o urse it is to be exp ected and—one hop es—later prov en that this standalone deﬁnition of external equiv alence, should beco me equiv alen t to the a bove deﬁnition once w e hav e nC AT in hand. A morphism of n -catego ries f : A → B s hould induce a morphism of s ets Ob( A ) → Ob( B ) (usually denoted just b y x 7→ f ( x )), and for any x, y ∈ Ob( A ) it should induce a mo rphism of n − 1- c a tegories Mor A ( x, y ) → Mor B ( f ( x ) , f ( y )). Just as was the ca s e for comp osition, the morphism part of f needn’t necessa rily be very well deﬁned, but it should be well-deﬁned up to an a ppropriate kind of equiv a lence. It is now pos sible to state, b y induction on n , the second or “s tan- dalone” deﬁnition of externa l equiv alence. A morphism f : A → B is said to b e ful ly faithful if for every x, y ∈ Ob( A ) the morphism Mor A ( x, y ) → Mor B ( f ( x ) , f ( y )) is a n ex ter nal equiv alence betw een n − 1-catego ries. And f is said to b e essential ly surje ctive if it induces a sur- jection Ob( A ) ։ Ob( B ) / ∼ . Then f is s a id to be an external e quivalenc e if it is fully faithful and essentially surjective. W e can state the r equired compatibility b etw een the tw o notio ns: (EXEQ)—a morphism f : A → B , a n element of Mor 1 nC A T ( A , B ), is an external equiv alence (fully fa ithful and essentially surjective), if and only if it is a n inner equiv alence in nC AT (i.e. has a n essential inv erse g such that f g and g f are equiv alent to the identities). Note that the fully faithful condition implies (b y an inductive consid- eration and comparison with the tr uncation op era tion) that if f : A → B is an equiv alence in either of the tw o equiv alen t senses, and if a, b ar e i -morphisms of A wit h the same source and tar get, then the set of inner equiv alence classes of i + 1-morphis ms fr om a to b in A , is isomo rphic via f to t he set of inner equiv alence classes of i + 1- morphisms from f ( a ) to f ( b ) in B . When developing a theory of n -ca teg ories, we therefore exp ect to be in the following situation: having ﬁrs t obtained a 1 -categor y of n - categoric al structure s denoted nC at , we obtain a notion of when a mor- phism f in this ca tegory , or ﬁrst kind of functor , is an external equiv- alence. O n the o ther hand, in the full n + 1-c ategory f should have an essential inv erse g . So, one of the ma in steps towards co ns truction of nC AT is to formally inv ert the external equiv a lences. This is a typical lo calization problem. F urthermor e nC AT should b e closed under limits and co limits, so it is very natural to us e Quillen’s theory of mo del cat- egories, a nd a ll o f the lo calization machinery tha t is now known to go along with it, as our main technical too l for going to w ards the construc- tion o f nC AT . 52 F undamental elements of n -c ate gories T o ﬁnish this section, note one of the interesting and imp ortant fea- tures o f nC AT : it is, in a c e r tain sense, enriche d over itself . In other words, f or tw o ob jects A , B ∈ Ob( nC AT ), we get an n - category of mor- phisms Mor nC A T ( A , B ) w hich, since it is an n - c ategory (and furthermore in the sa me universe level as A and B ), is itself an ob ject of nC AT : Mor nC A T ( A , B ) ∈ Ob( nC AT ) . This is the motiv ation f or using the theor y o f cartesian model categories with in ternal Hom as a preliminary model for nC AT . 3.6 P oincar´ e n -group oids An n -ca tegory is said to be an n -gr oup o id if a ll i -morphisms ar e in- ner equiv alences. Mor e generally , we s ay that A is k -gr oupic if a ll i - morphisms are inner equiv alences for i > k . Lur ie intro duce s the nota- tion ( n, k ) -c ate gory for a k -gr oupic n -category , mostly used in the limit- ing case n = ∞ . F undamen tal to Gro thendiec k’s vision in “P ursuing stacks” was the Poinc ar´ e n -gr oup oid of a sp ac e . If X is a top ologic al space, this is to b e an n -group oid denoted by Π n ( X ), with the following pro per ties: Ob(Π n ( X )) = X , for 0 ≤ i < n Mor i [Π n ( X )] = C 0 glob ([0 , 1] i , X ) where the right hand side is the subset of maps of the i -cub e into X satisfying ce rtain globularity c onditions (explained b elow), a nd at i = n we hav e Mor n [Π n ( X )] = C 0 glob ([0 , 1] n , X ) / ∼ where ∼ is an equiv a lence relation similar to the o ne considered ab ov e (and indeed, it is the same in the context of Π k ( X ) for k > n ). The globular it y condition is automa tic for i = 1, th us the 1-mor phisms in Π n ( X ) are contin uous paths p : [0 , 1] → X with source s 1 ( p ) := p (0) and targ et t 1 ( p ) := p (1). In the limiting ca se n = 1, the 1 - morphisms in Π 1 ( X ) are homotopy classes of paths with ho mo topies ﬁxing the sourc e and targ et, and Π 1 ( X ) is just the classical Poincar´ e group oid of the space X . The globularity c o ndition is most easily understo o d in the case i = 2: 3.7 Interiors 53 a 2-morphism in Π n ( X ) should b e a homotopy b etw een paths, that is to say it should b e a map ψ : [0 , 1] 2 → X such that ψ (0 , t ) and ψ (1 , t ) are indep endent of t . F or 2 ≤ i ≤ n , the globularity co ndition on a map ψ : [0 , 1] i → X says that for any 0 ≤ k < i and a n y z 1 , . . . , z k − 1 ∈ [0 , 1], the functions ( z k +1 , . . . , z i ) 7→ ψ ( z 1 , . . . , z k − 1 , 0 , z k +1 , . . . , z i ) and ( z k +1 , . . . , z i ) 7→ ψ ( z 1 , . . . , z k − 1 , 1 , z k +1 , . . . , z i ) are constant in ( z k +1 , . . . , z n ). The sour c e and tar ge t of ψ are deﬁned b y s i ψ : ( z 1 , . . . , z i − 1 ) 7→ ψ ( z 1 , . . . , z i − 1 , 0) , t i ψ : ( z 1 , . . . , z i − 1 ) 7→ ψ ( z 1 , . . . , z i − 1 , 1) . Grothendieck’s fundamen tal prediction was that this globular set should hav e a natural s tr ucture of n -group oid denoted Π n ( X ); that there should be a r e a lization c onstruction tak ing an n -gro upo id G to a top olo gical space | G | ; and that these tw o cons tr uctions should set up a n equiv a- lence of homotopy theo ries betw een n -truncated space s (i.e. spaces with π i ( X ) = 0 for i > n ) and n - g roup oids. This w ould ge neralize the cla s- sical co rresp ondence b e t ween 1- group oids and their cla s sifying spaces which are disjoint unions of Eilenberg-Ma cLane K ( π , 1 )-spaces. 3.7 In teriors A useful notion which should ex is t in a theo r y of n -categ ories is the notion of k -gro upic interior denoted Int k ( A ). This is the largest sub- n - category o f A which is k -gro upic, and we should hav e Mor i [ Int k ( A )] = Mor i [ A ] , i ≤ k whereas for i > k the i -mor phisms of the interior Mor i [ Int k ( A )] s hould consist only of those i -morphisms of A which ar e eq uiv a lences, i.e. in- vertible up to equiv alence. Sp ecifying ex actly the structure of Int k ( A ) will depend o n the par ticular theo ry o f n -ca teg ories, but in any case it should be a k - groupic n -categor y i.e. an ( n, k )-categor y . The usual ca se is fo r k = 1, a nd sometimes the index 1 will then be for gotten. Thus, if A is a n n - c ategory then we get a 1-g roupic n - category I nt ( A ). As will be discussed b elow in the nex t section and 54 F undamental elements of n -c ate gories more extens ively in Chapter 5, the notion of 1-gr oupic n -categor y is well mo dele d by the notions of simplicia l category , Segal catego ry , Rezk complete Sega l s pace, or quasicatego ry . Simplicial categor ies typically arise a s Dwyer-Kan lo c a lizations L ( ) of mo del categ o ries, a nd one feature of the model catego ry PC n ( Set ) constr ucted in this bo o k is that Int ( nC AT ) = L ( PC n ( Set )) . This adds to the motiv ation for why it is interesting and impo rtant to use mo del categories as a substrate fo r the theo ry of n -categories: we get a calcula tory mo del for an imp o rtant piece of nC AT na mely it s in terior. 3.8 The case n = ∞ Constructing a theory of ∞ -categor ies in gener al, represents a new level of diﬃculty a nd to do this in deta il would g o b eyond the sco pe of the present b o ok. W e include here a few c o mmen ts ab out this problem, largely fo llowing Cheng’s obs erv ation [64]. The example co ns idered in her pap er shows that in a n algebr aic sens e, any ∞ -c a tegory whose i - morphisms have dua ls at all levels, lo oks like an ∞ -group oid. How ev er, it is clear th at w e don’t w an t to identify such ∞ -catego ries with duals, and ∞ -group oids. Indeed, they o ccupy almost dual p os itio ns in the g eneral theory as predicted by Baez and Dolan. F ro m this paradox one can conclude tha t the notion of “equiv alence” in a n ∞ -category is not merely an algebra ic one. One wa y around this problem would be to include the notion of equiv a lence in the initial structure o f an ∞ -categor y: it would be a glo bular set with additional structure s imilar to the structure used for the ca se of n -categor ies; but a lso with the infor mation of a subset of the s et of i -morphisms whic h a re to b e des ig nated as “equiv alences”. These subsets would b e requir ed to satisfy a compatibility condition similar to (EXEQ) abov e. Then, in Cheng ’s example [64] there would b e t wo compa tible choices: e ither to designate everybo dy as an equiv alence, in whic h cas e we get an ∞ -gr oup oid; or to designa te only some (or po ten tially no ne other than the identities) as equiv a le nces, yielding an “ ∞ -catego ry with duals” mor e like what Baez a nd Dola n are lo oking for. In view o f these pr oblems, it is tempting to take a shor tcut tow ards consideratio n of certain types of ∞ -categ ories. The sho rtcut is moti- v a ted by Grothendieck’s Poincar ´ e n -group oid corres po ndence, which he 3.8 The c ase n = ∞ 55 says should also extend to an equiv alence b etw een ∞ -group o ids and the homotopy theo ry of all CW-complexes. T ur ning this on its head, we can use that idea to deﬁne the notion of ∞ -g roup oid as s imply being a homo topy type of a spa ce. The itera tive enrichmen t pro cedure yields the no tion of n -ca tegories when star ted from 0-catego ries b e ing sets. If instead we start with ∞ - group oids b eing spaces, then iterating gives a notion of ( ∞ , n )-categ o ries which are n -gro upic ∞ - categorie s , i.e. ones whose i -mor phisms are supp osed and declar ed to b e inv ertible for all i > n . A t the ﬁr st stage , an ( ∞ , 1)-categor y is ther efore a categ ory enriched ov er spaces. I n Part I I we will consider in detail many of the diﬀerent current appro aches to this theory . At the n - th stage, in the Segal-type theory purs ue d here, we obtain the notion o f Se gal n - c ate gory . The po ssi- bilit y of doing an iterative deﬁnition in v arious diﬀerent cases, motiv ates our presentation here of a genera l iterativ e construction of the theory of categorie s weakly enr ich ed ov er a cartesian mo del catego ry . The carte- sian co nditio n corresp o nds to the fact tha t the comp osition morphis m go es out from a pro duct, so the mo del str ucture s ho uld hav e a go o d co m- patibilit y pr op erty with r esp ect to direc t pr o ducts. When the iteration starts from the mo del category of sets, we get the theory o f n -catego ries; and starting from the mo del categor y of spaces leads to the theory o f Segal n - categorie s whic h is one approa ch to ( ∞ , n )-categor ies. Lo oking only at ( ∞ , n )-c ategories rather than all ∞ - categories is com- patible with the notion of nC AT , in the sens e that ( ∞ , n ) C AT , the col- lection of all ( ∞ , n )-categ ories, is exp ected to hav e a natural s tr ucture of ( ∞ , n + 1)-c a tegory . In the theory pre sented here, this will b e achieved by constructing a ca rtesian mo del category for Seg a l n -catego r ies. The cartesian pro pe rty thus shows up at the o utput s ide o f the iteration, and at the input side beca use we need to handle pro ducts in or der to talk abo ut weak compos ition morphisms. Th us, one of our main goals is to cons truct an itera tion step star ting with a car tesian mo del ca tegory M and yielding a ca r tesian mo del catego r y PC ( M ) representing the homotopy theory of w eakly M -enriched categorie s . 4 The need for w eak comp osition In this c hapter, we take a brea k fr o m the general theory to consider the pro blem o f rea lization o f homoto p y 3-types. It is here that the phe- nomenon of “weak co mp os ition” shows up ﬁr s t, in that str ict 3-gro upo ids are not suﬃcient to mo del all 3-truncated homoto py type s . The nota- tions here refer and co nt inue those of Chapter 2. The classical Eckmann-Hilton argument, originally used to show that the homotopy groups π i are ab elian for i ≥ 2, applies in the context of strict n -categor ies to give a v a nishing of certain homotopy op erations. Indeed, not only th e π i but the i -th loo p spaces ar e abelia n ob jects, and this forces the Whitehead pro ducts to v anish. This observ ation, whic h I learned from G. Maltsiniotis and A. Brug ui` eres, ha d been used b y many peo ple to argue that s trict n -categor ies do not cont ain suﬃcient infor- mation to mo del homo topy n -t yp es, a s so o n as n ≥ 3. See for example Brown [5 6], with Gilb ert [57] a nd with Higg ins [58] [59]; Grothendieck’s discussion o f this in v a rious places in [108], and the pap er of Be r ger [29]. In R. Br own’s terminology , strict n -gro upo ids cor r esp ond to cr osse d c omp lexes . While a nontrivial action o f π 1 on the π i can o ccur in a crossed co mplex, the higher Whitehead op erations such a s π 2 ⊗ π 2 → π 3 m ust v a nish. The Eckmann-Hilton a r gument for s tr ict n -catego ries is also known as the “interc hange rule” or “Go dement rela tion”. This eﬀect o ccurs when one takes t wo 2-morphisms a a nd b b oth with source and target a 1 -identit y 1 x . Ther e are v arious wa ys of co mpo sing a a nd b in this situatio n, and comparis on of these comp ositions lea ds to the conclusion that all of the comp ositions are commutativ e. In a weak n - category , this commut ativity would only ho ld up to higher homotopy , which leads to the notio n of “br aiding”; and in fact it is exactly the braiding which leads to the Whitehead op era tion. How ever, in a s tr ict This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 4.1 Re alization functors 57 n -group oid, the comm utativit y is strict and applies to all higher arrows, so the Whitehead o per ation is tr ivial. The same may b e said in the setting of 3-catego ries no t nec essarily group oids: there ar e some exa mples (which G. Ma ltsiniotis p ointed out to me) in Gordon-Po wer-Street [10 4] of weak 3-ca tegories not equiv a lent to stric t ones. This in turn is related to the diﬀerence b etw een bra ided monoidal catego ries and symmetric monoidal c ategories , see for exa mple the nice dis c ussion in Baez-Dolan [6]. This chapter gives a v a riant on these obser v a tions; it is a mo diﬁed version of the preprint [1 97]. W e will show that one ca nnot obtain all homotopy 3-types by a ny r easonable rea lization functor fro m strict 3- group oids (i.e. gr o upo ids in the sense of [136]) to spa ces. More precis ely we show that one do es not obtain the 3 -type of S 2 . This constitutes a small generalization of Berger’s theorem [29], which concerned the stan- dard r ealization functor. W e deﬁne the notion of p ossible “reaso nable realization functor” in Deﬁnition 4.1.1 to b e an y functor ℜ from the category of s trict n -group oids to Top , provided with a natural transfor- mation r from the set of ob jects of G to the po ints of ℜ ( G ), and natural isomorphisms π 0 ( G ) ∼ = π 0 ( ℜ ( G )) and π i ( G , x ) ∼ = π i ( ℜ ( G ) , r ( x )). This ax- iom is fundament al to the question of whether one can r ealize ho mo topy t yp es b y strict n -group oids, because one want s to read oﬀ the homotopy groups of the space from the strict n -group oid. The standard realization functors satisfy other prop erties b eyond this minimal one. In order to apply Deﬁnition 4.1 .1, the interc hange arg umen t is written in a particular wa y . W e g et a picture of strict 3-group oids having only one ob ject and one 1-morphism, as being equiv alent to a belia n monoidal ob- jects ( G , +) in the categ ory o f group oids, such that ( π 0 ( G ) , +) is a gr oup. In the case in question, this gro up will b e π 2 ( S 2 ) = Z . Then comes the main part of the arg ument . W e show t hat, up to inv erting a few equiv a- lences, such a n ob ject has a morphism giv ing a s plitting of the P ostniko v tow er (Pr op osition 4.4 .1 ). It follows that for any rea lization functor re- sp ecting homotopy g r oups, the Postniko v tow er of the r e alization (which has tw o stages corr esp onding to π 2 and π 3 ) splits. This implies that the 3-type o f S 2 cannot o ccur as a r ealization, Theorem 4.4.2. 4.1 Realization functors Recall that nS tr Gpd is the categor y o f strict n -gro upo ids as deﬁned in Chapter 2. Let Top b e the ca tegory of top ologica l spaces. The fol- 58 The ne e d for we ak c omp osition lowing deﬁnition enc o des the minimum of what o ne would exp ect for a reasona ble realization functor from strict n -group oids to spaces. Deﬁnition 4.1 .1 A realization functor for strict n -gro upo ids is a fu n c- tor ℜ : nS trGpd → Top to gether with t he fol lowing natur al tr ansformations: r : Ob ( A ) → ℜ ( A ); ζ i ( A , x ) : π i ( A , x ) → π i ( ℜ ( A ) , r ( x )) , the latter including ζ 0 ( A ) : π 0 ( A ) → π 0 ( ℜ ( A )) ; su ch that t he ζ i ( A , x ) and ζ 0 ( A ) ar e isomorphisms for 0 ≤ i ≤ n , such that ζ 0 takes the iso- morphism class of x to the c onne cte d c omp onent of r ( x ) , and su ch that the π i ( ℜ ( A ) , y ) vanish for i > n . Theorem 4 .1.2 Ther e exists a r e alization functor ℜ for st rict n - gr oup oids. Kapranov a nd V o evodsky [136] c onstruct such a functor . Their co n- struction pr o ceeds by ﬁrs t deﬁning a no tion o f “dia grammatic set” ; they deﬁne a re a lization functor from n -group oids to dia grammatic sets (de- noted N er v ), and then deﬁne the top ological r ealization of a diag ram- matic set (denoted | · | ). The comp osition of these tw o constructions gives a realization functor G 7→ ℜ K V ( G ) := | N er v ( G ) | from str ict n -group oids to spaces. Note that this functor ℜ K V satisﬁes the a x ioms of 4.1.1 as a consequence of Prop ositions 2 .7 and 3.5 of [136]. One obtains a diﬀerent constr uction by co nsidering strict n -group oids as w eak n -group oids in the sense of [206] (m ultisimplicial sets) and then taking the realizatio n of [206]. This co nstruction is actually due to the Australian school many years b eforehand—see [29]—and we call it the standar d r e aliza tion ℜ std . The prop er ties of 4.1.1 can b e extracted fro m [206] (although again they a re classical results). W e don’t cla im here that a ny t wo rea lization functors must be the same. This is wh y we sha ll work, in wha t follows, with an arbitrary realization functor satisfying the axioms of 4.1.1. Prop ositio n 4.1.3 If C → C ′ is a morphism of st rict n -gr oup oids in- ducing isomorphisms on the π i then ℜ ( C ) → ℜ ( C ′ ) is a we ak homotopy 4.2 n -gr oup oids with one obje ct 59 e quiva lenc e. Conversely if f : C → C ′ is a morphism of strict n - gr oup oids which induc es a we ak e quivalenc e of r e alizations then f was an e quiva- lenc e. Pr o of Apply version (a) of the equiv alent conditions in Deﬁnition 2 .2 .3, together w ith the pr op erty of Deﬁnition 4.1.1 . 4.2 n -group oids wit h one ob ject Let C b e a str ict n -categor y with only one o b ject x . Then C is a n n - group oid if a nd only if C ( x, x ) is an n − 1-group oid and π 0 C ( x, x ) (whic h has a structure o f mono id) is a g roup. This is version (3) of the deﬁnition of gro upo id in Theorem 2.2.1 . Iterating this r emark one more time we get the following statemen t. Lemma 4.2.1 The c onstruction of 2.1.2 establishes an e quivalenc e of c ate gories b etwe en t he strict n -gr o up oids havi ng only one obje ct and only one 1 -morphism, and t he ab eli an monoid -obje cts G in ( n − 2) S tr Gpd such that the monoid π 0 ( G ) is a gr oup. Pr o of Lemma 2.1.2 gives an equiv alence be tw een the categor ies of ab elian monoid-ob jects in ( n − 2) S tr C at , and the str ict n -c a tegories having only one o b ject and one 1- mo rphism. The gr o upo id condition for the n -categor y is equiv alent to saying that G is a gr oup oid, and that π 0 ( G ) is a g r oup. Corollary 4.2.2 Supp ose C is a s trict n -c ate gory having only one ob- je ct and only one 1 -morphism, and let B b e the strict n + 1 -c ate gory of 2.1.3 with one obje ct b and B ( b, b ) = C . Then B is a st rict n + 1 -gr oup oid if and only if C is a strict n -gr oup oid. Pr o of: K e ep the notation U of the pro of of 2 .1.3. If C is a gro upo id this mea ns that G s atisﬁes the condition that π 0 ( G ) b e a gr oup, which in turn implies that U is a gr o upo id. Note that π 0 ( U ) = ∗ is a utomatically a g r oup; so applying the o bserv ation 4.2.1 once aga in, we get tha t B is a gr oup oid. In the other dir e ction, if B is a g roup oid then C = B ( b, b ) is a group oid by versions (2) and (3 ) of the deﬁnition of group oid.  60 The ne e d for we ak c omp osition 4.3 The case of the standard realization Before getting to our main res ult which concerns an ar bitrary realiza- tion functor satisfying 4.1 .1, we ta ke note of an ea sier a rgument which shows that the standard rea lization functor c a nnot g ive ris e to arbitr a ry homotopy t yp es. Deﬁnition 4.3.1 A c ol le ction of r e ali zation functors ℜ n for n -gr oup oids ( 0 ≤ n < ∞ ) satisfying 4.1.1 is said to b e c o mpatible with lo oping if ther e exist tr ansfo rmations natur al in an n -gr oup oid A and an obje ct x ∈ Ob( A ) , ϕ ( A , x ) : ℜ n − 1 ( A ( x, x )) → Ω r ( x ) ℜ n ( A ) (wher e Ω r ( x ) me ans the s p ac e of lo ops b ase d at r ( x ) ), such that for i ≥ 1 the fol lowing diagr am c ommutes: π i ( A , x )= π i − 1 ( A ( x, x ) , 1 x ) → π i − 1 ( ℜ n − 1 ( A ( x, x )) , r (1 x )) π i ( ℜ n ( A ) , r ( x )) ↓ ← π i − 1 (Ω r ( x ) ℜ n ( A ) , cst ( r ( x ))) ↓ wher e the top arr ow is ζ i − 1 ( A ( x, x ) , 1 x ) , the left arr ow is ζ i ( A , x ) , the right arr ow is induc e d by ϕ ( A , x ) , and the b ottom arr ow is the c ano nic al arr ow fr om top ol o gy. (When i = 1 , suppr ess t he b asep oints in the π i − 1 in the diagr am.) R emark: The arr ows on the top, the b ottom and the left are isomor- phisms in the ab ov e diagra m, so the ar row on the right is an isomo r phism and we obtain as a coro lla ry of the deﬁnition that the ϕ ( A , x ) are actu- ally weak equiv a lences. R emark: The co llection of standard realizatio ns ℜ n std for n -group oids, is c o mpatible w ith lo o ping. W e leave this as an exercise. Recall the statements o f 2.1 .3 and 4.2.2: if A is a strict n -categ ory with only o ne ob ject x a nd only one 1-morphism 1 x , then there exists a strict n + 1- category B with one ob ject y , and with B ( y , y ) = A ; and A is a str ict n -group oid if and o nly if B is a strict n + 1 -group oid. Corollary 4.3 . 2 Su pp ose {ℜ n } is a c ol le ction of r e alizatio n fun ctors 4.1.1 c omp atible with lo oping 4.3. 1. Then if A is a 1 -c onne cte d strict n -gr oup oid (i.e. π 0 ( A ) = ∗ and π 1 ( A , x ) = { 1 } ), the sp ac e ℜ n ( A ) is we ak -e quivalent to a lo op sp ac e. 4.4 Nonexist enc e of st rict 3 -gr oup oids giving rise to the 3 -t yp e of S 2 61 Pr o of: Let A ′ ⊂ A b e the sub- n -catego ry having one o b ject x and one 1-morphism 1 x . F or i ≥ 2 the inclusio n induces is omorphisms π i ( A ′ , x ) ∼ = π i ( A , x ) , and in view of the 1-c o nnectedness of A this means (accor ding to the Deﬁnition 2.2.3 (a)) that the morphism A ′ → A is an equiv alence. It follows (by deﬁnition 4.1.1 ) that ℜ n ( A ′ ) → ℜ n ( A ) is a weak equiv alence. Now A ′ satisﬁes the h ypothesis of 2.1.3, 4.2.2 as r ecalled abov e, so ther e is an n + 1-gr oup oid B having one ob ject y s uch that A ′ = B ( y , y ). By the deﬁnition of “compatible with lo oping” and the subsequent rema rk that the morphism ϕ ( B , y ) is a weak eq uiv a lence, we get that ϕ ( B , y ) induces a weak equiv a lence ℜ n ( A ′ ) → Ω r ( y ) ℜ n +1 ( B ) . Thu s ℜ n ( A ) is weak-equiv alent to the lo op-space of ℜ n +1 ( B ).  The following cor ollary is due to C. Berg er [29] (although the s a me statement a ppea rs without pro of in Gro thendieck [1 08]). See also R. Brown and c oauthors [56] [57] [58] [59]. Corollary 4.3.3 (C. Berger [29]) Ther e is no strict 3 -gr oup oid A su ch that the st andar d r e alization ℜ std ( A ) is we ak-e quival ent t o t he 3 - typ e of S 2 . Pr o of: The 3-type of S 2 is not a lo op-space. By the previous coro llary (and the fact that th e sta ndard realiza tions a r e compatible with looping, which we have ab ov e left as an exer cise for the r eader), it is impo ssible for ℜ std ( A ) to be the 3-type of S 2 .  4.4 Nonexistence of st rict 3 -group oids giving r ise to the 3 -type of S 2 The pres en t discussion a ims to extend Be r ger’s negative result to any realization functor satisfying the minimal deﬁnition 4.1.1. The ﬁrst step is to prov e the following sta tement (which contains the main part o f the argument). It basica lly says that the Postniko v tow er of a simply connected stric t 3 -group oid C , splits. The intermediate B is not rea lly necessar y for the statement but corres po nds to the technique of pr o of. Prop ositio n 4.4 . 1 Supp ose C is a strict 3 -gr oup oid with an obje ct c 62 The ne e d for we ak c omp osition such t hat π 0 ( C ) = ∗ , π 1 ( C , c ) = { 1 } , π 2 ( C , c ) = Z and π 3 ( C , c ) = H for an ab elian gr oup H . Then ther e exists a diagr am of st rict 3 -gr oup oid s C ← g B ← f A h → D with obje cts b ∈ O b( B ) , a ∈ Ob( A ) , d ∈ O b( D ) such t hat f ( a ) = b , g ( b ) = c , h ( a ) = d . The diagr am is such that g and f ar e e quivalenc es of strict 3 -gr oup oi ds, and such that π 0 ( D ) = ∗ , π 1 ( D , d ) = { 1 } , π 2 ( D , d ) = { 0 } , and such that h induc es an isomorphism π 3 ( h ) : π 3 ( A , a ) = H ∼ = → π 3 ( D , d ) . Pr o of Start with a strict gr oup oid C and ob ject c , satisfying the hy- po theses of 4 .4 .1. The ﬁrst step is to construct ( B , b ). W e let B ⊂ C be the sub-3- category having only one ob ject b = c , and only one 1 - morphism 1 b = 1 c . W e set Hom B ( b,b ) (1 b , 1 b ) := Hom C ( c,c ) (1 c , 1 c ) , with the same comp osition law. The ma p g : B → C is the inclusion. Note ﬁrst of a ll that B is a strict 3-g roup oid. This is ea sily seen using version (1 ) of the deﬁnitio n in Theor em 2.2.1 (but one has to lo ok a t the conditions in [136]). W e c a n also verify it us ing condition (3 ). Of course τ ≤ 1 ( B ) is the 1- category with only one ob ject and only o ne mo rphism, so it is a group oid. W e have to verify that B ( b, b ) is a s trict 2 -group oid. F or this, we again a pply co ndition (3) of 2.2.1. Her e we note that B ( b, b ) ⊂ C ( c, c ) is the full sub-2- category with only one ob ject 1 b = 1 c . Therefor e, in view of the deﬁnition o f τ ≤ 1 , we hav e that τ ≤ 1 ( B ( b, b )) ⊂ τ ≤ 1 ( C ( c, c )) is a full sub catego ry . A full sub ca teg ory of a 1-gr oup oid is aga in a 1- group oid, so τ ≤ 1 ( B ( b, b )) is a 1 -group oid. Finally , Hom B ( b,b ) (1 b , 1 b ) is a 1-gr o upo id since by co nstruction it is the same as Ho m C ( c,c ) (1 c , 1 c ) (whic h is a g roup oid by c ondition (3) applied to the stric t 2 -group oid C ( c, c )). This shows that B ( b, b ) is a s trict 2-gro up oid an hence that B is a strict 3 -group oid. Next, note that π 0 ( B ) = ∗ and π 1 ( B , b ) = { 1 } . On the other hand, for i = 2 , 3 we hav e π i ( B , b ) = π i − 2 ( Hom B ( b,b ) (1 b , 1 b ) , 1 2 b ) 4.4 Nonexist enc e of st rict 3 -gr oup oids giving rise to the 3 -t yp e of S 2 63 and s imilarly π i ( C , c ) = π i − 2 ( Hom C ( c,c ) (1 c , 1 c ) , 1 2 c ) , so the inclusion g induces an equality π i ( B , b ) = → π i ( C , c ). Ther efore, by deﬁnition (a) of equiv alence 2.2.3 , g is an equiv alence of strict 3- group oids. This completes the cons truction and v eriﬁcatio n for B a nd g . Before getting to the co nstruction of A and f , we analyze the strict 3-gro upo id B in terms of the discussion o f 2.1.2 and 4.2.1. Let G := Hom B ( b,b ) (1 b , 1 b ) . It is a n ab elia n monoid- ob ject in the categor y o f 1-g roup oids, with ab elian op era tion deno ted by + : G × G → G a nd unit element de- noted 0 ∈ G which is the same as 1 b . The ope ration + co rresp onds to bo th of the co mpo sitions ∗ 0 and ∗ 1 in B . The hypotheses on the homoto p y gro ups o f C also ho ld for B (since g was an equiv alence). These translate to the sta temen ts tha t ( π 0 ( G ) , +) = Z and G (0 , 0 ) = H . W e now constr uc t A and f via 2.1.2 and 4.2 .1, by co nstructing a morphism ( G ′ , +) → ( G , +) of ab elian monoid-ob jects in the catego ry of 1-gro upo ids. W e do this by a t yp e of “base- change” on the monoid of ob jects, i.e. we will ﬁr st deﬁne a mo r phism Ob( G ′ ) → Ob( G ) a nd then deﬁne G ′ to be the groupo id with ob ject se t Ob( G ′ ) but with morphisms corres p o nding to those of G . T o acco mplish the “ base-change”, sta rt with the following constr uc - tion. If S is a set, let co ds c ( S ) denote the group oid with S as set of ob jects, and with exactly one morphism betw een each pair of ob jects. If S has an ab elian monoid structure then co dsc ( S ) is an a belia n monoid ob ject in the ca tegory of gr oup oids. Note tha t fo r a n y group oid U there is a morphism of gro upo ids U − → codsc (Ob( U )) , and by “base change” we mea n the fo llowing op era tion: take a set S with a map p : S → Ob( U ) a nd lo o k a t V := co ds c ( S ) × cod sc (Ob( U )) U . This is a group oid with S a s set of o b jects, and with V ( s, t ) = U ( p ( s ) , p ( t )) . 64 The ne e d for we ak c omp osition A simila r co nstruction will be used later in Chapter 12 under the nota- tion V = p ∗ ( U ). F or the pr esent purp os es, note that if U is an ab e lian monoid ob ject in the categor y of group oids, if S is an ab elian monoid and if p is a map of mono ids then V is again an ab elian monoid ob ject in the ca tegory of group oids. Apply this as follows. Star ting with ( G , +) corres p o nding to B via 2.1.2 and 4.2 .1 a s ab ove, choos e ob jects a, b ∈ Ob( G ) s uch that the imag e of a in π 0 ( G ) ∼ = Z co r resp onds to 1 ∈ Z , and s uch that the image o f b in π 0 ( G ) corresp onds to − 1 ∈ Z . Let N denote the ab elian monoid, pro duct of tw o copies o f the na tur al num bers, with o b jects deno ted ( m, n ) for nonnegative integers m, n . Deﬁne a map of ab elian monoids p : N → Ob( G ) by p ( m, n ) := m · a + n · b := a + a + . . . + a + b + b + . . . + b. Note that this induces the surjection N → π 0 ( G ) = Z given by ( m, n ) 7→ m − n . Deﬁne ( G ′ , +) as the base-change G ′ := co ds c ( N ) × cod sc (Ob( G )) G , with its induced a belia n monoid op eration +. W e hav e Ob( G ′ ) = N , and the seco nd pro jection p 2 : G ′ → G (whic h induces p on ob ject sets) is fully fa ithful i.e. G ′ (( m, n ) , ( m ′ , n ′ )) = G ( p ( m, n ) , p ( m ′ , n ′ )) . Note that π 0 ( G ′ ) = Z via the map induced by p or equiv alently p 2 . T o prov e this, say that: (i) N surjects onto Z so the map induced by p is surjective; and (ii) the fact that p 2 is fully faithful implies that the induced map π 0 ( G ′ ) → π 0 ( G ) = Z is injective. W e le t A b e the strict 3-gro upo id corresp onding to ( G ′ , +) via 2.1.2, and let f : A → B b e the map c o rresp onding to p 2 : G ′ → G ag ain via 2.1.2. Let a be the unique ob ject o f A (it is mapp ed b y f to the unique ob ject b ∈ Ob( B )). The fact that ( π 0 ( G ′ ) , +) = Z is a gro up implies that A is a strict 3-gro upo id (4.2.1). W e hav e π 0 ( A ) = ∗ and π 1 ( A , a ) = { 1 } . Also , π 2 ( A , a ) = ( π 0 ( G ′ ) , +) = Z 4.4 Nonexist enc e of st rict 3 -gr oup oids giving rise to the 3 -t yp e of S 2 65 and f induces a n isomo rphism fro m here to π 2 ( B , b ) = ( π 0 ( G ) , +) = Z . Finally (using the notation (0 , 0) for the unit o b ject of ( N , + ) and the notation 0 for the unit o b ject of Ob( G )), π 3 ( A , a ) = G ′ ((0 , 0) , (0 , 0)) , and s imilarly π 3 ( B , b ) = G (0 , 0) = H ; the map π 3 ( f ) : π 3 ( A , a ) → π 3 ( B , b ) is an isomorphism because it is the same as the map G ′ ((0 , 0) , (0 , 0)) → G (0 , 0) induced b y p 2 : G ′ → G , a nd p 2 is fully faithful. W e hav e now co mpleted the v eriﬁcatio n that f induces isomor phisms o n the homotopy gro ups, so by version (a) of the deﬁnition of equiv alence 2 .2.3, f is a n e quiv a lence of s trict 3-gr o upo ids. W e now construct D a nd deﬁne the ma p h by a n explicit calculation in ( G ′ , +). First of all, let [ H ] denote the 1-gr o upo id with o ne ob ject denoted 0 , a nd with H as gro up o f endomo r phisms: [ H ](0 , 0) := H. This ha s a str uctur e of ab elian monoid- o b ject in the categor y of gr oup oids, denoted ([ H ] , +), b ecause H is a n a belia n g roup. Let D b e the s trict 3- group oid corres po nding to ([ H ] , +) via 2.1.2 and 4.2.1 . W e will cons truct a morphism h : A → D via 2.1.2 b y constructing a morphism of abelia n monoid ob jects in the ca tegory of g r oup oids, h : ( G ′ , +) → ([ H ] , +) . W e will construct this morphism s o tha t it induces the identit y morphism G ′ ((0 , 0) , (0 , 0)) = H → [ H ](0 , 0) = H. This will ins ur e that the morphism h has the proper t y required for 4.4.1. The ob ject (1 , 1) ∈ N go es to 0 ∈ π 0 ( G ′ ) ∼ = Z . Thus we may choose an iso morphism ϕ : (0 , 0) ∼ = (1 , 1) in G ′ . F o r a n y k let k ϕ denote the isomorphism ϕ + . . . + ϕ ( k times ) going from (0 , 0) to ( k , k ). On the other ha nd, H is the automo r phism gro up of (0 , 0) in G ′ . The op eratio ns + and comp ositio n coincide o n H . Finally , for any ( m, n ) ∈ N let 1 m,n denote the identit y automorphism of the o b ject ( m, n ). Then any arrow α in G may b e uniquely written in the form α = 1 m,n + k ϕ + u 66 The ne e d for we ak c omp osition with ( m, n ) the so urce of α , the ta rget being ( m + k , n + k ), a nd wher e u ∈ H . W e hav e the follo wing formulae for the comp o sition ◦ of ar rows in G ′ . They all co me fro m the basic rule ( α ◦ β ) + ( α ′ ◦ β ′ ) = ( α + α ′ ) ◦ ( β + β ′ ) which in turn co mes simply from the fac t that + is a morphism of group oids G ′ × G ′ → G ′ deﬁned on the cartesian pr o duct of t wo copies of G . Note in a similar vein that 1 0 , 0 acts as the iden tity for the op eratio n + on a rrows, and also tha t 1 m,n + 1 m ′ ,n ′ = 1 m + m ′ ,n + n ′ . Our ﬁr s t equation is (1 l,l + k ϕ ) ◦ l ϕ = ( k + l ) ϕ. T o pr ov e this note that l ϕ + 1 0 , 0 = l ϕ and our basic formula says (1 l,l ◦ l ϕ ) + ( k ϕ ◦ 1 0 , 0 ) = (1 l,l + k ϕ ) ◦ ( l ϕ + 1 0 , 0 ) but the left side is just l ϕ + k ϕ = ( k + l ) ϕ . Now our basic fo rmula, for a co mpo sition starting with ( m, n ), going ﬁrst to ( m + l , n + l ), then going to ( m + l + k , n + l + k ), g ives (1 m + l,n + l + k ϕ + u ) ◦ (1 m,n + l ϕ + v ) = (1 m,n + 1 l,l + k ϕ + u ) ◦ (1 m,n + l ϕ + v ) = 1 m,n ◦ 1 m,n + (1 l,l + k ϕ ) ◦ l ϕ + u ◦ v = 1 m,n + ( k + l ) ϕ + ( u ◦ v ) where of co urse u ◦ v = u + v . This formula shows tha t the mo r phism h fro m a rrows of G ′ to the group H , deﬁned by h (1 m,n + k ϕ + u ) := u is co mpatible with co mpo s ition. This implies that it pr ovides a morphism of gro upo ids h : G → [ H ] (recall from ab ov e that [ H ] is deﬁned to be the group oid with one ob ject whose a utomorphism group is H ). F urthermore the mor phism h is obviously compatible with the op eration + since (1 m,n + k ϕ + u ) + (1 m ′ ,n ′ + k ′ ϕ + u ′ ) = 4.4 Nonexist enc e of st rict 3 -gr oup oids giving rise to the 3 -t yp e of S 2 67 (1 m + m ′ ,n + n ′ + ( k + k ′ ) ϕ + ( u + u ′ )) and once aga in u + u ′ = u ◦ u ′ (the o per ation + on [ H ] b eing given b y the co mm utative op eration ◦ on H ). This completes the co ns truction of a morphism h : ( G , +) → ([ H ] , +) which induces the identit y on Hom (0 , 0). This cor resp onds to a mor- phism of strict 3-group oids h : A → D as requir e d to complete the pro of of P rop osition 4.4.1. W e can now give the nonrealization statement. Theorem 4.4. 2 L et ℜ b e any r e aliza tion functor satisfying the pr op - erties of Deﬁnition 4.1.1. Then ther e do es not exist a strict 3 -gr oup oid C such that ℜ ( C ) is we ak-e quivalent t o the 3 -trunc ation of t he homotopy typ e of S 2 . Pr o of Supp o se for the moment that we know Pro po sition 4.4.1; with this we will pr ove 4.4.2. Fix a realiza tion functor ℜ f or strict 3-gro up oids satisfying the axioms 4.1.1 , and as s ume that C is a s trict 3-gro upo id such that ℜ ( C ) is weak homoto p y-equiv alent to the 3-type o f S 2 . W e shall derive a c o nt radiction. Apply Propo sition 4.4.1 to C . Cho ose an ob ject c ∈ Ob ( C ). Note that, bec ause of the isomorphisms betw een homotopy sets or groups 4.1.1, w e hav e π 0 ( C ) = ∗ , π 1 ( C , c ) = { 1 } , π 2 ( C , c ) = Z and π 3 ( C , c ) = Z , so 4.4.1 applies with H = Z . W e obtain a seq uence o f str ic t 3 -group oids C ← g B ← f A h → D . This g ives the diagram of spaces ℜ ( C ) ← ℜ ( g ) ℜ ( B ) ← ℜ ( f ) ℜ ( A ) ℜ ( h ) → ℜ ( D ) . The a xioms 4.1.1 for ℜ imply that ℜ tra nsforms eq uiv alences of strict 3-gro upo ids in to w eak ho motopy equiv alences o f spaces. Thus ℜ ( f ) and ℜ ( g ) are weak homotopy equiv alences a nd we get that ℜ ( A ) is weak homotopy equiv alen t to the 3- t yp e of S 2 . On the other ha nd, a gain by the axio ms 4.1 .1, we hav e that ℜ ( D ) is 2-connected, and π 3 ( ℜ ( D ) , r ( d )) = H (via the isomo r phism π 3 ( D , d ) ∼ = H induced by h , f and g ). By the Hurewicz theorem there is a cla ss η ∈ H 3 ( ℜ ( D ) , H ) which induces a n iso morphism Hur ( η ) : π 3 ( ℜ ( D ) , r ( d )) ∼ = → H . Here Hur : H 3 ( X, H ) → H om ( π 3 ( X, x ) , H ) 68 The ne e d for we ak c omp osition is the Hurewicz ma p for a ny po in ted space ( X , x ); and the co homol- ogy is singular cohomolo gy (in par ticular it only depends on the w eak homotopy t yp e of the space). Now lo ok at the pullba ck of this class ℜ ( h ) ∗ ( η ) ∈ H 3 ( ℜ ( A ) , H ) . The hypo thesis that ℜ ( u ) induces an isomorphism on π 3 implies that Hur ( ℜ ( h ) ∗ ( η )) : π 3 ( ℜ ( A ) , r ( a )) ∼ = → H . In par ticular, Hur ( ℜ ( h ) ∗ ( η )) is nonzero so ℜ ( h ) ∗ ( η ) is nonzer o in H 3 ( ℜ ( A ) , H ). This is a contradiction bec a use ℜ ( A ) is weak homotopy-equiv alen t to the 3-type o f S 2 , and H = Z , but H 3 ( S 2 , Z ) = { 0 } . This c o nt radiction completes the pro of of the theorem. As was discussed in [108], this r esult motiv ates the s earch for a notio n of higher category weak er than the notion of strict n -category . F ollowing the yoga describ ed by Lewis [151], it app ears to b e suﬃcient to weaken any single pa rticular asp ect. 5 Simplicial approac hes There a r e a num b er of a ppr oaches to w eak higher categories based on the s implicial categor y ∆, inc luding the Segal approach and its itera - tions which ar e the main sub ject of o ur bo o k. W e also discuss several other a ppr oaches, which co ncern ﬁrst and foremost the theory of ( ∞ , 1)- categorie s. 5.1 Strict simplicial categories A simplicial c ate gory is a K -enric hed ca tegory . It has a set of ob jects Ob( A ), and for each pair x, y ∈ Ob( A ) a simplicial se t A ( x, y ) tho ught of a s the “ space o f morphisms” from x to y . The comp osition maps are morphisms of simplicia l sets A ( x, y ) × A ( y , z ) → A ( x, z ) satisfying the asso ciativity condition strictly , that is for a ny x, y , z , w the dia gram of simplicial sets A ( x, y ) × A ( y , z ) × A ( z , w ) → A ( x, y ) × A ( y , w ) A ( x, z ) × A ( z , w ) ↓ → A ( x, w ) ↓ commutes. The identities of A ar e po int s (i.e. vertices) of the simpli- cial sets A ( x, x ) sa tis fying left and right identit y conditions which are equalities of ma ps A ( x, y ) → A ( x, y ). A functor of simplicial categories f : A → B c o nsists of a map f : Ob( A ) → Ob( B ) a nd for each x, y ∈ Ob( A ), a ma p of simplicial sets f x,y : A ( x, y ) → B ( f ( x ) , f ( y )), co mpatible with the compo sition maps and identities in a n obvious wa y . I n keeping with our genera l no tation This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 70 Simplicia l appr o aches for enriched categor ie s, the ca tegory o f simplicial ca tegories is deno ted Ca t ( K ). Given a simplicial categor y A , we deﬁne its trun c atio n τ ≤ 1 ( A ) to b e the ca tegory whose set of ob jects is the same as Ob( A ), but for any x, y ∈ Ob( A ) τ ≤ 1 ( A )( x, y ) := π 0 ( A ( x, y )) . The comp osition maps a nd identities for A deﬁne composition maps and ident ities for τ ≤ 1 ( A ), and we obtain a functor τ ≤ 1 : Ca t ( K ) → Ca t . A functor f : A → B b etw een simplicial categor ies is s a id to b e ful ly faithful if for ev ery x , y ∈ Ob( A ) the map f x,y : A ( x , y ) → B ( f ( x ) , f ( y )) is a weak equiv alence of s implicia l s e ts, in other words a w eak equiv a- lence in the standard model structure of K . A functor f is said to b e essential ly surje ctive if the functor τ ≤ 1 ( f ) b e t ween usual 1-c a tegories is essentially surjective, in other words it induces a surjection on se ts of isomorphism classe s Iso τ ≤ 1 ( A ) ։ Iso τ ≤ 1 ( B ). A functor f : A → B is said to b e a Dwyer-Kan e quivalenc e b etw een simplicial categories , if it is fully faithful and essentially s ur jective. In this cas e , τ ≤ 1 ( f ) is a lso an equiv alence of categ ories, in particula r it is bijective o n s ets of is omor- phism cla sses. Given a simplicial c a tegory A , its underlying c ate go ry is the category with ob jects Ob( A ), but the mo rphisms fro m x to y a re the set of po int s or vertices of the simplicia l set A ( x, y ). This is not to b e confused with τ ≤ 1 ( A ), but there is a na tural pro jection functor fro m the underlying category to the truncated categor y . An “arrow” in A fr om x to y means a ma p in this underlying categ ory; such an arrow is said to b e an int ern al e quiva lenc e if it pr o jects to an isomorphism in τ ≤ 1 ( A ). In these terms, the ess ent ial surjectivity condition for a functor f : A → B may b e rephrased as saying that any ob ject of B is internally equiv alen t to the image o f an o b ject o f A . Dwyer, Kan a nd Bergner have constr ucted a mo del ca tegory structure on Ca t ( K ) such that the weak equiv alences ar e the Dwyer-Kan equiv a- lences, and the ﬁbratio ns are the functors f : A → B such that ea ch f x,y is a ﬁbration of s implicial sets, and furthermore f s atisﬁes an additional lifting condition whic h basica lly says that an internal equiv alence in B should lift to A if one o f its endp oints lifts. It is in teresting to note that Dwyer and Kan star ted ﬁrst b y c o n- structing a mo del s tructure on Ca t ( X , K ), the catego ry of simplicial 5.2 Se gal’s delo op ing machine 71 categorie s with a ﬁxed set of ob jects X . Refer to their pap er s [89] [90] [91]. W e will a lso adopt this route, following a sugges tion b y Clark Ba r- wick. Simplicial categor ie s app ear in an imp ortant way in ho motopy the- ory . Quillen deﬁned the notion of simplicial mo del c ate gory , and of N is a simplicial mo del catego ry then we obtain a simplicial c a tegory N spl cf of ﬁbr ant and coﬁbrant o b jects, such that its tr uncation τ ≤ 1 ( N spl cf ) ∼ = ho( N ) is the ho motopy categ ory o f N . Dwyer and Kan then developped the theory o f simplicia l lo c alization whic h gives a go o d simplicial c ate- gory even when N do esn’t have a simplicial mo de l structure. If C is any category , and if we are given a sub collection of ar rows Σ ⊂ Arr ( C ), then Dwyer and Kan deﬁne a simplicial categ ory L ( C , Σ) whose trunca tio n is the class ical Gabriel-Zis ma n lo calizatio n: τ ≤ 1 ( L ( C , Σ)) ∼ = Σ − 1 ( C ). In the c a se wher e N is a simplicia l mo del catego ry , then the tw o options L ( N , N w ) (where N w denotes the cla ss of w eak eq uiv a le nces) and N spl cf , are Dwyer-Kan equiv alen t as simplicial ca tegories [9 0] [9 1]. Even though simplicial catego ries hav e str ictly asso ciative comp osi- tion, they are weak er than str ic t n -categor ies in the sense that the highe r categoric al structure is enco ded by the simplicial morphism sets r a ther than by stric t n − 1 -categor ies. Hence, the need for w eak comp osition de- scrib ed in the previous chapter, is not contradicatory with the fact that strict simplicia l categories mo del all ( ∞ , 1)-catego r ies. F o r the weak er versions to b e discussed next, one can rectify back to a strict simpli- cial category , as was orig inally shown by Dwy er-Ka n-Smith [92 ] and Sch w¨ anzl-V o gt [184], then extended to Quillen equiv alences b etw een the corres p o nding mo del structures by Bergner [34]. 5.2 Segal’s delo oping mach ine The b est-k nown version of Segal’s theor y is his notion of inﬁnite delo op- ing machine o r Γ-space. Gro thendieck mentioned so me cor r esp ondence from Larry Breen in 19 75 co ncerning this ide a : Dear Larry , . . . The construction which you prop ose for the n otion of a non-strict n - category , and of the nerve functor, has certainly th e merit of ex isting, and of b eing a ﬁrst precise approach . . . Otherwise, not having understoo d the idea of Segal in your last letter. . . In the ﬁrst letter of 19 83, Gr othendieck als o mentioned the notion of m ultisimplicial nerve of a (strict) n -categ ory . So it would seem that the 72 Simplicia l appr o aches idea of applying Segal’s delo oping machine muc h as was done in [206], was present in some sense at the time. The s ta rting po in t is Segal’s 1-delo oping machine. Recall from top ol- ogy that for a p o inted space ( X, x ) the lo op sp ac e Ω X is the space of po int ed lo o ps ( S 1 , 0) → ( X , x ). These ca n b e c o mpo sed by repara metriz- ing the lo ops in a well-known way , a lthough the res ulting comp osition is only as so ciative and unital up to homotopy . It is p os sible to replace Ω X by a top olo gical gro up, for ex ample with Quillen’s realiz a tion of homotopy types as coming from simplicial gro ups. A clas sifying space construction usually denoted B ( · ) allows one to get back to the or iginal space: B (Ω X ) ∼ X . A po pular question in top olog y in the 1960 ’s was how to deﬁne v arious t yp es of structure on spaces homotopy equiv alen t to Ω X , which would be weak er than the strong structure o f topo logical g roup, but which would include suﬃciently muc h homotopical data to let us get back to X by a classifying spa ce constructio n B ( · ). Such a kind of structure was known as a “deloo ping m achine”. Ther e w ere a num ber of examples including A ∞ -algebra s (in the linearized case), PROP’s, op era ds, a nd the one whic h w e will be considering: Segal’s simplicial delo oping machine. An y of these delo oping mac hines should lea d to one or s e veral notions of higher category , indeed this has b een the case as we shall disc us s elsewhere. Let ∆ denote the simplicial catego ry who se ob jects are denoted m for po sitive integers m , and where the morphisms p → m are the (not- necessarily s trictly) order- preserving maps { 0 , 1 , . . . , p } → { 0 , 1 , . . . , m } . A morphism 1 → m sending 0 to i − 1 and 1 to i is ca lled a princip al e dge of m . A morphism which is not injective is called a de ge ner acy . A s implicia l set A : ∆ o → Set such that A 0 = ∗ and such that the Se gal maps obta ined by the principal edges (01) , (12) , . . . , (( n − 1 ) n ) ⊂ (0123 · · · n ) = [ n ] A m → A 1 × · · · × A 1 5.2 Se gal’s delo op ing machine 73 are is omorphisms of se ts , cor resp onds to a structure of monoid on the set A 1 . Indeed, the dia gram A 2 ∼ = → A 1 × A 1 A 1 ↓ where the horizontal map is the Segal map and the vertical map is given by the third edg e (02) ⊂ (01 2), provides a co mpo sition law A 1 × A 1 → A 1 . The degenera cy map A 0 → A 1 provides a unit—proved using the degeneracy maps for A 2 —and consideration of A 3 gives the pro of of asso ciativity . In Segal’s 1 -delo oping theo ry , this character ization of mono ids is weak- ened by replacing the condition of iso mo rphism by the condition of weak homotopy equiv alences (i.e. maps inducing isomor phisms o n the π i ). Thu s, a lo op s pace is deﬁned to be a simplicia l space A · : ∆ o → Top such that A 0 is a single po in t, suc h that the Segal maps, ag ain using the principal edges (01) , (12) , . . . , (( n − 1) n ) ⊂ (01 23 · · · n ) = [ n ] A m → A 1 × · · · × A 1 are weak homotopy equiv a le nces, and which is gr ouplike in that the monoid which results when we comp ose π 0 ◦ A · : ∆ o → Top → S et should be a gr oup. Segal explains in [187] ho w to deloop such an ob ject: if Top is replaced b y the categ ory of simplicia l s ets then the structure A · is a bisimplicial set, and its de lo oping B ( A· ) is just the dia gonal realization. As was well known at the time, the characteriz a tion of monoids gen- eralizes to give a characterization of the nerve of a catego ry in terms of Segal maps b e ing isomorphisms. Indeed, a monoid ca n b e viewed a s a categor y with a sing le o b ject, and a s mall change in the deﬁnition makes it apply to the case of categor ies with an arbitrary set of ob jects : a simplicial s et A · : ∆ o → Set 74 Simplicia l appr o aches is the nerve o f a 1-catego ry if and only if the Sega l maps ma de using ﬁber pr o ducts are isomo rphisms A m ∼ = → A 1 × A 0 · · · × A 0 A 1 . Here the ﬁb er pro ducts are taken over the tw o maps A 1 → A 0 corre- sp onding to (0) ⊂ (01 ) a nd (1) ⊂ (01), alternating ly and starting w ith (1) ⊂ (0 1 ). These cor r esp ond to the inclusions of the intersections of adjacent principal edges. 5.3 Segal categories A Se gal pr e c ate go ry is a bisimplicial set A = {A p,k , p, k ∈ ∆ } in other words a functor A : ∆ o × ∆ o → Set s a tisfying the globular c ondi tion that the s implicial set k 7→ A 0 ,k is c o nstant equa l to a set which w e denote by A 0 (called the set of obje cts ). If A is a Sega l pr ecategor y then for p ≥ 1 we obta in a simplicial set k 7→ A p,k which w e denote by A p/ . This yields a simplicial collectio n of s implicial sets, o r a functor ∆ o → K to the Kan-Quillen mo del categor y K of simplicial sets. One could instead lo o k at s implicia l sp ac es , tha t is functors ∆ o → Top suc h tha t A 0 is a discr ete spa ce t hought of as a set. This gives an equiv alent theor y , although there are degeneracy pr o blems which appar ently need to b e treated in an app endix in that ca se [16 7] [208]. W e often use the “ simplicial s pa ce” p oint of view when sp eaking informally , as it is more intuitiv ely comp elling; howev er, we do n’t wan t to g et into deta ils o f deﬁning a model structure o n Top , and instead use K for technical statements. F or each m ≥ 2 there is a mor phism o f simplicial s ets whose co m- po nent s are g iven by the principal edges o f m , which we call the Se gal map : A m/ → A 1 / × A 0 . . . × A 0 A 1 / . The morphisms in the ﬁb er pro duct A 1 / → A 0 are alterna tively the inclusions 0 → 1 sending 0 to the ob ject 1, or to the o b ject 0 . W e w ould like to t hink of the in verse ima ge A 1 / ( x, y ) of a pair ( x, y ) ∈ 5.3 Se gal c ate gories 75 A 0 × A 0 by the tw o maps A 1 / → A 0 referred to abov e, as the simplicial set of maps fr om x to y . W e say that a Se g al precatego ry A is a Se gal c ate gory if for all m ≥ 2 the Seg a l ma ps A m/ → A 1 / × A 0 . . . × A 0 A 1 / . are weak e q uiv a lences o f s implicia l sets. This no tion was intro duced by Dwyer, Kan a nd Smith [92] and Sch w¨ anzl and V ogt [184]. Given a strict simplicia l categor y A , we obtain a corr esp onding Segal precatego r y by setting A n/ := a ( x 0 ,...,x n ) ∈ Ob( A ) n +1 A ( x 0 , x 1 ) × · · · × A ( x n − 1 , x n ) . This is a Segal catego ry , b ecause the Segal maps are isomorphisms . In the o ther direction, a Sega l categ ory such that the Segal maps ar e is o- morphisms co mes from a unique strict simplicial category . The “generato rs a nd r e lations” op eration int ro duced in Cha pter 16 is a wa y of starting with a Se g al pr ecategor y and enfor cing the condition of bec oming a Segal category , by forcing the condition of weak equiv alence on the Sega l maps. As a general ma tter we will ca ll op eratio ns of this t yp e A 7→ Seg ( A ). Suppo se A is a Segal catego r y . Then the simplicial set p 7→ π 0 ( A p/ ) is the nerve of a category which we call τ ≤ 1 A . W e say that A is a Se gal gr oup oid if τ ≤ 1 A is a group oid. This means that the 1-mo rphisms of A are invertible up to equiv a lence. In fact we can make the sa me deﬁnition even for a Segal preca tegory A : we deﬁne τ ≤ 1 A to b e the simplicia l set p 7→ π 0 ( A p/ ). W e can now describ e ex a ctly the situa tio n envisaged in [3] [18 7]: a Segal catego ry A with o nly one o b ject, A 0 = ∗ . W e ca ll this a Se gal monoid . If A is a gr oup oid then the homotopy theoris ts ’ terminolo gy is to s ay that it is gr oupli ke . 5.3.1 Equiv alences of Segal categories The ba sic in tuition is to think o f Segal ca tegories as the na tur al w eak version of the notion of top olog ical (i.e. Top -enriched) category . One of the main concepts in ca tegory theory is that of a functor which is an “equiv alence of categories”. This ma y b e generalized to Segal categ ories. F or simplicial (i.e. K -enric hed) categor ies, this notion is due to Dwy er and Kan, and is often called DK-e quivale nc e . The sa me thing in the 76 Simplicia l appr o aches context o f n -c a tegories is w ell-known (see Kapranov-V oe vodsky [13 6] for example); in the w eak case it is describ ed in T amsamani’s pa pe r [2 06]. W e say that a mor phis m f : A → B of Segal categ ories is an e quiva- lenc e if it is ful ly faithful , mea ning that for x, y ∈ A 0 the map A 1 / ( x, y ) → B 1 / ( f ( x ) , f ( y )) is a weak e quiv a lence of s implicial sets; and essent ial ly s u rje ctive , mean- ing that the induced functor o f categorie s τ ≤ 1 ( A ) τ ≤ 1 f → τ ≤ 1 ( B ) is surjective o n iso morphism classe s of ob jects. Note that this induced functor τ ≤ 1 f will be an equiv alence of categories as a consequence of the fully faithful condition. The homotop y theory that w e a re in terested in is that of the category of Segal ca tegories mo dulo the a b ove notion of equiv alence. In particular, when w e search for the “ right answer” to a question, it is only up to the ab ov e type of equiv a lence. Of course when dealing with Sega l categories having only one ob ject (as will actua lly b e the case in what follows) then the essen tially surjective co ndition is v ac uo us and th e fully faithful condition just amounts to eq uiv a le nc e on the level of the“underlying space” A 1 / . In order to have an appropriately rea s onable po int of view on the ho- motopy theory of Segal ca tegories o ne should lo ok at the clo sed mo del structure (which is one o f our main goals, sp ecialized to the mo del ca t- egory K , see Cha pter 17): the rig h t notion of weak mor phism from A to B is that of a mo r phism from A to B ′ where B ֒ → B ′ is a ﬁbra nt replacement of B . 5.3.2 Segal’s theorem W e deﬁne the r e ali zation of a Sega l catego ry A to b e the space |A| which is the realizatio n of the bis implicial set A . Supp ose A 0 = ∗ . Then we hav e a morphism |A 1 / | × [0 , 1 ] → |A| giving a mor phism |A 1 / | → Ω |A| . The notation |A 1 / | means the realiza tio n of the simplicial set A 1 / and Ω |A| is the lo op space base d at the ba sep oint ∗ = A 0 . 5.3 Se gal c ate gories 77 Theorem 5.3.1 (G. Segal [187], Pr op osition 1.5) Supp ose A i s a Se ga l gr oup oid with one obje ct. Then the morphism |A 1 / | → Ω |A| . is a we ak e quivalenc e of sp ac es. Refer to Segal’s pap er, o r also May ([16 6] 8.7), for a pro o f. T a msamani noted that the same works in the case o f many ob jects, and indeed this was a key step in his pro of of the top o lo gical rea lization theorem for n -categor ies. Corollary 5.3. 2 Su pp ose A is a Se gal gr oup oi d. Then the morphism |A 1 / | → Ω |A| . is a we ak e quivalenc e in K . Pr o of [206]. In order to do these things inside the world of simplicia l spaces, the additional coﬁbr ancy conditions in Top w ould necess ita te a dis cussion of “whiskering” as is s tandard in deloo ping and classifying space construc- tions (cf [187] [165], [167], [208]). This is why w e ha ve repla ced “spaces” by “simplicia l sets” in the ab ov e discuss io n, and corresp onds a lso to our use of Reedy mo del s tructures in the ma in ch apters. 5.3.3 ( ∞ , 1) -categories Simplicial categories and Segal categories are tw o models for what Lurie calls the no tio n o f ( ∞ , 1) -c ate gory , meaning ∞ -catego ries where the i - morphisms are inv ertible (ana logous to b eing an inner e q uiv a lence) fo r i ≥ 2 . The Dwy er-Ka n simplicia l lo ca lization may b e viewed as the lo calization in the ( ∞ , 2)-ca tegory of ( ∞ , 1)-ca tegories. Part of o ur goal in this bo ok is to dev elop algebraic formalism useful fo r lo ok ing at these situations, as well as their iter ative counterparts for ( ∞ , n )-categ ories. 5.3.4 It e ra tion A s ubtle p oint is that simplicial catego ries don’t b ehave w ell under di- rect pro ducts: the Dwyer-Kan- B ergner mo del structure o n Ca t ( K ) is not cartesia n for the direct pro duct b ecause a pro duct o f tw o coﬁbrant ob jects is no longer coﬁbrant. Thus, if we try to co n tinue by lo oking at Ca t ( Ca t ( K )) the resulting theo ry do e s n’t hav e the rig h t pro pe r ties. 78 Simplicia l appr o aches This ho oks up with what we have seen in Cha pter 4, that iterating the strict enriched categor y constructio n, do esn’t lead to enough ob jects. If we use Segal’s metho d, o n the other hand, one can iterate the con- struction with a b etter eﬀect. This lea ds to T amsama ni’s itera tive deﬁ- nition of n -categor ies. See Chapter 7. It is also related to Dunn’s theory of iter ated n -fold Segal deloo ping mac hines [85], and it will undoubtedly be proﬁtable to compare [206] a nd [85]. The next tw o sections will be devoted to brief descriptions of t wo other ma jo r p oints of view on ( ∞ , 1)-categ ories. After that, we discuss the co mparison b etw een the v arious theorie s. 5.3.5 Strictiﬁcation and Bergner’s comparison result The v arious mo dels of ( ∞ , 1)-ca teg ories discussed ab ove all furnis h essen- tially the same homo topy theory . Such a rectiﬁcation res ult was known very early for homotopy mono id structures. F or Seg a l catego ries, the ﬁrst r esult of this kind was due to Dwyer, Kan and Smith who show ed how to r ectify a Seg al categ ory into a strict simplicial ca tegory in [92]. Similarly , Scw¨ anzl a nd V o gt show ed the same thing in their pap er in- tro ducing Segal categor ies [1 84]. T he full homotopy e q uiv a lence result, stating that the rectiﬁcation op e r ation is a Quillen equiv alence b etw een mo del structur e s, was s hown by B ergner [3 4] at the same time as s he constructed the mo del structure s in question. The mo del structure for Segal catego ries is the sp ecial ca se M = K o f the globa l construction we are do ing in the present b o ok. With respe ct to the mo dels we ar e g oing to discuss next, Berg ner also gav e a Quillen e q uiv a lence with Rezk’s mo del catego ry of complete Segal spaces, and the compa r ison c an b e extended to q uasicatego ries to o , as shown b y Joy al and Tierney in [13 0]. 5.4 Rezk categories Rezk has given a diﬀerent wa y of using the Sega l maps to sp ecify a n ( ∞ , 1)-catego r ical structure. Bar wick showed how to itera te this con- struction, and this iteration has now also been ta ken up b y Lurie and Rezk. Their itera tion is philos ophically similar to wha t we a re doing in the main part of this bo ok. In the pr esent section w e discuss Rezk’s orig- inal case, which he called “complete Seg al spaces” . These ob jects enter int o Bergner’s three-way c o mparison [34]. 5.4 Re zk c ate gories 79 It will b e co n venien t to start our discussion b y refering to a g eneric notion of ( ∞ , 1)-catego ry , which co uld be concretized by simplicial cat- egories, or Sega l categor ies. Recall that an ( ∞ , 1)-ca tegory A ha s a n ( ∞ , 0)-catego r y or ∞ -gro upo id, a s its interior denoted A int . The inte- rior is the universal ∞ -gr oup oid mapping to A . F o r any x, y ∈ Ob( A ), the mapping spa ce is the subspace A int ( x, y ) ⊂ A ( x, y ) union of a ll the connected co mpo nen ts corr esp o nding to ma ps which are inv ertible up to equiv a le nce. By Segal’s theorem (which will b e dis cussed further in Chapter 17), this co rresp onds to a space which we can denote b y |A int | . It is the “mo duli spa ce of ob jects of A up to equiv alence”: there is a separate co nnected co mpo ne nt for eac h equiv alence c la ss of ob jects. The vertices c o ming fro m the 0- simplices co rresp ond to the or iginal ob jects Ob( A ), and within a connected comp onent the space of paths fr o m one vertex to another, is the spa c e A int ( x, y ) of eq uiv a le nces b etw een the corres p o nding o b jects. In Rezk’s theory , our ( ∞ , 1)-catego r y is r e presented by a simplicial space A R with A R 0 = |A int | in degr ee z ero. The homotopy ﬁb er of the map A R 1 → A R 0 × A R 0 ov er a p oint ( x, y ) is (canonically eq uiv alent t o) the space of morphis ms A ( x, y ). The categor ical structure is deﬁned by imp o sing a Segal con- dition on homotopy ﬁb er pr o ducts: for any n , there is a version o f the Segal ma p g oing to the homotopy ﬁber pro duct A R n → A R 1 × h A R 0 A R 1 × h A R 0 · · · × h A R 0 A R 1 and this is require d to b e a weak equiv alence. A c omplete Se gal sp ac e is a simplicial space satisfying these Segal conditions, and also the c om- pleteness c o ndition which cor resp onds to the r equirement A R 0 = |A int | . W e had for m ulated that r equirement by ﬁrst considering a generic the- ory of ( ∞ , 1)-catego ries. Internally to Rez k ’s theor y , the co mpleteness condition sa ys that A R, int 1 → A R 0 × A R 0 should be equiv alent to the path space ﬁbr a tion, where A R, int 1 ⊂ A R 1 denotes the union of connected com- po nent s co rresp onding to morphisms which are in vertible up to equiv- alence. This co ndition is shown to b e equiv alen t to a more abstra ct condition useful f or manipulating the model structure, see [178, 6 .4] [34, 3.7] [1 30, Section 4]. Rezk’s theory is a little bit more complica ted in its initial stages than the theo ry of Segal categor ies. The Segal maps g o to a ho motopy ﬁber pro duct, which nevertheless ca n be a ssumed to b e a regula r ﬁb er pr o d- 80 Simplicia l appr o aches uct by imp osing a Reedy ﬁbrant condition on the s implicial space, for example. Since the set o f ob jects is not really to o well-deﬁned, the kind of reasoning which w e are cons idering here ( and whic h was also follow ed by Dwyer a nd Kan in their series of pap ers), break ing up the pro blem int o ﬁr st a problem for higher categories with a ﬁxed set o f ob jects, then v a rying the s et of o b jects, is les s av a ilable. On the other hand, Rezk’s theory has the adv a n tage that A R is a canonical mo del for A up to levelwise homo topy equiv alence in the cat- egory of diagra ms ∆ o → Top . Thus, a map A R → B R of complete Segal space s , is an equiv alence if and only if each A R n → B R n is a weak equiv alence o f spaces (and it suﬃces to chec k n = 0 and n = 1 b ecause of the Seg al conditions). This contrasts with the case o f Segal categories, where the set of ob jects A 0 = Ob ( A ) is not in v a riant under equiv alences of ca tegories. As Berg ner has p ointed out [40], the canonica l nature of the spaces in- volv ed makes Rezk’s theory particularly a menable to calculating limits. F or exa mple, if A R → B R ← C R are tw o arrows betw een complete Segal spa ces, then the levelwise homo- topy ﬁbe r pr o duct U R n := A R n × h B R n C R n is again a complete Segal space, and it is the right homotopy ﬁb er pr o d- uct in the world of ( ∞ , 1 )- categories . This aga in contrasts with the case of Seg al categ ories, or indeed even usual 1- categories . F or example, letting E deno te the 1 -categor y with t wo isomorphic ob jects υ 0 and υ 1 , the inclusion maps { υ 0 } → E ← { υ 1 } are e q uiv a lences of categories, so the homotop y ﬁb er pro duct in an y rea- sonable mo del str ucture for 1-c ategories , should also b e equiv alen t to a discrete singleton categ ory . How ever, the ﬁb er pr o duct o f categories , or of s implicial sets (the nerves) is empt y . In Rezk’s theory , the degr ee 0 space E R 0 will again be c o nt ractible, since ther e is only a single eq uiv a - lence c lass of o b jects of E a nd the have no no nt rivial automorphisms. As usual, for treating the technical asp ects o f the theory it is b etter to lo ok a t bisimplicial sets rather than s implicial spaces. Rezk constructs a mo del structure on the c a tegory of bisimplicial sets, such that the ﬁbrant ob jects are complete Seg al spaces which ar e Reedy ﬁbr ant as 5.5 Qu asic ate gories 81 ∆ o -diagra ms and levelwise ﬁbra n t [178]. Bergner cons iders further this theory and shows the e q uiv a lence with simplicial catego ries and Seg a l categorie s [34]. Barwick has sug gested to iterate this construc tio n to a Rezk-style theory of ( ∞ , n ) catego r ies for a ll n , and Rezk has taken this up in [179]. He sho ws th at the resulting m o del categories are ca rtesian, in pa r ticular this gives a constr uctio n of the ( ∞ , n + 1)-catego r y of ( ∞ , n )-ca teg ories. 5.5 Quasicategories Joy al and Lurie ha ve developped extensively the theory of quasic ate- gories . These ﬁrst a ppea red in the b o ok of Boa rdman and V og t [42] under the na me “restricted Kan co mplexes”. An important example ap- pea red in work of Co rdier a nd Porter [74]. A goo d place to start is to rec all Kan’s o riginal horn-ﬁlling conditions for the categ ory of simplicial sets K . As K is a ca tegory of diag rams ∆ o → Set , we hav e in particular the r epr esentable diagr ams which we shall denote R ( n ), deﬁned by R ( n ) m := ∆([ m ] , [ n ]). T his is the “s tan- dard n -simplex” , classically denoted by R ( n ) = ∆[ n ]. F or our purp oses this classical notation would seem to risk some confusion with to o many symbols ∆ around, so w e call it R ( n ) instead. Now, R ( n ) has a stan- dard s implicial subset deno ted ∂ R ( n ), which is the “b ounda ry”. It can be deﬁned as the n − 1-skeleton of R ( n ), or as the union of the n − 1 - dimensional faces of R ( n ). The faces are indexed by 0 ≤ k ≤ n ; in terms o f linearly order ed sets, the k - th face corres po nds to the linear ly ordered subset of [ n ] o btained by cr ossing out the k -th element. Now, the k -th horn Λ( n, k ) is the subset of ∂ R ( n ) which is the union of all the n − 1- dimensional face s ex cept the k -th one. If X ∈ K is a simplicial s et, then the universal pro per ty o f the rep- resentable R ( n ) says that X n = H om K ( R ( n ) , X ). Kan’s horn-ﬁl ling c ondi tion sa ys that a n y map Λ( n, k ) → X extends to a map R ( n ) → X . The simplicial sets X satisfying this co nditio n ar e the ﬁbrant ob jects of the mo del str ucture on K . Boardman and V ogt intro duced the r estricte d Kan c ondition , satis- ﬁed by a simplicial set X whenever any map Λ ( n, k ) → X extends to R ( n ) → X , for ea ch 0 < k < n . In other words, they co nsider only the horns o btained by taking out any except for the ﬁrst and last faces. This c ondition corresp onds to keeping a directio na lit y of the 1-cells in X . T his may be seen most clearly by looking at the case n = 2. A 2-cell 82 Simplicia l appr o aches may be drawn as R (2) : r r r h g f where h , g and f are the 1-cells cor resp onding to e dges (01), (1 2 ) and (02) resp ectively . Such a 2-cell is thought of as the r elation f = g h . In the usua l Kan condition, there are three ho rns which need to b e ﬁlled: Λ(2 , 0) : r r r h ? f Λ(2 , 1) : r r r h g ? Λ(2 , 2) : r r r ? g f How ev er, in the restr icted Kan co nditio n, only the middle horn Λ(2 , 1) is required to be ﬁlled. This corresp onds to saying that for any comp osa ble arrows g and h , there is a compo sition f = gh . On the other hand, ﬁlling the horn Λ(2 , 0) w ould c orresp ond to saying that given f and h , there is g such that f = g h , which essentially means we lo o k for g = f h − 1 ; and ﬁlling Λ(2 , 2) would corresp ond to saying tha t given f and g ther e is h such that f = g h , that is h = g − 1 f . When we lo o k at t hings in this w ay , it is clear that the full Kan condi- tion c o rresp onds to imp osing , in addition to the categ orical comp os ition of arrows, some kind of gro upo id condition of existence o f in verses. It isn’t sur prising, then, that Kan complexes corresp ond to ∞ -gro upo ids. F ollowing through this philoso ph y has le d Joyal to the theory o f qua - sicategor ies, which a re simplicial sets sa tis fying the restricted Kan co n- 5.6 Going b etwe en Se gal c ate go ries and n -c ate gories 83 dition, but viewed as ( ∞ , 1)-catego ries with arrows which a re not nec- essarily inv ertible. Making the translation from restricted Ka n simplicial s ets to ( ∞ , 1 )- categorie s is not altog ether trivia l, mos t notably for any tw o vertices x, y of a q uasicatego ry X we nee d to deﬁne the simplicial mapping sp ac e X ( x, y ); o ne p oss ibilit y is to say that it is the Kan simplicial set k 7→ Hom x,y ( R (1) × R ( k ) , X ) where the supers cript indicates maps s ending 0 × R ( k ) to x and 1 × R ( k ) to y . Ther e is also a wa y of describing directly a simplicia l categor y which is the rectiﬁcation o f the co r resp onding ( ∞ , 1 )- category ; see [18 0] for a detailed discussion. Joy al constructs a mo del category str uc tur e whose underlying cate- gory is that of simplicial sets, in for which the ﬁbr a nt ob jects a re ex a ctly those s atisfying the restric ted Kan condition. The pas sage from a g eneral simplicial set to its ﬁbran t repla cement, done b y enfor cing the res tr icted Kan horn ﬁlling conditions using the small ob ject a rgument, is a version of the “calculus o f gener ators and relations” very similar to what w e will be discussing in Cha pters 16 and 1 7 for the case of Seg al categories. In the three basic kinds of simplicial ob jects which we now hav e repre- senting ( ∞ , 1)-catego ries with weak comp os itio n, we ca n see a trade-oﬀ betw een infor mation conten t and simplicity . The simplest mo del is that of quasica tegories, which are just simplicial sets satis fying a very classi- cal hor n-ﬁlling condition; but in this c ase it isn’t ea sy to ge t back some of the main pieces o f information in an ( ∞ , 1)-ca tegory suc h as the sim- plicial mapping sets . A t the other end, in Rezk’s c o mplete Segal spa ces, the full information of the ∞ -group oid interior is contained within the ob ject, to the extent that the ho mo topy type of the ∆ o -diagra m is an inv ariant of the ( ∞ , 1)-ca tegory up to equiv alence; on the other hand, the initial steps of the theory ar e more complicated. The theo ry of Sega l categorie s ﬁts in b etw een: a Segal category has mo re informa tion r eadily at hand than a quasicategory , but less than a complete Segal space; a nd the initia l theo ry is more complicated than for quasica tegories but less than for co mplete Segal spaces. 5.6 Going b et w een Segal categories and n -categories W e mention br ie ﬂy the rela tionship between the no tions of Sega l cate- gory and n -catego r y . T amsamani’s deﬁnition of n -catego ry is rec ur sive. 84 Simplicia l appr o aches The basic idea is to use the same deﬁnition as a bove for Segal ca tegory , but where the A p/ are themselves n − 1-ca tegories. The appropriate condition on the Segal maps is the co ndition of equiv a lence of n − 1- categorie s, which in turn is deﬁned (inductiv ely) in the sa me w ay as the notion of equiv alence of Segal categor ies explained ab ove. T amsamani shows that the homo topy category of n -gr oup o ids is the same as that of n -truncated spaces. The tw o r elev a nt functors a re the realization a nd Poincar´ e n -gro up oid Π n functors. Applying this to the n − 1-categor ies A p/ we obtain the following rela tio nship. An n -category A is s aid to b e 1 - gr oupi c (notation introduced in [194]) if the A p/ are n − 1-gro upo ids. In this cas e, r eplacing the A p/ by their realizatio ns |A p/ | w e obtain a simplicial space which satisﬁes the Segal condition. Conv ersely if A p/ are s paces or simplicial sets then replacing them by their Π n − 1 ( A p/ ) we obtain a s implicia l co llection of n − 1-catego ries, again satisfying the Sega l condition. These constr uctio ns are no t quite inv erses be c ause | Π n − 1 ( A p/ ) | = τ ≤ n − 1 ( A p/ ) is the Postniko v t runcation. If we think (heuristically) o f setting n = ∞ then we get in verse constr uctions. Thus—in a s ense which I will not currently make more pr ecise than the above discussion—o ne can say that Seg al categories are the s ame thing as 1-gro upic ∞ -categor ies. The passa ge from simplicia l sets to Seg al categor ies is the same a s the inductiv e passag e from n − 1-categor ies to n -ca tegories. In [193] w as int ro duced the no tio n of n - pr e c at , the analog ue of the ab ov e Seg al precat. Noticing that the results and a rguments in [193] are basic ally orga nized int o one gigantic ind uctive step passing from n − 1- precats to n -precats, the s ame step applied only o nce works to give the analogo us results in the pas sage fr om s implicial sets to Segal precats. The notion of Sega l categor y thus pres ent s, from a technical p oint o f view, an a spe ct of a “ba b y” v ersion of the notion of n -catego ry . On the other hand, it allows a ﬁrst intro duction of homo topy going all the w ay up to ∞ (i.e. it allows us to av oid the n -truncation inherent in the no tion of n -categor y ). One can easily imag ine combining the tw o into a notion of “ Segal n - category ” which would be an n - simplicial simplicial se t satisfying the globular condition at ea ch stage. It is interesting and histor ically im- po rtant to note that the notion of Sega l n - category with only o ne i - morphism for each i ≤ n , is the same thing as the notio n of n -fold delo op- ing machine . This translation comes out of Dunn [85], whic h apparent ly 5.7 T owar ds we ak ∞ -c ate gori es 85 dates essentially ba ck to 19 84. In retros pect it is not to o ha rd to see how to go fr om Dunn’s notio n of E n -machine, to T a msamani’s notion of n -categor y , s imply by relaxing the conditions of having only one ob ject. Metaphorically , n - fo ld delo oping mac hines corr esp ond to the Whitehead tow er, whereas n -gro upo ids corr esp ond to the Postnik ov to wer. There are other prop o s als for simplicial mo dels for n -categories whic h we hav en’t b een able to discuss. F or e x ample, Street prop osed a mo del based on simplicial sets with c e rtain distinguished simplicial subsets which he calls “thin s ubco mplexes”. 5.7 T o w ards w eak ∞ -categories W e mention here some ideas for going to wards a theory of ∞ -catego r ies. As the iterative approach makes cle a r, ther e is no direct generaliza tion of our theory to the case n = ∞ (which means the ﬁrst inﬁnite ordinal). The notion of equiv alence of Segal categor ies, crucia l to everything, is deﬁned by a top-down induction, so by its nature it is related to some kind of n -categ o ries. As our pro cedure makes clear , a nd as came out in [117] and [171], this iteratio n can sta rt with any mo del catego ry such as K , a llowing us to deﬁne Sega l n -categories whic h in Lurie’s notation corres p o nd to ( ∞ , n )-ca tegories i.e. ∞ -categor ies where all morphisms are invertible starting from degree n + 1. Cheng’s argument [64] shows that if A is to b e an ∞ -categor y with du- als at a ll levels then, in an alg ebraic sense, a ll morphisms lo ok inv ertible. How ev er, it is clear fr om Ba ez-Dolan’s predictions ab out the theory , that we don’t wan t to identify ∞ -categor ies with duals and ∞ -gro upo ids, in- deed they repres ent so mehow complement ary p oints o f view, the ﬁrst being related to quantum ﬁeld theory and the second to top ology . Her- mida, Ma kk ai and Po w er have discussed these issues in [114]. F rom these obs erv ations and thinking ab out sp eciﬁc kinds o f examples , the following idea emerges: in a true ∞ -category A , the information ab out which i -mor phisms should b e considered as inv ertible, should b e thought o f as an additional structu r e b eyond the algebraic s tructure of some kind of weak multiplication op erations. Going back to the stric t case, o ne ca n well imag ine a strictly asso cia tiv e ∞ -ca tegory A in which any i - morphism u has a morphism going in the o ther dir ection v and i + 1-morphisms uv → 1 and v u → 1. Then, w e co uld either dec la re all mor phisms to b e inv ertible, in which ca se v would b e the inv erse of u since the i + 1 -morphisms going to the identities would b e inv ertible; 86 Simplicia l appr o aches or alternatively we could dec la re that no morphisms (other tha n the ident ities) are in vertible. Both choices would b e rea sonable, and w ould lead to diﬀer ent ∞ - categorie s sharing the same underlying algebraic structure A . So, if we imagine a theory of weak ∞ -catego ries in which the infor- mation of which mor phisms are inv ertible is somehow present then it bec omes re a sonable to deﬁne the trunc ation op er ations τ ≤ n as in [206] but go ing from weak ∞ -categ ories to weak n -ca teg ories. Thus a n ∞ - category A would lead to a compatible system of n -ca teg ories τ ≤ n ( A ) for all n . This suggests a deﬁnition: it app ears that one should get the right the- ory by taking a homo topy inv erse limit of the theorie s of n -c ategories . Jacob Lurie had men tioned something like this in corresp ondence some time ago. Given the compa tibilit y of the Rezk-Ba rwick theory with ho- motopy limits [40], that might b e sp ecially adapted to this task. One migh t alternatively b e able to view the theory of ∞ -categor ie s as some k ind o f ﬁrst “ﬁxed p oint” of the op eration M → PC ( M ) which we will construct in the ma in chapters. W e will leav e these cons iderations on a sp eculative le vel for now, hop- ing only that the techniques to b e develop ed in the main par t o f the bo o k will be us e ful in attacking the problem of ∞ -c a tegories later. 6 Op eradic approac hes Apart fro m the s implicial appr oaches, the other ma in dir ection is com- prised o f a num ber of op er adic appr o ches , deﬁnitions of higher catego ries based on Peter May’s no tion of “oper ad”. This dic hotomy is not s ur pris- ing, given that o per ads and simplicia l ob jects are the tw o main wa ys of do ing delo oping machines in algebraic topo logy . The oper adic ap- proaches are not the main sub ject of this b o ok, so our presentation will be more succ inc t des igned to inform the reader o f what is out ther e. T om Leins ter’s bo o k [1 49] ha s a very complete discussion of the rela- tionship b etw een op erads and higher categorie s . His pap er [148] gives a brief but detailed exp osition of numerous diﬀerent deﬁnitions of higher categorie s, including several in the op era dic direction, and tha t was the ﬁrst app earence in print of some deﬁnitions such as T r imb le’s for exam- ple. 6.1 Ma y’s delo oping mach ine W e start b y rec alling Peter May’s delo oping machine. An o p er ad is a col- lection of sets O ( n ), thoug ht of as the “ set of n -ary op erations ”, tog ether with some maps w hich a re thought of as the result of substitutions: ψ : O ( m ) × O ( k 1 ) × · · · × O ( k m ) → O ( k 1 + · · · + k m ) for any uplets of integers ( m ; k 1 , . . . , k m ). If we think of an element u ∈ O ( n ) a s representing a function u ( x 1 , . . . , x n ) then ψ ( u ; v 1 , . . . , v m ) is the function o f k 1 + . . . + k m v a riables u ( v 1 ( x 1 ,...,x k 1 ) ,v 2 ( x k 1 +1 ,...,x k k 1 + k 2 ) ,...,v m ( x k 1 + ··· + k m − 1 +1 ,...,x k 1 + ··· + k m )) . This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 88 Op er adic appr o aches The substitution op era tio n ψ is required to satisfy the appr opriate ax- ioms [165]. A top olo gic al op er ad is the s ame, but where the O ( n ) are top olog ical spaces; it is said to b e c ontr actibl e if each O ( n ) is co nt ractible. Mor e generally , we can co ns ider the notion o f op era d in any categor y a dmitting ﬁnite pro ducts. There is a notion of a ction of an op er a d O on a set X , whic h mea ns an asso ciatio n to each u ∈ O ( n ) of a n actua l n -ary function X × · · ·× X → X such that the substitution functions ψ map to function substitution as describ ed ab ov e. If X is a spa c e a nd O is a top olog ical op erad, we ca n r equire that the a ction consist of co nt inuous functions O ( n ) × X n → X , or more generally if O is an op erad in M then an action o f O on X ∈ M is a collec tion of morphisms O ( n ) × X n → X satisfying the a ppropriate compatibility conditions. A delo opi ng structur e on a space X , is an action of a contractible op erad o n it. The typical example is that of the little intervals op er ad : here O ( n ) is the space of inclusio ns of n consecutive int erv als into the given interv a l [0 , 1], and substitution is given b y pasting in. A v ariant is used in Sec tio n 6.4 b elow. 6.2 Baez-Dolan’s deﬁnition In Ba ez and Dolan’s appro ach, the notion of opera d is ﬁrst and foremost used to deter mine the shap es of higher-dimensio na l cells. They in tro duce a ca tegory of op etop es and a notion of op etopic set which, like the case of simplicial sets, just mea ns a pr esheaf on the c ategory o f op etop es. They then impose ﬁller conditions. Their scheme of ﬁller conditions is inductive o n the dimension of the op etop es, but is ra ther intricate. W e describ e the categor y of o p e to pes by drawing some of the standa rd pic- tures, a nd then give an infor mal discuss ion of the ﬁller conditions . I n addition to the o riginal pap ers [6] [9], Leinster’s [148] was one of our main sources. Rea ders may also consult [63] [142] for o ther appr o aches to deﬁning and calculating with op etop es. An n -dimensional op e to pe should b e thought of as a ro ughly globular n -dimensional ob ject, with an output face whic h is an n − 1-dimensio nal op etop e, a nd an input face which is a p asting diagr am o f n − 1 -dimensional 6.2 Baez-Dolan ’s deﬁnition 89 op etop es. T o paste op etop es tog e ther , match up the output faces with the diﬀerent pieces of the input faces. The only 0-dimensiona l op etop e is a p oint. The only 1-dimensiona l op etop e ha s a s input and output a s ing le p oint, so it is a single arrow r r ✲ A pasting diagram of 1-dimensio na l op etop es can therefore be composed of s everal arrows joined head to tail: r r r r ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ✯ ✲ ❥ A 2-dimensiona l op etop e can then ha ve such a pasting diag ram as input, with a single 1-dimensional opeto pe or a r row as output: r r r r ⇓ ✸ ③ ❫ ✿ A pas ting diagra m of 2-dimensional op etop es would ar ise if we add on so me other op etop es, with the output e dg es a ttached to the input edges. This can b e done r ecursively . A picture where we add three more op etop es on the three inputs, with 2, 1 and 3 input edges r esp ectively; then a fourth o ne on the seco nd input edge of the ﬁr st new one, would lo ok like this: r r ✗ r ✸ r ✸ ✲ s r ❘ ✲ r ③ r ❯ r ✎ ❫ ✿ ⇓ ⇓ ⇓ ⇓ ⇓ Now, a 3 - dimensional op etop e co uld have the ab ove pasting diag r am as input; in that case, the output opetop e is supposed to ha ve the shap e 90 Op er adic appr o aches of the b ounda ry of the pas ting dia gram: r r ✗ r ✸ r s r ❘ r ③ r ❯ r ✎ ⇓ That is to say , it is a 2 -dimensional op etop e with 7 input edges. W e don’t dr aw the 3-dimensional op etop e; it should lo ok like a “cushion”. Baez and Dolan deﬁne in this wa y the c ate go ry of op etop es Otp . The reader is r efered to the ma in r eferences [6 ] [9 ] [1 14] [6 3 ] [1 42] [1 48] for the precise deﬁnitions. A Baez-Dolan n - c ate gory is an op etop ic set , that is a functor A : Otp → Set , which is req uir ed to satisfy so me conditions. Refer a gain to [6] [9 ] [114] [63] [14 2] [148] for the precise statements of these co nditions. see also [217] for o ne o f the most r ecent treatments. Included in the conditio ns ar e things having to do with truncation so that the cells of dimension > n + 1 don’t matter, when sp eak ing of an n -categor y . Beyond this truncation, one o f the main features of the Baez-Do lan viewp oint is that an n - dimensional o peto pe represents an n -morphism which is not necessa rily inv ertible, from the “comp osition” of the op etop es in the input face, to that of the output face. In particula r, the output face of an op etop e is not nec e s sarily the s a me as the comp osition of the input faces; this distinguishes their setup fr o m the Segal-st yle picture we are mos tly consider ing in the pre sent b o ok, in which a simplex repres e n ts a co mpo s ition with the outer edge being e quiv a lent to the comp osition of the pr incipal edges. The idea that op etop es re pr esent arbitrary morphisms from the co m- po sition of th e inputs, to the o utput, mak es it s omewhat complicated to collect together and write down all of the appropria te conditions which an ope to pic set should satis fy in order to b e an n -c ategory . The main step is to designa te certain cells as “univ ersa l” , meaning that the output edge is equiv alen t to the co mpos ition of the input edges, which is basi- cally a homotopic initiality pr op erty . The necessar y uniqueness has to be treated up to eq uiv a le nce, whence the need for a deﬁnition of eq uiv - 6.3 Batanin ’s deﬁnition 91 alence in the induction. All of this leads to Bae z and Dola n’s notions of niches , b alanc e d c el ls and so for th. As a r ough a pproximation, we ca n say that t hese conditions are a form of hor n-ﬁller conditions, g eneralizing the res tricted Ka n condition used in the deﬁnition of quasica tegory [42] [12 6] but a dapted to the op etopic context. T o close out this sectio n, here are a f ew thoughts o n the poss ible rela- tionship b etw een this theo ry and the many other theories o f n -categor ies. F or one thing, the fact that the op etopic cells represe nt explicitly the n -morphisms which are not necessa rily inv ertible, w ould se e m to render this theory particularly well a dapted to lo oking a t things like lax func- tors (what Benab ou calls “mor phisms” as o ppo s ed to “ho momorphisms” [28]). On the other hand, for a co mparison with other theories, it would b e int eresting to inv estigate functors F : Otp → P where P is a clos ed mo del category of “ n -catego ries” (for example, the P = PC n ( Set ) which we are going to b e constr ucting in the r est of the bo o k). Given such a functor , if B ∈ P is a ﬁbrant ob ject, we would obtain an opetopic set F ∗ ( B ), and conversely g iven an op etopic set A we could construct its r ealization F ! ( A ) in P . Under the righ t hypothes es, o ne ho pe s , these should set up a Quillen adjunction betw een a mo del categor y o f op etopic sets, and the o ther mo del category P . If b oth of the ab ov e remark s could be rea lized, it would lead to the int ro duction of p owerful new tec hniques for trea ting lax functors in any of the o ther theories of n - categor ie s. 6.3 Batanin’s deﬁnition Batanin’s deﬁnition is certainly the clo sest to Gro thendieck’s origina l vi- sion. Recall the passage that we hav e quo ted from “Pursuing Stacks” o n p. 17 in Chapter 1 ab ov e, saying that whenever tw o morphisms which ar e naturally obtained as some kind o f comp os ition, have the same source and tar get, there should b e a homotopy b et ween them a t o ne level up. Batanin’s deﬁnition puts this into place, by carefully studying the notion of p os sible compo sition of a r rows in a higher catego ry . It was recently po int ed out by Maltsiniotis that Grothendieck ha d in fact given a def- inition of higher g roup oid [161] and that a small mo diﬁcation of that approach yields a deﬁnition of higher categor y which is very close to Batanin’s [1 62]. 92 Op er adic appr o aches T om Leinster and Eugenia Cheng hav e r eﬁned Batanin’s orig inal work, and our discussio n will b e informed b y their ex p o s itions [148] [14 9] [65], to which the reader should refer for mor e details. Leinster has also intro- duced some related deﬁnitions of w eak n -ca tegory ba sed on the notion of multicategory , again for this we refer to [149]. One of Batanin’s innov ations was to intro duce a notion of op erad adapted to higher categor ies, based on the notio n o f globular set , a presheaf o n the ca tegory G which has o b jects g i for each i , and maps s ! , t ! : g i → g i +1 . The ob jects g i are supp osed to repr e sent globular pictures of i -morphisms, for example g 2 may be pictured a s r r ❘ ✒ ⇓ and the maps s ! , t ! are viewed as inclusions of smaller-dimensio nal glob- ules on the bo undary . Th us, the inclusion maps ar e sub ject to the rela - tions s ! s ! = t ! s ! , t ! s ! = t ! t ! . Dually , a globular set is a co lle ction of sets A i := A ( G i ) with so urce and ta rget maps A i s → t → A i − 1 · · · A 1 s → t → A 0 sub ject to the r elations s ◦ s = s ◦ t and s ◦ t = t ◦ t . This deﬁnition diﬀers slightly fr om the one which was sugg e sted in Chapter 2 ; there we lo o ked a t wha t should proba bly b e ca lled a unital globular set having iden tity maps going back in the other direction i : A i → A i +1 . F or the presen t pur po ses, w e conside r globular s ets witho ut ident ity maps. Batanin’s idea is to use the notion of globular s e t to gene r ate the appropria te kind o f “collection” for a notion of higher op erad. One can think o f a n op erad as sp ecifying a collectio n of op eratio ns of each p os- sible arity; and an arity is a p oss ible conﬁguration of the collection of inputs. F or Batanin’s globular op era ds, an input conﬁgura tion is rep- resented by a globular p asting diagr am P , and the family of o per ations of a rity P sho uld itself for m a globular set. A glo bular pas ting dia gram 6.3 Batanin ’s deﬁnition 93 in dimension n is an n -cell in the free strict ∞ -ca tegory gener ated b y a single non-identit y cell in each dimension. Batanin gives an explicit description of the n -c e lls in this free ∞ - category , using planar trees. The no des of the trees come from the source and target maps b etw een sta ges in a globular set, which is not to b e confused with the o ccur ence o f trees in parametriza tio ns of ele- men ts of a free no nasso ciative algebr a (where the no des corr esp o nd to parenthetizations)—for the free ∞ -categor y inv olv ed her e, it is strictly asso ciative and go ing up a level in the tree co rresp onds rather to g oing from i -morphisms to i + 1- morphisms. As Cheng describ es in a detailed exa mple [65], a glo bular pasting diagram is re ally just a pictur e of how one might comp ose tog ether v a rious i - morphisms. Given any glo bular set A , and a globular pa sting diagram P , we get a set denoted A P consisting o f all the wa ys o f ﬁlling in P with la bels from the globular set A , consis ten t with sources and targets. Now, a globula r op era d B consists o f specifying , for each glo bula r pasting dia gram P , a globular s et B ( P ) of op eratio ns of ar it y P . This globular set consists in pa rticular of sets B ( P ) n for each n calle the set P -ary op er atio ns of level n , together with so ur ces and ta rgets B ( P ) n +1 s → t → B ( P ) n satisfying the globular ity r elations. An action of B on a globula r set A consists of sp ecifying for each globula r pas ting diagr am P o f deg ree n , a map o f s e ts B ( P ) n × A P → A n such that the source diagram B ( P ) n × A P → A n B ( sP ) n − 1 × A sP s ↓ → A n − 1 s ↓ commutes and similarly for the target diagr am. Here sP and tP are the source and target of th e pasting diagra m P , whic h are pasting diagrams of degree n − 1. Of co urse we need to describ e an additional o pe radic structure on B and the a c tion should b e compatible with this to o , but it is e a sier to ﬁrst consider what data a n algebra s hould hav e. 94 Op er adic appr o aches T o descr ib e wha t an op er a dic str ucture should mea n, notice that there is a n o per ation o f substitution of glo bular pasting diagra ms, indeed the free ∞ -catego ry o n one cell in each dimens ion can b e deno ted G P , and if P ∈ G P m is a globular pasting dia gram in degree m then we get a map o f globular sets G P P → G P , that is to say that g iven a lab eling of the c ells o f P where the lab els L j are themselves g lobular pasting dia grams, we ca n substitute the lab els int o the cells and obtain a big r esulting globular pasting diagr a m. Now the op eradic structure, which is in addition to the s tructure of B describ ed ab ov e, should say that for a ny e le ment of G P P consisting of lab els denoted L j ∈ G P n j for the cells of P a nd yie lding an output pasting diag r am S , if we are given elements of the B ( L j ) plus a n element of B ( P ) then there should be a big output element of B ( S ). This needs to b e compatible with so urce and target op era tions, a s well as compatible with iteration in the s tyle of usual o pe rads. Batanin describ es explicitly the combinatorics of this us ing the iden- tiﬁcation betw een pasting diagr a ms and trees, whereas Leinster takes a more abstract a pproach using monads and m ulticategories. W e refer the reader to the r eferences for further details. The main p oint is that th is discussion establishes a language in which to say that the s ystem of coherenc ie s s hould satisfy a glo bular con- tractibility condition. Observe tha t contractibilit y is a very ea sily de- ﬁned pr op erty o f a globula r set, a nd it do esn’t dep end on any k ind of comp osition law: a globular set A is contractible if it is nonempty , and if for any tw o f , g ∈ A m with s ( f ) = s ( g ) and t ( f ) = t ( g ), there e x ists an h ∈ A m +1 with s ( h ) = f a nd t ( h ) = g . Note that this deﬁnition is r e ally only appro priate if we a re working with globular sets which are tr unca ted a t some lev el (i.e. trivial ab ove a certain degree n ), or o nes which a r e suppo sed to r epresent ( ∞ , n )- categorie s, that is ones in which all i -morphisms are declar e d inv ertible for i > n . F or the purp oses of Bata nin’s deﬁnition of n -categ ories, this is the c a se. Now, a Ba tanin n -ca tegory is a globular se t A pr ovided with an ac- tion of a globular op erad B , such that B is contractible. Batanin con- structs a universal g lobular o p er ad, but it can also b e conv enien t to work with other contractible glo bular o p er ads, as in Cheng’s compa r ison with T rimble’s work which we discuss next. The element s of B ( P ) n are the “natura l op era tions ta king a collection 6.4 T rimble’s deﬁnition and Cheng’s c omp arison 95 of morphisms of v arious degrees, and combining tog ether to get an n - morphism”. Contractibility of B rea lly puts int o eﬀect Gr othendieck’s dictum that, giv en t w o natural o pe r ations f and g with the same source and targ et, there should b e an op eration h one level hig her whos e source is f and whose target is g . So , Batanin’s deﬁnition is the closest to what Grothendieck was asking for. 6.4 T rim ble’s deﬁnition and Cheng’s comparison T rimble’s deﬁnition o f hig her ca tegory ha s acq uir ed a central role b e- cause of Cheng’s r ecent work comparing it to Batanin’s deﬁnition [6 5]. This has also b een taken up by Batanin, Cisinski and W eb er in [25]. T rimble’s framework has the a dv a nt age that it is iterative. In the future it should b e po ssible to establish a compar ison with the itera tive Seg al approach we are discuss ing in the rest of the b o ok. Such a comparis on result would b e very in teresting, but we don’t discuss it here. Instead we just give the basic outlines o f T rimble’s appro ach and state C he ng ’s compariso n theorem. W e are fo llowing very closely her article [65]. Consider the fo llowing op erad O T in Top : O T ( n ) ⊂ C 0 ([0 , 1] , [0 , n ]) is the subset o f endp oint-preserving co n tinuous maps. The op erad struc- ture ψ T is given b y ψ ( f ; g 1 , . . . , g m )( t ) := g j ( f ( t ) − ( j − 1)) , f ( t ) ∈ [ j − 1 , j ] ⊂ [0 , m ] . The spaces O T ( n ) ar e contractible. This top ological o per ad is particu- larly a dapted to lo op spaces and path spaces . If Z ∈ Top , and x, y ∈ Z let Path x,y ( Z ) denote the space of pa ths γ : [0 , 1] → Z with γ (0) = x and γ (1) = y . F or any sequence o f p oints x 0 , . . . , x m ∈ X we hav e a “substitution” map O T ( m ) × Path x 0 ,x 1 ( Z ) × · · · × Path x n − 1 ,x n ( Z ) , and these are compatible with the o per ad structur e. In particular when the points are all the same, this g ives an actio n of O T on the lo op space Ω x ( Z ). A T rimble top olo gic al c ate gory co nsists of a s et X “of ob jects”, to- gether with a collection of spa ces A ( x, y ) for any x, y ∈ X , and collec tion of ma ps φ x · : O T ( n ) × A ( x 0 , x 1 ) × · · · × A ( x n − 1 , x n ) → A ( x 0 , x n ) 96 Op er adic appr o aches for any sequence ( x 0 , . . . , x n ) in X . Thes e s ho uld satisfy a c ompatibil- it y co nditio n with the op era d structure ψ T for O T : given a sequenc e x 0 , . . . , x m and s equences y i 0 , . . . , y i k i with y i 0 = x i − 1 and y i k i = x i , φ ( ψ T ( f ; g 1 , . . . , g m ); u 1 1 , . . . , u 1 k 1 , . . . , u m 1 , . . . , u m k m ) = φ ( f ; φ ( g 1 ; u 1 1 , . . . , u 1 k 1 ) , . . . , φ ( g 1 m ; u m 1 , . . . , u m k m ) . Similarly , for any catego ry M admitting ﬁnite pro ducts, and any op- erad ( O, ψ ) in M , we can deﬁne a notion of ( M , O, ψ )-categor y; this consists of a set X of ob jects, together with a collectio n o f A ( x, y ) ∈ M for any x, y ∈ X , and a co llection of maps φ as abov e satisfying the sa me compatibility condition. Suppo se we are g iven a category M with ﬁnite pro ducts, and a functor Π : To p → M , then we o btain an op erad Π( O T ) in M . W e get the notion of ( M , Π( O T ) , Π( ψ T ))-category . If contractibilit y mak es sense in M and Π( O T ( n )) is contractible, then this is a generaliza tion due to Cheng [65], of T rimble’s no tion of higher category enriched in M . T rimble’s orig inal deﬁnition, w hich ﬁrst appeared publicly in [148], in- cluded an inductive co nstruction o f the Poincar´ e n -gr oup oid functor Π n . He deﬁnes inductively a sequence o f ca tegories which Cheng denotes by V n , starting with V 0 = Set ; to gether with pro duct-compatible Poincar´ e n -group oid functors Π n : Top → V n . The inductive deﬁnition is tha t V n +1 is the categor y of ( V n , Π n ( O T ) , Π n ( ψ T ))-categorie s , and Π n +1 ( Z ) = ( X , A , φ ) where X := Z disc is the disc r ete set of p oints of Z , wher e A is deﬁned by A ( x, y ) := Π n (Path x,y ( Z )), and φ is deﬁned using the action describ ed ab ov e of O T on the pa th spa ces. See [65] for further details , a s well as for the generaliza tion to the ca se where an arbitrary contractible operad P n replaces Π n ( O T ). Cheng go es on to compare this family of deﬁnitions, with Batanin’s deﬁnition: she shows how to combine the P 0 , P 1 , . . . , P n − 1 together to form a co n tractible globular opera d Q ( n ) such that the category of glob- ular alge bras of Q ( n ) is V n . This expresses V n as a categor y o f Batanin n -categor ies for this particula r choice of c ontractible glo bular op erad. The rea der is refer ed to [65], as well as to [3 0], [25] a nd [69] for related asp ects of this k ind o f co mparison. 6.5 We ak units 97 6.5 W eak units In the course of inv estigating the nonre alization of ho motopy 3-types b y strict 3- group oids ([19 7], see C ha pter 4), the main obstruction seemed to b e the strict unit condition in a strict n -ca tegory . This is one of the main a sp ects whic h a llows the Ec kmann-Hilton a rgument to w ork. That was explained to me by Georges Maltsiniotis a nd Alain Brugui` eres, but has of cours e b een well-known for a long time. This led to the conjec- ture that maybe it would b e suﬃcien t to keep the strict asso ciativit y of comp osition, but to weaken the unit condition. Recall fro m homo to p y theor y (cf [15 1]) the yoga that it s uﬃce s to weak en an y o ne of the principal structures in volv ed. Most w eak notions of n -ca teg ory inv olv e a weak ening of the ass o ciativity , or even tually of the Go dement in terchange conditions. O. Leroy [150] a nd a ppa rently , indep endantly , Joyal and Tierney [129] were the ﬁrst to do this in the context of 3 -types. See a lso Gordon, Po wer, Street [104] and Berger [29] for weak 3-ca tegories and 3- t yp es. B aues [26] show ed that 3-types co rresp ond to quadr atic mo dules (a generaliza tio n of the notion o f cros s ed co mplex) [26]. Then come the mo dels for weak higher ca tegories which w e are considering in the rest of the b o ok. It seems lik ely that the arguments o f [13 6] w ould sho w that one could instead w eaken the c o ndition o f being u n ital , that is having identities, and keep asso cia tivity and Go dement. W e give a pr op osed deﬁnition of what this w ould mean and then state t wo conjectures. This can be motiv ated by lo oking at the Mo or e lo op sp a c e Ω x M ( X ) of a s pace X based at x ∈ X , cited in [136] a s a motiv a tio n for their construction. Recall that Ω x M ( X ) is the spa c e o f p airs ( r , γ ) where r is a r e al num ber r ≥ 0 and γ = [0 , r ] → X is a path star ting and ending at x . This has the adv an tage of being a strictly as so ciative monoid. On the other side o f the coin, the “length” function ℓ : Ω x M ( X ) → [0 , ∞ ) ⊂ R has a sp ecial behavoir ov er r = 0. Note that ov er the o pen half-line (0 , ∞ ) the length function ℓ is a ﬁbration (even a ﬁber- space) with ﬁb er homeomorphic to the usual lo o p space. How ever, the ﬁber ov er r = 0 consists o f a single p oint, the co ns tant path [0 , 0] → X based at x . This additional p oint (whic h is the unit element of th e mono id Ω x M ( X )) do esn’t a ﬀect the top olo gy o f Ω x M (at lea st if X is lo cally contractible at x ) b ecause it is glued in as a limit of paths whic h ar e more and mor e concentrated in a neighborho o d o f x . How ever, the map ℓ is no long er 98 Op er adic appr o aches a ﬁbra tio n ov er a neig h b orho o d of r = 0 . This is a bit of a problem bec ause Ω x M is not compatible with direct pro ducts of the space X ; in order to obtain a compatibility one has to take the ﬁber pro duct over R via the length function: Ω ( x,y ) M ( X × Y ) = Ω x M ( X ) × R Ω y M ( Y ) , and the fact tha t ℓ is not a ﬁbration could end up ca using a problem in an a ttempt to iteratively apply a construction like the Mo or e lo op-s pa ce. Things seem to get better if we r estrict to Ω x M ′ ( X ) := ℓ − 1 ((0 , ∞ )) ⊂ Ω x M ( X ) , but this asso c ia tive monoid no lo ng er has a strict unit. Even so, the constant path of any pos itiv e length gives a weak unit. A motiv ation c o ming from a diﬀerent directio n was an obser v a tion made b y T amsamani early in the cour se of his thesis work. He was try- ing to deﬁne a strict 3-catego ry 2 C at whose ob jects would b e the strict 2-catego ries and whose mor phis ms would b e the weak 2- functors b e- t ween 2 -categor ies (plus notions of weak natural tra nsformations and 2-natural transfor mations). At so me p oint he ca me to the conclusion that one c ould adequately deﬁne 2 C at as a strict 3- category e x cept that he co uldn’t get str ict iden tities. Because of this problem we abandonned the idea and lo o ked tow ard weakly asso ciativ e n -ca tegories. In retrosp ect it would b e interesting to purs ue T ams a mani’s co ns truction of a s trict 2 C at but with only w eak identities. In [197] was in troduce d a preliminary deﬁnition of we akly un it al strict n -c ate gory (called “snucategory” there), including a notion o f direct pro duct. The prop osed deﬁnition wen t as follows. Supp ose we know wha t these ar e for n − 1 . Then a weakly unital str ict n - c ategory C cons is ts of a set Ob( C ) o f ob jects toge ther with, for every pair of ob jects x, y ∈ Ob( C ) a weakly unital strict n − 1-ca tegory C ( x, y ) and compos itio n mor phisms C ( x, y ) × C ( y , z ) → C ( x, z ) which are s tr ictly a sso ciative, such that a we ak u nital c ondition holds . W e now explain this condition. An element e x ∈ C ( x, x ) is called a we ak identity if: —comp osition with e induces equiv a lences of weakly unital strict n − 1- categorie s C ( x, y ) → C ( x, y ) , C ( y , x ) → C ( y , x ); —and if e · e is equiv alen t to e . It would b e b est to complete this last 6.5 We ak units 99 condition to the fuller colle c tio n o f c o herence conditions introduced by Ko ck [140]. In order to complete the r ecursive deﬁnition w e must deﬁne the no tion of when a mo rphism of weakly unital stric t n -ca tegories is a n equiv alence, and we must deﬁne wha t it means for tw o ob jects to b e equiv alen t. A morphism is s aid to be an eq uiv a lence if the induced mo rphisms on the C ( x, y ) a re equiv alences of weakly unital s tr ict n − 1-categ o ries and if it is essentially surjective on ob jects: e a ch ob ject in the target is equiv alent to the image of an ob ject. It thus remains just to b e seen what int ernal equiv alence of o b jects means. F or this we in tro duce the trunc ations τ ≤ i C of a weakly unital strict n -categor y C . Again this is done in the sa me wa y as usua l: τ ≤ i C is the w eakly unital s tr ict i -catego r y with the same ob jects a s C and whose morphism i − 1-categ ories ar e the tr unca tions H om τ ≤ i C ( x, y ) := τ ≤ i − 1 C ( x, y ) . This works for i ≥ 1 by recurr e nce, and for i = 0 we deﬁne the trunca- tion to b e the set o f isomorphis m classes in τ ≤ 1 C . Note that truncation is compatible with direct pro duct (direct pro ducts are deﬁned in the obvi- ous w ay) and tak es equiv alences to equiv alenc e s . These statements used recursively allow us to show that the truncations themselves satisfy the weak unary condition. Finally , w e say that tw o ob jects are equiv alen t if they ma p to the s a me thing in τ ≤ 0 C . Pro ceeding in the s ame w ay as in Chapter 2, w e can deﬁne the no tion of weakly unital s trict n -gro upo id. Conjecture 6.5. 1 Ther e ar e functors Π n and ℜ b etwe en t he c ate- gories of we a kly u nital st rict n -gr oup o ids and n -tr u nc ate d sp ac es (going in the usu al dir e ctions) to gether with adjunction m orphisms inducing an e quiva lenc e b etwe en the lo c alization of we akly unital st rict n -gr oup oid s by e quivalenc es, and n -t runc ate d sp ac es by we ak e quivalenc es. Joachim Ko ck has develop ed the rig h t de ﬁnitio n of weakly unitary strict n -categ ory which takes into a ccount the full collection of higher coherence relations [140] [141] rather than just asking that e ∼ e · e ; we refer the reader ther e fo r his deﬁnition which sup e rsedes the preliminary version describ ed ab ov e. Joy al and Ko ck ha ve proven Conjecture 6 .5.1 for the case n = 3 in [128]. F o r general n , one could ho pe to a pply the argument of [136]. These results concern the case o f group oids , how ev er we might also exp ect that weakly unita l strict n -ca teg ories ser ve to model all weak n -categor ies: 100 Op er adic appr o aches Conjecture 6. 5.2 The lo c alization of the c ate gory of we akly u n ital strict n -c ate gories by e quivale nc es, is e quivalent t o the lo c a lizations of the c ate gories of we ak n -c ate gori es of T amsamani and/or Baez-Dolan and/or Batanin by e quivalenc es. While we’re discussing the sub ject o f unitalit y conditions, the follow- ing remark is in order. The ro le o f strict unitality conditions in the int erchange or Eckmann-Hilton relations, and the consequent nonreal- ization of homotopy types with nontrivial Whitehead bracket, sugge s ts that we need to take some care about this p oint in the g eneral a rgument which will b e developped in Parts I I I and IV. It turns out that, in or - der to insure a go od car tesian prop erty , o ur Sega l- style weakly enriched categorie s s ho uld nonetheless b e endow ed with strict units in a certa in sense. These co rresp ond to the degenera cies in the simplicial categor y ∆, and are imp ortant for the Eilenber g-Zilb er ar gument which yields the cartesian prop erty . They don’t corre spo nd to full s trict units in the maximal po ssible wa y , b ecause the co mpo sition o p e ration will not even be well-deﬁned; that is why we will be able to imp ose the unitalit y con- dition in P art II I, without running up against the pro blems iden tiﬁed in Chapter 4. 6.6 Other notions The theor y o f n -ca tegories is an essentially globular theory : an i -mo rphism has a single i − 1-morphism a s sour ce, a nd a single o ne as target. This ba- sic shap e can be relaxed in many w ays. F or example, Leinster and others hav e investigated notio ns of multic ate gory where the input is a collection of ob jects rather than just one ob ject. This is somewhat related to the op etopic shap es in tro duced b y Baez and Dolan. Another wa y of r elaxing the globula r sha pe is to iterate the int ernal category constr uction. Brown and Lo da y cons tr ucted the ﬁrst a lgebraic representation for homoto p y n -types , with the notio n of C at n -gr oup . Let C at 1 ( Gp ) denote the categ ory of internal ca tegories in the categ o ry Gp , then C at n +1 ( Gp ) is the catego ry of in ternal c ategories in C at n ( Gp ). There is a natur al realiza tion functor fr om C at n ( Gp ) to homotopy n - t yp es, and Brown and Lo day prov e that all homo topy n -types are real- ized. Paoli has r e c en tly r eﬁned this mo del to g o back in the g lobular di- rection, by introducing a notion of sp e cia l C at n -gr oup [170]. If we think 6.6 Other notions 10 1 of an in ternal ca tegory as being a pair o f ob jects connected by several morphisms, then an internal n -fold category ma y be seen a s a collec tio n of ob jects arr anged at the vertices of an n -cube. The sp ecia lit y conditio n requires that certain faces of the c ub e b e co n tractible. The sp eciality condition is a sor t of weak g lobularity c ondition. An int ernal n -fold ca tegory is not in and o f itself a glo bular ob ject, because the ob ject of ob jects may b e nontrivial. A strict globular co ndition would hav e the ob ject of ob jects be a discrete set; the sp eciality c ondition requires only that it be a disjoint union o f contractible C at n − 1 -groups. Paoli shows that the spec ial C at n -groups mode l ho motopy n -types , and s he re la tes this mo del to T amsama ni’s mo del. This provides a semistric- tiﬁcation r esult saying that we can hav e a strictly asso ciative comp osi- tion at a ny one s tage of T amsa mani’s mo del. An alternative pr o of of th is result for semistrictiﬁca tion a t the last stage, ma y b e obtained using the fact that Seg al categories a re equiv alen t to str ic t simplicial categories . Paoli’s mo del relates to Lewis’s principle cited ab ov e [151] in an in- teresting wa y: in a C at n -group all the str uctures—asso cia tivit y , units, inv erses, interc hange—ar e strict; the sp ecia l C at n -groups weak en instea d the glo bularity condition itself. Penon’s deﬁnition [1 72] is completely algebraic in the sense that a weak n -categor y is a n a lgebra ov er a monad. F or a ra pid description of the mona d, o ne can refer to [148, pp 14–1 7], see a lso [23] [66 ] [67] [99]. Penon introduces a category Q consisting of ar r ows of “ ω -magmas” M → S where S is a strict ω -ca tegory , to gether with a contractive structure on π . The monad is a djoint to the functor “underlying g lob- ular set of A ♯ ”. If A is a g lobular set then it go es to an element of Q with S being the free strict ω -c ategory gener ated by A . Note that this fre e catego ry is the set o f globular pasting diagrams a s in Section 6.3, and M may be viewed as some sort of family of elements over S with a contractible structure. In this sense Penon’s deﬁnition uses ob- jects o f the same sor t a s Batanin’s deﬁnition, indeed Batanin has made a more pr ecise compar is on in [23]. Penon’s deﬁnition uses globular sets with identities (“ reﬂexive g lobular se ts ” ) wherea s Ba ta nin’s were with- out them, so [23] pr op oses a mo diﬁed version of Penon’s deﬁnition with non-reﬂexive globular sets. Cheng and Mak k a i have p o int ed out tha t it is better to use the non-r eﬂexive version, s ince the reﬂexive v ersion do esn’t lead to all the ob jects one would wan t [67], essentially b ecaus e of the Eckmann-Hilton argument. F utia prop os e s a generalized family of Penon-st yle deﬁnitions in [9 9]. In these deﬁnitions, o ne could say that 102 Op er adic appr o aches globally the goa l is to b e able to pa rametrize higher comp ositio ns sorted according to their s hap es which are globular pasting diagr a ms. 7 W eak enric hmen t ov er a cartesian mo del category: an in tro duction T o close out the ﬁrst par t o f the b o ok, w e descr ibe in this chapter the basic outlines of the theory which will o ccup y the rest of the w ork. The basic idea, alrea dy consider ed in Pelissier’s thesis, is to abstract T am- samani’s iter ation pro ces s to obtain a theor y of M -enr iched categories , weak in Segal’s sense, for a model category M . 7.1 Simplicial ob jects in M The origina l deﬁnitions of Segal category , T amsama ni n -categor y , and Pelissier’s enriched categorie s, too k a s ba sic ob ject a functor A : ∆ o → M . The ﬁrst condition is that the image A 0 of [0] ∈ ∆ should be a “discrete ob ject”, that is the imag e of a set under the natural inc lus ion Set → M which sends a s e t X to the c olimit of ∗ over the discr e te ca tegory cor- resp onding to X . This version of the theory therefore r equires, at lea s t, some axioms saying tha t the functor Set → M is fully faithful and compatible with disjoint unions. Thus A 0 may b e v ie w ed as a set and the expres sion x ∈ A 0 means that x is an ele men t of the corresp o nd- ing set, equiv alently x : ∗ → A 0 . The hig her ele ments of the simplicial ob ject will b e denoted A m/ . Then, the Seg al category condition says that the Segal maps A m/ → A 1 / × A 0 · · · × A 0 A 1 / are s uppo s ed to be weak equiv alences. The pair of structural maps ( ∂ 0 , ∂ 1 ) : A 1 / → A 0 × A 0 serves to decomp ose A 1 / = a x,y ∈A 0 A ( x, y ) This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 104 We ak enrichment over a c artesian mo del c ate gory : an intr o duction where A ( x , y ) is the in verse imag e o f ( x, y ) ∈ A 0 × A 0 , or mor e precisely A ( x, y ) := A 1 / × A 0 ×A 0 ∗ with the right map of the ﬁber pro duct b eing given by ( x, y ) : ∗ → A 0 × A 0 . W e can s imilarly decomp ose A m/ = a ( x 0 ,...,x m ) ∈A m +1 0 A ( x 0 , . . . , x m ) and the Se g al condition may be expressed equiv a len tly as saying that A ( x 0 , . . . , x m ) → A ( x 0 , x 1 ) × · · · × A ( x m − 1 , x m ) is a weak equiv a lence in M . 7.2 Diagrams o v er ∆ X Upo n clo ser insp ection, mos t of the arg ument s ab out M -Sega l ca te- gories can really b e phras ed in terms of the ob jects A ( x 0 , . . . , x m ); and in these terms, the Seg al condition inv olves only a pro duct r ather than a ﬁb er pr o duct. So, it is natural and useful to consider the ob jects A ( x 0 , . . . , x m ) as the primar y ob jects of study ra ther than the A m/ . This economizes some hypo theses and arguments ab out discrete ob jects and ﬁb er pro ducts. This po int of view has b een introduced by Lurie [155]. F or any set X , deﬁne the ca tegory ∆ X whose o b jects are ﬁnite linearly or dered sets decorated by e lemen ts of X , that is to say an ob ject of ∆ X is an ordered set [ m ] ∈ ∆ plus a map of se ts x · : [ m ] → X . This pair will b e denoted ( x 0 , . . . , x m ), that is it is an m + 1-tuple of elements of X . The morphisms in the categor y ∆ X are the morphisms of ∆ which res p ect the decor ation, so for example the three standard mor phisms [1] → [2] yield mo rphisms of the form ( x 0 , x 1 ) → ( x 0 , x 1 , x 2 ) , ( x 1 , x 2 ) → ( x 0 , x 1 , x 2 ) , ( x 0 , x 2 ) → ( x 0 , x 1 , x 2 ) . Now, an M - Segal ca tegory will be a pair ( X , A ) where X is a set, ca lled the set of obje cts , and A : ∆ o X → M is a functor deno ted b y ([ m ] , x · ) = ( x 0 , . . . , x m ) 7→ A ( x 0 , . . . , x m ) or just ( x 0 , . . . , x m ) 7→ A ( x 0 , . . . , x m ) if there is no danger of co nfusio n, such that the Segal ma ps A ( x 0 , . . . , x m ) → A ( x 0 , x 1 ) × · · · × A ( x m − 1 , x m ) 7.3 Hyp otheses on M 1 05 are weak equiv alences in M . At m = 0 the Segal co ndition s ays that A 0 ( x 0 ) → ∗ is a weak equiv alence. This is a sor t of weak unitality condition, but for our purpose s it is genera lly s pea king better to imp ose the strict unitality c onditio n that A 0 ( x 0 ) = ∗ for any x 0 . This condition bec omes essential when w e consider direct pro ducts. A t m = 1, the morphism space b etw een tw o elements x, y ∈ X is A ( x, y ). At m = 2 the usual diag ram using the three standard mor - phisms, s erves to deﬁne the comp osition op eratio n in a weak sense: A ( x, y ) × A ( y , z ) ← A 2 ( x, y , z ) → A ( x, z ) (7.2.1) with the left ward arr ow b eing a weak equiv alence in M . F o r higher v a lues of m w e get the higher homoto p y co herence co nditions starting with asso ciativity at m = 3. 7.3 Hyp ot heses on M In the weak enrichmen t, the comp o sition o pe ration is given by a diagr am of the form (7.2.1) ab ov e, using the us ua l direct pro duct × in M a nd where the left ward a rrow is the Seg al map which is r equired to b e a weak equiv alence. Therefore, the main condition w hich we need to impo se up on M is that it b e c artesian , that is to say a monoidal mo del categor y whose monoidal op era tio n is the direct pro duct. This insures that direct pro d- uct is compatible with the coﬁbr ations and w eak equiv alences. Mono idal mo del catego ries hav e b een considered by many authors, see Hovey [120] for exa mple, a nd the ca rtesian theory is a sp ecial case. This condition will b e discus sed in Cha pter 10. F or conv enience w e also imp ose the conditions that M b e left pr op er , and tr actable . T r a ctability is Barwick’s slight mo diﬁcation of J . Smith’s notion of c ombinatorial mo del c ate go ry . Reca ll that a combinatorial mo del category is a coﬁbra n tly gener ated one whos e underlying categor y is lo- cally pre sentable—locally presentable categor ie s are the most appro pri- ate en vironment for using the small ob ject a rgument, one of our sta ple s . Barwick’s tractability adds the condition that the doma ins of the ge n- erating coﬁbr a tions and trivial coﬁbrations, b e themselves coﬁbrant ob- jects. This is useful at so me technical places in the small o b ject argu- men t. Our discussio n of these topics is put together in Chapters 8 and 9. In Sectio n 12 .7 we consider some additional hypotheses on M saying 106 We ak enrichment over a c artesian mo del c ate gory : an intr o duction that disjo in t unions b e have like we think they do; if M satisﬁes these hypotheses then the discrete- set ob jects in M work well, and we ca n use the notatio n A m/ for the disjoint union o f the A ( x 0 , . . . , x m ). This reduces to consideration of simplicial ob jects in M ra ther than functors from ∆ o X . If M is a categor y of presheav es over a connected ca tegory Φ then it satis ﬁes the additional hypotheses, and the catego ry of M - precatego r ies discussed next will aga in b e a category of presheaves (ov er a quotient of ∆ × Φ). The fact tha t iteratively we s tay within the world of pr esheaf categorie s , is co nv enien t if one wan ts to think of the small ob ject ar gument in a simpliﬁed wa y . 7.4 Precategories A tra ctable left pro per car tesian mo del catego ry M is ﬁx ed. F or the original case of n -categories, M would be the mo del categ ory for ( n − 1)- categorie s c o nstructed a ccording to the inductive h yp othesis. In o rder for the induction to work, the main go al is to construct fro m M a new mo del ca tegory , who s e o b jects r epresent up to homotopy the M -enriched Segal categories, and whic h sa tisﬁes the same hypothes e s of tractabilit y , left prop erness, and the ca rtesian condition. If M satis ﬁes the additional hypothese s on disjoint unions, then an M -enriched categor y is a functor A : ∆ o → M such tha t A 0 is a discrete set a lso called Ob( A ), and such that the Segal ma ps A n → A 1 / × A 0 · · · × A 0 A 1 / are weak equiv alences in M . How ev er, lo oking at the catego ry of all such functors, the Segal co n- dition is no t preserved by limits or co limits of diag rams. It would be preserved by homo topy limits, but not even b y homotopy co limits, and indeed the problem of taking a homoto p y colimit of diagra ms a nd then impo sing the Segal condition is o ur ma in technical diﬃculty . So, in o r der to obta in a mo del ca tegory structure, we hav e to relax the Segal conditio n. This lea ds to the bas ic notion of M -pr e c ate gory . Our utilisation of the w ord “precateg ory” is similar to but not the same as that of [122]. The r eader may r efer to the introduction o f [193] for a discussion o f this notion in the original n -categor ical context. If M sa tisﬁes the additional co ndition a bo ut disjoint unions, then an M -precategor y may be deﬁned as a functor A : ∆ o → M such that A 0 7.5 Unitality 1 0 7 is a discrete set, tha t is to say a disjoint sum o f copie s of the co initia l ob ject ∗ ∈ M . In the mor e g eneral case, an M -enriched precategory is a pair ( X , A ) where X is a set (often denoted by Ob( A )), and A : ∆ o X → M is a functor s atisfying the un itality c ondition that A ( x 0 ) = ∗ for a ny sequence of leng th zero (i.e. having only a single elemen t). In either of these s ituations, the categor y PC ( M ) of such diagra ms is closed under limits a nd colimits, a nd further more if M is loc a lly pre- sentable then PC ( M ) is lo cally presentable to o. The catego r y PC ( M ) will thus ser ve a s a suita ble substra te for our mo del structure. The ﬁ- brant ob jects of the mo del structure should additionally satisfy the Segal conditions. An additional be neﬁt of the no tation ∆ X is that it allows us to break down the a rgument into tw o pieces, a sugg estion of Clark Bar wick [14]. Indeed, w e obtain t wo diﬀerent categ o ries, PC ( X ; M ) a nd PC ( M ). The ﬁrst cons is ts of all M -preca tegories with a ﬁxed set of o b jects X . It is just the full sub category of the diagra m ca tegory Func (∆ o X , M ) consisting of diagr ams satisfying the strict unitality condition A 0 ( x 0 ) = ∗ . The study of PC ( X ; M ), considered ﬁr st, is ther efore almo st the sa me as the study of the categ ory of M -v alued dia g rams on a ﬁxed catego ry ∆ o X . The catego ry PC ( M ) is obtained b y letting X v ary , with a natural deﬁnition o f morphism ( X, A ) → ( Y , B ). O nce everything is well under wa y and the ob jects of PC ( M ) beco me our main ob jects of study , then we will drop the s e t X fr om the notation: an ob ject of PC ( M ) will b e denoted A and its set of o b jects by Ob( A ) , but with the same letter for the functor A : ∆ o Ob( A ) → M , in other words A denotes (Ob( A ) , A ). 7.5 Unitality The stric t unitality condition says that A ( x 0 ) = ∗ . The r e a son for impos - ing this co ndition, aside from its c o nv enience, is that it is needed to ob- tain the cartesia n condition on the mo del categ ory o f M -precategor ies. Indeed, if we do n’t imp ose the unitality co nditio n, then the precate- gories must b e allow ed to hav e A ( x 0 ) arbitra r y , even those would be forced to b e co nt ractible by the Segal condition of leng th 0. Pr o duct with a non-unital precateg ory such that B ( x 0 , . . . , x m ) = ∅ for all se- quences of ob jects, is not compatible w ith weak equiv alences (as will b e discussed in Section 19.3.1). 108 We ak enrichment over a c artesian mo del c ate gory : an intr o duction Given the fact tha t the Ec kmann-Hilton a rgument rules out a num ber of diﬀerent approa ch es to higher categ ories, a s we have seen in Chapter 4 but also as in Cheng and Makk ai’s remark in P enon’s original deﬁnition [67], we s hould justify why o ur version of the unitality condition which says that A ( x 0 ) = ∗ do esn’t a lso lea d to a n E ckmann-Hilton argument. The po in t is that in the Segal-style deﬁnitions, the co mp os ition is not a w ell-deﬁned op eration. So, even if th ere exist cells which are suppos ed to b e the “identities”, there is no t a single well-deﬁned compo sition with the identit y . The deg eneracies pr ovide 2 -cells which say that, for an i - morphism f , some p ossible comp ositio n of the fo rm 1 t ( f ) ◦ f or f ◦ 1 s ( f ) will b e equa l to f , but these choices (which we call resp ectively the left and right degenera cies) are not the only pos s ible ones. The main step of the E ckmann-Hilton argument going from r r r f ⇓ 1 ⇓ 1 ⇓ g ⇓ to r r 1 ◦ f = f ⇓ g ◦ 1 = g ⇓ inv olv es g lue ing the left degeneracy for f on top to the right degener acy for g on the b ottom, genera ting a copro duct of cells which do esn’t ﬁt int o any ca no nical global comp osition op era tion for the four 2- morphisms at once. And similar ly for the step inv olving vertical comp ositions. The information on compo sition with units which comes fro m the unitalit y condition and the degener acies of ∆, is luckily not enoug h to make the Eckmann-Hilton ar gument work. Becaus e we’re clos e to the bo rderline here, it is clear that some care should b e taken to verify everything re- lated to the unitality condition in the tec hnical parts of our co nstruction. Unitalit y will therefor e b e considered n the context o f more general up-to-homotopy ﬁnite pro duct theories, in Chapter 13. 7.6 R e ct iﬁc ation of ∆ X -diagr ams 109 7.6 Rectiﬁcation of ∆ X -diagrams The r eader s hould now b e asking the following question: wouldn’t it b e better to co nsider M as some kind of hig her category , and to lo ok at we ak functors ∆ o X → M ? This would certa inly seem like the most natur al thing to do . Unfortunately , this idea leads to “b o otstrapping” problems bo th philosophical as w ell as practica l. O n the philoso phical level, the really go o d version of M as a higher category , is to think of M a s being enriched ov er itself. W e exploit this p oint of view starting from Section 22.5 where M consider e d as an M -enriched ca tegory is called Enr ( M ). How ev er, if we ar e loo king to deﬁne a notion of M -enriched ca tegory , then we s houldn’t start with s omething which is itself a n M -enriched category . O ne can imag ine getting around this problem by no ting that M , considered as a catego ry enriched over itse lf, is actually strictly asso ciative; howev er for lo oking at functors to M we need to go to a weak er mo del, and we end up basically having at lea st to pa ss through the notio n of strict functors ∆ X → M . O ne c ould alterna tively say that instead o f requiring that M b e considered as an M -enr iched catego ry , we could lo o k a t a slightly easier structure such as the Dwyer-Kan simplicial lo calization a sso ciated to M . In this ca se, we would need a theory of weak functors from ∆ o X to a simplicial ca tegory . This theory has a lready bee n done by Bergner [33] [34], so it would be p ossible to go that ro ute. How ev er, it would seem to lead to many notational and ma thematical diﬃculties. Luckily , we do n’t need to w orry a bo ut this issue. It is w ell-known that any kind of weak functors from a usua l 1-catego ry , to a higher cate- gory such a s comes fro m a mo del categor y , ca n b e r e ctiﬁe d (or “stric- tiﬁed”) to actual 1-functors. This beca me apparent as ear ly a s [107] where Grothendieck p ointed out that ﬁb ered ca teg ories a re equiv alen t to strictly cartesian ﬁbe red categor ies. Since then it ha s bee n well-known to homotopy theorists working on dia gram c ategories , and indeed the v ar- ious mo del structur e s on the ca tegory of diagr ams Func (∆ o X , M ) serve to provide mo del catego r ies whose co rresp onding higher catego ries, in whatever s ense one would lik e, a re eq uiv alent to the higher catego r y of weak functors. In the c ontext of diagra ms tow ards Seg al categ ories, a n argument is given in [117]. Due to the philo sophical b o o tstrapping problem ment ioned ab ove, I don’t see any wa y of making the a rgument given in the previous para- graph in to anything other than the heur istic co nsideration that it is. 110 We ak enrichment over a c artesian mo del c ate gory : an intr o duction But, taking it a s a basic pr inciple, w e shall stic k to the notio n of a usual functor ∆ o X → M as b eing the underlying ob ject of study . 7.7 Enforcing the Segal condition W e relaxed the Seg al conditio n in o rder to get a go od lo ca lly present able category PC ( M ) of M -precatego r ies. The Segal condition sho uld then be built into the mo del structure, fo r exa mple it is supp o sed to b e sa t- isﬁed by the ﬁbr a nt ob jects. This g uides our construction of the mo del structure: a ﬁb rant replacemen t should imp os e or “force” the Segal con- dition, and such a pro cess in turn tells us how to deﬁne the notion o f weak equiv a lence. T o understand this, one should view an M -enriched pr ecategor y as being a presc r iption for co nstructing an M -e nr iched category by a co llec- tion of “genera tors and rela tio ns”. The notion of precatego ry w as ma de necessary by the need for colimits, s o one s hould think of a precatego ry as b eing a co limit o f smaller pieces . The asso cia ted M -enriched category should then b e seen a s the homoto p y colimit of the sa me pieces, in the mo del catego ry we are lo o king for. That is to say it is an o b ject sp eciﬁed by generator s and relations. This is explained in some detail for the case of 1 -categor ies in Section 16.8. The c alculus of gener ators and r elatio ns is the pro ce s s wher eby A may be replaced with Seg ( A ) w hich is, in a homotopical sense, the minimal ob ject satisfying the Segal c onditions with a ma p A → Seg ( A ). Another wa y o f putting it is that we enforce the Segal conditions using the small ob ject argument. In order to ﬁnd the mo del ca tegory , we should deﬁne and inv estigate closely this pro ces s of genera ting a n M -enriched category . The cons tr uction Seg ( A ) do esn’t in itself change the set of ob jects of A , so we ca n lo ok at it in the sma ller catego ry PC ( X, M ). There, it can be co nsidered as a case of left Bousﬁeld lo calization. This wa y of breaking up the pro ce dur e was s uggested by Barwick. Luckily , the left Bousﬁeld lo calizatio n which occ ur s here has a pa r ticular form whic h we call “ direct”, in which the the weak e quiv a lences may be characterized explicitly , and we develop that theo ry with gener al nota tions in Chap- ter 11 . Go ing to a more general situation helps to clar ify and simplify notations at each s ta ge; it isn’t clear tha t these discussions would hav e signiﬁcant other a pplications although that cannot be ruled o ut. Con tin- uing in this wa y , we discuss in Cha pter 13 the applica tion of dire ct left 7.7 Enfor ci ng the Se gal c onditio n 1 11 Bousﬁeld lo calization to algebra ic theories in diagra m catego ries. This formalizes the idea o f r e q uiring cer tain dir ect pr o duct maps to b e weak equiv alences, with the ob jectiv e o f a pplying it to the Seg a l maps. Here we refer implicitly to the theories of sketc hes and alg ebraic theories . Then, in Chapters 14 and 16 we apply the pr eceding genera l discus- sions to the case of M -enriched precategor ies, and deﬁne the op eratio n A 7→ Seg ( A ) whic h to an M enriched precatego ry A asso cia tes an M - enriched ca tegory , i.e. a precategor y satisfying the Segal condition. As a r ough approximation the idea is to “force” the Segal condition in a minimal wa y , an op era tion tha t can be acco mplished using a ser ies of pushouts along s ta ndard coﬁbrations. The passage from precatego ries to Segal M -catego ries is inspired by the workings of the theory of simplicia l presheaves a s developped b y Joy al and Jardine [125] [123]. Whereas their ultimate ob jects of interest were simplicial pres heav es satisfying a des cent c ondition, it was most conv enien t to consider all simplicial presheaves and imp ose a mo del structure such that the ﬁbra nt ob jects will s atisfy descent. The Sega l condition is very close to a descent condition as has bee n r emarked by Berger [3 0]. As in [123], we are tempted to us e the inje ctive mo del structu r e for diagrams, deﬁning the coﬁbra tions to be a ll maps of diagr ams A → B which induce co ﬁbrations at each stage A n → B n . It turns out that a slig htly b etter alternative is to use a Ree dy deﬁnition of co ﬁbration, see Chapter 15 . If M is itself an injective mo del categor y then they coincide. It can also b e helpful to maint ain a para lle l pr oje ctive mo del structur e where the coﬁbra tions a re generated by elementary coﬁbrations as originally done by Bousﬁeld. How ev er, the pro jective s tr ucture is not helpful at the iteration step: it w ill not gener ally g ive back a ca rtesian mo del ca tegory . Once we hav e a co ns truction A 7→ Seg ( A ) which enforces the Se- gal condition, a map A → B is said to b e a “weak equiv a le nce” if Seg ( A ) → Seg ( B ) satisﬁes the usual conditions for b eing an equiv a - lence of enriched categories , essential sur jectivit y and full faithfulness. A map of M -enriched categor ies f : A → B is ful ly faithful if, for any t wo ob jects x, y ∈ Ob( A ) the map A ( x, y ) → B ( f ( x ) , f ( y )) is a weak eq uiv a lence in M . T aking a homotopy class pro jection π 0 : M → Set gives a trunca tion o per ation τ ≤ 1 from M -enrich ed cate- gories to 1-categor ies, and we say that f : A → B is essential ly sur - je ct ive if τ ≤ 1 ( f ) : τ ≤ 1 ( A ) → τ ≤ 1 ( B ) is a n essentially sur jective map of 1-catego ries, i.e. it is surjective o n isomorphism cla sses. The isomor phism 112 We ak enrichment over a c artesian mo del c ate gory : an intr o duction classes of τ ≤ 1 ( A ) sho uld be thoug h t of a s the “eq uiv a lence classes” o f ob jects of A . Putting these tog ether, w e say that a map f : A → B is an e quiva lenc e of M -enriche d c ate gori es if it is fully faithful and essen tially surjective. Now, we say that a map f : A → B of M -enriched precategor ies, is a we ak e quivalenc e if the corresp onding map b etw een the M -enriched cate- gories obtained b y generator s a nd relations Seg ( f ) : Seg ( A ) → Seg ( B ) is an equiv alence in the ab ov e sense. With this deﬁnition a nd a n y one of the clas ses of coﬁbrations brieﬂy r eferred to ab ov e and considered in detail in Chapter 15, the sp eciﬁca tio n o f the mo del s tructure is com- pleted by deﬁning the ﬁbra tions to b e the morphisms sa tisfying right lifting with res pect to trivial c oﬁbrations i.e. coﬁbr ations which are weak equiv alences. 7.8 Pro ducts, in terv als and the mo del st ructure The intro duction of M -pre c ategories to gether with the o per ation Seg allows us to deﬁne pushouts of we akly M -enriche d c ate gories : if A → B and A → C a re morphisms of weak M -enr iched categ ories, then the pushout of diag rams ∆ o → M gives an M -precategor y B ∪ A C . The asso ciated pushout in the world of weakly M -enriched categorie s is sup- po sed to b e S eg ( B ∪ A C ). Pr oving that the collections o f maps we hav e deﬁned ab ove, really do deﬁne a closed model catego ry , may b e viewed as showing that this pushout op e ration b ehav es well. As came out pretty clearly in Jar dine’s construction [12 3] but was for malized in Smith’s r e c o gn ition principl e [27] [84] [16 ], the key step is to prov e that pus ho ut by a trivial coﬁbration is again a triv ial coﬁbration. Before getting to the pro of of this prop erty , one has to calcula te some- thing somewhere, which is what is done in the Chapters 18 and 19 lead- ing up to the theorem that the ca lculus of g enerator s and r elations is compatible with direct pr o ducts: Seg ( A ) × Seg ( B ) → Seg ( A × B ) is a weak equiv alence. Our pro of of this c o mpatibility really star ts in Chapter 18 abo ut free or dered M -enriched catego ries. These may be used as basic building blo cks for the gener ators deﬁning the model str uc- ture, so it suﬃces to chec k the pro duct condition on them, which is then done in Chapter 19. The compatibility b etw een Seg a nd direct pr o ducts leads to what will 7.8 Pr o ducts, intervals and t he mo del structur e 113 be the main part o f the car tesian property for the model catego ry which is b eing constructed. This is a categ orical analogue of the Eilenberg - Zilb er theorem for simplicial sets. It wouldn’t b e true if w e hadn’t kept the deg eneracy maps in ∆, and the s tr ict unitality condition seems to be essential too . F rom this result on direct pro ducts, a tr ick lets us conclude the main result for constructing t he mo del structure via a Smith-t ype reco gnition theorem: that trivial coﬁbrations ar e preser ved by pus ho ut. F or that trick, one r equires also a go o d notion of interv al, which was the sub ject of Pelissier’s correction [171] to an errof in [19 3]. Although Pelissier dis- cussed o nly the case of Seg a l ca teg ories enr iched ov er the mo del category K of simplicial sets, his cons tr uction transfers to PC ( M ) by functo- riality using a functor K → M . A somewhat similar cor rection was made by Berg ner in her co nstruction of the mo del category s tructure for simplicial categories o riginally suggested by Dwyer and Ka n [3 3]. The construction of a natura l “interv al categ ory” is describ ed in Chap- ter 2 0 . It is a sort o f versal replacement for the simple categ ory 0 ← → 1 with t wo isomo rphic o b jects. This is the po int where Pelissier’s co r- rection [171] of [193] comes in, a nd in order to make the pro cess fully iterative we just hav e to p oint out that an interv al for the ca se of the standard mo del categor y M = K of simplicial sets, lea ds by functo- riality to an interv al for any o ther M . The g o od version of the versal int erv al co nstructed by Pelissier [1 71] is similar to the interv al ob ject for dg-catego ries subsequently in tro duced b y Drinfeld [8 3]. W e mo dify slightly Pelissier’s construction, but one could use his orig inal one to o. Once all of these ingredients are in place, we can construct the mo del structure in Cha pter 2 1. W e obtain a mo del catego ry structure on PC ( M ) which a gain satisﬁes all of the hypotheses which w ere required of M , so the pro cess ca n b e itera ted. Starting with the tr ivial mo del categ o ry s tr ucture on Set , the n -th iterate PC n ( Set ) is the mo del catego ry str uctur e for n -pr ecategor ies as considered in [193]. If instead we s tart with the Ka n-Quillen mo del cate- gory K o f simplicial sets, then PC n ( K ) is the model categor y of Segal n -precateg ories which was used in the work [117] ab out n -stacks. W e discuss these iterations for weak n -catego ries (which a r e T amsamani’s n -nerves ) and Segal n -ca teg ories, tog e ther with a few v ariants where the initailzing ca tegory o f s ets is repla ced by a catego ry of gra phs or other things, in Chapter 22. The internal Hom op eration then leads to a categor y enriched over 114 We ak enrichment over a c artesian mo del c ate gory : an intr o duction our new mo del ca tegory , which in the iter ative scheme for n -categor ies gives a construction of the n + 1 -categor y nC AT . The last par t of the bo o k, not yet inc luded in the presen t v ersion, will be dedicated to co nsidering how to write in this language some basic elements of higher catego ry theo ry , such as in verting mo rphisms, and limits and colimits. W e also hop e to dis c uss the Breen-Ba ez-Dolan sta- bilization h yp othesis, ab out the behavior of the theor ies of n - c a tegories for diﬀerent v alues of n . P A R T I I CA TEGORICAL PRELIMINARIES 8 Some catego r y theory In this chapter, we regroup v arious things which ca n be said in the context of a bstract categ ory theo ry . O ur dis cussion is bas ed in large part o n the bo ok o f Adamek and Rosicky [2] ab out lo ca lly presentable and acces sible categorie s. Refer there for historical re ma rks ab out these notions. The a pplicability of this theory to mo del categories came out with J. Smith’s notio n o f c ombinatorial mo del c ate gory [198], slightly mo diﬁed by Barwick with his notion of tr actable mo del c ate gory [1 6]. One of our main goals is to provide a fair ly gener al discussion of the notion of c ell complex in a lo cally presentable category . Hirschhorn has formalized the use of c ell complexes for the s mall ob ject arg umen t and le ft Bousﬁeld lo ca lization, in [116]. How ever, he used an additional assumption of a monomor phism prop erty o f elements o f the g enerating set o f arrows I ⊂ Arr ( M ), enco ded in his no tion o f c el lular mo del c ate gory . W e would like to avoid this hypothesis. Indeed, one of the main examples whic h w e can use to start out our induction is the mo del category on Set , but as Hir s chhorn p ointed out this is not cellular. It turns out that a somewha t mor e abstract appro ach to cell co mplexes works prett y well. Our discussion covers muc h the sa me materiel as Lurie in the a pp endix to [153]. Lurie intro duces a notion of “tree” gener alizing the standard transﬁnite cell-a ddition pr o cess. The ba sic idea is that once we have attached a certain num ber of cells, the next cell is attached a long a κ -presentable sub complex, but this informatio n is lo s t under the usual indexation by an ordinal. In o ur discussion, just to b e diﬀerent, we’ll stick to the sta nda rd ordinal prese n tation, but we intro duce a categ o ry of “inclusio ns of cell complexes” , and s how that the catego ry of κ -small inclusions of cell complexes into a given o ne, is κ -ﬁltered. Roughly sp eak- ing, an inclusion of ce ll complexes cor resp onds to a down w ard-clos e d This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 118 Some c ate gory the ory subset of a tree. W e sketc h a pro of of Lur ie ’s theorem [153, Prop os ition A.1.5.12] that co ﬁbrations are cell complexes ov er κ -small coﬁbratio ns, rather than just retracts o f such. The main a pplica tion of this r esult is to construc t the genera ting s et for injectiv e coﬁbra tions. Again w e give a brief acc o unt of a pr o of of the main technical r esult in the present chapter, althoug h the reader can also r efer to [1 53] and [16]. This dis cussion pr epares t he w a y for the “ recognition principle ” intro- duced in Chapter 9, bas ed on Smith’s recognitio n principle a s rep orted by Barwick [16]. Our additio n is to give a statement which enco des the accessibility argument. The adv antage is that the notion of accessibility no longe r app ears in the statement, so we ca n then us e that in later chapters to construct mo del categorie s without ne e ding to disc us s the notion of a c cessibility anymore. So, in a certain sense wha t we are doing here is to ev ac ua te some of the mor e technical details in the theory of mo de l ca tegories, tow ards these ﬁr st tw o chapters. W e hop e that this will be helpful to the reader who wishes to av oid this kind of discussion: if willing to tak e for gr anted the recognition principle which will b e stated as Theor em 9 .9.7 in the next chapter, the reade r may la rgely skip over the mo st technical par ts of these ﬁrst t wo chapters. In o rder to av oid rep etitive languag e, we often apply the following conv en tions a b out universes. W e assume given a t least tw o universes U ∈ V . Recall that these ar e sets which themselves provide mo dels for ZFC se t theory . A c a te gory will mean a c ategory ob ject in V . An exa mple is the category Set U of s ets in U . A smal l c ate gory will b e a catego ry ob ject in U , which is also one in V . Often a ca tegory C will have smal l morphism s et s , that is for any x, y ∈ Ob( C ) the set H om C ( x, y ) ∈ V is isomorphic to a set in U . Depending on co nt ext, the word “ categor y” ca n so metimes mean “sma ll category ”, o r sometimes “ category wit h small morphism sets” . How ever, when we need to consider categ ories outside of V this will b e explicitly men tioned. Recall that an or dinal is a s et a , s uch tha t if x ∈ y and y ∈ a then x ∈ a ; a nd suc h that a is well-ordered b y the strict relation x < y ⇔ x ∈ y for x, y ∈ a . F or the co rresp onding no n-strict or der re lation we then hav e x ≤ y ⇔ x ⊂ y , and for any x ∈ a the successo r of x is x ∪ { x } . A c ar dinal is an or dinal a with the pro pe rty that for any b ∈ a , b is not is omorphic to a . Any set x has a unique c ar dinality | x | which is a cardinal such that x ∼ = | x | . 8.1 L o c al ly pr esentable c ate gories 119 An or dinal (r esp. c ar di nal) of U is an ordinal (r e sp. cardinal) whic h is an element of U . Thes e are the ordina ls (resp. c a rdinals) for the mo del of s et theory g iven b y U . In particular , for any x ∈ U we hav e | x | ∈ U . W e s ay that an ordinal α is appr o ache d by a se quenc e of c ar dinality λ if there is a subset x ⊂ α with | x | = λ , such that α is the lea st upper b ound of x . A car dina l κ is r e gular if it is not approached by any sequence o f cardinality < κ . 8.1 Lo cally presen table categories Fix a regula r cardinal κ . A categor y Φ is said to b e κ -ﬁlt ere d if for a n y collection of < κ ob jects X i ∈ C , there exists an o b ject Y and mo r phisms X i → Y ; a nd for any pair of ob jects X and Y , a nd any co llection of < κ morphisms f i : X → Y ther e exists a morphism g : Y → Z such that all the g f i are e q ual. Note that taking a n empt y set of o b jects in the ﬁr st condition implies that C is nonempty . A κ -ﬁlt er e d c o limit is a colimit over a κ -ﬁltered index categor y . See [2, Remark 1.21]. Let C b e a ca teg ory . W e assume that C a dmits κ -ﬁltered colimits. Then, say tha t an ob ject X ∈ C is κ -pr esentable if, for any κ -ﬁlter e d colimit c o lim i ∈ Φ Y i = Z , the map colim i ∈ Φ Hom C ( X, Y i ) → H om C ( X, Z ) is an isomor phism of sets. An ob ject Z ∈ C is said to b e κ - ac c essible if it can be ex pressed a s a κ - ﬁlter ed co limit of κ -presentable ob jects. Deﬁnition 8.1.1 L et κ b e a r e gular c ar di nal. A c ate gory C with smal l morphism s et s is c al le d κ -a ccessible if: (1)— C admits κ -ﬁlter e d c olimits; (2)—the ful l sub c ate gory of κ - pr esentable obje cts is e quivalent to a smal l c ate gory; (3)—every obje ct of C is κ -ac c essible. F urthermor e, if C admits al l smal l c olimits then we say that C is lo cally κ -presentable . A c ate gory which is lo c ally κ - pr esentable for some r e gular c ar dinal κ is c al le d lo cally presentable . F or the co unt able ca rdinal κ = ω , the terminolog y “lo cally ﬁnitely presentable” is in terchangeable with “lo c a lly ω -presentable”. Theorem 8.1.2 Supp ose C is lo c al ly pr esentable. Then it is c omplete, i.e. it admits smal l limits to o. Each obje ct has only a smal l set of s u b- obje cts u p to isomorphism. A l l κ -ﬁlter e d c o limits c ommute with κ -smal l 120 Some c ate gory the ory limits (i.e. limits over c ate gories c ar dinality < κ ). F or any X ∈ C , t he sub c ate gory C κ /X of κ - pr esentable obje cts of C , is κ -ﬁlter e d and X is c anonic al ly t he c olimit of the for getful functor on C κ /X . If C is lo c al ly κ -pr esentable then for any r e gular c ar d inal κ > κ it is lo c al ly κ ′ -pr esentable to o. Pr o of See [15 9], or [2], Prop osition 1 .22, Coro llary 1.28, Remar k 1.56 and P rop osition 1.59. F o r the last s e n tence see [2, page 22]. Lemma 8. 1.3 S u pp ose Ψ is a smal l c ate gory, and C is lo c al ly κ - pr esentable. Then the c ate gory Func (Ψ , C ) of diagr ams fr om Ψ t o C , is lo c ally κ -pr esentable. The κ -pr esentable diagr ams in Func (Ψ , C ) ar e exactly the functors F : Φ → C such that F ( a ) is κ -pr esentable in C for every a ∈ Φ . Pr o of See Mak k ai and Pare [159], or Adamek and Ro sicky [2, Cor ollary 1.54]. Corollary 8.1.4 Su pp ose Ψ is a smal l c ate gory , then the c ate gory Presh(Ψ) of pr eshe aves of set s on Ψ , is lo c al ly ﬁ nitely pr esentable. Pr o of The category of s ets is lo cally ﬁnitely presentable. If C is a catego ry , let A rr ( C ) be the catego ry of arrows of C , whose ob jects a re the diagra ms of sha pe X f → Y in C . The morphisms in Arr ( C ) from X f → Y to X ′ f ′ → Y ′ are the commutativ e square s X → X ′ Y f ↓ → Y ′ f ′ ↓ . Corollary 8. 1.5 Supp ose C is a lo c al ly κ - pr esentable c ate gory. Then Arr ( C ) is lo c ally κ -pr esentable. Pr o of Indeed, Arr ( C ) = Fun c ( E , C ) where E is the catego ry with t wo ob jects 0 , 1 and a s ingle morphis m 0 → 1 b esides the identities. Therefore 8.1.3 a pplies. In a similar wa y , any c a tegory o f commutativ e diagrams o f a given shap e in a lo cally presentable category , will again b e loca lly pr esentable. A functor q : α → β is c oﬁnal if: —for any ob ject i ∈ β there exists j ∈ α and an ar row i → q ( j ); 8.1 L o c al ly pr esentable c ate gories 121 —for a n y pa ir of a rrows i → q ( j ) and i → q ( j ′ ) in β there a re a rrows j → j ′′ and j ′ → j ′′ in α such that the diagra m i → q ( j ) j ′ ↓ → q ( j ′′ ) ↓ commutes (see [2, 0.11]). Recall that if q : α → β is a coﬁnal functor , then it induces an equiv alence betw een the theo ry of colimits indexed b y β and the theory of co limits indexed b y β , see [2, pag e 4 ]. A basic and mo tiv a ting example of a lo cally presentable categ ory is when Φ o is a site, and M ⊂ Set Φ U is the sub categ ory of sheav es. In this case H ∗ just deno tes the ident ity inclusion, whereas T = H ! is the sheaﬁﬁcation functor. This motiv a tes the following characterization. Prop ositio n 8. 1.6 A V -c ate go ry C with U -smal l morphism s et s is lo c al ly pr esentable if it has U -smal l limits and c olimits, and if ther e exists a U - s mal l c ate gory Φ and adjunction H ! : Set Φ U ← → C : H ∗ and a r e gular c ar dinal κ ∈ U such t hat H ! H ∗ is the identity, and t he c omp osition T := H ∗ H ! : Set Φ U → Set Φ U c ommutes with β -dir e cte d c ol- imits. Or e quivalently, that H ∗ itself pr eserves κ -dir e cte d c olimits. Pr o of This rephra ses [2], Theorem 1 .46, using an adjoint pair of func- tors such tha t one co mpo sition is the identit y , instea d of a full reﬂective sub c ategory . A lo c a lly κ - pr esentable categor y C will in ge neral have the pr op erty that | Hom ( X , Y ) | > κ for t wo κ -pr esentable ob jects. F or example, Set Φ U is lo cally κ -pres en table whatever the size of Φ, but if | Φ | > κ then there can b e > κ morphisms b etw een κ - presentable o b jects. This is r ectiﬁed by taking κ big enough, but the bo und has to b e exp onential in κ beca use we lo ok at ma ps fro m ob jects o f size < κ . Corollary 8.1.7 Supp ose C is a lo c ally pr esentable c ate gory. Ther e is a r e gular c ar d inal κ such that C is lo c ally κ -pr esentable, such that for any r e gular c ar dinals λ, µ > κ and obje cts X , Y such that X is λ - pr esentable and Y is µ -pr esentable, the set of morphisms Hom C ( X, Y ) has size < µ λ . 122 Some c ate gory the ory Pr o of Supp o se C is κ 0 -presentable to b e gin with. The total cardinal- it y o f a presheaf A ∈ Set Φ U is the sum of the ca rdinalities of the v a l- ues A ( x ) for x ∈ Φ. Use the c haracter isation o f the previous prop o- sition, and cho ose a new regula r car dinal κ 1 > sup( κ 0 , | Φ | ). F or any λ ≥ κ 1 the λ -presentable ob jects of C a re those of the fo rm H ! ( A ) for pr esheav es A : Φ → Set U of total cardinality < λ (se e [E xample 1.31]AdamekRo sicky). Choose a regular cardina l κ ≥ κ 1 so that the to- tal cardinality of H ∗ H ! ( B ) is < κ for any presheaf B of total cardinalit y < κ 1 . Now supp ose λ, µ > κ . By [2 , Remark 1.30(2 )], the λ -presentable ob jects o f C are κ 1 -ﬁltered colimits of size < λ , of κ 1 -presentable ob- jects (and the sa me for µ ). Supp ose X a nd Y are λ -pr esentable and µ -presentable ob jects resp ectively . W rite X = c o lim i ∈ I H ! ( A i ) (resp. Y = co lim j ∈ J H ! ( B j )) wher e I (res p. J ) is a κ 1 -ﬁltered ca tegory of size < λ (resp. o f size < µ ) , and A i and B j are presheaves of total cardinality < κ 1 . Then H ∗ H ! ( B j ) has total c ardinality < κ . Now Hom C ( X, Y ) = lim i ∈ I (colim j ∈ J Hom C ( A i , H ∗ H ! ( B j ))) . This ha s s ize < µ λ . One should als o b e able to prove this using the characteriza tio n of lo cally pr esentable c ategories as categor ies of models of limit theorie s [2, Theorem 5 .30]. The following lemma w ill b e useful in dea ling with unitality co nditions in Cha pter 13 . Lemma 8 .1.8 Supp ose M is a c ate gory with c oinitia l obje ct ∗ , and supp ose α is a nonempty c onne cte d smal l c ate gory (that is, a c ate gory whose nerve is a c onne cte d simplicia l set). Then the c olimit of the c on- stant fu n ctor C · : α → M deﬁne d by C i = ∗ for al l i ∈ α , exists and is e qual to ∗ . Pr o of There is a uniq ue compatible system of morphisms φ i : C i = ∗ → ∗ . W e claim that this makes ∗ in to a colimit of C · . Suppose U ∈ M and ψ i : C i → U is a compa tible s ystem of morphisms. P ick i 0 ∈ α and use f := ψ i 0 : C i 0 = ∗ → U . W e cla im that for any j ∈ α the comp osition f φ j is e q ual to ψ j . Let α ′ be the subset of ob jects of α for which this is true, no nempt y since it co nt ains i 0 . If j ′ ∈ α ′ and if g : j → j ′ is a n a rrow of α then f φ j = f φ j ′ C g = ψ j ′ C g = ψ j 8.2 Monadic pr oje ction 123 so j ∈ α ′ . Supp ose j ′ ∈ α ′ and if g : j ′ → j is an arrow of α . Then f φ j C g = f φ j ′ = ψ j ′ = ψ j C g but C g is an isomorphis m so f φ j = ψ j and j ∈ α ′ . These tw o steps imply inductively , using connectedness of α , that α ′ = Ob( α ). Thus f φ · = ψ · . Clearly f is unique, so we get the requir ed universal pr op erty . 8.2 Monadic pro jection Suppo se C is a catego r y , and R ⊂ C a f ull subcategor y . W e assume that R is sta ble under isomorphisms. A monadic pr oj e ction from C to R is a functor F : C → C together with a natural transformatio n η X : X → F ( X ), such that: (Pr1)— F ( X ) ∈ R for all X ∈ C ; (Pr2)—for any X ∈ R , η X is a n isomorphism; and (Pr3)—for any X ∈ C , the map F ( η X ) : F ( X ) → F ( F ( X )) is an isomorphism. Lemma 8.2.1 Su pp ose ( F , η ) is a monadic pr oje ction. Then the two isomorphi sms F ( η X ) and η F ( X ) fr om F ( X ) to F ( F ( X )) ar e e qual. Pr o of Naturality of η with resp ect to the morphism η X gives the com- m utative diagr a m X η X → F ( X ) F ( X ) η X ↓ F ( η X ) → F ( F ( X )) η F ( X ) ↓ . F or any X ∈ R , η X is an isomo rphism so comp osing with its inv erse we get F ( η X ) = η F ( X ) . On the other hand, w e ca n also apply F to the ab ov e diagram. By (Pr3), for any X ∈ C we have that F ( η X ) is a n isomor phism s o again comp osing with its in verse we co nclude that F ( F ( η X )) = F ( η F ( X ) ) for any X ∈ C . 124 Some c ate gory the ory F or arbitrary X ∈ C , consider the dia g ram F ( X ) F ( η X ) → F ( F ( X )) F ( F ( X )) η F ( X ) ↓ F ( F ( η X )) → F ( F ( F ( X ))) η F ( F ( X )) ↓ . It commutes by naturality of η with res pec t to the mor phism F ( η X ). On the other hand, b y the ﬁrs t statemen t w e pr ov ed ab ove, and noting that F ( X ) ∈ R by (Pr 1), we hav e η F ( F ( X )) = F ( η F ( X ) ). On the other hand, by the seco nd statement we prov ed ab ove, F ( F ( η X )) = F ( η F ( X ) ). W e hav e now shown that b oth second maps in the tw o equal comp ositions of this diagr a m, are the same is omorphism. It follows that the tw o ﬁrst maps alo ng the top and the left side, are the same. This prov es the lemma. Prop ositio n 8.2.2 Supp ose R ⊂ C ar e as ab ove, and ( F, η ) and ( G, ϕ ) ar e t wo monadic pr oje ctions fr om C to R . Then, for any X ∈ M the maps F ( ϕ X ) : F ( X ) → F ( G ( X )) and G ( η X ) : G ( X ) → G ( F ( X )) ar e isomorphi sms. The diagr am of isomorphi sms F ( X ) F ( ϕ X ) → F ( G ( X )) G ( F ( X )) ϕ F ( X ) ↓ ← G ( η X ) G ( X ) η G ( X ) ↑ c ommutes. Pr o of Deﬁne the functor H ( X ) := G ( F ( X )), with a natural transfor - mation ψ X : X → H ( X ) deﬁned as the comp osition X ϕ X → G ( X ) G ( η X ) → G ( F ( X )) . Naturality of the tr ansformation ϕ with r esp ect to the morphism η X 8.2 Monadic pr oje ction 125 gives a comm utative diagr a m X ϕ X → G ( X ) F ( X ) η X ↓ ϕ F ( X ) → G ( F ( X )) G ( η X ) ↓ giving the the a lternative express ion for ψ , ψ X = G ( η X ) ϕ X = ϕ F ( X ) η X . (8.2.1) W e claim th at ( H, ψ ) is a gain a monadic pro jection from C to R . The ﬁrst (Pr1) is a dire ct consequence of the same conditions for F a nd G . F or the second co ndition, supp ose X ∈ R . Then η X is an iso mo rphism by (Pr2) for ( F, η ), and ϕ F ( X ) is an isomorphism by (Pr2 ) for ( G, ϕ ) plus (Pr 1 ) for F . By the expre s sion (8.2.1) we get tha t ψ X = ϕ F ( X ) η X is an isomor phism which is (Pr2) for ( H, ψ ). One c o uld instead use the expression ψ X = G ( η X ) ϕ X and the fact that a functor G prese r ves isomorphisms. F or the third condition, note from (8.2.1) aga in, that H ( ψ X ) is ob- tained by a pplying G to the comp osed map F ( X ) F ( η X ) → F ( F ( X )) F ( ϕ F ( X ) ) → F ( G ( F ( X ))) . (8.2.2) The ﬁrst ar row F ( η X ) is a n is omorphism by (Pr 3 ) for ( F, η ). Consider the diag ram F ( X ) η F ( X ) → F ( F ( X )) G ( F ( X )) ϕ F ( X ) ↓ η G ( F ( X )) → F ( G ( F ( X ))) F ( ϕ F ( X ) ) ↓ . It commu tes b y naturality of η with resp ect to the morphism ϕ F ( X ) . The right vertical arrow is the the seco nd arr ow in the co mpos ition (8.2.2). The top arr ow is an iso morphism by (Pr1) and (Pr2 ) for ( F , η ). The left arrow is an isomorphis m by (Pr1) for F a nd (Pr2) for ( G, ϕ ). The bo ttom a rrow is an isomor phism by (Pr1) for G and (Pr 2) for ( F, η ). It follows that the right vertical arrow F ( ϕ F ( X ) ) is an isomorphism. 126 Some c ate gory the ory Now apply the ab ov e co nclusions together with the fact that G pr e- serves isomorphis ms , in the express ion H ( ψ X ) = G  F ( ϕ F ( X ) ) ◦ F ( η X )  to conclude that H ( ψ X ) is an isomor phism. This completes the pro of of (Pr3) to show that ( H, ψ ) is a mona dic pr o jection. Contin ue now the pro of of the prop osition. W e hav e a morphism G ( X ) G ( η X ) → G ( F ( X )) = H ( X ) . Consider the dia gram G ( X ) G ( ϕ X ) → G ( G ( X )) H ( X ) G ( η X ) ↓ H ( ϕ X ) → H ( G ( X )) G ( η G ( X ) ) ↓ H ( G ( η X )) → H ( H ( X )) . The square c o mm utes b ecause it is o btained by applying G to the nat- urality diagram for η with resp ect to the morphism ϕ X . The to p arrow is an iso morphism by (Pr 3) for ( G, ϕ ). The middle vertical arr ow is an isomorphism b y applying G to (Pr2) for ( F, η ) and using (Pr 1 ) for G . This shows that the comp os itio n in the s q uare is a n isomor phism, s o we deduce that the comp osition of the left b ottom a rrow with the left vertical arrow is an isomorphism. The comp osition along the bo ttom is equal to H ( ψ X ), indeed ψ X = G ( η X ) ϕ X by deﬁnitio n. Hence, by (Pr3) fo r ( H , ψ ) w hich w as proven ab ov e, the comp osition along the bo ttom is an isomorphism. The morphism H ( ϕ X ) at the b ottom o f the squar e, now has ma ps which comp ose on the left a nd the rig ht to isomorphisms. It follows that this map is an isomorphism, and in tur n that the left vertical arr ow G ( η X ) is an isomorphism. This is one of the statements to be prov en in the pro p o sition. The other statement, that F ( ϕ X ) is an isomo rphism, is obta ine d by symmetry with the roles of F and G reversed. T o ﬁnish the pro of, we have to show that the s quare diagra m of is omor- phisms commutes (which means that it commutes as a usually shap ed diagram when the inv erses of the isomorphis ms are included). Apply F G to the diag ram in question, and add o n a no ther square to 8.2 Monadic pr oje ction 127 get the diagram o f isomorphisms F GF ( X ) F GF ( ϕ X ) → F GF G ( X ) F GGF ( X ) F G ( ϕ F ( X ) ) ↓ ← F GG ( η X ) F GG ( X ) F G ( η G ( X ) ) ↑ F GF ( X )) F G ( ϕ F ( X ) ) ↑ ← F G ( η X ) F G ( X ) . F G ( ϕ X ) ↑ The b ottom squar e commutes by F G applied to the naturality square for ϕ with r e spe c t to η X . On the left side, we have the same isomo rphism going in both dir ections. Consider the o uter square F GF ( X ) F GF ( ϕ X ) → F GF G ( X )) F GF ( X )) w w w w w w w w w ← F G ( η X ) F G ( X ) . F G ( η G ( X ) ϕ X ) ↑ It is o btained by applying F G to the squar e X η X → F ( X ) G ( X ) ϕ X ↓ η G ( X ) → F G ( X ) F ( ϕ X ) ↓ which commut es by natura lit y of η with resp ect to ϕ X . Since the o uter square, and the b ottom sq ua re of the ab ov e diagr a m of iso morphisms commute, it follows that the upp er square commutes. The upper s quare was obtained by applying F G to the square of isomorphisms in R in question, and F G is a functor which is naturally isomorphic to the iden- tit y functor on R . Therefore the diagram of the prop osition, which is a diagram of is o morphisms in R , commutes. 128 Some c ate gory the ory 8.3 Miscellan y ab out limits and colimits Here is a n element ary observ ation ab out limits. Lemma 8.3. 1 Supp ose F : I → J is a fun ctor b etwe en smal l c ate- gories, supp ose M is c o c omplete, and supp ose B : J → M is a diagr am. Pul lb ack along F induc es the diagr am B ◦ F which is note d F ∗ B . Ther e is an induc e d m ap in M colim I F ∗ B → colim J B Pr o of Let C I and C J be the “constant diagr a m” constructio ns. One wa y of deﬁning colimits is by adjunction with C · . W e ha ve a universal map in M J B → C J (colim J B ) , and a pplying restriction along F w e get a map F ∗ B → F ∗ ( C J (colim J B )) = C I (colim J B ) . By universalit y of the map B ◦ F → C I (colim I F ∗ B ) w e get a facto riza- tion F ∗ B → C I (colim I ( B ◦ F )) → C I (colim J B ) for a unique map colim I F ∗ B → colim J B which is the map in question for the lemma. Here is a nother fact useful in the co nstruction of adjoints. Lemma 8 .3.2 Supp ose p : α → β is a functor b etwe en smal l c ate- gories, and supp ose A : β → M is a diagr am with values in a c o c om- plete c ate gory M . Then ther e is a natur al map c olim α p ∗ ( A ) → colim β A , satisfying c omp atibi lity c onditions in c ase of c omp osi tions of functors. Pr o of Indeed, ther e is a ta uto logical natura l tra nsformation o f β -diagra ms from A to the cons tant diagra m with v alues colim β A , and the pull- back of this natura l trans formation to α is a natura l trans formation of α -diagra ms fro m p ∗ ( A ) to the constant diagram with v alues colim β A , which gives the map c olim α p ∗ ( A ) → co lim β A in question. If q : δ → α is a nother functor then the compo sition of the maps for p and fo r q colim α q ∗ ( p ∗ ( A )) → co lim α p ∗ ( A ) → colim β A is the natural ma p for pq . Similarly , if p is the identit y functor then the asso ciated natural map is the identit y . 8.4 Diagr a m c ate gories 129 8.4 Diagram categories Suppo se Φ is a small ca tegory and M a ca tegory . Consider the diagr am c ate gory Func (Φ , M ) of functors Φ → M . If M is complete (res p. c o- complete) then so is Func (Φ , M ) a nd limits (res p. colimits) o f diagra ms are co mputed ob ject wise, that is ov er each ob ject of Φ. Suppo se f : Φ → Ψ is a functor b e t ween small catego ries. Given any diagra m A : Ψ → M then the comp osition A ◦ f is a diagram Φ → M whic h will also be denoted f ∗ ( A ). This gives a functor f ∗ : Func (Ψ , M ) → Func (Φ , M ). If M is complete (resp. co complete) then the functor f ∗ preserves limits (resp. colimits) since they are computed ob ject wise in b oth Func (Φ , M ) and Func (Ψ , M ). W e consider the left and rig h t adjoints of f ∗ . This parallels the similar discussion worked out with A. Hirschowitz in in [117, Chapter 4] but of course these things ar e of a nature to hav e b een well-known mu ch earlier. See a lso [153, A.2.8.7]. If M is co complete then we can construct a left adjoint f ! : Func (Φ , M ) → Func (Ψ , M ) as follows. F or an y ob ject y ∈ Ψ consider the category f / y of pa irs ( x, a ) where x ∈ Φ and a : f ( x ) → y is an arr ow in Ψ . Ther e is a forgetful functor r f ,y : f /y → Φ sending ( x, a ) to x . Suppo se A ∈ Fun c (Φ , M ). Put f ! ( A )( y ) := colim f /y r ∗ f ,y ( A ). Suppose g : y → y ′ is an arr ow. Then we obtain a functor c g : f /y → f /y ′ sending ( x, a ) to ( x, g a ). F urthermore this commutes with the forgetful functors in the s ense that r f ,y ′ ◦ c g = r f ,y . Thus r ∗ f ,y ( A ) = c ∗ g ( r ∗ f ,y ′ ( A )). Applying the a bove remark ab out colimits, we get a natural map f ! ( A )( y ) := colim f /y c ∗ g ( r ∗ f ,y ′ ( A )) → co lim f /y ′ r ∗ f ,y ′ ( A ) =: f ! ( A )( y ′ ) . Using the last par t o f the par agraph ab out colimits ab ove, we see that this collectio n of maps turns f ! ( A ) into a functor from Ψ to M , that is an ob ject in Func (Ψ , M ). The construction is functorial in A so it deﬁnes a functor f ! : Func (Φ , M ) → Func (Ψ , M ) . The str uctural maps for the colimit deﬁning f ! ( A )( y ) are maps A ( x ) → f ! ( A )( y ) for any a : f ( x ) → y . In particular when y = f ( x ) and a is the iden- tit y we get maps A ( x ) → f ! ( A )( f ( x )) = f ∗ f ! ( A )( x ). This is a natur a l transformatio n from the identit y on Func (Φ , M ) to f ∗ f ! . On the other hand, s uppo s e A = f ∗ ( B ) for B ∈ Func (Ψ , M ). Then, 130 Some c ate gory the ory for any ( x, a ) ∈ f /y w e get a map r ∗ f ,y ( f ∗ ( B ))( x, a ) = B ( f ( x )) B ( a ) → B ( y ). This gives a map fro m r ∗ f ,y ( f ∗ ( B )) to the constant diagram with v alues B ( y ), hence a map on the colimit f ! f ∗ ( B )( y )co lim f /y r ∗ f ,y ( f ∗ ( B )) → B ( y ) . It is functor ia l in y and B so it giv es a na tural transformation from f ! f ∗ to the identit y . Lemma 8.4.1 Supp osing that M is c o c omple te and with these natu ra l tr ansformatio ns, f ! b e c omes left adjoint to f ∗ and one has the formula f ! ( A )( y ) = colim f /y r ∗ f ,y ( A ) . Pr o of Supp o se A ∈ Func (Φ , M ) a nd B ∈ Fun c (Ψ , M ). A map A → f ∗ ( B ) consists of g iving, for e ach x ∈ Φ, a map A ( x ) → B ( f ( x )). This is e quiv- alent to giv ing, for ea ch y ∈ Ψ a nd ( x, a ) ∈ f /y , a map A ( x ) → B ( y ) sub ject to some natura lit y constraints a s x, a, y v ary . This in turn is the sa me as giving a map of f / y -diag rams from r ∗ f ,y ( A ) to the constant diagram with v alues B ( y ), which in turn is the same as g iving a map from f ! ( A ) = colim f /y r ∗ f ,y ( A ) to B . It is le ft to the reader to verify that these ide ntiﬁcations are the same as the ones giv en by the ab ov e-deﬁned adjunction ma ps. Lemma 8. 4.2 Supp ose t hat M is c omp lete. Then f ∗ has a right ad- joint denote d f ∗ given by the formula f ∗ ( A )( y ) = lim f \ y s ∗ f ,y ( A ) wher e f \ y is t he c ate gory of p airs ( z , u ) wher e z ∈ Φ and f ( z ) u → y is an arr ow in Ψ , and s f ,y : f \ y → Φ is the for getful fun ctor. Pr o of Apply the previous lemma to the functor f o : Φ o → Ψ o for diagrams in the o ppo site ca tegory M o . 8.5 Enric hed categories Enriched catego ries have b een familiar ob jects for quite a while, see Kelly’s b o o k [139]. Our overall goal is to discuss a homotopica l a nalogue susceptible of b eing iterated, so it is worthwhile to re call the cla ssical theory . F urther more, this will provide an impo rtant in termediate step of our ar gument: a weakly enriched catego ry over a mo del categ ory M gives r ise to a ho( M )-enriched ca tegory in the clas sical sense, and this 8.5 Enriche d c ate gories 131 construction is co nserv ativ e for w eak equiv a lences. This is used notably for Prop osition 14.6.4. Suppo se E is a categor y admitting ﬁnite direct pro ducts. This includes existence of the co initial ob ject ∗ which is the empty direct pro duct. If X is a set, then a E -enriche d c ate gory on obje ct set X is a collection of ob jects A ( x, y ) ∈ E for x, y ∈ X , together with morphis ms A ( x, y ) × A ( y , z ) → A ( x, z ) a nd ∗ → A ( x, x ), suc h that for an y x, y the compos ed map ∗ × A ( x, y ) → A ( x, x ) × A ( x, y ) → A ( x, y ) is the identit y; the comp osed map A ( x, y ) × ∗ → A ( x, y ) × A ( y , y ) → A ( x, y ) is the identit y; and for any x, y , z , w the dia g ram A ( x, y ) × A ( y , z ) × A ( z , w ) → A ( x, y ) × A ( y , w ) A ( x, z ) × A ( z , w ) ↓ → A ( x, w ) ↓ commutes. An E -enriche d c ate go ry is a pair ( X , A ) a s ab ove, but often this will be deno ted just by A with X = Ob( A ). A functor betw een E -enriched categorie s f : A → B co ns ists of a map of sets f : Ob( A ) → Ob( B ), and for each x, y ∈ Ob( A ) a mor phism f x,y : A ( x, y ) → B ( f ( x ) , f ( y )) in E , such that the diagr ams ∗ → A ( x, x ) ∗ ↓ → B ( f ( x ) , f ( x )) ↓ and A ( x, y ) × A ( y , z ) → A ( x, z ) B ( f ( x ) , f ( y )) × B ( f ( y ) , f ( z )) ↓ → B ( f ( x ) , f ( z )) ↓ commute. 132 Some c ate gory the ory Let Ca t ( E ) denote the categor y of E - enriched categor ies. It admits direct pro ducts to o : if A and B ar e E -enriched categor ies the Ob( A × B ) = O b( A ) × Ob( B ) , and fo r ( x, x ′ ) and ( y , y ′ ) in Ob( A ) × Ob( B ) we hav e ( A × B )(( x, x ′ ) , ( y , y ′ )) = A ( x, y ) × B ( x ′ , y ′ ) . Hence this co nstruction can b e iterated and we can obta in Ca t n ( E ), the category o f strict n -categor ies enriched in E at the top level. Starting with E = S et yields Ca t ( Set ) = Ca t , and Ca t n ( Set ) is the category of strictly asso c iative and strictly unital n -categ ories. These ob jects have b een studied a gr eat deal. How ever, as we have seen in Chapter 4, thes e ob jects do not hav e a suﬃcien tly rich ho motopy theory , in particular the gro upo id o b jects therein do not model homotopy types in an y reasonable way . This oberv ation is the motiv ation for considering weakly asso ciative ob jects as in the remainder o f this work. If ϕ : E → E ′ is a functor compatible with direct pro ducts, apply- ing it to the morphism ob jects of an E -enriched category A gives an E ′ -enriched ca tegory Ca t ( ϕ )( A ) with the sa me set of ob jects, and mor- phism o b jects deﬁned b y Ca t ( ϕ )( A )( x, y ) := ϕ ( A ( x, y )) . W e apply this in pa rticular to the functor τ ≤ 0 : E → Set deﬁned by τ ≤ 0 ( E ) := Hom E ( ∗ , E ). Deﬁne τ ≤ 1 := Ca t ( τ ≤ 0 ), i.e. τ ≤ 1 A ( x, y ) = Hom E ( ∗ , A ( x, y )) . With th is, τ ≤ 1 A ∈ Ca t ( Set ) is a usual category . Let Iso τ ≤ 1 A denote its set of isomo r phism classes. A functor A → B of E -enr iched categor ie s is s aid to be essential ly su rje ctive if the induced map Iso τ ≤ 1 A → Iso τ ≤ 1 B is surjectiv e. A functor f : A → B o f E -enriched categories is said to be ful ly faithful if, for each x, y ∈ O b( A ) the mor phism f x,y : A ( x, y ) → B ( f ( x ) , f ( y )) is a n isomor phism in E . A functor is an e quiva lenc e of E -enriche d c ate- gories if it is essentially surjective and fully faithful. These deﬁnitions ar e useful b ecause, a s we sha ll see in Cha pter 14, a morphism of M -enriched prec a tegories is a global weak equiv alence if and only if the asso ciated mor phis m of ho( M )-enrich ed categ ories is an 8.5 Enriche d c ate gories 133 equiv alence in the pr esent sense. So, we c an alre ady obtain versions of some of the ma in closure pr op erties: closure under retracts and 3 for 2. Lemma 8. 5.1 Supp ose ϕ : E → E ′ is a functor c ommuting with dir e ct pr o ducts. Su pp ose f : A → B is an e quival enc e of E -enriche d pr e c ate gories. Then Ca t ( ϕ )( f ) : Ca t ( ϕ )( A ) → Ca t ( ϕ )( B ) is an e quivalenc e of E ′ -enriche d pr e c ate gories. This applies in p articular to t he fun ctor ϕ = τ ≤ 0 , to c onclude that τ ≤ 1 ( f ) : τ ≤ 1 ( A ) → τ ≤ 1 ( B ) is an e quivalenc e of c ate gories, henc e f induc es an isomorph ism of s et s Iso τ ≤ 1 A ∼ = Iso τ ≤ 1 B . Pr o of The functor ϕ sends ∗ E to ∗ E ′ so it induces a map Hom E ( ∗ , A ) → Ho m E ′ ( ∗ , ϕ ( A )) . This natural trans fo rmation induces a natural transfor mation of functors τ ≤ 1 ,ϕ : τ ≤ 1 , E → τ ≤ 1 , E ′ ◦ Ca t ( ϕ ) from Ca t ( E ) to C a t , which is the iden tity on underlying sets of o b jects. Therefore the resulting natural transforma tion Iso τ ≤ 1 ,ϕ ( A ) : Iso τ ≤ 1 , E ( A ) → Iso τ ≤ 1 , E ′ ( Ca t ( ϕ ) A ) is surjective. It follows that if f : A → B is essentially surjective in Ca t ( E ) then Ca t ( ϕ ) f : Ca t ( ϕ ) A → Ca t ( ϕ ) B is also essentially sur- jective. O n the other hand, ϕ tak es isomorphisms to isomorphisms, so if f is fully faithful then for any x, y ∈ Ob( Ca t ( ϕ ) A ) = Ob( A ), the map Ca t ( ϕ )( f ) x,y = ϕ ( f x,y ) : ϕ ( A ( x, y )) → ϕ ( B ( f ( x ) , f ( y ))) is an isomorphism in E ′ . This shows tha t if f is a n equiv alence in Ca t ( E ) then Ca t ( ϕ )( f ) is an e quiv a lence in Ca t ( ϕ )( E ′ ). F or the second part of the s tatement , apply this to ϕ := τ ≤ 0 which preserves pro ducts. Lemma 8.5 .2 In any c ate gory E , the class of isomorphi sms is close d under r etr acts and s atisﬁ es 3 for 2. Pr o of It is ea sy to see for the catego ry of sets. A r etract o f ob jects in E gives a retrac t o f representable functors to se ts; so if we hav e a retract of a n isomor phism then the res ulting retracted na tural tra nsfor- mation is a natural iso morphism, and a morphism in E whic h induces an isomorphism b etw een repres e n table functors , is an is omorphism. 134 Some c ate gory the ory The 3 for 2 prop erty is ea sy to see using the inverses of the isomor- phisms in q uestion. Theorem 8.5. 3 The notion of e quival enc e of E -enriche d c ate gories is close d un der r etr acts and satisﬁes 3 for 2. If A f → B and B g → A ar e functors b etwe en E -enriche d c a te gories su ch that f g and g f ar e e quiva- lenc es, then f and g ar e e quivalenc es. Pr o of Supp o se f : A → B is a retract of an equiv alence of E -enriched precatego r ies, by a commutativ e diagr am A → U → A B f ↓ → V ↓ → B f ↓ such that the horizontal comp ositions are the iden tity and the middle vertical ar row is an eq uiv a lence. W e get a corresp onding diagram of s ets Iso τ ≤ 1 ( A ) → Iso τ ≤ 1 ( U ) → Iso τ ≤ 1 ( A ) Iso τ ≤ 1 ( B ) ↓ → Iso τ ≤ 1 ( V ) ↓ → Iso τ ≤ 1 ( B ) ↓ where the middle v ertical arrow is an isomorphism by Lemma 8.5.1 . It follows that Iso τ ≤ 1 ( A ) → Iso τ ≤ 1 ( B ) is an isomo rphism, in particular f is e s sentially surjective. If x 0 , x 1 ∈ Ob( A ) is a pair of ob jects, denote the imag e ob jects in U , V and B res pectively by u i , v i and y i . Then we get a co mm utative diagram A ( x 0 , x 1 ) → U ( u 0 , u 1 ) → A ( x 0 , x 1 ) B ( y 0 , y 1 ) ↓ → V ( v 0 , v 1 ) ↓ → B ( y 0 , y 1 ) ↓ in which the hor izontal comp ositio ns are the ident ity and the middle ver- tical map is an equiv alence. The clas s o f isomorphisms in E is closed un- der retr a cts, by the pr evious lemma. It follows that A ( x 0 , x 1 ) → B ( y 0 , y 1 ) is an isomor phism in E . This proves that f is fully faithful, completing 8.5 Enriche d c ate gories 135 the proof that the class of equiv alences betw een E -enriched categories is closed under retracts. T urn to the pro o f of the 3 for 2 prop erty which says that if A f → B g → C is a compo sable pair of morphisms and if any tw o of f , g and g f a r e equiv alences, then the third one is to o. Suppo se tha t so me tw o of f , g and g f a re equiv alences of E -enriched categorie s. Applying the truncation functor gives a comp o sable pair o f morphisms o f sets Iso τ ≤ 1 ( A ) → Iso τ ≤ 1 ( B ) → Iso τ ≤ 1 ( C ) and the corresp onding tw o of the maps a re isomorphisms by Lemma 8.5.1. F rom 3 for 2 for isomor phisms of sets, it follows that the third map is also a n isomorphism, hence the third ma p among the f , g and g f is es s ent ially s urjective. F or the fully faithful condition, c onsider ﬁrst the tw o easy cas es. If f and g are global weak equiv alences then for a ny pair of ob jects x 0 , x 1 ∈ Ob( A ) w e hav e a factoriz a tion A ( x 0 , x 1 ) → B ( f ( x 0 ) , f ( x 1 )) → C ( g f ( x 0 ) , g f ( x 1 )) where both maps are w eak equiv alences in E . The previous lemma gives 3 for 2 for isomor phisms in E , so the comp osed map is an isomor phism, which shows that g f is fully faithful. Similarly , if w e assume kno wn that g and g f ar e equiv alences , then in the same fa ctorization we know that the comp osed map and the second map are is omorphisms in E , so again b y Le mma 8 .5.2 it f ollows that the ﬁrst map is a weak equiv alence, showing that f is fully faithful. More work is nee de d for the third ca s e: with the a ssumption that f and g f ar e equiv alences, to show that g is an equiv a lence. Applying Lemma 8.5.1 w e get that Iso τ ≤ 1 ( f ) and Iso τ ≤ 1 ( g ) ◦ Iso τ ≤ 1 ( f ) ar e iso morphisms of s ets, s o Iso τ ≤ 1 ( g ) is an isomor phism. The problem is to show the fully faithful condition. Supp ose x, y ∈ O b( B ). Cho o se x ′ , y ′ ∈ Ob( A ) a nd isomorphisms in τ ≤ 1 ( B ), u ∈ τ ≤ 1 ( B )( f ( x ′ ) , x ) and v ∈ τ ≤ 1 ( B )( f ( y ′ ) , y ) plus their inv erses denoted u − 1 and v − 1 . These are r eally maps u : ∗ → B ( f ( x ′ ) , x ) and v : ∗ → B ( f ( y ′ ) , y ) and simila rly for u − 1 and v − 1 . The co mpo sition map B ( f ( x ′ ) , x ) × B ( x, y ) × B ( y , f ( y ′ )) → B ( f ( x ′ ) , f ( y ′ )) 136 Some c ate gory the ory comp osed with u × 1 × v − 1 : ∗ × B ( x, y ) × ∗ → B ( f ( x ′ ) , x ) × B ( x, y ) × B ( y , f ( y ′ )) gives B ( x, y ) → B ( f ( x ′ ) , f ( y ′ )) . Using u − 1 and v gives B ( f ( x ′ ) , f ( y ′ )) → B ( x, y ), and the asso ciativity axiom, deﬁnition o f inv erses and unit a xioms combine to say that these are inv erse is omorphisms of ob jects in E . The action o f g on morphism ob jects r esp ects a ll o f these o per ations s o w e get a diagram B ( x, y ) → C ( g ( x ) , g ( y )) B ( f ( x ′ ) , f ( y ′ )) ↓ → C ( g f ( x ′ ) , g f ( y ′ )) . ↓ The ﬁrst v ertical map is an isomorphism as descr ibed abov e. The second vertical map is a n isomo rphism for the sa me reason applied to C a nd noting that g ( u ) and g ( v ) are is omorphisms in τ ≤ 1 ( C ). The b ottom map is an is omorphism by Lemma 8.5 .2 and b eca use of the hypotheses that f and g f are fully faithful. Therefor e the top map is an isomorphism, showing that g is fully faithful. F or the la s t par t, supp os e given functor s b etw een E -enriched cate- gories A f → B and B g → A such that f g and g f are equiv alences. On the le vel of sets τ ≤ 0 we g et a pair of maps whose comp ositions in b oth directions a re iso morphisms, it follows that τ ≤ 0 ( f ) and τ ≤ 0 ( g ) are is o - morphisms. It also ea sily follows that for any ob jects x, y ∈ Ob( A ), the map B ( f ( x ) , f ( y )) → A ( g f ( x ) , g f ( y )) has both a left a nd right inv erse, so it is in v ertible. But since any ob ject of B is isomorphic to some f ( x ), a n argument similar to the prev io us one shows that B ( u, v ) → A ( g ( u ) , g ( v )) is an isomor phism for any u, v ∈ Ob( B ). Doing the same in the other direction we see that b oth f and g are eq uiv a lences. 8.5 Enriche d c ate gories 137 8.5.1 Interpretation of enric hed categories as functors ∆ o X → E Lurie [153] has used an imp or tant v ariant on the notion o f nerve o f a category (undoubtedly well-known in the 1-categor ical context). This makes the usual nerve construction apply to an enric hed category , with - out needing to assume anything further ab out E . In this p oint o f view, the set of ob jects is singled out as a set while the mor phis m ob jects of an E -enriched categ ory are consider ed as ob jects of E . The nerve is then neither a functor from ∆ o to Set , nor a functor to E , but rather a mixture of the tw o. W e will adopt this po int o f view when deﬁning M -enriched precategor ies later on. It has the adv antage of allowing us to avoid consideratio n of disjoint unions, s idestepping some of the diﬃ- culties o f [171]. If X is a set, deﬁne the c a tegory ∆ X to consis t of all se q uences of elements of X denoted ( x 0 , . . . , x n ) with n ≥ 0. O ne should think of such a sequence as a dec o ration of the basic ob ject [ n ] ∈ ∆. The morphisms of ∆ X are deﬁned in an obvious way generalizing the mo rphisms of ∆, by just requiring compatibility of the deco rations on the source and target. This will be dis cussed further in Chapter 12. If A is an E -e nr iched categ o ry , let X := Ob( A ). The nerve of A is the functor ∆ o X → E , denoted a lso b y A , deﬁned b y A ( x 0 , . . . , x n ) := A ( x 0 , x 1 ) × · · · × A ( x n − 1 , x n ) . Here the notations A ( x, y ) used o n the r ig ht side ar e those of the enr iched category , but after having made the deﬁnition they a re see n to b e the same as the notations for the nerve, so there is no co n tradiction in this notational shortcut. The tr a nsition maps for the functor A : ∆ o X → E are o btained us ing the comp osition maps of A , for example A ( x 0 , x 1 , x 2 ) = A ( x 0 , x 1 ) × A ( x 1 , x 2 ) → A ( x 0 , x 2 ) in the main c ase that was discussed in the in tro duction. Theorem 8.5. 4 The c ate gory Ca t ( E ) of E -enriche d c ate gories b e- c omes e quivalent, via the ab ove c onstruction, to the c ate gory of p airs ( X, A ) wher e X is a s et and A : ∆ o X → E is a functor satisfying the Se- gal condition that for any se quenc e of elements x 0 , . . . , x n the morphism A ( x 0 , . . . , x n ) → A ( x 0 , x 1 ) × · · · × A ( x n − 1 , x n ) is an isomorphism in E . 138 Some c ate gory the ory Understanding this theor em is crucial to understanding the weakly enriched version whic h is the ob ject of this bo ok. It is left as an exercise for the r eader. Note that the deﬁnition of morphisms b etw een pairs ( X, A ) → ( Y , B ) is done in an obvious wa y , but the re ader may consult the co rresp onding discussion in Chapter 12 b elow. 8.6 Internal H om An impor tant as pec t of the theory is the c artesian condition on the mo del categories inv olved. In this section, we explain how compatibilit y with pro ducts induces an in ternal Hom . Suppo se M is a lo cally presentable catego ry . Say that dir e ct pr o du ct distributes over c olimits if for any small diagr am A : η → M and any ob ject B ∈ M the na tural map colim i ∈ η ( A i × B ) → (colim i ∈ η A i ) × B is an isomorphism. If this is the case, then for an y pair of dia g rams A : η → M a nd B : ζ → M , the natur al map colim ( i,j ) ∈ η × ζ ( A i × B j ) → (co lim i ∈ η A i ) × (colim j ∈ ζ B j ) is an isomorphism, so these t wo w ays of stating the co ndition are equiv- alent. W e say that M admits an internal Hom if, for any A, B ∈ M the functor E 7→ H om M ( A × E , B ) is representable by a n ob ject H om ( A, B ) contra v a riantly functor ial in A and cov arian tly functorial in B together with a natural transformation Hom ( A, B ) × A → B . That is to sa y that a map E → H om ( A, B ) is the same thing as a map A × E → B . Prop ositio n 8. 6.1 Supp ose M is a lo c al ly pr esent able c ate gory such that dir e ct pr o duct distributes over c olimits. Then M admits an internal Hom . Pr o of See [1], Theorem 27 .4, applying the fact that lo cally pr esentable categorie s are co-wellpow ered ([2, Remark 1.56 (3)]). Corollary 8 . 6.2 Supp ose Φ is a smal l c ate go ry. Then the c ate go ry of pr eshe aves of sets Pres h(Φ) = Func (Φ p , Set ) admits an int ernal Hom . This may b e c al culate d as fol lows: if A, B ar e pr eshe aves of set s on Φ then Hom ( A, B ) is the pr eshe af which to x ∈ Φ asso ciates the set Hom Presh(Φ /x ) ( A | Φ /x , A | Φ /x ) . 8.7 Cel l c omplexes 139 Pr o of Note that Presh(Φ) is loc ally presentable. Direct pro ducts and colimits ar e calcula ted ob ject wise, so direct pro duct distributes over co l- imits since the sa me is true for the catego ry Set . The ex plicit descriptio n of H om ( A, B ) is c lassical. Corollary 8.6.3 Su pp ose M is a lo c ally pr esentable c ate gory such that dir e ct pr o duct distributes over c olimits, and supp ose Φ is a smal l c ate gory. Then dir e ct pr o duct distributes over c olimits in Func (Φ , M ) and this c ate go ry admits an internal Hom . Pr o of Again, direct pro duct a nd colimits a re calculated ob jectwise in Func (Φ , M ). 8.7 Cell complexes F or the basic trea tmen t we follow Hirschhorn [1 16]. Fix a lo ca lly pre- sentable categor y M . If α is an ordina l, deno te by [ α ] the set α + 1 , tha t is the set of all ordinals j ≤ α . This notation ex tends the usual notatio n [ n ] used in designating ob jects o f the simplicial category ∆. Note that b y deﬁnition [ α ] is again an ordinal. W e can wr ite interchangeably i ≤ α or i ∈ [ α ]. A se quenc e is a pair ( β , X · ) where β is an or dina l, a nd X : [ β ] → M is a functor. W e us ually denote this b y the collectio n o f ob jects X i for i ≤ β , with morphisms φ ij : X i → X j whenever i < j ≤ β . A sequence is c ontinuous if for any j ≤ β such that j is a limit ordinal, the map colim j 0 , q − ( i ) = sup j i j then h ([ k ]; B )( y 0 , . . . , y p ) = ∅ . F or any precateg o ry A ∈ PC ( M ), a map h ([ k ]; B ) → A is the same thing as a collection of ele ments x 0 , . . . , x k ∈ Ob( A ) together with a map B → A ( x 0 , . . . , x k ) in M , so we can think of h ([ k ] , B ) as being a “represe n table” ob ject in a certain sense. It has as “b oundar y” the pr ecategor y h ( ∂ [ k ] , B ), deﬁned using the skeleton construction in Chapter 15, with the following descr iption (se e Lemma 15.2.3): —if ( y 0 , . . . , y p ) is increasing but not constant i.e. i j − 1 ≤ i j but i 0 < i p , and if ther e is any 0 ≤ m ≤ k s uc h that i j 6 = m for all 0 ≤ j ≤ k , then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = B ; —if ( y 0 , . . . , y p ) is constant i.e. i 0 = i 1 = . . . = i p then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = ∗ ; and otherwise, that is if either there exists 1 ≤ j ≤ p such that i j − 1 > i j or els e if the map j 7→ y j is a sur jection fr o m { 0 , . . . , p } to [ k ], then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = ∅ . More g enerally , the pusho uts h ([ k ] , ∂ [ k ]; A f → B ) := h ([ k ]; A ) ∪ h ( ∂ [ k ]; A ) ∂ h ( ∂ [ k ]; B ) are a lso useful, b eing the gener ators of the Reedy coﬁbr ations in PC ( M ). In Section 1 6.1 and later in Chapter 18 we co nsider precatego ries Υ( B 1 , . . . , B k ) with the sa me set of ob jects [ k ], dep ending on B 1 , . . . , B k ∈ M . The bas ic idea is to put B i in a s spa ce of morphisms from υ i − 1 to υ i . Thus, the main part of the structure of precatego ry is given by Υ( B 1 , . . . , B k )( υ i − 1 , υ i ) := B i . 12.6 Limits, c o limits and lo c al pr esentability 237 This is extended whenever there is a constant string of p oints on either side: Υ( B 1 , . . . , B k )( υ i − 1 , . . . , υ i − 1 , υ i , . . . , υ i ) := B i . The unitality condition on the diag ram ∆ { υ 0 ,...,υ k } → M means that for 0 ≤ i ≤ k we have Υ( B 1 , . . . , B k )( υ i , . . . , υ i ) := ∗ . In a ll other cases, Υ( B 1 , . . . , B k )( x 0 , . . . , x n ) := ∅ . If A ∈ PC ( M ), a map Υ ( B 1 , . . . , B k ) → A is the s ame thing as a c o llec- tion of ob jects x 0 , . . . , x k ∈ Ob( A ) together with maps B i → A ( x i − 1 , x i ) in M . The ca tegoriﬁca tion of Υ ( B 1 , . . . , B k ) ca n b e describ ed explicitly; it is denoted by e Υ( B 1 , . . . , B k ) in Section 18.1. F or any sequence υ i 0 , . . . , υ i n with i 0 ≤ . . . ≤ i n , we put e Υ k ( B 1 , . . . , B k )( υ i 0 , . . . , υ i n ) := B i 0 +1 × B i 0 +2 × · · · × B i n − 1 × B i n . (12.5.1) F or any o ther sequence, that is to say a ny sequence which is not increas- ing, the v alue is ∅ . The v alue o n a constant sequence is ∗ . As a r ough approximation, the calc ulus of ge nerators and r elations can be understo o d as b eing the s mall o b ject a rgument a pplied using the inclusions Υ( B 1 , . . . , B k ) ֒ → e Υ( B 1 , . . . , B k ). Starting with a preca t- egory A , for every map Υ( B 1 , . . . , B k ) → A , ta ke the pushout with e Υ( B 1 , . . . , B k ). Keep do ing this and eventually we get to a precatego ry which satisﬁes the Segal conditions. 12.6 Limits, colimits and lo cal presentabilit y It will b e useful to have ex plicit des criptions of limits and colimits in PC ( M ). W e can then show lo ca l pre s ent ability . Suppo se { A i } i ∈ α is a diagr am of ob jects A i ∈ PC ( M ), that is a functor α → PC ( M ). Let X i := Ob( A i ) denote the o b ject sets so we can consider A i ∈ PC ( X i ; M ). F o r any f : i → j in α denote by φ f : X i → X j the transition map o n ob ject sets, then ρ f : A i → φ ∗ f A j the tra nsition maps on the level of precatego ries. 238 Pr e c ate gories Start b y constr uc ting the limit in PC ( M ). The ob ject set o f the limit will b e X := lim i ∈ α X i . W e hav e maps p i : X → X i , hence p ∗ i ( A i ) ∈ PC ( X ; M ). These are provided with tra nsition maps, indeed fo r f : i → j in α , φ f p i = p j so p ∗ i ( φ ∗ f A j ) = p ∗ j A j and p ∗ i ( ρ f ) : p ∗ i ( A i ) → p ∗ j A j provide the transition maps for the sys tem o f p ∗ i ( A i ) considered as a diagr am α → PC ( X ; M ). Put A := lim PC ( X ; M ) i ∈ α p ∗ i ( A i ) ∈ PC ( X ; M ) . W e claim tha t this is the limit of the dia gram { A i } i ∈ α in PC ( M ). Suppo se B ∈ PC ( M ) with Ob( B ) = Y and supp ose given a sys- tem of maps B → A i . Thes e corr esp ond to maps r i : Y → X i and ϕ i : B → r ∗ i ( A i ). The collection of r i gives a uniquely deter mined map r : Y → X , and r ∗ p ∗ i ( A i ) = r ∗ i ( A i ) so the ϕ i corres p o nd to a collec- tion of maps B → r ∗ ( p ∗ i ( A i )). This gives a uniquely determined map ϕ : B → r ∗ ( A ) whose co mp os ition with the pro jections of the limit ex- pression for A , are the ϕ i . The pair ( r , ϕ ) is a ma p B → A in PC ( M ) uniquely solving the universal problem to show that ( X , A ) is the limit of the A i . The limit A ca n be describ ed explicitly as a n element of PC ( X ; M ), by the discussion preceding Lemma 14.4.2 : for any x 0 , . . . , x n ∈ X , A ( x 0 , . . . , x n ) = lim i ∈ α A i ( p i x 0 , . . . , p i x n ) . T urn now to the cons truction of the colimit. The ob ject s et will b e Z := colim i ∈ α X i with maps q i : X i → Z . These give q i, ∗ ( A i ) ∈ PC ( X ; M ), with tra nsi- tion ma ps deﬁned as follows. If f : i → j is an arr ow in α then q j φ f = q i so q j, ! ( φ f , ! ( A i )) = q i, ! ( A i ). The adjunction b etw een φ f , ! and φ ∗ f means that the transition map ρ f for the system of A i may b e view ed as a map denoted ˜ ρ f : φ f , ! ( A i ) → A j , in particular we get q j, ! ( ˜ ρ f ) : q i, ! ( A i ) = q j, ! ( φ f , ! ( A i )) → q j, ! ( A j ) . These pro vide t ransitio n maps for the dia gram { q i, ! ( A i ) } i ∈ α with v alues in PC ( Z ; M ) and set C := co lim PC ( Z ; M ) i ∈ α q i, ! ( A i ) ∈ PC ( Z ; M ) . The ob ject ( Z, C ) is the colimit of t he diag ram of A i , for the s ame forma l reason a s in the prev ious discussion of the limit. 12.6 Limits, c o limits and lo c al pr esentability 239 The colimit to deﬁne C is taken in the categor y PC ( Z ; M ) whic h presupp oses in g eneral applying the op eration U ! if α is disconnected. How ev er, when the deﬁnition is unw ound explicitly this phenomenon disapp ears, being absor bed in the ca lculation of the v alue of C o n a sequence of p oints ( z 0 , . . . , z n ) v ia the introduction of a new category α/ ( z 0 , . . . , z n ). Suppo se ( z 0 , . . . , z n ) is a s equence of elemen ts o f Z . Le t α/ ( z 0 , . . . , z n ) denote the set of pairs ( i, ( x 0 , . . . , x n )) where i ∈ α and ( x 0 , . . . , x n ) is a sequence o f p oints in X i such that q i ( x k ) = z k for k = 0 , . . . , n . The asso ciation ( i, ( x 0 , . . . , x n )) 7→ A i ( x 0 , . . . , x n ) is a diagram from α/ ( z 0 , . . . , z n ) to M . Lemma 12.6.1 In the ab ove situ ation, the value of the c olimiting ob- je ct C on t he se quenc e ( z 0 , . . . , z n ) is c alculate d as a c olimi t of the dia- gr am α/ ( z 0 , . . . , z n ) → M : C ( z 0 , . . . , z n ) = colim ( i, ( x 0 ,...,x n )) ∈ α/ ( z 0 ,...,z n ) A i ( x 0 , . . . , x n ) . Pr o of Let q wu i, ! : Func (∆ o X i , M ) → Func (∆ o Z ; M ) deno te the push- forward in the world o f no n- unital preca teg ories. It is the pushforward for diagrams v alued in M alo ng the functor ∆ o X i → ∆ o Z . If ( z 0 , . . . , z n ) is a sequence of p oints in Z , then q wu i, ! ( A i )( z 0 , . . . , z n ) is the colimit of the A i ( x 0 , . . . , x k ) ov er the categor y o f pairs ( u, ( x 0 , . . . , x k )) where u : ( q i x 0 , . . . , q i x k ) → ( z 0 , . . . , z n ) is a map in ∆ o Z or equiv a lently ( z 0 , . . . , z n ) → ( q i x 0 , . . . , q i x k ) is a map in ∆ Z . Such a map factors through a unique map ( z 0 , . . . , z n ) → ( q i x i 0 , . . . , q i x i n ) where ( i 0 , . . . , i n ) is a multiindex repr e s ent ing a map [ n ] → [ k ] in ∆. Hence the category of pairs in ques tion is a disjoint union of categories having coinitial ob jects. This yields the expressio n of q wu i, ! ( A i )( z 0 , . . . , z n ) a s the disjoint sum of A i ( x 0 , . . . , x n ) ov er all sequences ( x 0 , . . . , x n ) of ob jects in X i such that q i ( x j ) = z j . Putting these together ov er all i ∈ α gives colim i ∈ α ( q wu i, ! ( A i )( z 0 , . . . , z n )) = colim ( i, ( x 0 ,...,x n )) ∈ α/ ( z 0 ,...,z n ) A i ( x 0 , . . . , x n ) . Now U ! is a left adjoint so it preserves colimits, in particular C = colim PC ( Z ; M ) i ∈ α q i, ! ( A i ) = colim i ∈ α U ! ( q wu i, ! ( A i )) = U !  colim i ∈ α q wu i, ! ( A i )  = U ! ( C ′ ) 240 Pr e c ate gories where C ′ ∈ Func (∆ o Z ; M ) is the o b ject deﬁned by C ′ ( z 0 , . . . , z n ) := colim ( i, ( x 0 ,...,x n )) ∈ α/ ( z 0 ,...,z n ) A i ( x 0 , . . . , x n ) . T o ﬁnish the pro of note that C ′ is already in PC ( Z ; M ). Indeed for a single element z 0 ∈ Z the ca tegory α/ ( z 0 ) of pairs ( i, x 0 ) with q i ( x 0 ) = z 0 is connected. This is factoid abo ut colimits of sets. Since A i is unital we hav e A i ( x 0 ) = ∗ , thus C ′ ( z 0 ) = colim ( i,x 0 ) ∈ α/ ( z 0 ) ∗ = ∗ , in other words C ′ is unital. Therefor e C = U ! ( C ′ ) = C ′ which is the statement of the lemma. An a lternative pro of would b e to note that C ′ satisﬁes the r equired universal prop erty for deﬁning a colimit. Corollary 12. 6.2 If α is a c onne cte d c ate gory and A · : α → PC ( X ; M ) is a diagr am of M -pr e c ate gories with a c ommon obje ct set X , t hen the natur al map colim PC ( M ) i ∈ α A i → colim PC ( X ; M ) i ∈ α A i is an isomorphism. Pr o of The explicit descriptions of bo th sides are the same. It is worth while to discuss explicitly so me sp ecial ca ses. F or exa mple Ob( A× B ) = Ob( A ) × Ob( B ) a nd for any s equence (( x 0 , y 0 ) , . . . , ( x n , y n )) of ele men ts of Ob( A ) × Ob( B ) we hav e A × B (( x 0 , y 0 ) , . . . , ( x n , y n )) = A ( x 0 , . . . , x n ) × B ( y 0 , . . . , y n ) . This e x tends to ﬁber pro ducts: if A → C and B → C are maps, then Ob( A × C B ) = Ob( A ) × Ob( C ) Ob( B ) and aga in for any s equence (( x 0 , y 0 ) , . . . , ( x n , y n )) o f elements in the ﬁb er pro duct of ob ject sets we have A × C B (( x 0 , y 0 ) , . . . , ( x n , y n )) = A ( x 0 , . . . , x n ) × C ( z 0 ,...,z n ) B ( y 0 , . . . , y n ) where the z i are the common images of x i and y i in O b( C ). F or colimits, the disjoint sum A ⊔ B has ob ject set Ob( A ) ⊔ Ob( B ), and fo r a se quence o f elements ( z 0 , . . . , z n ) in here w e hav e ( A ⊔ B )( z 0 , . . . , z n ) =    A ( z 0 , . . . , z n ) if all z i ∈ Ob ( A ) B ( z 0 , . . . , z n ) if all z i ∈ Ob ( B ) ∅ otherwise . Lo ok at the copro duct or pushout of tw o maps u : C → A and v : C → B , 12.6 Limits, c o limits and lo c al pr esentability 241 suppo sing tha t one of the maps say v is injective on the set of ob jects. This w ill usua lly be the case in our applications b eca use w e usua lly look at pus ho uts a long coﬁbrations. The copro duct A ∪ C B has ob ject se t Ob( A ) ∪ Ob( C ) Ob( B ) = Ob( A ) ⊔ (Ob( B ) − Ob( C )) . The categor y α indexing the colimit has three ob jects denoted a, b, c with arrows a ← c → b . Given a s equence ( z 0 , . . . , z n ) of elements in here , the catego ry α/ ( z 0 , . . . , z n ) has ob jects o f three kinds denoted a , b and c , and ob jects o f a giv en kind corresp ond to sequences in Ob( A ), Ob( B ) or Ob( C ) r e spe ctively mapping to the g iven z · . Our general for m ula of Lemma 12.6.1 expr esses ( A ∪ C B )( z 0 , . . . , z n ) as the pushout of disjoint sums a x · 7→ z · A ( x 0 , . . . , x n ) ∪ ` w · 7→ z · C ( w 0 ,...,w n ) a y · 7→ z · B ( y 0 , . . . , y n ) . Another impo rtant case is that of ﬁlter ed co limits. Supp ose α is a ﬁltered (resp. κ -ﬁltered) categor y a nd {A i } i ∈ α is a diagram in PC ( M ). Put X i := Ob( A i ) and Z := colim i ∈ α X i . Then for a ny seq ue nce ( z 0 , . . . , z n ) of elements of Z , the categ ory α/ ( z 0 , . . . , z n ) is again ﬁltered (r esp. κ - ﬁltered), s o A ( z 0 , . . . , z n ) = colim ( i, ( x 0 ...,x n ) ∈ α/ ( z 0 ,...,z n ) A ( x 0 . . . , x n ) is a ﬁltere d (resp. κ -ﬁltered) colimit in M . Lemma 12 .6.3 Su pp ose M is lo c al ly pr esentable. An obje ct A ∈ PC ( M ) is κ -pr esentable if and only if X := Ob( A ) is a set of c ar- dinality < κ , and e ach A ( x 0 , . . . , x p ) is a κ -pr esentable obje ct of M . Pr o of Supp o se X := Ob( A ) is a set o f cardinality < κ , and each A ( x 0 , . . . , x p ) is a κ -presentable ob ject of M . Suppo se {B i } i ∈ β is a dia- gram in PC ( M ) indexed by a κ -ﬁltered ca tegory β , a nd suppo se given a map A → B := colim i ∈ β B i . In pa r ticular we g et a map Ob( A ) → Ob( B ) = co lim i ∈ β Ob( B i ) and the co ndition | Ob( A ) | < κ implies that this ma p factor s through a map Ob( A ) → Ob( B j ) for some j ∈ β . Given a sequence o f ob jects ( x 0 , . . . , x n ) ∈ Ob( A ), let ( y 0 , . . . , y n ) denote the c orresp onding sequence of ob jects in Ob( B j ) a nd ( z 0 , . . . , z n ) the seq uence in Ob( B ). Let j \ α denote the catego ry o f ob jects under j in α . Given j → i the image 242 Pr e c ate gories of ( y 0 , . . . , y n ) is a s equence denoted ( y i 0 , . . . , y i n ) in Ob( B i ) mapping to ( z 0 , . . . , z n ). This gives a functor j \ α → α/ ( z 0 , . . . , z n ) , and the κ -ﬁltered prop erty o f α implies that this functor is co ﬁnal. Hence by Lemma 1 2.6.1 and the inv a r iance of c o limits under coﬁnal functors, B ( z 0 , . . . , z n ) = colim ( j → i ) ∈ j \ α B i ( y i 0 , . . . , y i n ) . The ca tegory j \ α is κ -ﬁltered, so the map A ( x 0 , . . . , x n ) → B ( z 0 , . . . , z n ) factors through one o f the B i ( y i 0 , . . . , y i n ) for j → i in α . The cardi- nality of the set of p ossible sequences ( x 0 , . . . , x n ) is < κ , so the κ - ﬁltered prop erty s ays that we c a n ch o ose a single i . Then the stan- dard kind of ar gument shows that by going further along, the maps A ( x 0 , . . . , x n ) → B i ( y i 0 , . . . , y i n ) can b e assumed to a ll ﬁt together into a natur al tra nsformation in terms of ( x 0 , . . . , x n ) ∈ ∆ o X . Thus we get a facto rization of our map A → B through some A → B i . This shows that A is κ -presentable. Suppo se o n the o ther hand tha t A is κ -presentable. W e note ﬁrst of a ll that | X | < κ . Indeed, if | X | ≥ κ then w e c ould consider a κ - ﬁltered system of s ubsets Z i ⊂ X with | Z i | < κ , but colim Z i = X . Let co dsc ( Z i ) and co dsc ( X ) denote the co discrete precatego ries o n these ob ject sets, that is the preca tgories whose v alue is ∗ o n a ny se- quence of ob jects. Then c olim i co dsc ( Z i ) = co dsc ( X ) in PC ( M ) (see Lemma 8.1.8), but the identit y map on underlying o b ject sets gives a map A → co dsc ( X ) not factor ing thro ugh a ny co dsc ( Z i ), contradicting the assumed κ -presentabilit y of A . Hence we may assume that | X | < κ . W e claim that A is κ - presentable when co ns idered as an o b ject in PC ( X ; M ). Indeed, if { B i } i ∈ β is a κ -ﬁltere d diagr a m in PC ( X ; M ) then since β is co nnected, a n y map A → co lim PC ( X ; M ) i ∈ β B i is a lso a map A → colim PC ( M ) i ∈ β B i by Corollar y 12 .6 .2. By the assumed κ -presentabilit y of A this would have to factor throug h o ne of the B i , necessarily a s a ma p inducing the identit y on underlying ob ject sets X . This shows that A is κ -pre s ent able when considered as a n o b ject in PC ( X ; M ). Now Cor ollary 14.4.3 tells us that the A ( x 0 , . . . , x p ) are κ -presentable ob jects of M . 12.7 Interpr etations as pr eshe af c ate gori es 243 Prop ositio n 12.6. 4 Supp ose M is lo c ally pr esentable. Then the c at- e gory of M -pr e c ate gories PC ( M ) is also lo c al ly pr esen t able. Pr o of Let κ b e a r egular ca r dinal suc h that M is lo cally κ - presentable. Note that, for a n y set X the categ ory PC ( X , M ) is lo cally κ -pr esentable by its adjunction with the dia gram categ ory Func (∆ o X , M ) which in turn is lo cally κ -presentable by Lemma 8.1.3. Existence of arbitrar y colimits in PC ( M ) was s hown a t the s tart of the s ection. It is clea r fro m the descr iptio n of κ -pres ent able o b jects in L emma 12.6.3 tha t the iso morphism cla sses of κ -presentable ob jects of PC ( M ) form a set. F urthermore, any ob ject ( Z , A ) is κ -acces sible. Indeed, consider the c a tegory of triple s ( X , B , u ) w he r e X ⊂ Z is a subs e t of c a rdinality | X | < κ , B ∈ PC ( X , M ) is a κ -present able ob ject, and u : B → A| X is a mor phism to the pullback of A a long the inclusion X ⊂ Z . Using the lo cal κ -pr esentabilit y of ea ch PC ( X , M ) and the expres sion of Z a s a κ -ﬁltered union o f the subsets X , the ca teg ory of triples is κ -ﬁltered and the colimit of the ta uto logical functor is ( Z, A ). 12.7 Interpretations as presheaf categories With some additio na l hypotheses on M , the tric k of introducing the categorie s ∆ X bec omes unnecess a ry . This corresp onds more closely with some previo us references [206] [193] [194], and will b e useful in e stab- lishing notation for iterated n -prec atgories. Starting from this ﬁrst dis- cussion w e show later on that if M itself is a pre sheaf categor y then PC ( M ) is a pr esheaf categ ory . This considera tion will not usually enter into our argument a ltho ugh it do e s provide a conv enien t change of notation for Cha pter 1 7 . Never- theless, many arguments b ecome consider ably simpler in the cas e o f a presheaf ca tegory—a s we hav e already seen for cell co mplexes in Chap- ter 8. As we shall no w s e e, the passage from M to PC ( M ) preserves the condition of b eing a presheaf categ ory , a nd many important initial cases such as the mo del ca tegory of simplicia l se ts K s atisfy this c o ndition. So it should b e helpful throughout the bo o k to b e able to think of the case o f presheaf categorie s. The ﬁr st part of o ur discussion follows Pelissier [171]. Suppo se M is a lo cally presentable categor y . Deﬁne a functor disc : 244 Pr e c ate gories Set → M by disc ( U ) := a u ∈ U ∗ . If nece ssary we shall denote by ∗ u the term corr esp onding to u ∈ U in the copr o duct. The discr ete ob ject functor has a right adjoint di s c ∗ : M → Set deﬁned by disc ∗ ( X ) := Hom M ( ∗ , X ), with Hom M ( disc ( U ) , X ) = Hom Set ( U, disc ∗ ( X )) . In pa r ticular, disc commut es with colimits. Given an ex pression X = ` u ∈ U X u the universal prop erty o f the copro duct applied to the maps X u → ∗ u yields a map X → disc ( U ). On the other hand, given a map f : X → disc ( U ) w e can put f − 1 ( u ) := X × disc ( U ) ∗ u . W e have a natural map ` u ∈ U f − 1 ( u ) → X co mpatible with the maps to di sc ( U ). Consider the following hypothesis on M , saying that dis jo in t unions (i.e. co limits over discrete categor ies) be have w ell. This kind of hypoth- esis was introduced for the same purp os e b y Pelissier [171, Deﬁnition 1.1.4]. Condition 12. 7.1 (DISJ) (a)—If f : X → disc ( U ) is a map, t hen t he natu r al map is an isomor- phism ` u ∈ U f − 1 ( u ) ∼ = X . If X = ` u ∈ U X u is a c op r o duct expr ession then X u = f − 1 ( u ) for the c orr esp onding map f : X → disc ( U ) . (b)—The c oinitia l obje ct ∗ is inde c omp osable, t hat is to say it c annot b e written as a c opr o duct of t wo nonempty obje cts. (c)—The c ate gory M has mor e than just a single obje ct up to isomor- phism. T o see why conditio n (b) is neces s ary , note for e x ample that if M = Presh(Φ) is the categ ory of presheav es on a category Φ which has more than one connected comp onent (for example the discrete catego ry with t wo ob jects) then ∗ is decomp osable. Condition (c) is r equired to rule out the trivial category M = ∗ which satisﬁes a ll of our other h yp otheses. Lemma 12. 7.2 Assume that M is lo c al ly pr esentable, and satisﬁes Condition (DCL) of the c artesia n c ondition 10.0. 9. Assume that M sat- isﬁes Cond ition (DISJ) 12. 7.1 ab ove. Then we have the fol lowing furt her pr op erties: 12.7 Interpr etations as pr eshe af c ate gori es 245 (1)—F or any obje ct X ∈ M , giving an expr ession X ∼ = ` u ∈ U X u is e quiva lent to giving a map X → disc ( U ) . (2)—If Y → ` u ∈ U X u is a map fr om a single obje ct to a c opr o duct, then sett ing Y u := Y × ` u ∈ U X u X u the n atur al map ` u ∈ U Y u → Y is an isomorphism. (3)—The map ∅ → ∗ is n ot an isomorphism. (4)—F or any set U the adjunction map U → disc ∗ disc ( U ) is an iso- morphism. (5)—The functor di sc is ful ly faithful, and d i sc ∗ gives an inverse on its essential image. (6)—If X ∼ = ` u ∈ U X u and Y ∼ = ` u ∈ U Y u ar e de c omp ositions c orr e- sp ondi ng to maps X , Y → disc ( U ) t hen X × disc ( U ) Y = a u ∈ U X u × Y u . (7)—Copr o ducts ar e disjoint: if { X u } u ∈ U is a c ol le ctio n of obje cts and u 6 = v then X u × ` u ∈ U X u X v = ∅ . (8)—The functor dis c pr eserves ﬁn ite limits. Pr o of Condition (1) is just a restatement of the ﬁrst part of (DISJ). F or (2), supp ose Y → X = ` u ∋ U X u is a map. Comp ose with the map X → di sc ( U ) given by (1), to get Y → disc ( U ). In turn this corres p o nds to a decompo sition Y = ` u ∈ U Y u with Y u = Y × disc ( U ) ∗ u , but X u = X × disc ( U ) ∗ u so Y u = Y × X X u which is the desired statement. F or (3 ), if ∅ → ∗ were an isomorphism, then we would hav e ∅ × X = X for all ob jects X , but in v iew of Le mma 10.0.10 following from (DCL) this would imply that all ob jects a r e ∅ contradicting the nontrivialit y hypothesis (DISJ) (c) on M . F or (4) supp ose ﬁrst we are g iven a map f : ∗ → disc ( U ). By (1) this co rresp onds to a decomp osition ∗ = ` u ∈ U ∗ × disc ( U ) ∗ u . By Con- dion (DISJ)(b), a ll but one of the summands must b e ∅ . F r om (3) it follows that one of the summands is diﬀerent, so ther e is unique one of the summands which is ∗ . W e get tha t our map fa ctors through a unique ∗ u . This shows that the a djunction map U → disc ∗ disc ( U ) is an isomorphism. F or (5), supp ose giv en a map f : disc ( U ) → disc ( V ). By prop erty (4) 246 Pr e c ate gories there is a unique map U g → V compatible with disc ∗ ( f ) by the adjunc- tion iso morphisms. Uniqueness shows that disc is faithful. Applying disc to this comptibility diagra m a nd comp osing with the natura lit y squar e for the other adjunction ma p gives a s quare disc ( U ) disc ( g ) → disc ( V ) disc ( U ) ↓ f → disc ( V ) ↓ where the vertical maps are the adjunction comp ositions disc ( U ) → discdis c ∗ disc ( U ) → disc ( U ) and the same for V . These ar e the iden tities so f = di s c ( g ). This shows that di sc is fully faithful, and the disc ∗ gives an essential inverse by (4). In the situation of (6 ), given Z h → disc ( U ) cor resp onding to Z = ` u ∈ U Z u , a map Z → X compatible with h is the same thing as a collection of maps Z u → X u by (2). Similarly for a map to Y . It follows that ` u ∈ U X u × Y u satisﬁes the univ ersal proper t y for the ﬁb er product, giving (6 ). F or (7), it suﬃces to show that ∗ u × disc ( U ) ∗ v = ∅ for v 6 = u , in v iew of the co nsequence Lemma 1 0.0.10 of Co ndition (DCL) s aying that no nonempty ob ject can map to ∅ . But ∗ u = ` v ∈ U ( ∗ u ) × disc ( U ) ∗ v and, as was s een in the pr o of of (4), co ndition (DISJ) (b) implies tha t a ll but one of these terms must be ∅ . Since there is a diagonal map from ∗ to the term v = u , the terms for v 6 = u must b e ∅ . F or (8), the functor dis c preserves ﬁnite direct pro ducts: g iven maps Z → disc ( U ) and Z → disc ( V ) they corres po nd to decomp ositions as in (1). The maps Z u → di sc ( V ) corr esp ond to deco mpo s itions Z u = a v ∈ V Z u,v , Z u,v = Z u × disc ( V ) ∗ v . Putting thes e to gether ov er all u ∈ U we get a decompo sition o f Z corres p o nding to a unique map Z → disc ( U × V ). This shows that disc ( U × V ) satisﬁes the univ ersal pro pe r ty to be the pro duct dis c ( U ) × disc ( V ). A similar ar gument shows that disc preserves equalizer s. Suppo se now that M satisﬁes co ndition (DISJ) in a ddition to b eing tractable left pr op er a nd ca rtesian, so the pro p e rties o f the preceding 12.7 Interpr etations as pr eshe af c ate gori es 247 lemma apply . Given A ∈ PC ( M ) deﬁne a functor ∆ o → M de no ted [ n ] 7→ A n/ by A n/ := a ( x 0 ,...,x n ) ∈ Ob( A ) n +1 A ( x 0 , . . . , x n ) . The functoriality maps are deﬁned using those of A . This ha s the prop- erty that A 0 / is a discr e te o b ject, indeed the unitalit y conditions s ay that A ( x 0 ) = ∗ so there is a natural isomor phism A 0 / ∼ = disc (Ob( A )), and we identify A 0 / with the set Ob( A ), sometimes using the notation A 0 . The pr o p e r ty that A 0 / is a discre te ob ject is called the c onstancy c ondi tion [206], closely related to the g lobular nature of the theory of n -categor ies. Let Func ([0] ⊂ ∆ o , Set ⊂ M ) denote the full subcatego ry of Func (∆ o , M ) cons isting of functors whic h satisfy this constancy con- dition. Suppo se on the other ha nd that n 7→ A n/ is a functor ∆ o → M which satisﬁes the c o nstancy condition. Set X := Ob( A ) := disc ∗ ( A 0 / ), then A 0 / = disc (Ob( A )). F or any s e quence x 0 , . . . , x n ∈ X = Ob( A ), w e get a map ( x 0 , . . . , x n ) : ∗ → A 0 / × · · · × A 0 / and de ﬁne A ( x 0 , . . . , x n ) := A n/ × A 0 / ×···×A 0 / ∗ where the ma p A n/ → A 0 / × · · · × A 0 / is o btained us ing the n + 1 vertices of the simplex [ n ]. Notice that A 0 / × · · · × A 0 / = disc (Ob( A ) × · · · × Ob( A )) since di sc preser ves ﬁnite pro ducts (Lemma 1 2.7.2 (8)). The simplicial maps for A · / provide tr ansition maps to make A ( · · · ) into a functor ∆ p X → M . Theorem 12.7.3 Supp ose M satisﬁes Condition (DISJ ) 12.7.1. The ab ove c onstructions pr ovi de essential ly inverse functors PC ( M ) ← → Func ([0] ⊂ ∆ o , Set ⊂ M ) . The ful l sub c ate gory Func ([0] ⊂ ∆ o , Set ⊂ M ) ⊂ Func (∆ o , M ) is close d under limits and c olimits, and the ab ove essent ial ly inverse fu n c- tors pr eserve limits and c olimits. 248 Pr e c ate gories Pr o of This follo ws fr om L e mma 12.7.2: suppose ﬁxed the set of ob jects X , then giving A n/ with map to the direct pro duct A n/ → di sc ( X × · · · X ) = d i sc ( X ) × · · · × dis c ( X ) is the s ame as g iving the pieces of the decomp osition A n/ = a ( x 0 ,...,x n ) A ( x 0 , . . . , x n ) . F or iteratio n of our basic construction, the following lemmas show that the a bove notational reinterpretation is very rea sonable. Lemma 12.7.4 Su pp ose M satisﬁes the hyp othesi s (DCL) of Deﬁni- tion 10.0.9. Then the c ate gory PC ( M ) satisﬁ es c ondition (DISJ). Pr o of The discrete pr ecategor ies are constructed as follows. The coini- tial ob ject ∗ ∈ PC ( M ) has a single element O b( ∗ ) = ∗ = { x } and ∗ ( x, . . . , x ) = ∗ ∈ M . Thus, if U is a ny set then disc ( U ) calculated in PC ( M ) has ob ject s et disc ( U ) ∼ = U , and disc ( U )( x 0 , . . . , x n ) = ∗ if x 0 = · · · = x n , otherwise disc ( U )( x 0 , . . . , x n ) = ∅ if the sequence is no t constant. F or part (a), supp ose A ∈ PC ( M ) and A → disc ( U ) is a map. Put X := Ob( A ), with X → U which corres p o nds to a deco mpo sition X = ` u ∈ U X u . Let A u ⊂ A b e the pullback of A along X u → X , that is it is the “full sub-precategor y” with ob ject set X u ⊂ X . This is indeed A × disc ( U ) ∗ u . The ma p a u ∈ U A u → A is a n isomor phism. This fo llows from the fact that dis c ( U )( x 0 , . . . , x n ) = ∅ for an y no nconstant seq ue nc e , plus the c onsequence Lemma 10.0.10 of (DCL). On the other hand g iven a decomp ositio n in copro duct A = ` u ∈ U A u we get a map A → disc ( U ) for which the comp onents A u are the ﬁb ers. Conditions (b) and (c) are ea sy . Lemma 12. 7.5 Supp ose Ψ is a c onne cte d c ate gory (i.e. its nerve is a c onne cte d sp ac e, or e qu ivalently any two obje cts ar e joine d by a zig-zag of arr ow s). Then the c ate gory Presh(Ψ) = Func (Ψ o ; Set ) satisﬁes c ondi - tion (DISJ), with the discr ete obje cts b eing those e quivalent to c onstant functors. In p articular, t he mo del c ate gory Set wher e al l morphisms ar e 12.7 Interpr etations as pr eshe af c ate gori es 249 ﬁbr ations and c oﬁbr atio ns, and t he we ak e quivalenc es ar e isomorphisms, satisﬁes (DIS J ). And t he mo del c ate gory K of simplicia l sets satisﬁes (DISJ). Pr o of If U is a set, the discr e te pres he a f dis c ( U ) is the co nstant presheaf x 7→ U on all x ∈ Ψ. F or (a), a map A → disc ( U ) is the same thing as a decompo sition A = ` u ∈ U A u , as can be seen levelwise. Condition (b) follows from the connectedness of Ψ (indeed it is equiv alent ), and Condition (c) is easy . W e now tu rn to the further situation where M is a preshea f category , say M = Presh(Ψ) = Func (Ψ o ; Set ). In this case Func (∆ o X ; M ) = Func (∆ o X × Ψ o ; Set ) = Presh(∆ X × Ψ ) is a gain a presheaf c a tegory . W e construct a new c ategory denoted C (Ψ), which lo ok s somewhat like a “co ne” on Φ. It is deﬁned by contracting { [0] } × Ψ ⊂ ∆ × Ψ to a single ob ject denoted 0. Thus, the ob jects of C (Ψ) a re of the for m ([ n ] , ψ ) for n ≥ 1 and ψ ∈ Ψ, or the ob ject 0. The mo rphisms are as follows: —there is a single ident ity morphism b etw een 0 and itself; —for n ≥ 1 and ψ ∈ Ψ there is a unique morphisms from any ([ n ] , ψ ) to 0; —the morphisms from 0 to ([ n ] , ψ ) a re the same as the morphisms [0] → [ n ] in ∆ (i.e. there are n + 1 of them); and —the morphisms from ([ n ] , ψ ) to ([ n ′ ] , ψ ′ ) are of tw o kinds: either ( a, f ) where a : [ n ] → [ n ′ ] is a mor phism in ∆ such tha t a does n’t factor through [0 ] and f : ψ → ψ ′ is a morphism in Ψ; or e ls e ( a ) wher e a : [ n ] → [ n ′ ] is a morphism in ∆ which factors as [ n ] → [0] → [ n ′ ]. Comp osition of morphisms is deﬁned in an obvious wa y . N otice that the comp osition of anything with a morphism which factors through [0] will again factor through [0], which allows us to deﬁne comp ositions o f the form ( a ) ◦ ( a ′ , f ′ ) or ( a, f ) ◦ ( a ′ ) in the las t case. The division of the morphisms from ([ n ] , ψ ) to ([ n ′ ] , ψ ′ ) into tw o cases is necessa ry in o rder to deﬁne comp osition. Prop ositio n 1 2.7.6 Supp ose M = Presh(Ψ) is a pr eshe af c ate gory. Ther e is a natur al isomorphism b etwe en Pre s h( C (Ψ)) and the c ate gory of unital M -pr e c ate gori es PC ( M ) . Thus, PC ( M ) is again a pr eshe a f c ate gory. Pr o of Supp o se F : C (Ψ) o → Set is a preshea f. Let X := F (0). F or 250 Pr e c ate gories n ≥ 1 and any ( x 0 , . . . , x n ) ∈ ∆ X , let A ( x 0 , . . . , x n ) be the preshea f on Ψ which assigns to ψ ∈ Ψ the subset o f elemen ts of F ([ n ] , ψ ) which pr o ject to ( x 0 , . . . , x n ) under the n + 1 pro jectio n maps F ([ n ] , ψ ) → F ( ι ) = X corres p o nding to the n + 1 ma ps 0 → ([ n ] , ψ ). A t n = 0, set A ( x 0 ) := { x 0 } . The pair ( X, A ) is an element of PC ( M ). Conv ersely , g iven any ( X, A ) ∈ PC ( M ), deﬁne a pres heaf F : C (Ψ) o → Set by setting F (0) := X and F ([ n ] , ψ ) := a ( x 0 ,...,x n ) ∈ X n +1 A ( x 0 , . . . , x n )( ψ ) . These constructions are in verses. Let c Ψ : ∆ × Ψ → C (Ψ ) denote the pro jectio n. If Ψ is c o nnected, then the catego r y of presheav es Presh( C (Ψ)) may b e identiﬁed, via c ∗ Ψ , with the full sub categ ory of Pr esh(∆ × Ψ) consis ting of presheaves A which satisfy T amsamani’s c onstancy c ondition that A (0 , ψ ) is a constant set independent of ψ ∈ Ψ. The construction Ψ 7→ C (Ψ) w as the iterativ e step in the construction of the seq uence of catego ries denoted Θ n in [193]. In that notation, Θ 0 = ∗ and Θ n +1 = C (Θ n ). Ho wev er, t he nota tio n Θ n was subse quent ly used by Joyal [127] to denote a rela ted but diﬀerent sequence of categories ; in o r der to avoid confusion w e will use the cone notation C . F or the theor y of non-unital M -precateg ories, one can a lso co nstruct a category deno ted C + (Ψ) with the prop er t y that P resh( C + (Ψ)) is the catego ry o f non-unital M -pr e c ategories . This is in fact even more straightforward than the construction of C for the unital theory . The fol- lowing discussion is optiona l, but ser ves to put the previo us discussion of C in to a better p er spe ctive. Recall that PC ( M ) was deﬁned as a ﬁb ered category o v er Set whose ﬁber o ver a set X was PC ( X , M ). In the s ame wa y , the c ate gory of non-unital M -pr e c ate gories is the ﬁb ere d category o ver S et whose ﬁber ov er X is Func (∆ o X , M ). The co nstruction of C + (Ψ) is to fo r mally add an ob ject denoted ι to ∆ × Ψ . The ob jects of C + (Ψ) are of the form either ι or ([ n ] , ψ ) for n ∈ ∆ a nd ψ ∈ Ψ. The morphisms of C + (Ψ) are deﬁned as follows: —there is a single ident ity morphism b etw een ι and itself; —there are no morphisms from ([ n ] , ψ ) to ι ; —the mor phisms fro m ([ n ] , ψ ) to ([ n ′ ] , ψ ′ ) are the same as the mor - phisms in ∆ × Ψ ; a nd —the mor phis ms from ι to ([ n ] , ψ ) are the same as the morphisms 12.7 Interpr etations as pr eshe af c ate gori es 251 [0] → [ n ] in ∆ (i.e. there are n + 1 of them). Comp osition o f mor phisms is deﬁned so that the single automo rphism of ι is the ident ity , so that comp osition o f morphisms within ∆ × Ψ is the sa me as fro m that categor y , a nd a co mpo sition o f the for m ι → ([ n ] , ψ ) → ([ n ′ ] , ψ ′ ) is g iven by the corres po nding comp osition [0] → [ n ] → [ n ′ ]. Prop ositio n 1 2.7.7 Supp ose M = Presh(Ψ) is a pr eshe af c ate gory. Ther e is a natur al isomo rphism b etwe en P resh( C + (Ψ)) and t he c ate gory of non-unital M -pr e c ate gories. In p articular, the latter is a pr eshe af c a t- e gory . Pr o of Supp o se F : C + (Ψ) o → Set is a pr esheaf. Let X := F ( ι ). F or any ( x 0 , . . . , x n ) ∈ ∆ X , let A ( x 0 , . . . , x n ) b e the presheaf on Ψ which assigns to ψ ∈ Ψ the subset of elements of F ([ n ] , ψ ) which pro ject to ( x 0 , . . . , x n ) under the n + 1 pr o jection maps F ([ n ] , ψ ) → F ( ι ) = X corres p o nding to the n + 1 maps ι → ([ n ] , ψ ). Then pair A is an element of Func (∆ o X , M ). Conversely , given any A ∈ Func (∆ o X , M ), deﬁne a presheaf F : C + (Ψ) o → Set by setting F ( ι ) := X and F ([ n ] , ψ ) := a ( x 0 ,...,x n ) ∈ X n +1 A ( x 0 , . . . , x n )( ψ ) . These constructions are in verses. The inclus io n functor U ∗ from PC ( M ) to the catego ry of non-unital precatego r ies, can be viewed as a pullback. Indeed, there is a pro jection functor π : C + (Ψ) → C (Ψ ) deﬁned by π ([ n ] , ψ ) = ([ n ] , ψ ) if n ≥ 1, π ([0] , ψ ) = 0 and π ( ι ) = 0 . The pullback U ∗ is just the pullback functor π ∗ : PC ( M ) ∼ = Presh( C (Ψ)) → Pres h( C + (Ψ)) . The left adjoint of the pull back or inclusion, denoted pushforward U ! ab ov e, is also the pushforward π ! for the functor π . Using the op era tions C + (Ψ) and C (Ψ) w e may stay ess ent ially en- tirely within the realm o f presheaf categ ories. The only place where we go outside of there is when we sp eak of the ca tegory of unital M - precatego r ies ov er a ﬁxed set of ob jects X ; even if M is a preshea f category , PC ( X ; M ) will not gener ally b e a presheaf categ ory . On the 252 Pr e c ate gories other hand, the non-unital version Func (∆ o X , M ) = Presh(∆ X × Ψ) remains a pre sheaf c ategory . 13 Algebraic theori es in mo del categories In this chapter we consider a lgebraic dia gram theories consisting of a collection of ﬁnite pro duct conditions imp ose d on diag rams Φ → M . This is motiv a ted by the situation considered in the previous Chapter 12. There we deﬁned the notion of precateg ory on a ﬁxed set of ob jects, which is a diagram A : ∆ o X → M . The Segal co nditions req uir e that ce r- tain maps b e weak equiv a lences. Imp o sing these co nditions amounts to a homotopical a nalogue of the “ ﬁnite pro duct theor ies” often considered in category theo ry [146] [147], see further historical remark s in [2, pp 171- 172]. The ho motopical ana logue, whose or ig ins go ba ck to the v arious theories of H -spaces and lo o p spaces, was considered by Ba dz io c h in [5], and his treatment was use d by Bergner for Segal catego ries in [34] [3 9], and also genera lized to mo dels in simplicia l ca tegories in [3 7]. Rosicky carried these idea s further in [182] and he po in ts out more references. Let ǫ ( n ) denote the categor y ξ 0 q q ξ 1 q ξ 2 q ξ 3 . . . q ξ n − 1 q ξ n ✸ ✶ ✿ q s The Sega l ma ps ar e obtained by pulling back A a long functors ǫ ( n ) → ∆ 0 X . This sets up a lo calization pr oblem which can b e phra sed in mor e general terms. W e treat the gener al situation in the present chapter. Even ignoring any p oss ible other applications, that simpliﬁes no ta tions for the general asp ects of the pro blem of enforcing the g iven co lle ction of ﬁnite pr o duct conditions. This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 254 Alge br aic the ories in mo del c ate gorie s By a n algebr aic the ory we mean a catego ry Φ provided with a collection of “direct pro duct dia grams” , that is diagrams w ith the sha pe of a direct pro duct, which are functors ǫ ( n ) P → Φ. A realiza tion o f the theor y in a classical 1-ca tegory C is a functor Φ → C which sends these diagra ms to direct pr o ducts in C . Many of the easiest kinds o f struc tur es c an be written this w ay , a lthough it is w ell understo o d that to get mo re complicated str uctures, one needs to go to the notion o f sketch which is a category pro vided with more gener ally shaped limit diagra ms. F or our purp oses, it will s uﬃce to consider direct pro duct diagra ms. Suppo se M is a n a ppropriate k ind of mo del categor y . Then a homo - topy realiza tion of the theory in M is a functor Φ → M which sends the dire ct pro duct diagra ms, to homotopy direct pro ducts in M . These no tio ns lead to a “ calculus of generator s and relations ” where we s tart with an arbitrar y functor Φ → C (r esp. Φ → M ) and try to enforce the dir ect pro duct (resp. homo topy direct pro duct) condition. The main work of this chapter will be to do this for the case o f ho motopy realizations in a mo del ca tegory M . 13.1 Diagrams o v er t he categories ǫ ( n ) The ﬁrst task is to take a c lose lo ok at the ca teg ories indexing direct pro ducts. Let ǫ ( n ) denote the category with ob jects ξ 0 , ξ 1 , . . . , ξ n ; and whose o nly morphisms apart from the iden tities, are s ing le mo rphisms ρ i : ξ 0 → ξ i . W e also le t ξ i denote the functors ξ i : {∗} → ǫ ( n ) sending the p oint ∗ to the ob ject ξ i ( ∗ ) = ξ i . This includes the case n = 0 wher e ǫ (0) is the discrete categor y with one ob ject ξ 0 . Suppo se C is so me other catego ry . A functor A : ǫ ( n ) → C corr e- sp onds to a collection of o b jects a 0 , a 1 , . . . , a n ∈ ob ( C ) together with maps p i : a 0 → a i for 1 ≤ i ≤ n . W e sometimes write A = ( a 0 , . . . , a n ; p 1 , . . . , p n ) . Say that A is a dir e ct pr o duct diagr am if the collec tio n of maps p i ex- presses a 0 as a direct pr o duct of the a 1 , . . . , a n in C . Suppo se M is a mo del ca tegory , in par ticular dire c t pro ducts exist. W e say that a diag ram A = ( a 0 , . . . , a n ; p 1 , . . . , p n ) from ǫ ( n ) to M is a homotopy dir e ct pr o duct diagr am if the mor phism ( p 1 , . . . , p n ) : a 0 → a 1 × · · · × a n 13.1 D iagr ams over the c ate gories ǫ ( n ) 255 is a weak equiv a lence. Consider no w the diagram catego ry Func ( ǫ ( n ) , M ). W e describ e ex- plicitly its injective and pro jective mo del structures, which are well- known to exist (se e Hirschhorn [116], Bar wick [1 6]). A mor phism of dia grams A = ( a 0 , . . . , a n ; p 1 , . . . , p n ) → B = ( b 0 , . . . , b n ; q 1 , . . . , q n ) is a c ollection of maps g = ( g 0 , . . . , g n ) with g i : a i → b i such that q i g 0 = g i p i . W e can also think o f a mor phism g a s consisting of the maps ( g 1 , . . . , g n ) plus a map g P : a 0 → b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) such that the second pr o jection is the structural ma p for A . W e then write g = h g 1 , . . . , g n ; g P i . A mor phism g is ﬁbrant in the pr o jective structure, if and o nly if each g 0 , . . . , g n is ﬁbra n t in M . Similar ly , g is coﬁbrant in the injective structure, if and o nly if ea ch g 0 , . . . , g n is coﬁbrant in M . T o s tudy ﬁbrant maps in the injectiv e structure, supp ose C = ( c 0 , . . . , c n ; r 1 , . . . , r n ) , D = ( d 0 , . . . , d n ; s 1 , . . . , s n ) are tw o diagrams , with morphisms forming a square A u → C B g ↓ v → D h ↓ . These give diagrams a i u i → c i b i g i ↓ v i → d i h i ↓ . W e lo ok fo r a lifting f : B → C such that f g = u a nd hf = v . This amounts to asking for liftings f i : b i → c i such that f i g i = u i and h i f i = v i , and also such that s i f 0 = f i r i . 256 Alge br aic the ories in mo del c ate gorie s Suppo se given already the liftings f 1 , . . . , f n . Then to s pecify a full lifting f as ab ov e we need to ﬁnd f 0 : b 0 → c 0 such that the diagram a 0 → c 0 b 0 g 0 ↓ v P → f 0 → d 0 × d 1 ×···× d n ( c 1 × · · · × c n ) ↓ commutes. Lemma 13. 1.1 A m orphism h : C → D with notations as ab ov e, is ﬁbr ant in the inje ctive mo del st ructur e on Func ( ǫ ( n ) , M ) if and only if e ach h 1 , . . . , h n is ﬁ br ant in M , and the map h P : c 0 → d 0 × d 1 ×···× d n ( c 1 × · · · × c n ) is ﬁ br ant in M . Pr o of If h satisﬁes the conditions stated in the lemma, and f is a levelwise coﬁbra tion i.e. each f i is a coﬁbration, then we can ﬁrst cho ose the liftings f i for i = 1 , . . . , n , then choose f 0 lifting h P . Therefo r e h is ﬁbrant. Supp ose on the other hand that h is ﬁbrant. F or any coﬁbration a → b and any i = 1 , . . . , n we have an injective coﬁbra tion b et ween ob jects with a (resp. b ) placed at i a nd the r e maining places ﬁlled in with ∅ . Since h satisﬁes liftin g along an y suc h c o ﬁbration, it follo ws t hat h i is ﬁb rant in M . Similarly , given a coﬁbration a → b with a → c 0 and b → d 0 × d 1 ×···× d n ( c 1 × · · · × c n ), w e get a square dia gram as a bove with A = ( a, b, b . . . , b ) and B = ( b, b, . . . , b ). The map A → B is a levelwise coﬁbration, so the ﬁbr a nt condition for h implies existence of a lifting, which gives a lifting for the map h P . Consider now the sa me questio n in the other dir ection: cho ose ﬁr st the lifting f 0 . F or any i ≥ 1 w e get a map b 0 ⊔ a 0 a i → c i , a nd the choices of f i for i = 1 , . . . , n corres po nd to choices of lifting in the diagra ms b 0 ⊔ a 0 a i → c i b i ↓ v i → f i → d i h i ↓ . Lemma 13. 1.2 A morphism g : A → B with notations as ab ove, is c oﬁbr ant in the p r oje ctive mo del structur e on Func ( ǫ ( n ) , M ) if and only 13.1 D iagr ams over the c ate gories ǫ ( n ) 257 if g 0 is c oﬁbr ant in M , and for e ach i = 1 , . . . , n the map b 0 ⊔ a 0 a i → b i is c oﬁbr ant in M . In p articular an obje ct B is c oﬁbr ant if and only if b 0 is c oﬁbr ant and e a ch b 0 → b i is a c oﬁbr atio n. Pr o of Similar to the previous pro of. One can describ e explicit gener ating sets for th e coﬁbrations and triv- ial c o ﬁbrations in b oth struc tur es Func inj ( ǫ ( n ) , M ) a nd Func pro j ( ǫ ( n ) , M ). Recall the s tandard a djunctions for the functors ξ i : ∗ → M . If X ∈ M is a n ob ject, we obtain a diagr am ξ i, ! ( X ) : ǫ ( n ) → M . Explicitly , if i = 0 then ξ 0 , ! ( X ) is the cons tant diag ram ( X , X , . . . , X ; 1 X , . . . , 1 X ) with v alues X . If i ≥ 1 then ξ 0 , ! ( X ) is the diag ram ( ∅ , . . . , ∅ , X , ∅ . . . ; ι, . . . , ι ) where ι denote the unique maps from ∅ to anything else, and her e X is at the -th pla ce. The adjunction says that for any diagram A = ( a 0 , . . . , a n ; p 1 , . . . , p n ), a morphis m ξ i, ! ( X ) → A is the sa me thing as a morphism X → ξ ∗ i ( A ) = a i . Suppo se given g enerating sets I for the coﬁbratio ns o f M and J for the triv ia l co ﬁbrations. F or f : X → Y in I , we obtain a co ﬁbration ξ 0 , ! ( f ) : ξ 0 , ! ( X ) → ξ 0 , ! ( Y ) in the pro jective mo del structure. T o see this, use Lemma 1 3.1.2 o n g = ξ 0 , ! ( f ) and note that g 0 is just f so it is coﬁbrant; and b 0 ⊔ a 0 a i = Y ⊔ X X = Y maps to b i = Y by a coﬁbr a tion. If f ∈ J then ξ 0 , ! ( f ) is a trivial coﬁbr ation: ov er each ob ject o f ǫ ( n ) we just g et back the ma p f so it is a n levelwise w eak equiv alence. F or f ∈ I a s ab ove, at any i ≥ 1, ξ i, ! ( f ) is a coﬁbration, aga in using Lemma 13.1.2 on g = ξ 0 , ! ( f ). The map g 0 is the identit y of ∅ , and the maps b 0 ⊔ a 0 a j → b j are either the ident ity of ∅ for j 6 = i , or f when j = i , so these are coﬁbrations. Again, if f ∈ J then ξ i, ! ( f ) is a trivial coﬁbration: ov er each o b ject o f ǫ ( n ) it gives either the identit y o f ∅ which is a utomatically a weak equiv a lence, or else f . Deﬁne I ǫ ( n ) (resp. J ǫ ( n ) ) to b e the set consisting o f diagrams of the form ξ i, ! ( f ) for 0 ≤ i ≤ n and f ∈ I (resp. f ∈ J ). Prop ositio n 13. 1.3 The sets I ǫ ( n ) and J ǫ ( n ) ar e gener ators for t he pr oj e ctive mo del c ate go ry stru ctur e Func pro j ( ǫ ( n ) , M ) . Pr o of By the adjunction, a morphism g in Func ( ǫ ( n ) , M ) satisﬁes th e right lifting prop erty with res pect to I ǫ ( n ) (resp. J ǫ ( n ) ) if and only if ξ ∗ i ( g ) satisﬁes the r ight lifting pr o pe rty with r esp ect to I (resp. J ) for all 0 ≤ i ≤ n . Since I (resp. J ) is a s e t of g enerator s for the coﬁbratio ns (re sp. trivial coﬁbr ations) of M , this lifting pro p er t y is equiv alen t to s aying that each ξ ∗ i ( g ) is a trivial ﬁbration (resp. a ﬁbration). By the deﬁnition 258 Alge br aic the ories in mo del c ate gorie s of the pro jective mo del str ucture, this is equiv alen t to saying that g is a trivial ﬁbration (resp. a ﬁbration). Hence inj ( I ǫ ( n ) ) is the class of tr ivial ﬁbrations and inj ( J ǫ ( n ) ) is the class of ﬁbrations, so cof ( I ǫ ( n ) ) is the class of co ﬁbrations and cof ( J ǫ ( n ) ) is the cla ss of trivial coﬁbratio ns. T o g et g enerators for the injective mo del str ucture, w e need to a dd a new kind o f injectiv e coﬁbration. If f : X → Y is a coﬁbration, consider the diag rams  ( f ) := ξ 0 , ! ( X ) ∪ ` i ξ i, ! ( X ) a i ξ i, ! ( Y ) = ( X, Y , . . . , Y ; f , . . . , f ) (13.1 .1) and ξ 0 , ! ( Y ). W e hav e a ma p  ( f ) → ξ 0 , ! ( Y ) which is f at the o b ject ξ 0 ( ∗ ) and 1 Y at the o ther ob jects. Denote this ma p by ρ ( f ) = ( f , 1 , . . . , 1). Prop ositio n 13 .1.4 Le t I + ǫ ( n ) denote the union of I ǫ ( n ) with the set of maps of the form ρ ( f ) = ( f , 1 , . . . , 1) for f ∈ I . L et J + ǫ ( n ) denote the union of J ǫ ( n ) with the set of maps o f the form ρ ( f ) fo r f ∈ J . Then I + ǫ ( n ) and J + ǫ ( n ) ar e gener ating sets for t he inje ctive mo del c ate gory str u ctur e Func inj ( ǫ ( n ) , M ) . Pr o of If f : X → Y is a coﬁbratio n, then ( f , 1 , . . . , 1 ) : U → V is a coﬁbration with the no tations as a bove. Supp o se that g : A → B satisﬁes right lifting with re spe ct to f , where g = ( g 0 , . . . , g n ) go es from A = ( a 0 , . . . , a n ; p 1 , . . . , p n ) to B = ( b 0 , . . . , b n ; q 1 , . . . , q n ). The lifting prop erty says that for any map u 0 : X → a 0 and ma ps u i : Y → a i for i ≥ 1, and maps v i : Y → b i for i ≥ 0 such tha t p i u 0 = u i f , v i = g i u i for i ≥ 1, and v 0 f = g 0 u 0 , then ther e s hould exist a ma p u ′ 0 : Y → a 0 such tha t u ′ 0 f = u 0 , g 0 u ′ 0 = v 0 , and p i u ′ 0 = u i . This is the s ame as the right lifting prop erty for the square X → a 0 Y ↓ → b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) ↓ so the co nditio n that g s atisﬁes rig h t lifting with r esp ect to any ( f , 1 , . . . , 1) for f ∈ J is equiv a lent to the condition that a 0 → b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) is a ﬁbration. T hus, inj ( J + ǫ ( n ) ) consists o f maps whic h are lev- elwise ﬁbrations (b ecaus e o f lifting with res p ect to J ǫ ( n ) ) and s uch that the map o f Lemma 13.1.1 is a ﬁbration. By Lemma 13.1 .1, inj ( J + ǫ ( n ) ) is the cla ss of ﬁbrations. 13.2 Imp osing the pr o duct c ondition 259 Similarly , the fact that g sa tisﬁes right lifting with resp ect to any ( f , 1 , . . . , 1) for f ∈ I is equiv alen t to the co nditoin that a 0 → b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) be a trivia l ﬁbratio n. Thus, inj ( I + ǫ ( n ) ) co nsists o f maps which are level- wise trivial ﬁbrations (be c ause of lifting with resp ect to I ǫ ( n ) ) and s uch that the map of Lemma 13.1.1 is a triv ial ﬁbration. W e claim that this is eq ual to the class of ﬁbra tio ns. If g ∈ inj ( I + ǫ ( n ) ), then it is a ﬁbra tion for the injective structure since J + ǫ ( n ) ⊂ I + ǫ ( n ) , and also levelwise a weak e quiv a lence, so it is a trivial ﬁbration. If g is a trivial ﬁbration, then it is levelwise a trivia l ﬁbr a tion, and the map a 0 → b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) is a ﬁbration. How ever, the maps a i → b i are trivial ﬁbrations, so a 1 × · · · × a n → b 1 × · · · × b n is a trivial ﬁbration. Th us, the map b 0 × b 1 ×···× b n ( a 1 × · · · × a n ) → b 0 is a trivial ﬁbration, and by 3 for 2 we conclude that a 0 → b 0 × b 1 ×···× b n ( a 1 ×· · · × a n ) is a weak eq uiv a le nc e . Hence it is a triv ia l ﬁbration as required for the claim. This identiﬁes inj ( I + ǫ ( n ) ) a nd inj ( J + ǫ ( n ) ) with the cla s ses of trivia l ﬁ- brations and ﬁbrations resp ectively , so cof ( I + ǫ ( n ) ) and cof ( J + ǫ ( n ) ) are the classes of co ﬁbrations a nd trivial coﬁbratio ns resp ectively . Sc holium 13.1.5 If ( M , I , J ) is a c oﬁbr antly gener ate d (resp . c ombi- natorial, tr actable, left pr op er) m o del c ate gory, then Func inj ( ǫ ( n ) , M ) and Func pro j ( ǫ ( n ) , M ) ar e c oﬁbr antly gener ate d (r esp. c ombinatorial, tr actable, left pr op er) mo de l c ate go ries. See Theo rem 9.8.1. 13.2 Imp osing the pro duct condition Assume that M is a tracta ble left prop er ca rtesian mo del catego r y . Recall that the cartesia n condition 1 0 .0.9 implies that for any X ∈ M and weak equiv alence f : Y → Z the induced map X × Y → X × Z is a w eak e quiv a lence. W e are going to a pply the direct left Bousﬁeld loca lization theory of Chapter 11. Say that an ob ject A ∈ Func ( ǫ ( n ) , M ) is pr o duct- c omp atible if the map A ( ξ 0 ) → A ( ξ 1 ) × . . . × A ( ξ n ) is a weak equiv alence. Let R ǫ ( n ) ⊂ Func ( ǫ ( n ) , M ) de no te the full sub categor y of pro duct- compatible ob jects. 260 Alge br aic the ories in mo del c ate gorie s Lemma 13.2. 1 S u pp ose M is c artesian. The sub c ate gory R ǫ ( n ) is in- variant under we ak e quivalenc e: if A is pr o duct-c omp atible and its image in ho Func ( ǫ ( n ) , M ) is isomorphic to the image of another o bje ct B , then B is also pr o duct- c omp atibl e. Pr o of If A → B is a levelwise weak equiv alence, then the horizontal arrows in the dia gram A ( ξ 0 ) → B ( ξ 0 ) A ( ξ 1 ) × . . . × A ( ξ n ) ↓ → B ( ξ 1 ) × . . . × B ( ξ n ) ↓ are weak e q uiv a lences, using the cartesian condition for the b ottom ar- row. Hence, o ne of the vertical arr ows is a w eak equiv a lence if a nd only if the other one is. 13.2.1 Direct lo calization of the pro jectiv e structure W e now deﬁne the dire ct loc a lizing system whic h goes with the full sub- category of product- c o mpatible ob jects in the pro jective mo del category structure. F or each generating coﬁbra tion f : X → Y in I , reca ll the morphism ρ ( f ) = ( f , 1 , . . . , 1) deﬁned ab ov e (see (13 .1.1))  ( f ) = ( X, Y , . . . , Y ; f , . . . , f ) ρ ( f ) → ξ 0 , ! ( Y ) = ( Y , . . . , Y ; 1 , . . . , 1) . It is obviously an injective co ﬁbr ation, and the do main  ( f ) is pro jec- tively coﬁbrant (apply Lemma 13 .1.2). Howev er, ρ ( f ) will not in general be a pro jective coﬁbration. Cho os e a factor ization Y ∪ X Y a ( f ) → Z b ( f ) → Y such that a ( f ) is a coﬁbratio n and b ( f ) is a trivial ﬁbration. Denoting by i 2 the ﬁrst inclusion Y → Y ∪ X Y we ge t a trivial co ﬁbr ation a ( f ) i 2 : Y → Z . Le t  ( f ) = ( X, Y , . . . , Y ; f , . . . , f ) ζ ( f ) → ψ ( f ) := ( Y , Z , . . . , Z ; a ( f ) i 2 , . . . , a ( f ) i 2 ) be the map given by f ov er ξ 0 and a ( f ) i 1 ov er ξ 1 , . . . , ξ n . Then the map o ccuring in Lemma 13.1.2 is exactly a ( f ), so ζ ( f ) is a pro jective coﬁbration. 13.2 Imp osing the pr o duct c ondition 261 Recall the explicit generating set J ǫ ( n ) for the trivial coﬁbr a tions in the pro jectiv e model structure. Put K pro j ǫ ( n ) := J ǫ ( n ) ∪ { ζ ( f ) } f ∈ I . Theorem 13.2. 2 The p air ( R ǫ ( n ) , K pro j ǫ ( n ) ) is a dir e ct lo c alizing system for t he mo del c ate gory Func pro j ( ǫ ( n ) , M ) of ǫ ( n ) -diagr ams in M with the pr oje ctive mo del stru ctur e. L et Func pro j , Π ( ǫ ( n ) , M ) denote the left Bousﬁeld lo c alize d mo del structure c onstructe d in Chapter 11. A mor- phism A → B is a we ak e quivalenc e if and only if it induc es we ak e qu iv- alenc es levelwise over the obje cts ξ 1 , . . . , ξ n . An obj e ct A is ﬁ br ant in the lo c alize d stru ct ur e if and only if it is pr o duct-c omp atible and e ach A ( ξ i ) is ﬁ br ant. Pr o of W e verify the prop er ties (A1)–(A6). P r op erties (A1) and (A2) are immedia te. F o r (A3) we hav e chosen ζ ( f ) so as to b e a pro jectively coﬁbration, and its domain is pro jectively co ﬁbrant. Condition (A4) is given b y Lemma 13.2.1 . Suppo se a diagram A is in inj ( K pro j ǫ ( n ) ). In particular it is J ǫ ( n ) -injective, which is to say ﬁbrant in the pr o jectiv e mo del structure. This means that each A ( ξ i ) is a ﬁbrant ob ject in M . The lifting pr op erty along the ζ ( f ) implies the following homotopy lifting prop erty of the map p : A ( ξ 0 ) → A ( ξ 1 ) × · · · × A ( ξ n ) . along a gener ating coﬁbr ation f : X → Y in I . Recall that Y ∪ X Y a → Z b → Y w as c hosen above. If we are given a diagram X u → A ( ξ 0 ) Y f ↓ v → A ( ξ 1 ) × · · · × A ( ξ n ) p ↓ then there is a map Y w → A ( ξ 0 ) such that wf = u , a nd a map Z h → A ( ξ 1 ) × · · · × A ( ξ n ) such that hai 2 = v but hai 1 = pw . This homo topy lifting prop erty implies that p is a w eak equiv alence, see Lemma 9.4 .1 . Thu s, A ∈ R ǫ ( n ) which gives (A5). F or condition (A6), s upp os e A is in R ǫ ( n ) and A → B is a pushout along an element of K pro j ǫ ( n ) . Applying the small o b ject a rgument we can ﬁnd a map B → C in cel l ( K pro j ǫ ( n ) ) such that C ∈ inj ( K pro j ǫ ( n ) ). Then A → C 262 Alge br aic the ories in mo del c ate gorie s is also in cell ( K pro j ǫ ( n ) ). Notice, how ev er, that the elements of K pro j ǫ ( n ) are levelwise trivial coﬁbrations ov er the ob jects ξ 1 , . . . , ξ n ∈ ǫ ( n ) (but not ov er ξ 0 ). Therefore the map A → C induces weak eq uiv alences ov er ea ch ξ 1 , . . . , ξ n . Using the car tes ian condition on M , this implies that the right v ertical map in the square A ( ξ 0 ) → A ( ξ 1 ) × · · · × A ( ξ n ) C ( ξ 0 ) ↓ → C ( ξ 1 ) × · · · × C ( ξ n ) ↓ is a weak equiv a lence. The hypothesis that A ∈ R ǫ ( n ) says that the top map is a w eak equiv a lence, a nd the fact that C ∈ inj ( K pro j ǫ ( n ) ) and part (A5) pr ov ed ab ov e say that C ∈ R ǫ ( n ) , so the bo ttom map is a weak equiv alence. By 3 fo r 2 the left vertical a rrow is a weak equiv alence, showing tha t A → C is a levelwise weak equiv alence of diagra ms. This shows (A6). Our dir ect lo ca lizing system leads to a left Bousﬁe ld lo caliza tion by Theorem 1 1.7.1. W e now lo ok a t the characterizations of new w eak equiv alences. As seen ab ov e, the elements of cell ( K pro j ǫ ( n ) ) are weak equiv alences levelwise ov er the ξ 1 , . . . , ξ n . Using the characteriza tin of Co rollar y 11.4.3 we see that a ll new weak equiv alences are levelwise w eak equiv a lences over the ξ 1 , . . . , ξ n . Supp ose A f → B is a morphism inducing a weak equiv alence ov er each ξ 1 , . . . , ξ n . Cho ose B b → B ′ in cell ( K pro j ǫ ( n ) ) such that B ′ is in inj ( K pro j ǫ ( n ) ), in particula r it is pro duct-compatible. F actor the co mpos ed map a s A a → A ′ g → B ′ where a ∈ cell ( K pro j ǫ ( n ) ) a nd g ∈ inj ( K pro j ǫ ( n ) ). All of the ab ov e maps ar e levelwise weak equiv alences ov er the ob jects ξ 1 , . . . , ξ n , how ev er A ′ and B ′ are pr o duct-compatible. It follows that g is a levelwise weak equiv- alence. The criter ion of Coro llary 11.4.3 implies that f is a new weak equiv alence. The ch ara cterization of ﬁbrant ob jects is a ﬁrst version in the sim- pliﬁed situation o f a single pro duct diag ram, o f Bergner ’s characteri- zation of ﬁbra n t Sega l c a tegories [36]. The new ﬁbrant ob jects ar e in inj ( K pro j ǫ ( n ) ) so they a re in R ǫ ( n ) i.e. pro duct-co mpatible, and levelwise ﬁbrant. Supp ose A is pro duct-compatible and levelwise ﬁbrant. Supp ose 13.2 Imp osing the pr o duct c ondition 263 U f → V is a new trivia l coﬁbr ation, and we are given a map U u → A . By the pr evious para g raph it induces a levelwise trivial coﬁbration over the ob jects ξ 1 , . . . , ξ n . Hence the components u 1 , . . . , u n extend to maps V ( ξ i ) v ′ i → A ( ξ i ). Putting these together, the comp osition V ( ξ 0 ) → V ( ξ 1 ) × · · · × V ( ξ n ) → A ( ξ 1 ) × · · · × A ( ξ n ) gives the bo ttom arrow of the diagram U ( ξ 0 ) u 0 → A ( ξ 0 ) V ( ξ 0 ) f 0 ↓ → A ( ξ 1 ) × · · · × A ( ξ n ) . ↓ The rig h t vertical a rrow is a weak equiv alence b etw een ﬁbr ant o b jects, so by Lemma 9.4.1 there is a homotopy lifting relative U ( ξ 0 ), in other words a map V ( ξ 0 ) v 0 → A ( ξ 0 ) such that rf 0 = u 0 , a nd the other tr iangle commutes up to a homo to p y relative U ( ξ 0 ). Lemma 9.4.2 says w e can change the maps v ′ i to maps v i , still restricting to u i on U ( ξ i ), but compatible with v 0 . W e have now constr uc ted the requir ed extension V → A , showing that A is a new ﬁbant o b ject. 13.2.2 Direct lo calization of the injectiv e structure When p ossible, it is more c o nv enien t to use the injectiv e mo del structure on Func ( ǫ ( n ) , M ). Consider the explicit genera ting s et J + ǫ ( n ) for the trivial coﬁbra tions in the injective mo del structure, g iven by P rop osition 13.1.4. W e can deﬁne tw o diﬀerent s e ts of co ﬁbrations, the ﬁrst extending K pro j ǫ ( n ) : K inj+ ǫ ( n ) := K pro j ǫ ( n ) ∪ J + ǫ ( n ) = J + ǫ ( n ) ∪ { ζ ( f ) } f ∈ I ; and the sec ond deﬁned using the simpler maps ρ ( f ) which w ere alrea dy injectiv e coﬁbrations: K inj ǫ ( n ) := J + ǫ ( n ) ∪ { ρ ( f ) } f ∈ I . Theorem 13.2.3 The p airs ( R ǫ ( n ) , K inj+ ǫ ( n ) ) and ( R ǫ ( n ) , K inj ǫ ( n ) ) ar e b oth dir e ct lo c al izing s ystems for the mo del c ate gory Func inj ( ǫ ( n ) , M ) of ǫ ( n ) -diagr ams in M with the inje ctive mo del structure . L et Func inj , Π ( ǫ ( n ) , M ) denote t he left Bousﬁeld lo c alize d mo de l stru ctur e, which is the same in 264 Alge br aic the ories in mo del c ate gorie s b oth c ases. The we ak e quivalenc es ar e the same as for the pr oj e ctive structur e. An obje ct A is ﬁbr ant in the lo c alize d structu r e if and only if it is pr o duct -c omp atible and s atisﬁ es the ﬁbr ancy criterion of L emma 13.1.1 for the inje ctive mo del st ructur e. The identity functor is a left Quil len functor Func pro j , Π ( ǫ ( n ) , M ) → Func inj , Π ( ǫ ( n ) , M ) fr om the new pr oje ctive to the new inje ctive mo del st ructur e. Pr o of The functor Func pro j ( ǫ ( n ) , M ) → Func inj ( ǫ ( n ) , M ) is a left Quillen functor, whose co rresp onding right Q uillen functor (both b e- ing the ident ity on underlying catego ries) preser ves the class R ǫ ( n ) of pro duct-compatible diagr ams. In other words the transfer ed class is the same. The s ubset K inj+ ǫ ( n ) is the transfered subset given in Theorem 11.8 .1, so by that theorem ( R ǫ ( n ) , K inj+ ǫ ( n ) ) is a direct lo calizing system and the ident ity functor is a left Quillen functor from the previous new pro jec- tive mo del structure to the res ulting left Bousﬁeld lo ca lization of the injectiv e structure Func pro j , Π ( ǫ ( n ) , M ) → Func inj , Π ( ǫ ( n ) , M ) . The pro of that ( R ǫ ( n ) , K inj ǫ ( n ) ) is a direct lo ca lizing system is the same as in the pro o f o f the prev ious Theorem 1 3.2.2, but in fact easier since an ob ject whic h satis ﬁe s lifting with resp ect to the ρ ( f ) has the stronger prop erty that A ( ξ 0 ) → A ( ξ 1 ) × · · · × A ( ξ n ) is in inj ( I ), that is it is a trivial ﬁbration. So in this case w e do n’t need to r ely o n the notion of homotopy lifting pro per t y as was done in the previous pro of. W e get conditions (A1)–(A6) a nd a lso the same description o f weak e q uiv a lences, and the co rresp onding description of ﬁbrant ob jects. The tw o mo del str uctures given by Theo rem 1 1.7.1 applied to ( R ǫ ( n ) , K inj ǫ ( n ) ) and ( R ǫ ( n ) , K inj+ ǫ ( n ) ) are the same, by Prop ositio n 11.7.2 . Suppo se A ∈ Func ( ǫ ( n ) , M ). Supp ose given a factorizatio n A ( ξ 0 ) e 0 → E 0 p → A ( ξ 1 ) × · · · × A ( ξ n ) in M . Let E i := A ( ξ i ) for i = 1 , . . . , n . The structural map p gives a structure o f ǫ ( n )-dia gram to the collection ( E 0 , . . . , E n ), ca ll it E . The map e gives a map e : A → E . If e 0 is a coﬁbra tion in M then e is a coﬁbration in Func inj ( ǫ ( n ) , M ). 13.2 Imp osing the pr o duct c ondition 265 Lemma 13 . 2.4 In the ab ove situation, if e 0 is a c oﬁbr atio n and p is a we ak e quivalenc e in M then e : A → E is a trivial c oﬁbr ation in Func inj , Π ( ǫ ( n ) , M ) . Pr o of The map e is levelwise c o ﬁbrant b y constructio n. It is a weak equiv alence since it induces a weak e quiv a lence levelwise ov er the o b jects ξ 1 , . . . , ξ n . 13.2.3 T ransfering these structures Putting together the ab ov e a nalysis of diagrams ov er ǫ ( n ) with the transfer a long a Q uillen functor g ives the following g eneral picture . Suppo se we are given a set Q , integers n ( q ) ≥ 0 for q ∈ Q , a fam- ily of tr actable left prop er ca rtesian mo del categor ie s M q for q ∈ Q , a tractable left prop er mo del categor y N , and a family of Quillen functors F q : Func pro j ( ǫ ( n ( q )) , M q ) ← → N : G q . Let R ′ ⊂ N b e the full sub catego ry of ob jects Y such that, for a ﬁbrant replacement Y → Y ′ , the diagr ams G q ( Y ′ ) : ǫ ( n ( q )) → M q are pro duct-compatible. Let ( I q , J q ) b e generating s ets for M q , and ( I ′ , J ′ ) generator s for N . Corollary 13.2.5 L et K ′ b e the union of J ′ , of t he set of morphisms of the form F q ( g ) for g ∈ J q ǫ ( n ( q ) ) , and of t he set of morphisms of the form F q ( ζ ( f )) for f ∈ I q . Then ( R ′ , K ′ ) is a dir e ct lo c alizing system for N . If fur t hermor e F q ar e left Quil len functors fr om Func pro j ( ǫ ( n ( q )) , M q ) to N , then we c an c onsider K inj , the union of J ′ with the set of mor- phisms of the form F q ( ρ ( f )) for f ∈ I q , and ( R ′ , K inj ) is a dir e ct lo c al - izing system for N giving the same m o del stru ctur e as ( R ′ , K ′ ) . Pr o of Let K ′′ be the union of K ′ with the set o f mor phisms of the form F q ( g ) for g ∈ J q ǫ ( n ( q ) ) . Then ( R ′ , K ′′ ) is a direct lo calizing sys tem for N , by Theo rem 1 1.8.3 applied to the direct lo calizing systems of Theorem 13.2.2. How ev er, the F q ( g ) for g ∈ J q ǫ ( n ( q ) ) are trivial co ﬁbrations b etw een coﬁbrant ob jects in the or ig inal model struc tur e of N , so they could b e included in a bigger generating set J ′′ for the original trivial coﬁbrations of N . But one can no te that in the constructio n of a direct lo calizing system by a dding on some new morphisms to the or iginal generating set, the prop erties are indep endent of the choice of orig inal genera ting set. So K ′ works as well as K ′′ . Suppo se now that F q remain left Quillen functors when w e use the in- jective mo del structures on their sources. Then, with a similar dis cussion 266 Alge br aic the ories in mo del c ate gorie s for leaving out the images of the morphisms in J q, + ǫ ( n ( q ) ) , Theo rem 11.8.3 applies to the direct lo calizing systems of Theor em 1 3 .2.3 to conclude that ( R ′ , K inj ) is a direct lo calizing system. As p ointed out in P rop osi- tion 11.7.2, the re s ulting mo del structure is the sa me as for ( R ′ , K ′ ). 13.3 Algebraic diagram theories Classically , a n “ algebra ic theor y” is g iven by a sma ll c a tegory Φ a nd a collection of pr o duct diagra ms P q : ǫ ( n ( q )) → Φ. The ob jects of the theory a re the functors A : Φ → S et with the prop erty that p ∗ q ( A ) is a dir e ct pr o duct, tha t is pr o duct-c omp atib le in the ab ove ter minology . Of cour se this theo r y has since b een muc h generalized, to include the notion of “ﬁnite limit s ketc hes” amo ng other things. How ev er, for our purp oses it will b e suﬃcient to co nsider just the basic version of the theory , and to give it a weak-enriched c o unt erpar t using the notion of direct left B ousﬁeld lo c alization we hav e developped so far . So, supp ose Φ is a small categ ory , Q is a small set, we hav e integers n ( q ) ≥ 0 for q ∈ Q , and supp ose given functors P q : ǫ ( n ( q )) → Φ for q ∈ Q . F or the co eﬃcien ts, ﬁx a tra ctable left prop er mo del catego ry M satisfying condition (PROD). Let ( I , J ) b e a set of genera tors fo r M . Playing the role of the mo del categ ory N will b e the ca tegory of Φ- diagrams in M with its pro jective o r injectiv e mo del str ucture, N = Func pro j (Φ , M ) (resp. N = Func pro j (Φ , M )). Let ( I Φ , pro j , J Φ , pro j ) and ( I Φ , inj , J Φ , inj ) be the sets of generator s for the pro jective a nd injectiv e mo del s tr uctures resp ectively , see the discuss ion of re fer ences for The- orem 9.8 .1 . Recall that the identit y functor is a Quillen adjunction b e- t ween the pro jective and injective diagra m categor ie s 1 : Func pro j (Φ , M ) ← → Func inj (Φ , M ) : 1 . Lemma 13.3.1 With the ab ove notations, for e ach q ∈ Q we get a Quil len adjunction P q, ! : Func pro j ( ǫ ( n ( q )) , M ) ← → Func pro j (Φ , M ) : P ∗ q . This c omp oses with the identity functor to give a Q uil len adjunction 1 P q, ! : Func pro j ( ǫ ( n ( q )) , M ) ← → Func inj (Φ , M ) : P ∗ q 1 . Pr o of This is the standard Quillen adjunction b etw een Bo usﬁeld pro- jective model s tructures co ming from the functor P q : ǫ ( n ( q )) → Φ. 13.3 Algebr ai c diagr am the ories 267 In the abov e situation, let R (Φ , P · , M ) denote the full sub categ ory of Func (Φ , M ) cons is ting of diagra ms A such that for a ﬁbrant replace- men t A → A ′ in the injective mo del structure (which is also a ﬁbrant replacement in the pr o jective mo del structure), for all q ∈ Q , P ∗ q ( A ′ ) ∈ Func ( ǫ ( n ( q )) , M ) is pro duct-compatible. Let K pro j / inj (Φ , P · , M , I , J ) be the sets given by Corolla ry 13.2.5 for the pro jective/injective s truc- ture, consisting of the elements of J Φ , pro j / inj , of the P q, ! ( J ǫ ( n ( q ) ) ), and of the P q, ! ( ρ ǫ ( n ( q ) ) ( f )) for f ∈ I . Here a choice of subscript pro j or inj is prop osed when necessary . Theorem 13 .3.2 The p air ( R (Φ , P · , M ) , K pro j / inj (Φ , P · , M , I , J )) is a dir e ct lo c alizing system. Deﬁne t he mo del catego ry of weak (Φ , P · )- algebras in M denote d by Alg pro j / inj (Φ , P · ; M ) , to b e the dir e ct left Bousﬁeld lo c alization of the pr oje ctive or inje ctive diagr am mo del c ate- gory Func pro j / inj (Φ , M ) with r esp e ct to the ful l sub c ate gory R (Φ , P · , M ) and sets K pro j / inj (Φ , P · , M , I , J ) . The c oﬁbr ations ar e levelwise c oﬁbr a- tions in the inje ctive structu r e, and pr oje ctive diag r am c oﬁbr ations in the pr oj e ctive structu r e. The ﬁbr ant obje cts of Alg pro j / inj (Φ , P · ; M ) ar e the diagr ams A : Φ → M such that A is levelwise ﬁbr ant (for the pr o- je ct ive structur e) or ﬁbr ant in the inje ctive diagr am structu re , and for e ach q ∈ Q the pul lb ack P ∗ q ( A ) : ǫ ( n ( q )) → M is pr o duct-c omp atible. Given a map f : A → B of Φ -diagr ams in M , the fol lowing c onditions ar e e quivale nt: — f is a we ak e quiva lenc e in A lg pro j / inj (Φ , P · ; M ) —for any squar e diagr am A ← A ′ → A ′′ B ↓ ← B ′ ↓ → B ′′ ↓ such that the left horizontal arr ows ar e pr oj e ctive/l evelwise c oﬁbr ant r e- plac ements , and A ′′ , B ′′ ∈ R (Φ , P · , M ) , then the morphism A ′′ → B ′′ is an levelwise we ak e quivalenc e; —ther e ex ist s a squar e diagr am as ab ov e with A ′′ → B ′′ an levelwise we ak e quiva lenc e; —ther e ex ist s a squar e diagr am as ab ov e with A ′′ → B ′′ an levelwise we ak e quiva lenc e, but without the r e qu ir ement A ′′ , B ′′ ∈ R (Φ , P · , M ) . These pr oj e ctive and inje ctive mo del c ate gories of we ak alge br as ar e tr actable and left pr op er. 268 Alge br aic the ories in mo del c ate gorie s Pr o of Apply the constr uc tio n of the dir ect left Bo usﬁeld lo ca liz a tion given in the previous chapter, starting with either Func pro j (Φ , M ) or Func inj (Φ , M ) of Theorem 9.8 .1, and tra nsfering the direct lo c alizing systems as in Co r ollary 13.2.5 . F or the characterization of ﬁbrant ob jects, see the succes sive state- men ts of Prop os itio n 9.9.8 , Theorem 11.7 .1, Remar k 11.8 .2 and Theo - rem 1 1.8.3. Apply these together with the characteriza tions of ﬁbrant ob jects in the pro duct-co mpatible ǫ ( n )- dia gram model structures. 13.4 Unitality Suppo se given a full sub categ ory Φ 0 ⊂ Φ. Typically , these will b e the P q ( ξ 0 ) for q ∈ Q such that n ( q ) = 0. W e would like to consider diagrams A : Φ → M s uch that A ( x ) = ∗ for x ∈ Φ 0 . Call such a diagra m unital along Φ 0 . Let Func (Φ / Φ 0 , M ) ⊂ Func (Φ , M ) denote the full sub catego ry of diagrams which are unital along Φ 0 . Denote by U ∗ Φ 0 : Func (Φ / Φ 0 , M ) → Func (Φ , M ) the identit y inclusion functor. The idea fo r this nota tion is that Φ / Φ 0 represents the contraction of Φ 0 to a p oint, the r esult b eing a p ointed categ ory i.e. a categor y with distinguished ob ject; and Func (Φ / Φ 0 , M ) is the categ ory o f p ointed functors from her e to the categ ory M p o inted b y dis tinguishing the coinitial ob ject ∗ . The present discussio n plays an imp ortant role in the theo r y of weakly enriched pr ecategor ies: the unitality condition corr esp onds to T a msamani’s constancy condition in the case o f n -pr ecategor ies, corresp o nding to th e idea of having a glo bular theor y in which the ob jects f orm a discr ete s et. The motiv a ting example, ﬁrst in tro duced in Section 12.1 above, is when Φ = ∆ X and Φ 0 = X is the sub categor y of sequences of length 0. F or any x ∈ Φ de no te by Φ 0 /x the categ ory of arr ows z → x with z ∈ Φ 0 . Theorem 13 .4.1 If M is a lo c al ly pr esentable c ate gory, t hen the c at- e gory Func (Φ / Φ 0 , M ) is lo c ally pr esentable and U ∗ Φ 0 has a left adjoint U Φ 0 , ! . The left adjoint is given as fol lows: if A ∈ Func (Φ , M ) then U Φ 0 , ! A is t he diagr am which sends an obje ct x ∈ Φ to t he c opr o d- uct of A ( x ) and co lim Φ 0 /x ∗ over colim z ∈ Φ 0 /x A ( z ) . The adjunction map 13.4 U nitality 269 U Φ 0 , ! U ∗ Φ 0 B → B is the identity for any B ∈ Func (Φ / Φ 0 , M ) , s o U Φ 0 , ! is a monadic pr oje ction in the terminolo gy of Se ction 8.2. The ful l sub c ate gory Func (Φ / Φ 0 , M ) ⊂ Func (Φ , M ) is close d un- der smal l limits and over c ol imits with smal l nonempty c onne cte d index sets, in p articular it is close d under c opr o ducts, ﬁlter e d c olimits, and tr ansﬁnite c omp osition. F or any r e gular c ar dinal κ with M b eing lo c al ly κ -pr esentable and | Φ | < κ , an obje ct A ∈ Func (Φ / Φ 0 , M ) is κ -pr esentable if and only if e ach of t he A ( x ) ar e κ - pr esen t able in M . Pr o of Put U Φ 0 , ! ( A )( x ) := A ( x ) ∪ colim z ∈ Φ 0 /x A ( z ) colim Φ 0 /x ∗ . Given a morphism x → y , we o btain mor phisms colim Φ 0 /x ∗ → colim Φ 0 /y ∗ and colim z ∈ Φ 0 /x A ( z ) → colim z ∈ Φ 0 /y A ( z ) . These ar e compatible with the maps in the above copro duct so they g ive U Φ 0 , ! ( A ) a structure of diagr a m (i.e. functor). If x ∈ Φ 0 then x is the coinitial ob ject o f Φ 0 /x , and we get U Φ 0 , ! ( A )( x ) = A ( x ) ∪ A ( x ) ∗ = ∗ . Thu s U Φ 0 , ! ( A ) ∈ Func (Φ / Φ 0 , M ). T o show adjunction, supp ose B ∈ Func (Φ / Φ 0 , M ). Given a map A → U ∗ Φ 0 B then for any z ∈ Φ 0 the map A ( z ) → B ( z ) factors through ∗ (since indeed B ( z ) = ∗ ). F or any x this gives a factoriza tion colim z ∈ Φ 0 /x A ( z ) → colim z ∈ Φ 0 /x ∗ A ( x ) ↓ → B ( x ) ↓ so our map o f diag rams fa c tors through a unique map U Φ 0 , ! ( A ) → B . If A = U ∗ Φ 0 B is a diagr am with A ( z ) = ∗ for z ∈ Φ 0 already , then the second map in the copr o duct de ﬁning U Φ 0 , ! ( A ) is the identit y , so U Φ 0 , ! ( A ) = A , i.e. the a djunction is a mona dic pro jection. Closure under a rbitrary sma ll limits is automa tic since U ∗ Φ 0 is a r ight adjoint. F or closure under co nnec ted colimits, supp ose α is an index category with connected ner ve. Then colim α ∗ = ∗ w he r e ∗ is the coini- tial ob ject of M and the colimit is taken ov er the co nstant functor 270 Alge br aic the ories in mo del c ate gorie s α → M (Lemma 8.1.8 ). As colimits in Func (Φ , M ) are calculated levelwise, it follows that colimits over α pr e s erve the condition for in- clusion in Func (Φ / Φ 0 , M ) which is that the diag ram take v alues ∗ lev - elwise over Φ 0 . Note that this prop erty says that co nnected colimits in Func (Φ / Φ 0 , M ) a re ca lculated levelwise. That wouldn’t b e true, how- ever, for disconnected colimits such as disjoint sums. W e no w ident ify th e κ - pr esentable ob jects of Func ( Φ / Φ 0 , M ). If ea ch A ( x ) is a κ -pres ent able ob ject of M , then by Lemma 8 .1.3 A is κ - presentable in Func (Φ , M ). If we are given a κ -ﬁltered system {B i } i ∈ β in Func (Φ / Φ 0 , M ), any map A → colim Func (Φ / Φ 0 M ) i ∈ β B i is also a map to co lim Func (Φ , M ) i ∈ β B i by the closure under co nnected colimits; hence it factor s throug h one o f the B i . This factorization is a mor phis m in the full sub categ ory Func (Φ / Φ 0 , M ), which shows that A is κ -presentable in Func (Φ / Φ 0 , M ). Suppo se o n the other ha nd that A is κ -pr esentable in Func (Φ / Φ 0 , M ). Given our assumption that M is lo ca lly κ -presentable and | Φ | < κ , the category Func ( Φ , M ) is lo cally κ -pres ent able, a nd its κ -presentable ob- jects are exactly the diagra ms B such that B ( x ) is κ -presentable in M . This was s tated as Lemma 8.1.3 with r eference to [2]. In pa rticular, we can express A a s a co limit in the category Func (Φ , M ) A = colim i ∈ β B i with B i ( x ) b eing κ -prese ntable, and indexed by a κ -ﬁlter ed categor y β . The unitaliz a tion functor U Φ 0 , ! being a left adjoint, w e get A = colim i ∈ β U Φ 0 , ! ( B i ) in Func (Φ / Φ 0 M ) . (13.4.1) The hypothesis that A is κ -pres ent able in Func (Φ / Φ 0 , M ), a pplied to the identit y map of A , says that the identit y fa ctors thro ugh a map A → U Φ 0 , ! ( B i ). On the o ther ha nd, the explicit des cription of U Φ 0 , ! shows that each U Φ 0 , ! ( B i )( x ) is κ -presentable. W e hav e a retra c tion A ( x ) → U Φ 0 , ! ( B i )( x ) → A ( x ) the comp osition b eing the identit y of A ( x ). It easily follows that A ( x ) is κ -presentable in M . This completes the pr o of of the ident iﬁcation of κ -presentable ob jects of Func (Φ / Φ 0 M ). It is clear fro m this des cription that the κ - presentable o b jects form a small set. The ab ov e argument shows that any A ∈ Func (Φ / Φ 0 M ) 13.4 U nitality 271 is a κ - ﬁltered colimit o f κ - presentable ob jects, indee d we obta ined the expression (13.4.1). W e can construct the injectiv e mo del category structure. Prop ositio n 13. 4 .2 Supp ose M is t ra ctable. Ther e exists a tr actable inje ct ive mo del c ate go ry struct u r e Func inj (Φ / Φ 0 , M ) wher e t he we ak e quiva lenc es ar e levelwise we ak e quivale nc es, and c oﬁbr ations ar e lev- elwise c oﬁbr ations. If M is left pr op er then so is the mo del c ate gory Func inj (Φ / Φ 0 , M ) . Pr o of W eak equiv alences a re clearly clo sed under retra cts and satisfy 3 for 2. The class of trivial coﬁbrations, that is the in tersection of the classes o f co ﬁbrations and weak eq uiv alences, is closed under pushout and transﬁnite composition since these colimits are calculated le velwise. The sets of injective coﬁbratio ns and injective trivial co ﬁbrations hav e generating sets, as w as shown in T he o rem 8.9.3 using Lur ie’s technique of Theorems 8 .9.1 and 8.9.2. This gives the necess ary a ccessibility argument which allows to apply Smith’s r ecognition lemma to obtain the mo del structure, such as describ ed in [1 6]. If M is left pr op er, the colimits inv olv ed in this condition are connected so they a re computed lev elwise, hence the s ame condition ho lds for Func inj (Φ / Φ 0 , M ). And the pro jective structure. Prop ositio n 13.4.3 If Φ 0 ⊂ Φ and M is a tr actable left pr op er mo d el c ate gory, we get a pr oje ctive mo del structu r e Func pro j (Φ / Φ 0 , M ) which is a tr actable left pr op er mo del c ate gory. F urthermor e the identity on t he underlying c ate gory c onstitutes a left Quil len functor Func pro j (Φ / Φ 0 , M ) → Func inj (Φ / Φ 0 , M ) , and the un italization c onstruction is a left Quil len funct or Func pro j (Φ , M ) U Φ 0 , ! → Func pro j (Φ / Φ 0 , M ) . Pr o of The weak equiv alences in Func pro j (Φ / Φ 0 , M ) ar e deﬁned to be the levelwise weak equiv alences. These satisfy 3 for 2 and a re clos e d under retracts. The ﬁbra tio ns a re deﬁned to b e the levelwise ﬁbrations. The trivial ﬁbrations ar e the intersection o f these c la sses. The co ﬁbra- tions are determined b y the left lifting prop erty w ith resp ect to trivia l ﬁbrations. W e construc t explicitly a generating set, b y a small v ariant of B o usﬁeld’s original construction. Cho ose a ge ne r ating se t I for the co ﬁbrations of M . It leads to the 272 Alge br aic the ories in mo del c ate gorie s set I Φ of generator s for coﬁbrations in the pro jective mo del struc tur e Func pro j (Φ , M ) discussed in Theorem 9.8 .1. Recall that I Φ consists of all mo rphisms of the form i x, ! ( f ) wher e f : A → B is in I , where i x : { x } → Φ is the inclusion of a discrete single ob ject and where i x, ! : M = Func ( { x } , M ) → Func (Φ , M ) is the corres po nding left adjoint functor. This is just Bousﬁeld’s cla ssic generating set fo r pr o jective diagram coﬁbrations. Set I Φ / Φ 0 := U Φ 0 , ! ( I Φ ) . Notice tha t U Φ 0 , ! i x, ! : M = Fun c ( { x } , M ) → Func (Φ / Φ 0 , M ) is the left adjo int functor for inducing unital diagr ams from o b jects of M placed over x ∈ Ob(Φ). The set I Φ / Φ 0 consists o f all U Φ 0 , ! i x, ! ( f ) where f runs through the set I of generating coﬁbra tions for M and x runs through Ob(Φ). F or x a nd f : A → B ﬁxed, U Φ 0 , ! i x, ! ( f ) : U Φ 0 , ! i x, ! ( A ) → U Φ 0 , ! i x, ! ( B ) has the following explicit de s cription. F or an y o b ject y ∈ Φ, let Φ nf ( x, y ) denote the set of arrows fro m x to y which don’t factor through a n ob jects o f Φ 0 , and let Φ f ( x, y ) denote the set of arr ows whic h factor through a n element of Φ 0 . Thus Φ( x, y ) = Φ nf ( x, y ) ⊔ Φ f ( x, y ). Then, U Φ 0 , ! i x, ! ( A )( y ) = a Φ nf ( x,y ) A, if Φ f ( x, y ) = ∅ ; U Φ 0 , ! i x, ! ( A )( y ) = ∗ ⊔ a Φ nf ( x,y ) A, if Φ f ( x, y ) 6 = ∅ . F or an arrow y → z , comp osition induces Φ f ( x, y ) → Φ ( x, z ) but only Φ nf ( x, y ) → Φ nf ( x, y ) ⊔ Φ f ( x, y ). The mo rphisms of functoriality for the diagram U Φ 0 , ! i x, ! ( A ) are either the identit y on A or on ∗ , or else the pro jection A → ∗ in the case o f a n a rrow in Φ nf ( x, y ) which comp os e s with y → z to give a n a rrow in Φ f ( x, z ). Note that if u ∈ Φ 0 then Φ nf ( x, u ) = ∅ and U Φ 0 , ! i x, ! ( A )( u ) = ∗ so the above for m ula deﬁnes a unital diag ram. O ne can chec k by ha nd that the explicit construction describ ed ab ove is a djoint to the functor Func (Φ / Φ 0 , M ) → M o f ev a luation at x , which s erves to show that the explicit c o nstruction is indeed U Φ 0 , ! i x, ! . 13.4 U nitality 273 The same des cription ho lds for U Φ 0 , ! i x, ! ( B ) and the map U Φ 0 , ! i x, ! ( f ) is o btained by applying either f or 1 ∗ on the v arious factor s . A few things a re immediate fro m this description: (1) if f is any coﬁbration in M then U Φ 0 , ! i x, ! ( f ) is a n injectiv e i.e. levelwise coﬁbration in Func (Φ / Φ 0 ; M ), indeed the U Φ 0 , ! i x, ! ( f )( y ) are disjoint unions of copies of f and of the isomo rphism 1 ∗ ; and (2) if f is a trivia l coﬁbr a tion in M then U Φ 0 , ! i x, ! ( f ) is an injectiv e i.e. levelwise trivial coﬁbration in Func (Φ / Φ 0 ; M ), for the same rea son. On the other ha nd, the adjunction formula says that U Φ 0 , ! i x, ! is left adjoint to the r estriction i ∗ x : Func (Φ / Φ 0 , M ) → M i.e. the ev aluation at x ∈ Φ. Hence, a morphism g of diagrams in Func (Φ / Φ 0 , M ) sa tis ﬁe s right lifting with r esp ect to U Φ 0 , ! i x, ! ( f ), if and only if i ∗ x ( g ) = g ( x ) satisﬁes the r ight lifting prop erty with resp ect to f . The previous par agraph implies that inj ( I Φ / Φ 0 ) is equa l to the cla ss of levelwise trivial ﬁbrations, he nce cof ( I Φ / Φ 0 ) is the class of coﬁbrations. Thu s I Φ / Φ 0 is a set of generator s for the coﬁbrations. If we had started with a set J ge nerating the triv ia l coﬁbrations of M , then deﬁning J Φ / Φ 0 to be the set o f a ll U Φ 0 , ! i x, ! ( f ) for f ∈ J , gives b y the sa me a rgument inj ( J Φ / Φ 0 ) equal to the class of lev elwise ﬁbrations. W e claim that cof ( J Φ / Φ 0 ) is then equal to the class of triv ial coﬁbra- tions. By prop erty (2) ab ov e, J Φ / Φ 0 and hence cof ( J Φ / Φ 0 ) consist of levelwise w eak equiv a lences, so they are contained in the cla ss of trivial coﬁbrations. Suppo se g : R → S is a trivial coﬁbration. By the s ma ll ob ject ar gument it can b e factored as g = p h wher e p ∈ inj ( J Φ / Φ 0 ) and h ∈ cell ( J Φ / Φ 0 ). In pa r ticular p is an levelwise ﬁbration, but it is also an levelwise w eak equiv alence by 3 for 2, so it is a trivial ﬁbration hence satisﬁes lifting with resp ect to coﬁbratio ns . As g is assumed to be a coﬁbr ation, there is a lifting which shows g to be a retract o f h . Thu s g ∈ cof ( J Φ / Φ 0 ). W e hav e now s hown conditions (CG1)–(CG3b) for I and J with resp ect to the given three classes of morphisms, and we know that weak equiv a lences are closed under retracts and satisfy 3 for 2 . These give a co ﬁbrantly generated mo del structure (see Prop osi- tion 9.2.1 ). It is tractable since the elements o f the g enerating sets hav e coﬁbrant domains. Left prop erness is chec k ed levelwise. F or the statements ab out left Quillen functor s, note that b oth functors in question are left a djoin ts. F urthermore, they pre s erve coﬁbratio ns and trivial coﬁbrations , indeed by (1) and (2) a bove the genera ting co ﬁbra- tions of Func pro j (Φ / Φ 0 , M ) are also injective co ﬁbrations; and b y our 274 Alge br aic the ories in mo del c ate gorie s construction the gener ating sets I Φ and J Φ for coﬁbrations and trivia l coﬁbrations in Func pro j (Φ , M ) ar e mapp ed by U Φ 0 , ! to the gener ating sets for coﬁbrations and trivial coﬁbr a tions in Fun c pro j (Φ / Φ 0 , M ). Remark 13.4. 4 Unfortun ately U ! is not n e c essarily a left Quil len functor b etwe en the inje ctive structu r es. In the next chapter when o ur discussion is applied to the sp ecial ca se ∆ o X /X w e can imp ose an additional co ndition 14.2 .2 o n M so that it works, o r alternatively use the Ree dy str ucture on the categor y of diagrams. 13.5 Unital algebraic diagram theories Combine the previous discuss ions: s uppos e Φ is a small catego r y , Φ 0 ⊂ Φ is a full subc a tegory , Q is a small set, we hav e int egers n ( q ) ≥ 0 for q ∈ Q , and supp ose given functors P q : ǫ ( n ( q )) → Φ for q ∈ Q . Suppo se that M is a tracta ble left prop er car tes ian mo del categor y with g enerating sets I and J . W e o btain left Quillen functors (the leftmost v arying in a family index ed by q ∈ Q ): Func pro j ( ǫ ( n ( q )) , M ) P q, ! → Func pro j (Φ , M ) Func pro j (Φ / Φ 0 , M ) U Φ 0 , ! ↓ 1 → Func inj (Φ / Φ 0 , M ) . By the prop erty of transfer of families of direct lo ca liz ing systems along Quillen f unctors (Theorem 11.8.3), we obtain left Bousﬁeld loca liz ations of Func pro j (Φ / Φ 0 , M ) and Func inj (Φ / Φ 0 , M ) along the images of the ζ ǫ ( n ( q ) ) ( f ) for f ∈ I . Denote these respectively by A lg pro j (Φ / Φ 0 , P · ; M ) and A lg inj (Φ / Φ 0 , P · ; M ). They are tra ctable left prop er mo del ca t- egories whose underlying categor ie s are the unital diagr am c ategory Func (Φ / Φ 0 , M ). In the pr o jectiv e structure, the co ﬁbrations ar e gener- ated by U Φ 0 , ! ( I Φ ) whereas in the injectiv e structure the coﬁbra tio ns are the levelwise coﬁbratio ns. The ﬁbr ant ob jects in the pro jective structure are the levelwise ﬁbrant ob jects whose pullback to each ǫ ( n ( q )) satisﬁes the pro duct co ndition. In the pro jective case this c ompares with the non-unita l a lgebraic 13.6 The gener ation op er ation 275 diagram theories by a Quillen adjunction U Φ 0 , ! : A lg pro j (Φ , P · ; M ) ← → Alg pro j (Φ / Φ 0 , P · ; M ) : U ∗ Φ 0 . Indeed, the dir ect left Bousﬁeld lo ca lization of the unita l theory ca n b e seen as coming from the lo calization of the non-unital theor y g iven in Theorem 13.3.2 , by tra nsfer along the left Quillen functor U Φ 0 , ! , and in the situatio n of Theo rem 11.8.1 we still get a left Quillen functor. Remark 13.5 .1 If furt hermor e we know that U Φ 0 , ! gives a left Qu il len functor on the inje ctive diagr am st ructur es, then this c omple tes to a Quil len adjunction U Φ 0 , ! : A lg inj (Φ , P · ; M ) ← → A lg inj (Φ / Φ 0 , P · ; M ) : U ∗ Φ 0 . Lemma 13.5.2 S u pp ose r : A → A ′ is a trivial c oﬁbr ation towar d s a ﬁbr ant obje ct , in either of t he pr oje ctive or inje ctive mo del structu r es on Alg (Φ / Φ 0 , P · ; M ) . If A satisﬁes the pr o duct c onditio n, then r is levelwise a we ak e quivalenc e, that is r ( x ) : A ( x ) → A ′ ( x ) is a we ak e quiva lenc e in M for any x ∈ Φ . Pr o of This follows from Cor ollary 11.4 .5 , noting that the mo del struc- tures on Alg (Φ / Φ 0 , P · ; M ) are obtained from dire ct left Bousﬁeld lo - calizing s ystems with R b eing the class of ob jects satisfying the pro duct condition. 13.6 The generation op eration Suppo se Φ is a small category , Φ 0 ⊂ Φ is a full sub ca teg ory , Q is a small set, we hav e integers n ( q ) ≥ 0 for q ∈ Q , and supp ose given functor s P q : ǫ ( n ( q )) → Φ for q ∈ Q . Suppo se that M is a tractable left pro p e r mo del ca tegory with generating sets I and J . make the fo llowing assumption: (INJ)—the functors U Φ 0 , ! P q, ! send coﬁbra tio ns (resp. triv ial co ﬁbrations) in Func inj ( ǫ ( n q ) , M ) to levelwise co ﬁbrations (r esp. le velwise triv ial coﬁbrations). In o ther words we hav e left Quillen functors Func inj ( ǫ ( n q ) , M ) U Φ 0 , ! P q, ! → diag inj (Φ / Φ 0 , M ) . In this case we can use the g e nerating set for the new model str ucture on Func inj ( ǫ ( n q ) , M ) ma de fro m the simpler coﬁbrations ρ ( f ). 276 Alge br aic the ories in mo del c ate gorie s Suppo se A ∈ Func (Φ / Φ 0 , M ) and q ∈ Q . Let n := n q . Deﬁne a trivial co ﬁbration A → Gen ( A ; q ) as follows: choose a factorization P ∗ q ( A )( ξ 0 ) e 0 → E 0 p → P ∗ q ( A )( ξ 1 ) × · · · × P ∗ q ( A )( ξ n ) with e 0 a coﬁbration and p a weak equiv alence, in M . This gives a trivia l coﬁbration P ∗ q ( A ) e → E in Func inj , Π ( ǫ ( n ) , M ). Using condition (INJ) we obtain a co ﬁbration A → Gen ( A ; q ) := A ∪ U Φ 0 , ! P q, ! ( A ) U Φ 0 , ! P q, ! ( E ) (13.6.1) in the injective mo de l structure Func inj (Φ / Φ 0 , M ). Note tha t Gen ( A , q ) do es n’t depend, up to equiv alence, on the choice of factor iz a tion E . If necessary we can include the factorization in the notation Gen ( A , q ; e 0 , p ). The weak monadic pro jection from Func (Φ / Φ 0 , M ) to the cla ss of ob jects satisfying the pro duct condition, may b e thought of as a tra ns- ﬁnite iteration of the o p e ration A 7→ Gen ( A , q ) ov er all q ∈ Q . 13.7 Reedy structures In the main s ituation where the theory of this chapter will be applied, the underlying category Φ is a Re e dy c ate gory , and we ca n giv e the category of diagra ms Φ → M the Reedy mo del categor y structure denoted by Func Reedy (Φ , M ). Assume that Φ 0 consists of o b jects of the bo ttom degree in the Reedy structure. Then ther e is a cor resp onding mo del structure denoted Func Reedy (Φ / Φ 0 , M ) on th e categor y of unital diagrams, suc h that ( U ! , U ∗ ) remains a Q uillen adjunction. The Reedy str uctures Func Reedy (Φ , M ) a nd Func Reedy (Φ / Φ 0 , M ) can ag ain by lo calize d by dire c t left B ousﬁeld localizatio n, to giv e m o del categorie s denoted Al g Reedy (Φ , P · ; M ) and A lg Reedy (Φ / Φ 0 , P · ; M ). These ﬁt in b etw een the pro jective and the injective structures ab ov e. 14 W eak eq ui v alences This chapter co ntin ues the study of weakly enriched ca tegories using Segal’s m etho d. W e use the mo del category for algebra ic theories , devel- opp ed in the previo us chapter, to get mo del structures for Seg al precat- egories o n a ﬁxed set of o b jects. This structure will be studied in deta il later, to deal with the passa ge fro m a Segal pr e category to the Sega l category it g enerates. Then we consider the full category of Sega l preca tegories, with mov- able sets of ob jects, giving v arious deﬁnitions and notations. Co nstruct- ing a mo de l structure in this case is the main sub ject of the subsequent chapters. The reader will no te that this div is ion of the glo bal ar gument int o t wo pieces, was present already in Dwyer-Kan’s treatment of the mo del category fo r simplicial categ ories. They dis cussed the mo del categ ory for s implicia l categories on a ﬁxe d s et o f o b jects in a series of pap ers [89] [90] [91]; but it wasn’t until some time later with their unpublished manuscript with Hirschhorn [87], which subsequently b eca me [8 8], and then Bergner’s pa per [33] that the global case was treated. F or the theory of weak enrichmen t fo llowing Segal’s metho d, the co r- resp onding division and introductio n of the notion o f left Bousﬁeld lo- calization for the ﬁr st part, was suggested in Barwick’s thesis [14]. Assume throughout that M is a tracta ble left pr op er cartesian model category . See Chapter 10 for a n explanatio n and ﬁrs t consequenc e s of the ca rtesian condition. This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 278 We ak e quivale nc es 14.1 The mo del structures on PC ( X , M ) The Segal conditions (Section 12.2) fo r M -precateg ories can b e ex- pressed in ter ms of the alg ebraic diagram theor y of the pr evious chapter, which w as the motiv ation for introducing that notion. Let Φ := ∆ o X , a nd let Φ 0 = disc ( X ) b e the discrete subca tegory on ob ject set X , co nsidered as a sub category by letting x ∈ X co rresp ond to the sequence ( x ). An M -precategor y A ∈ PC ( X, M ) is by deﬁnit ion the s ame thing as a functor Φ → M sending the o b jects o f Φ 0 to ∗ , which is to say PC ( X, M ) = Func (∆ o X /X , M ) . The Segal conditions a r e a collection of ﬁnite pro duct conditions a s was considere d in the previous chapter. The set of product conditions Q consists of the full set of ob jects of ∆ o X . F o r q = ( x 0 , . . . , x n ) the int eger n ( q ) is equal to n , and we deﬁne a functor P ( x 0 ,...,x n ) : ǫ ( n ) → ∆ o X by P ( x 0 ,...,x n ) ( ξ 0 ) := ( x 0 , . . . , x n ) , P ( x 0 ,...,x n ) ( ξ i ) := ( x i − 1 , x i ) for 1 ≤ i ≤ n. The images o f the pro jectio n maps in ǫ ( n ) are the opp osites of the in- clusion maps ( x i − 1 , x i ) ֒ → ( x 0 , . . . , x n ) in ∆ X . An M -precategor y is an M -enr iched Seg al category , if and o nly if it satisﬁes the pro duct condition with resp ect to the c o llection of functors P · , indeed the tw o conditions are identically the same. Unitality gives the pro duct condi- tion whenever n = 0, where as the pro duct condition is a utomatically true whenever n = 1 b ecause the Sega l maps ar e the identit y in this case. Thus, this condition needs only to be imp os ed for n ≥ 2. W e obtain adjoint functors P ( x 0 ,...,x n ) , ! : Func ( ǫ ( n ) , M ) ← → Func (∆ o X , M ) : P ∗ ( x 0 ,...,x n ) and the unita l versions U ! P ( x 0 ,...,x n ) , ! : Func ( ǫ ( n ) , M ) ← → PC ( X, M ) : U ! P ∗ ( x 0 ,...,x n ) where the right adjoint is just the pullback of a diagra m A : ∆ o X → M to the category ǫ ( n ). An M -pr ecategor y A satisﬁes the Segal co nditions, if and only if U ! P ∗ ( x 0 ,...,x n ) ( A ) is a pro duct-compa tible diag ram ǫ ( n ) → M , for each sequence ( x 0 , . . . , x n ). 14.1 The mo del structu r es on PC ( X , M ) 2 79 A dir ect applicatio n of t he construction o f Cha pter 13 gives tw o mo del structures (pro jective and injectiv e) on P C ( X , M ) such that the ﬁbrant ob jects satisfy the Se g al conditio n. W e add a third R e e d y structure since ∆ o X is a Reedy categor y . The consider ation of these mo del s tr uctures is in termediate with re- sp ect to our main goa l of constr ucting global mo del structures on PC ( M ): the maps in PC ( X, M ) ar e o nes whic h induce the identit y on the set of o b jects X . Nonetheless, these easier mo del structures on PC ( X , M ) will b e very useful in num erous arguments later. The idea of introducing the in termediate model categor y PC ( X , M ), and of ex pressing it as a left Bousﬁeld lo calizatio n, is due to Ba rwick [14]. Start by dis cussing the case o f non-unital dia grams. If we ﬁx gen- erating sets ( I , J ) for M , then w e o btain generating s ets for the pro- jective mo del structure Func pro j (∆ o X , M ) and injective mo del struc- ture Func inj (∆ o X , M ), recalle d in Theorem 9.8 .1 . Notice that if M is tractable and I and J consist o f a r rows with coﬁbrant domains, the ex- plicit g e nerators for the pro jective mo del structure also ha ve coﬁbrant domains. Thus Fun c pro j (∆ o X , M ) will aga in be tra ctable. F or the in- jective mo del structure the construction of gener ating sets of Barwick and Lurie [16] [153] (as discussed in Theorem 8.9 .2 a bove) was compli- cated, and it doe s n’t seem clear w hether we can cho ose g e nerators with coﬁbrant domains. T his problem c an b e bypassed later with the Reedy mo del str uctures where again the generato rs b ecome explicit. Within the pro jective or injectiv e mo del categor ie s of no n-unital di- agrams , we o btain a dire ct left Bousﬁeld lo calizing s ystem ( R nu , K nu ) where R is the class of non-unital diagra ms s atisfying the Segal condi- tions, and K nu is given by the genera ting trivial coﬁbrations plus the maps of the form P ( x 0 ,...,x n ) , ! ( ζ n ( f )) for f in the generating set I o f coﬁbrations of M . Her e ζ n ( f ) is the diagr am ǫ ( n ) → M considere d in Section 13.2.1 . This yields the direct left Bous ﬁe ld lo calized mo del struc- tures which were designated by the no tation Alg ( . . . ) in the previous chapter. Denote these mo del catego ries of weakly unital precategor ies now b y Alg pro j (∆ o X , P · ; M ) , Alg inj (∆ o X , P · ; M ) . The underlying ca tegories are b oth the sa me Func (∆ o X , M ). The ﬁbra n t ob jects are dia g rams which a r e ﬁbrant in the pr o jectiv e or injective mo del str uctures for diagrams, and which are Segal categories . The same will work for for M -precategor ies where the unitality con- 280 We ak e quivale nc es dition is imp osed. The mo del structure on M -pr ecategor ies fo r a ﬁxed set of ob jects is given by the following theo rem. Note th e in tro duction o f the notation A → Seg ( A ) fo r a choice of ﬁbra n t replacement in either of the mo del catego ries. This notation w ill be us ed later, but can mean that a c hoice is ma de at each usag e, rather than ﬁxing a global choice once and for all. Most constr uctions will b e indep endent of the choice, up to e q uiv a lence. Theorem 14 .1.1 Sup ose M is a tr actable left pr op er c artesia n m o del c ate gory, then ther e ar e left pr op er c ombina torial mo del c ate gory struc- tur es on t he un ital M -pr e c ate gories PC pro j ( X ; M ) := Al g pro j (∆ o X /X , P · ; M ) , PC inj ( X ; M ) := Al g inj (∆ o X /X , P · ; M ) . The ﬁ br ant obje cts ar e u nital ﬁbr ant diagr ams which satisfy t he Se gal c ondi tion. The c oﬁbr ations ar e the pr oje ctive or inje ctive c oﬁbr atio ns in the unital diagr am c ate gory Func (∆ o X /X ; M ) . The we ak e quivalenc es ar e the same in b oth st ructur es. L et A → Seg ( A ) deno te a trivial c oﬁbr ation towar ds a ﬁbr ant re plac e- ment of A in the pr oje ctive mo del st ructur e PC pro j ( X ; M ) ; this c an b e chosen functorial ly. A map A → B is a we ak e quivalenc e if and only if Seg ( A ) → Seg ( B ) is a levelwise we ak e quivalenc e when c onsider e d as a map of diagr ams ∆ o X → M . Pr o of Prop os itions 13 .4.3 and 13 .4.2 give pro jective and injective di- agram model structures Func pro j (∆ o X /X , M ) and Func inj (∆ o X /X , M ) on the categ ory of unital diag r ams, which is the same under lying cate- gory as PC ( X, M ). W e get direct left Bo usﬁeld lo calizing systems ( R , K pro j ) and ( R , K inj ) for these mo del structures , by transfer ing the dir ect lo calizing s ystems for ǫ ( n )-diag rams of Theo rem 13 .2.2, as w as discus s ed in Section 13 .5 using Theorem 11.8.3. In bo th ca ses R is the class of M -precatego ries which sa tisfy the Segal conditions; then K pro j (resp. K inj ) is the union o f the set of g enerators for trivial coﬁbrations in the pro jective (resp. injective) diagram structure, plus the morphisms o f the for m U ! P ( x 0 ,...,x n ) , ! ( ζ n ( f )) for f in the gener- ating s e t I o f coﬁbra tions of M . Note that the images P U ( x 0 ,...,x n ) , ! ( g ) of generating trivial co ﬁbrations for the diag ram ca tegories on ǫ ( n ( q )) are already trivial coﬁbrations in Func (∆ o X /X , M ) so w e do n’t need to include them again. 14.2 U nitalization adjunctions 2 81 Now Theo r em 1 1.7.1 a pplies to give the requir ed mo del structures . The characterization of weak equiv alences co mes from Le mma 11 .4.5. 14.2 Unitalization adjunctions The pro jective model struc tur e of Theorem 14 .1.1 is related to the non- unital version b y a Quillen adjunction of unitalization U ! : A lg pro j (∆ o X , P · ; M ) ← → PC pro j ( X ; M ) : U ∗ . This fo llows from the application of Theorem 11.7.1. It is us eful to desc rib e the unitalization op eration for ∆ o X /X . Lemma 14.2 .1 Supp ose A : ∆ o X → M is a fu n ctor. Then U ! A has the fol lowing explicit description: —if x 0 = . . . = x n = x is a c onstant se quenc e then the ful l de gener ac y gives a map A ( x ) → A ( x, . . . , x ) and ( U ! A )( x, . . . , x ) = A ( x, . . . , x ) ∪ A ( x ) ∗ ; —otherwise, if x 0 , . . . , x n is n ot a c onstant se quenc e then ( U ! A )( x 0 , . . . , x n ) = A ( x 0 , . . . , x n ) . Pr o of The o b ject explicitly co ns tructed in this wa y is aga in a functor ∆ o X → M b e cause if ( x 0 , . . . , x n ) is a co nstant seq ue nce and ( y 0 , . . . , y k ) → ( x 0 , . . . , x n ) is any map in ∆ X then y · m ust als o b e a co nstant sequence, so the map A ( x 0 , . . . , x n ) → A ( y 0 , . . . , y k ) pas ses to the quo tien ts. The res ulting functor satisﬁe s the required left adjunction prop erty with resp ect to U ∗ on the right, so it m ust b e U ! A . In the injective case, U ! will not in genera l b e a le ft Quillen functor; we need to impo s e a n additional hypothesis. Condition 14. 2.2 (INJ) Supp ose A → X → A B ↓ → Y ↓ → B ↓ is a diagr am in M such that the vertic a l arr ows ar e c oﬁbr ations (r esp. trivial c oﬁbr atio ns) and t he horizontal c omp ositions ar e the identity. Then X ∪ A ∗ → Y ∪ B ∗ is a c oﬁbr a tion (r esp. trivial c oﬁbr ation). 282 We ak e quivale nc es The fo llowing observ ations allow us to us e the injective mo del cate- gories in many cases. In fact, in these cas es the injective structure also coincides with the Reedy structure, how ev er it seems co mfor ting to b e able to use the injectiv e structure which is conceptually simpler , instead. Lemma 14. 2.3 Supp ose M is a pr eshe af c ate gory, is left pr op er, and the class of c oﬁbr ations is the class of monomorphisms of pr eshe av es. Then Condition 14.2.2 holds. Pr o of Note ﬁrst that given a diagr am of sets as in Condition 14.2.2 where the vertical maps ar e injections, then the map X ∪ A ∗ → Y ∪ B ∗ is an injection o f sets. Indeed if x ∈ X maps to a n element of B then the image of x by the pro jection to A , ma ps to the same element of B . By injectivity of the map X → Y it follows that x ∈ A , which shows injectivit y of the map on quotients. Now if M is a pr e s heaf ca tegory and the coﬁbra tions are the monomor- phisms, a pplying the previous paragraph levelwise we obtain the desired result for coﬁbrations. Supp ose given a diag ram who se vertical arr ows are trivia l coﬁbrations. The split injections A → X a nd B → Y ar e monomorphisms, hence co ﬁbr ations, so Cor ollary 9.5.2 applies to con- clude that the pushout map is a w eak equiv alence, hence it is a trivial coﬁbration. Lemma 14.2.4 If M satisﬁes Condition 14.2.2 then the unitalization functors give a Quil len adjunction b etwe en inje ctive diagr am st ru ctur es U ! : Alg inj (∆ o X , P · ; M ) ← → PC inj ( X ; M ) : U ∗ wher e U ∗ is just the identity inclusion of unital pr e c ate gories in al l pr e- c ate gories. Pr o of W e ﬁrst show that unitalization is a Quillen adjunction b etw een levelwise injectiv e diagr a m catego ries U ! : Func inj (∆ o X , M ) ← → Func inj (∆ o X /X , M ) : U ∗ . Suppo se A → B is a levelwise coﬁbra tion (r esp. trivial coﬁbration) of diagrams, the claim is that U ! A → U ! B is a levelwise coﬁbratio n (re s p. trivial co ﬁbration). In view of the description of Lemma 14 .2.1 it suﬃces to lo ok at the v a lues ov er a constant seq ue nce ( x, . . . , x ). W e hav e a 14.3 The Re e dy structur e 283 diagram A ( x ) → A ( x, . . . , x ) → A ( x ) B ( x ) ↓ → B ( x, . . . , x ) ↓ → B ( x ) ↓ where the vertical arr ows ar e co ﬁbrations (resp. trivial coﬁbr ations), and the s econd horizontal ar rows are , s ay , the pro jections corre s po nding to the ﬁrs t ob ject of the se q uence. Co ndition 1 4.2.2 now says exactly that U ! A ( x, . . . , x ) = A ( x, . . . , x ) ∪ A ( x ) ∗ U ! B ( x, . . . , x ) = B ( x, . . . , x ) ∪ B ( x ) ∗ ↓ is a coﬁbr ation (r esp. trivia l coﬁbration). This shows the Quillen a djunc- tion fo r the dia g ram ca tegories. Now, the categorie s in question for the lemma ar e obtained by direct left Bousﬁeld lo caliza tion us ing the sets of genera tors plus the morphisms of the form P q, ! ( ζ n ( q ) ( f )) for q ∈ Q and f in a generating set for co ﬁ- brations of M . The left Quille n functor passes to a left Q uillen functor betw een lo calizations by Theore m 11.8.1 . 14.3 The Reedy structure The ca tegory ∆ o X is a Re e dy ca tegory , using the sub ca teg ories of in- jective and surjective ma ps o f ﬁnite ordered sets, as direc t and inv erse sub c ategories , a nd the length function ( x 0 , . . . , x n ) 7→ n . This leads to a R e e dy mo del structur e on the category o f diag rams, us ing lev elwise w eak equiv alences, denoted Func Reedy (∆ o X , M ) (Pr o p o sition 9.8.3). The notion of Reedy structure on diagra m categ ories is a sligh tly mo re techn ical area of the theory of mo del categories, but these structur es are very natural and turn out to be the best ones for our theory of preca te- gories. In ma ny us e ful examples the Reedy s tr ucture coincides with the injectiv e structure. This is the case for example if M = Pres h(Ψ) is a presheaf catego ry and the coﬁbr ations of M are the mo nomorphisms of presheav es, see Prop os ition 15.7 .2. 284 We ak e quivale nc es There is a unital version of the Reedy mo del structure. Prop ositio n 1 4.3.1 F or any tr actable left pr op er mo del c ate gory M , the u nital diagr am c ate go ry has a tr actable left pr op er mo del stru ct ur e denote d Func Reedy (∆ o X /X , M ) , r elate d t o t he non-u nital R e e dy struc- tur e by a Q uil len adjunction U ! : Func Reedy (∆ o X , M ) ← → Func Reedy (∆ o X /X , M ) : U ∗ using the levelwise we ak e quiva lenc es. The ﬁ br ations (r esp. c oﬁbr ations) of Fun c Reedy (∆ o X /X , M ) ar e exactly the maps f such that U ∗ ( f ) is a R e e dy ﬁbr ation ( re sp. c oﬁbr ation) in Func Reedy (∆ o X , M ) , in p articular they have the same description in terms of latching and matching ob- je ct s. The gener ating sets for t he un ital R e e dy structur e ar e obtaine d by applyi ng U ! to the gener ating sets for the r e gular R e e dy diagr am stru c- tur e. The R e e dy structur e lies in b etwe en the pr oj e ctive and inje ctive struc- tur es with a diagr am of left Qu il len functors Func pro j (∆ o X , M ) → Func Reedy (∆ o X , M ) → Func inj (∆ o X , M ) Func pro j (∆ o X /X , M ) U ! ↓ → Func Reedy (∆ o X /X , M ) U ! ↓ → Func inj (∆ o X /X , M ) wher e the horizontal r ows ar e identity functors. If M satisﬁes Condition 14.2.2 then this c an b e c omple te d by putting in the rightmost vertic al arr ow . Pr o of The ﬁrs t thing to sho w is that if A f → B is a Reedy c o ﬁbration of diagrams on ∆ o X , then U ∗ U ! A U ∗ U ! f → U ∗ U ! B is again a Ree dy coﬁbra tion. The latching ob jects ov er a seq uenc e ( x 0 , . . . , x n ) ∈ ∆ o X inv olv e maps which corresp ond to sur jections ( x 0 , . . . , x n ) σ → ( y 0 , . . . , y k ) of seq uences, with σ : [ n ] ։ [ k ] with y σ ( i ) = x i . In particular, any ( y 0 , . . . , y k ) inv olved in the latching map at a non-constant sequence ( x 0 , . . . , x n ), is also not constant. Over these sequences the relative latching map for U ∗ U ! f is the same as for f so the Reedy co ndition is preser ved. W e may therefor e concentrate on the case o f a co nstant sequence. L e t x n := ( x 0 , . . . , x n ) with x i = x . L e t A ′ := U ∗ U ! A , B ′ := U ∗ U ! B , and f ′ := U ∗ U ! f . Thus A ′ ( x n ) = A ( x n ) ∪ A ( x ) ∗ , B ′ ( x n ) = B ( x n ) ∪ B ( x ) ∗ . 14.3 The Re e dy structur e 285 The la tc hing ob ject for A is expressed a s the pus ho ut a 0 i j then h ([ k ]; B )( y 0 , . . . , y p ) = ∅ . The pul lb ack maps giving h ([ k ]; B ) a stru ct ur e of diagr am ar e al l either the unique maps of the form ∅ → B , ∅ → ∗ , or B → ∗ , or t he identity B → B . Pr o of The functor B 7→ h ([ k ]; B ) is by co ns truction left adjoint to the functor i { t k } ∗ from PC ([ k ] , M ) to M which se nds A to A ( υ 0 , . . . , υ k ). On the other hand we can ch eck by hand (as in the pro o f of the next lemma b elow) that the functor s e nding B to the precateg ory deﬁned explicitly in the statement o f the lemma, is als o adjoint to the same functor. Lemma 15. 2.2 If B ∈ M and C ∈ PC ( M ) t hen a morphism f : h ([ k ] , B ) → C is the same thing as a se quenc e of obje cts x 0 , . . . , x k ∈ Ob( C ) to gether with a map ϕ : B → C ( x 0 , . . . , x k ) in M . Pr o of Use the explicit description o f the pr e vious lemma. Given f w e get x i := f ( υ i ) and the map ϕ is given by f υ 0 ,...,υ k . On the other hand, given x · and ϕ , the r estrictions C ( x 0 , . . . , x k ) → C ( x i 0 , . . . , x i p ) lea d to maps B → C ( x i 0 , . . . , x i p ) for any s equence as in the ﬁrst par t of the previo us lemma; if i 0 = · · · = i p then this map factor s through ∗ by the unitality condition for C , tr e ating the second par t of the pr evious lemma; a nd nothing is needed for deﬁning a map in the third ca s e o f the previo us lemm a. This deﬁnes the require d map f . Note that the a bove construction is functorial in B , that is a map 15.2 S ome natu r al pr e c ate gories 307 A → B induces h ([ k ]; A ) → h ([ k ]; B ). Deﬁne the “b oundary” of h ([ k ]; B ) by the skeleton op eratio n: h ( ∂ [ k ]; B ) := s k k − 1 h ([ k ]; B ) , with the natural inclusio n h ( ∂ [ k ]; B ) → h ([ k ]; B ) . It is a lso functor ia l in B . Lemma 1 5.2.3 This b oundary obje ct has the fol lowing c oncr ete de- scription: —if ( y 0 , . . . , y p ) is incr e asing but not c onstant i.e. i j − 1 ≤ i j but i 0 < i p , and if ther e is any 0 ≤ m ≤ k such t hat i j 6 = m for al l 0 ≤ j ≤ k , then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = B ; —if ( y 0 , . . . , y p ) is c onstant i.e. i 0 = i 1 = . . . = i p then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = ∗ ; and otherwise, that is if either ther e exists 1 ≤ j ≤ p such that i j − 1 > i j or else if the map j 7→ y j is a surje ctio n fr om { 0 , . . . , p } to [ k ] , then h ( ∂ [ k ]; B )( y 0 , . . . , y p ) = ∅ . Pr o of Use the description of Lemma 15.2.1 and the fo rmula h ( ∂ [ k ]; B )( y · ) = colim y · ։ z · h ( B )( z · ) . If f : A → B is a co ﬁbration in M , put h ([ k ] , ∂ [ k ]; A f → B ) := h ([ k ]; A ) ∪ h ( ∂ [ k ]; A ) h ( ∂ [ k ]; B ) . W e therefor e obtain tw o natural maps coming from f , the ﬁrs t is P ([ k ]; f ) : h ([ k ]; A ) → h ([ k ]; B ) and the s econd is R ([ k ]; f ) : h ([ k ] , ∂ [ k ]; A f → B ) → h ([ k ]; B ) . The maps P ([ k ]; f ) will form the gener ators for the pr o jective coﬁbra- tions, while the R ([ k ]; f ) form the genera tors for the Reedy coﬁbra tions. Unfortunately we don’t hav e an easy way to describ e g enerator s for the injectiv e coﬁbrations. 308 Coﬁbr ations 15.3 Pro jective coﬁbrations The diﬀerent mo del structures are characterized and diﬀerent iated b y their notions of coﬁbrations. In the pro jective str uc tur e, the g enerating set is e asiest to describ e but on the other hand there is no easy criterion for being a coﬁbr a tion. A map f : A → B is a pr oje ctive c oﬁbr atio n if on the se t of ob jects it is an injective ma p o f sets Ob( f ) : O b( A ) → Ob( B ), and if the map Ob( f ) ! ( A ) → B is a co ﬁbration in the pro jective model catego ry str uc - ture PC pro j (Ob( B ) , M ). Re c a ll that Ob( f ) ! ( A ) is the precategor y A transp orted to the subset f (Ob( A )) ⊂ Ob( B ), then extended b y adding on the discrete or initial pr ecategory over the co mplemen tary s ubset Ob( B ) − Ob( f )(Ob( A )). Say that f is a pr oje ctive t rivial c oﬁbr atio n if it is a pro jective coﬁ- bration, a nd a global weak equiv a lence. F or now, we say that a morphism u : U → V in PC ( M ) is a pr o je ctive ﬁbr ation if it sa tisﬁes the right lifting pro p e rty with resp ect to a ll pr o- jective tr ivial co ﬁbrations; we say that u is a pr oje ctive trivial ﬁbr atio n if it is a pro jective ﬁbr ation and a glo bal w eak e q uiv a lence. F or the time b eing we als o ne e d a separate notation: say that u : U → V in PC ( M ) is an app ar ent pr oje ctive trivial ﬁbr atio n if it satisﬁes the right lifting prop erty with res pect to all pro jective coﬁbr ations. One of o ur tasks in future chapters will b e to identify the class of pr o jective trivial ﬁbrations with the apparent ones . F o r now we can characterize and us e the a pparent ones. Lemma 15.3.1 A morphism u : U → V in PC ( M ) is an app ar- ent pr oje ctive trivial ﬁ br atio n if and only if Ob( u ) is surje ctive and U → Ob( u ) ∗ ( V ) is an obje ctwise trivial ﬁbr atio n. A morphism f : A → B is a pr oj e ctive c oﬁbr ation if and only if it sat isﬁes the left lifting pr op erty with r esp e ct to the app ar ent pr oje ctive trivial ﬁbr ations. Pr o of Consider the class A of mo r phisms u such that Ob( u ) is s ur - jective and U → Ob( u ) ∗ ( V ) is an ob jectwise trivial ﬁbra tion. If f is a pro jective coﬁbration then it break s down as a c omp o sition f = f ′ ◦ d where f ′ : O b( f ) ! ( A ) → B is a mor phism in PC (O b( B ) , M ) and d is the extension by adding o n the discr ete precategor y Ob( B ) − Ob ( f )(Ob ( A )). Similarly , if u ∈ A then u = p ◦ u ′ where u ′ : U → Ob( u ) ∗ ( V ) is an ob ject wise trivia l ﬁbratio n in PC (Ob( U ) , M ) a nd p is the ta utological map Ob( u ) ∗ ( V ) → V . Notice that p sa tisﬁes the right lifting pro per ty with re spe ct to any morphism which induces an isomorphism o n sets of 15.3 Pr oje ctive c oﬁbr ations 309 ob jects, and a lso with respe c t to extensions b y adding discrete sets; and dually f ′ satisﬁes the left lifting pro per ty with r esp ect to tauto logical maps such as p , while d satisﬁes the left lifting prop erty with resp ect to any map surjective on ob jects. All told, the lifting prop erty with f on the left a nd u on the r ight, is equiv alen t to the lifting pro pe rty with f ′ on the left a nd u ′ on the right. This holds whenever u ′ is a n ob jectwise trivial ﬁbr ation and f ′ is a pro jective coﬁbration in PC (Ob( B ) , M ). These consider ations show that A is contained in the class of appa rent pro jective triv ia l ﬁbrations. F urthermore, if u satisﬁes right lifting for any pro jective c o ﬁbration f then in particular Ob( u ) must be surjective (using for f a n y extension b y a no nempty discr ete s et); and u ′ m ust satisfy right lifting with resp ect to pr o jectiv e tr ivial coﬁbrations inducing isomorphisms on sets of ob jects, so u ′ is an ob jectwise triv ial ﬁbra tio n (b y the pro jective model str ucture on the PC ( X , M )). This shows that the a pparent pro jective trivial coﬁbra tions ar e contained in A so the t wo classes coincide. Then, s imila rly , if f satis ﬁes left lift ing with resp ect to A then ﬁrst of all it must b e injectiv e on sets o f ob jects, as seen by considering u which are surjective maps of co discrete ob jects (i.e. ob jects U with U ( y · ) = ∗ for all sequence s y · ). And then decomp os ing f = f ′ ◦ d wher e d is the extension by the complemen tary subset, we see that f ′ should b e a pro- jective co ﬁbration in PC (Ob( B ) , M ). Th us f is a pro jective coﬁbration by deﬁnitio n. But all pro jective coﬁbrations satisfy lifting with resp ect to the apparent pro jective triv ial ﬁbr ations, by the deﬁnition of this latter cla ss, a nd e q uality with the c lass A shows that the class of pro- jective coﬁbrations is exactly that which satisﬁes left lifting with r esp ect to A . W e hav e stability under pushouts and retr acts. Corollary 15.3.2 Assume that M is a tr actable left pr op er c artesia n mo de l c ate gory. Su pp ose f : A → B and g : A → C ar e m orphisms in PC ( M ) such that f is a pr oje ctive c oﬁbr ation. Then t he map u : C → B ∪ A C is a pr oje ctive c oﬁbr ation. F urthermor e the pr oj e ctive c oﬁbr ations ar e stable un der r etr acts and t r ansﬁnite c omp osition. The class of pr oj e ctive trivial c oﬁbr ations is close d under tr ansﬁnite c omp osition and r etr acts. Pr o of Stability under pushouts, r e tracts and transﬁnite comp osition come from the character ization of the pro jective coﬁbrations as dual to 310 Coﬁbr ations the class of appa rent pr o jective trivia l ﬁbrations. The la s t sen tence now follows immediately from Prop ositio n 14.6.4 . The a dv a n tage of the notion of pro jective coﬁbration is that the gen- erating set is very easy to describ e. Prop ositio n 15 .3.3 Fix a gener ating set I for the c oﬁbr ations of M . Then for any k and any f : A → B in I , c onsider the map P ([ k ]; f ) : h ([ k ]; A ) → h ([ k ]; B ) . The c o l le ction of these for inte gers k and al l f ∈ I , to gether with the map ∅ → ∗ , forms a g ener ating s et for the class of pr oje ctive c oﬁbr ations in PC ( M ) . Pr o of Any pro jective coﬁbratio n f : A → B factor s a s f = f ′ d where A d → A ′ ⊔ disc ( Z ) f ′ → B where Z = Ob( B ) = Ob( f )(Ob( A )), wher e A ′ = Ob( f ) ! ( A ) is the precatego r y on the set Ob( f )(Ob( A )) o btained by transp ort of str ucture, disc ( Z ) is the discr e te prec ategory o n the set Z , and d is the extension morphism. In this situation furthermore, f ′ is a pro jective coﬁbra tion in PC ( X, M ) wher e X = Ob( B ) = O b( A ′ ⊔ dis c ( Z )). Now d is obta ine d by successive pushout a long ∅ → ∗ , while the w ! P ([ k ]; f ) for v arious maps w : [ k ] → X form the set of g enerator s for the pro jective coﬁbra tions in PC ( X , M ). Thus, f ′ is a retract o f a transﬁnite pushout alo ng the w ! P ([ k ]; f ). Putting these together g ives the expression of f as a retrac t of a transﬁnite pushout of morphis ms in our g enerating s et. 15.4 Injectiv e coﬁbrations It is easier to see whether a given map is an injective c o ﬁbration, since this condition is deﬁned ob ject wise, but the o nly wa y to get a generating set in g eneral is b y an accessibility arg umen t. A morphism f : A → B is an inje ctive c oﬁbr ation if Ob( f ) is a n injectiv e map of sets, and if the map O b( f ) ! ( A ) → B is a coﬁbration in the injective mo del category struc tur e PC inj (Ob( B ) , M ). Since inj ective coﬁbrations are deﬁned ob ject wise, we can be more explicit ab out this condition: it means that for a n y seq ue nce ( x 0 , . . . , x p ) in O b( A ), the map A ( x 0 , . . . , x p ) → B ( f ( x 0 ) , . . . , f ( x p )) is a coﬁbr ation in M , fo r 15.4 I n je ct ive c oﬁbr ations 311 any constant sequence ( y , . . . , y ) at a p oint y ∈ Y − f ( X ) the ma p ∗ → B ( y , . . . , y ) is a co ﬁbr ation in M , and for any non-constant sequence ( y 0 , . . . , y p ) such that at lea st o ne o f the y i is no t in f ( X ), the map ∅ → B ( y 0 , . . . , y p ) is a co ﬁbration i.e. B ( y 0 , . . . , y p ) is a coﬁbrant ob ject. Say that f is an inje ctive trivial c oﬁbr atio n if it is an injective co ﬁbr a- tion, a nd a global weak equiv a lence. W e again hav e the stability prop o s ition: Prop ositio n 15.4.1 Assum e that M is a t r actab le left pr op er c arte- sian mo d el c ate gory. Supp ose f : A → B and g : A → C ar e morphisms in PC ( M ) such that f is an inje ctive c oﬁbr ation. Then the m ap C → B ∪ A C is an inje ct ive c oﬁbr ation. F urt hermor e the inje ctive c oﬁbr ations ar e sta- ble under r etr acts and t r ansﬁnite c omp osition. The c lass of inje ctive triv- ial c oﬁbr a tions is close d u n der tr ansﬁnite c omp osition and r etr acts. Pr o of The explicit conditions given in the para graph deﬁning the in- jective coﬁbr ations, a re deﬁned ob ject wise over ∆ Z where Z is the set of o b jects of the tar get pr ecategor y . T he conditions are preserved by pushouts, retracts and transﬁnite comp osition. Again, the la st s ent ence now follo ws immediately from Prop os ition 14.6 .4. Lemma 1 5.4.2 Supp ose M is a t r actabl e mo del c ate gory. Then the class of inje ct ive c oﬁbr a tions in PC ( M ) admits a smal l set of gener a- tors. Pr o of F ollow the argument given in Barwick [16] for the pro of us ing a general accessibility argument. Unfortunately , we don’t g et very m uch information o n the set of gen- erators . In Section 1 5.7 b elow, if M satisﬁes some further h yp otheses which hold for presheaf categorie s , then the injective coﬁbrations are the same as the Reedy coﬁbrations and the gener ators will be describ ed explicitly . A similar pro blem with the injective str ucture is that if M is tracta ble , we don’t know whether the ge nerating coﬁbra tio ns fo r PC ( M ) hav e coﬁbrant do mains. This question for diagra m categories isn’t treated b y Lurie in [153] and see ms to r emain a n o pen q uestion. F or our purp oses this question is one fur ther reaso n for introducing the Ree dy mo del structure which ha s an explicit set of genera to rs; in many cases (such as when M is a presheaf category , Propo sition 15.7.2) the Reedy a nd injective structures will coincide. 312 Coﬁbr ations 15.5 A pushout expression for t he sk eleta The main observ ation crucial fo r understa nding the Ree dy coﬁbra tions, is an expression fo r the success ive sk eleta a s pushouts along maps o f the form R ([ k ]; f ). If x 0 , . . . , x m is a sequence o f ob jects, recall the notation d ( A ; x 0 , . . . , x m ) δ ( A ,x · ) → A ( x 0 , . . . , x m ) from Section 1 5.2. The identit y map d ( A ; x · ) = = sk m − 1 ( A )( x 0 , . . . , x m ) corres p o nds by the universal prop erty of h ([ m ]; · ) to a map h ([ m ]; d ( A ; x · )) → sk m − 1 ( A ) in PC ( M ). On the other hand, by the deﬁnition of h ( ∂ [ m ]; · ) and func- toriality of the skeleton op eration the map h ([ m ]; A ( x · )) → A yields a ma p h ( ∂ [ m ]; A ( x · )) → sk m − 1 ( A ) . These tw o maps agree on h ( ∂ [ m ]; d ( A ; x · )) so they give a map deﬁned on the copro duct h ([ m ] , ∂ [ m ]; d ( A ; x · ) δ ( A ; x · ) − → A ( x · )) = h ([ m ]; d ( A ; x · )) ∪ h ( ∂ [ m ]; d ( A ; x · )) h ( ∂ [ m ]; A ( x · )) , giving the to p map in the co mm utative squar e h ([ m ] , ∂ [ m ]; d ( A ; x · ) → A ( x · )) → sk m − 1 ( A ) h ([ m ]; A ( x · )) R ([ m ]; δ ( A ; x · )) ↓ → sk m ( A ) . ↓ The ma p on the b otto m is given by a djunction from the equality A ( x 0 , . . . , x m ) = sk m ( A )( x 0 , . . . , x m ) . Putting these together ov er all se quences x 0 , . . . , x m of leng th m we get an expression for sk m ( A ). 15.6 R e e dy c oﬁbr atio ns 313 Prop ositio n 15 .5.1 F or any m we have an expr ession of sk m ( A ) as a pushout of sk m − 1 ( A ) by c opies of the standar d maps R ([ m ]; · ) indexe d by se quenc es x · = ( x 0 , . . . , x m ) : sk m ( A ) = sk m − 1 ( A ) ∪ ` x · h ([ k ] ,∂ [ k ]; δ ( A ; x · )) a x · h ([ m ] , A ( x · )) . This is a pushout, and uses c opr o ducts, in the c ate gory PC ( M ) . Pr o of This is a classical fact ab out s implicial ob jects. 15.6 Reedy coﬁbrations Consider a map f : A → B in PC ( M ), giving for ea ch k a map o n skeleta sk k ( A ) → sk k ( B ). Deﬁne the r elative skeleton of f A ∪ sk k ( A ) sk k ( B ) sk rel k ( f ) → B . W e s ay that f is a R e e dy c oﬁbr ation if the r elative skeleton maps a re injectiv e coﬁbrations for every k . Lemma 15 .6.1 The class o f R e e dy c oﬁbr ations is close d under pushout, tr ansﬁnite c omp osition, and r etr a cts. Pr o of The skeleton op eration is g iven by a pushforward which is a kind of c o limit, so it commutes with colimits ov er connected categorie s which are computed levelwise. The relative skeleton map of a retra c t is again a retr act s o clos ur e under r e tr acts co mes fro m the same prop erty in M levelwise. Theorem 1 5.6.2 Supp ose f : A → B is map su ch that Ob( f ) is inje ct ive and v iew Ob( A ) as a subset of Ob ( B ) . The fol lowing c onditions ar e e quivale nt: (a)— f is a R e e dy c oﬁbr atio n; (b)—for any m ≥ 1 the map sk m ( A ) ∪ sk m − 1 ( A ) sk m − 1 ( B ) → sk m ( B ) (15.6.1 ) is an inje ctive c oﬁbr atio n; (c)—for any se quenc e ( x 0 , . . . , x p ) of obje cts in O b( A ) , the map A ( x 0 , . . . , x p ) ∪ d ( A ; x 0 ,...,x p ) d ( B ; x 0 , . . . , x p ) → B ( x 0 , . . . , x p ) (15 .6.2) is a c oﬁbr ation in M , and for any se quenc e ( y 0 , . . . , y p ) of obje cts in Ob( B ) not al l in Ob( A ) , the map d ( B ; y 0 , . . . , y p ) → B ( y 0 , . . . , y p ) is a 314 Coﬁbr ations c oﬁbr ation in M ; (d)—letting X := Ob( B ) , the map Ob( f ) ! ( A ) → B is Re e dy c oﬁbr ant in the mo del st ructur e PC Reedy ( X, M ) of The or em 14.3.2. Pr o of First note that (a) implies (c), indeed (c) is the s ta temen t of (a) for k = p − 1 a t the seque nce of o b jects ( x 0 , . . . , x p ) or ( y 0 , . . . , y p ). Next we show that (b) implies the following mor e general statement: for any 0 ≤ n ≤ m the ma p sk m ( A ) ∪ sk n ( A ) sk n ( B ) → sk m ( B ) (15.6.3) is an injective coﬁbr a tion. This is tautological for m = n . Let n be ﬁxed, and s uppo se we k now this statement for some m ≥ n . Then sk m +1 ( A ) ∪ sk n ( A ) sk n ( B ) = sk m +1 ( A ) ∪ sk m ( A ) (sk m ( A ) ∪ sk n ( A ) sk n ( B )) . Injective coﬁbra tions ar e s table under pushout (Lemma 15.4.2), and o ur inductive h yp othesis says that sk m ( A ) ∪ sk n ( A ) sk n ( B ) → sk m ( B ) is an injective coﬁbra tion. T ake the pushout o f this by sk m +1 ( A ) ov er sk m ( A ) and use the previous ident iﬁcation, to get that sk m +1 ( A ) ∪ sk n ( A ) sk n ( B ) → sk m +1 ( A ) ∪ sk m ( A ) sk m ( B ) is a n injective coﬁbration. On the other hand condition (b) says that sk m +1 ( A ) ∪ sk m ( A ) sk m ( B ) → sk m +1 ( B ) is a n injective coﬁbration, so comp o sing these gives that sk m +1 ( A ) ∪ sk n ( A ) sk n ( B ) → sk m +1 ( B ) is a n injective coﬁbra tion. This proves by induction that the ma ps (15 .6.3) are injective coﬁbrations. This s tatement now implies c o ndition (a ), indeed if ( x 0 , . . . , x p ) is any sequence of o b jects in Ob( B ) then for any m ≥ p, m ≥ k we hav e B ( x 0 , . . . , x p ) = sk m ( B )( x 0 , . . . , x p ) . The s a me is true for A if the x i are all in Ob( A ). Hence A ∪ sk k ( A ) sk k ( B )( x 0 , . . . , x p ) = sk m ( A ) ∪ sk k ( A ) sk k ( B )( x 0 , . . . , x p ) , so the map A ∪ sk k ( A ) sk k ( B )( x 0 , . . . , x p ) → B ( x 0 , . . . , x p ) 15.6 R e e dy c oﬁbr atio ns 315 is the s ame as the ma p sk m ( A ) ∪ sk k ( A ) sk k ( B )( x 0 , . . . , x p ) → B ( x 0 , . . . , x p ) . This latter is just the v alue o f (15 .6.3) on the seque nc e ( x 0 , . . . , x p ) so it is a coﬁbration in M , a s needed to show the Reedy condition (a). This shows that (b) implies (a). T o complete the pro of we nee d to show that (c) implies (b). F or this, use the expr ession of P rop osition 15 .5.1 for sk m ( B ) as a pusho ut of sk m − 1 ( B ) and the s tandard inclusio ns R ([ m ] , δ ( B ; y 0 , . . . , y p )) and simi- larly fo r A . W e have a diagram sk m − 1 ( A ) → sk m ( A ) sk m − 1 ( B ) ↓ → sk m ( B ) ↓ where the to p ar row is pushout along the R ([ m ] , δ ( A ; x 0 , . . . , x p )) for sequences of ob jects ( x 0 , . . . , x p ) of A , and the bottom a r row is pushout along the R ([ m ] , δ ( B ; x 0 , . . . , x p )) for sequences of ob jects ( x 0 , . . . , x p ) of B . T a k ing the pushout o f the upp er left cor ner of the diagram gives the expression sk m ( A ) ∪ sk m − 1 ( A ) sk m − 1 ( B ) = sk m − 1 ( B ) ∪ ` x · h ([ m ] ,∂ [ m ]; δ ( A ; x · )) a x · h ([ m ] , A ( x · )) . The copro ducts are ov er sequences x · = ( x 0 , . . . , x m ) of le ngth m of ob jects o f A , how ever it ca n b e extended to a copro duct over sequences of ob jects of B by setting A ( x 0 , . . . , x m ) := ∅ as w ell as d ( A ; x 0 , . . . , x m ) := ∅ if a ny of the x i are not in Ob( A ). On the other hand, sk m ( B ) = sk m − 1 ( B ) ∪ ` x · ∂ h ([ m ] , ∂ [ m ]; δ ( B ; x · )) a x · h ([ m ] , B ( x · )) . Putting these t wo together , we conclude that sk m ( B ) =  sk m ( A ) ∪ sk m − 1 ( A ) sk m − 1 ( B )  ∪ ` x · C ( x · ) a x · h ([ m ] , B ( x · )) where C ( x · ) := h ([ m ] , A ( x · )) ∪ h ([ m ] ,∂ [ m ]; δ ( A ; x · )) h ([ m ] , ∂ [ m ]; δ ( B ; x · )) . Hence, to prov e that the ma p sk m ( A ) ∪ sk m − 1 ( A ) sk m − 1 ( B ) → sk m ( B ) 316 Coﬁbr ations is a n injective coﬁbra tion, us ing s tability of injective c oﬁbrations under pushouts, it suﬃces to show that each of the maps C ( x 0 , . . . , x m ) → h ([ m ] , B ( x 0 , . . . , x m )) (15.6 .4) is an injective co ﬁbration. F or b oth sides, the set of ob jects is now our standard set [ m ] = { υ 0 , . . . , υ m } . Consider the v alue on a sequence of ob jects ( y 0 , . . . , y p ) in [ m ]. There are several po ssible cases : (i)—If it is a constant sequence, then h ([ m ] , A ( x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( A ; x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( B ; x · ))( y · ) = h ([ m ] , B ( x · ))( y · ) = ∗ and the ma p (15 .6.4) is the identit y . (ii)—If the sequence is somewhere decreasing i.e. there is some j with y j < y j − 1 then h ([ m ] , A ( x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( A ; x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( B ; x · ))( y · ) = h ([ m ] , B ( x · ))( y · ) = ∅ and a gain the ma p (15.6.4) is the identit y . (iii)—If the se quence is nondecr easing, but there is some υ i not co ntained in y · then h ([ m ] , A ( x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( A ; x · ))( y · ) = A ( x · ) , and h ([ m ] , B ( x · ))( y · ) = h ([ m ] , ∂ [ m ]; δ ( B ; x · ))( y · ) = B ( x · ) , so in this ca se C ( x · )( y · ) = A ( x · ) ∪ A ( x · ) B ( x · ) = B ( x · ) and o nce again (15 .6.4) is the identit y . (iv)—If the sequenc e is nondecrea sing and surjects onto the full se t of ob jects, then h ([ m ] , A ( x · ))( y · ) = A ( x · ) , h ([ m ] , ∂ [ m ]; δ ( A ; x · ))( y · ) = d ( A ; x · ) , and h ([ m ] , B ( x · ))( y · ) = B ( x · ) , h ([ m ] , ∂ [ m ]; δ ( B ; x · ))( y · ) = d ( B ; x · ) , so C ( x · )( y · ) = A ( x · ) ∪ d ( A ; x · ) d ( B ; x · ) 15.6 R e e dy c oﬁbr atio ns 317 so the map C ( x · )( y · ) → h ([ m ] , B ( x · ))( y · ) is e x actly the map (15 .6.2) A ( x · ) ∪ d ( A ; x · ) d ( B ; x · ) → B ( x · ) which is kno wn to b e a co ﬁbration in M b eca use we are assuming c o ndi- tion (c) of the theorem. This co mpletes the pro of that the map (15.6.1) is an injective coﬁbr a tion, sho wing (c) ⇒ (b). This completes the pro of of the equiv alence of (a), (b) and (c). F or the equiv alence with (d), no te that sk m (Ob( f ) ! A ) = Ob( f ) ! sk m ( A ) by commutation of pushforwards. Now A ∪ sk m ( A ) B = Ob( f ) ! ( A ) ∪ Ob( f ) ! sk m ( A ) B from the deﬁnition of co limits in PC ( M ) (Section 12 .6), so A → B is Reedy coﬁbr a nt if and only if O b( f ) ! A → B is. Howev er, this latter induces an iso morphism o n sets of o b jects, and for such maps the cri- terion (c) ab ov e is the s ame as the Reedy condition that the relative latching maps b e coﬁbr a nt in M . This shows that (d) is equiv alen t to (a),(b),(c). The map (1 5 .6.4) o ccuring a bove is of the for m R ([ m ] , g ) as shown in the following lemma. Lemma 15. 6.3 Supp ose E a → F U u ↓ b → V v ↓ is a diagr am in M . Consider the induc e d map g : U ∪ E F → V . Then the two maps h ([ m ] , U ) ∪ h ([ m ] ,∂ [ m ]; E u → U ) h ([ m ] , ∂ [ m ]; F u → V ) → h ([ m ] , V ) and R ([ m ] , g ) : h ([ m ] , ∂ [ m ]; g ) → h ([ m ] , V ) ar e the same. 318 Coﬁbr ations Pr o of Lo ok at the v alues on any se q uence y · = ( y 0 , . . . , y p ) of ob jects in [ m ] = { υ 0 , . . . , υ m } . There are several p ossibilities: —if the sequence is co nstant then bo th maps are ∗ → ∗ ; —if the sequence is a nywhere decreasing then bo th maps are ∅ → ∅ ; —if the sequence is nondecrea sing but misse s some elemen t υ j , then we are in the b ounda ry ∂ [ m ] and the ﬁr st map is U ∪ U V → V and the s econd map is U → V , these are the same; —if the sequence is nondecrea sing and surjects onto [ m ] then both ma ps are U ∪ E F → V . W e nee d to p oint o ut that these ident iﬁcations of the ma ps, and in particular of their sources , a re functorial in the r estriction maps for diagrams over ∆ o [ m ] , so they g ive an iden tiﬁcation h ([ m ] , U ) ∪ h ([ m ] ,∂ [ m ]; E u → U ) h ([ m ] , ∂ [ m ]; F u → V ) ∼ = h ([ m ] , ∂ [ m ]; g ) and b o th ma ps fro m her e to h ([ m ] , V ) a re the same. Corollary 15.6.4 F or an obje ct A ∈ PC ( M ) , t he fol lowing ar e e qu iv- alent: (a)— A is R e e dy c oﬁbr ant; (b)—for any m ≥ 1 the m ap sk m − 1 ( A ) → sk m ( A ) is an inje ctive c oﬁ- br atio n; (c)—for any se quenc e of obje cts ( x 0 , . . . , x p ) t he map sk p − 1 ( A )( x 0 , . . . , x p ) → A ( x 0 , . . . , x p ) is a c oﬁbr ation in M ; (d)— A is a R e e dy c oﬁbr ant obje ct in PC Reedy ( X, M ) wher e X = Ob( A ) . Pr o of Apply the previous pro p os ition to the map ∅ → A . Note tha t Ob( ∅ ) = ∅ and sk m ( ∅ ) = ∅ . Condition (c) he r e is the pa rt of condition (c) o f the pro po sition, co ncerning sequence s of ob jects not all coming from the source. Corollary 15.6.5 If f : A → B is a c oﬁbr ation in M , then R ([ k ]; f ) (p ag e 307) is a R e e dy c oﬁbr ation. Pr o of Recall that h ([ k ] , ∂ [ k ] , f ) = h ([ k ] , A ) ∪ sk k − 1 h ([ k ] ,A ) sk k − 1 h ([ k ] , B ) 15.6 R e e dy c oﬁbr atio ns 319 and R ([ k ]; f ) is the map fro m here to h ([ k ] , B ). If m ≤ k − 1 then sk m h ([ k ] , ∂ [ k ] , f ) = sk m ([ k ] , B ) so the ma p o ccuring in condition (b) of Theorem 1 5.6.2 is the ident ity in this case. F or m ≥ k , sk m h ([ k ] , ∂ [ k ] , f ) ∪ sk m − 1 h ([ k ] ,∂ [ k ] ,f ) sk m − 1 h ([ k ] , B ) = sk m h ([ k ] , A ) ∪ sk m − 1 h ([ k ] ,A ) sk m − 1 h ([ k ] , B ) so the map o c c uring in co nditio n (b) is sk m h ([ k ] , A ) ∪ sk m − 1 h ([ k ] ,A ) sk m − 1 h ([ k ] , B ) → sk m ( h [ k ] , B ) . As for the equiv alence with condition (c), it suﬃces to note that this in- duces a c oﬁbration ov er se q uences x 0 , . . . , x m of length m in { υ 0 , . . . , υ k } (similar to the pr o of of Prop ositio n 15.7.1 b elow). Corollary 15.6 . 6 Many maps b etwe en t he h ([ k ] , B ) ar e Re e dy c oﬁ- br atio ns due to the pr evious c or ol lary. F or ex ample if f : A → B is a c oﬁbr ation in M then the map h ([ k − 1 ] , A ) → h ([ k ] , B ) induc e d by applying f at one of the fac es of t he k - simplex [ k − 1] ⊂ [ k ] , is a R e e dy c oﬁbr ation. Pr o of Either calculate dire ctly the s keleta, or use the previous co rollary inductively . The following statement is somewha t simila r to giving a se t o f gen- erators for the Reedy co ﬁbrations. On the o ne ha nd it refers to the full class of morphisms o f the form R ([ m ] , g ) for coﬁbratio ns g , while on the other hand g iving a stronger expression without refering to retracts. Prop ositio n 1 5.6.7 A morphism f : A → B is a R e e dy c oﬁbr ation if and only if it is a tr ansﬁnite c omp osition of a disjoi nt union with a discr ete set, and then pushouts along morphisms of the form R ([ m ] , g ) for c oﬁbr atio ns g in M . Pr o of A Reedy coﬁbration f : A → B ca n b e expr essed as the countable comp osition of the morphis ms A ∪ sk m − 1 ( A ) sk m − 1 ( B ) → A ∪ sk m ( A ) sk m ( B ) which are themselv es Reedy co ﬁbrations. At the start, A ∪ sk 0 ( A ) sk 0 ( B ) is the disjoint union of A with the discr ete set Ob( B ) − Ob( A ). Then, as we have seen in the pro of of Theore m 15.6.2 , at each stage 320 Coﬁbr ations the mor phism in question is o bta ined by simultaneous pushout along morphisms (1 5 .6.4) of the fo r m h ([ m ] , U ) ∪ h ([ m ] ,∂ [ m ]; E u → U ) h ([ m ] , ∂ [ m ]; F v → V ) → h ([ m ] , V ) (15.6 .5 ) where fr o m the notations of (15 .6.4) we put U := A ( x 0 , . . . , x m ), V := B ( x 0 , . . . , x m ), E := d ( A ; x 0 , . . . , x m ) and F := d ( B ; x 0 , . . . , x m ), with u := δ ( A ; x 0 , . . . , x m ) : E → U and v := δ ( B ; x 0 , . . . , x m ) : F → V . The maps u and v , as well as the maps U → V and E → F all ﬁtting int o a commutative s q uare. Condition (c) o f the theor em says that the map g : U ∪ E F → V is a coﬁbr ation. By Lemma 1 5.6.3, the map (15.6.5) is the s ame as R ([ m ]; g ). The the no tion of Reedy coﬁbration is similar to that of pro jective coﬁbration, in that we can give explicitly the set of genera tors. Lemma 15 .6.8 Supp ose I is a set of maps in M . L et R ( I ) ⊂ Arr ( PC ( M )) denote the set of al l arr ows of the form R ([ k ]; g ) for k ∈ N and g ∈ I . If f ∈ cell ( I ) and m ∈ N then R ([ m ]; f ) ∈ cell ( R ( I )) . Pr o of Lo ok at the behavior of R ([ k ]; f ) under pushouts and transﬁnite comp osition. Suppo se A f → B C g ↓ v → P u ↓ is a pushout diag ram in M , that is P = B ∪ A C . This induces a diagra m h ([ k ] , ∂ [ k ]; g ) → h ([ k ] , ∂ [ k ]; u ) h ([ k ]; C ) R ([ k ]; g ) ↓ → h ([ k ]; P ) . R ([ k ]; u ) ↓ W e claim that this sec ond diagr am is then also a pushout in PC ( M ). In fact it is a dia gram in PC ([ k ] , M ), and connected colimits of di- agrams in PC ([ k ] , M ) are the s ame a s the corr e s po nding colimits in 15.6 R e e dy c oﬁbr atio ns 321 PC ( M ), a lso in turn they ar e the same as the corr esp onding colimits in Func (∆ o [ k ] , M )(Section 12.6 ), so the pushout o f the second diag ram can b e computed o b ject wise. Then it is easy to see that it is a pushout, using the explicit descriptio n of the v alues of h (([ k ] , ∂ [ k ]); · ) and h ([ k ]; · ). The conclusion from this discussion, is that any pushout along R ([ k ]; u ) will a lso be a pusho ut along R ([ k ]; g ). Consider no w a transﬁnite comp osition: supp ose we hav e a ser ie s . . . → A i f i,i +1 → A i +1 → . . . in M indexed b y i ∈ β for some or dinal β . T o treat limit o rdinals we need also to consider the transition maps f i,j : A i → A j for any i < j . Assume that if j is a limit ordina l then A j = colim i m , a re marked with an × . The dots ab ov e these are also marked with a n × but no other dots are (newly) marked with an × . In the situation of Theorem 17.3.5 , we sta rt with green dots at ( p, k ) for p + k ≤ n . W e may as well assume that the rest of the dots a re colored re d. Start with ( m, k ) = ( n + 1 , 0) and apply the pro cedur e of the previous paragraph. The dot ( n + 1 , 0) becomes green, the dot ( n, 0) stays gr een, and the dots ( n, 0 ) , ( n + 1 , 0) , . . . a re marked with an × . Contin ue now at ( n, 1) and so on. At the end w e have made all of the dots ( p, k ) with p + k = n + 1 green, a nd we will hav e ma r ked with an × all of the dots ( p, k ) with p + k = n (including the dot (1 , n − 1); a nd also a ll o f the dots a bove this line). W e can now iterate the pro cedur e. W e succes sively get green dots on each of the lines p + k = n + j for j = 1 , 2 , 3 , . . . . F urthermore , no new dots will be marked w ith a × . After tak ing the union ov er all o f these iterations, we obta in a n A ′ which is ( p, k )-arra nged for all ( p, k ). Thus A ′ is a Seg al c ategory . Note tha t the mo rphism |A| → |A ′ | is a weak equiv alence of spaces. By lo oking a t which dots ar e marked with an × , we ﬁnd that the morphisms A p/ → A ′ p/ induce is o morphisms on π i whenever i < n − p . This completes the pr o of of the theo rem. 17.3.3 Itera t ion The following corollar y to Theorem 17.3 .5 sa ys that in order to calculate the n -t ype of Ω |A| we just hav e to change A by pushouts preserving the weak equiv a lence t ype of |A| in such a wa y that A is ( m, k )-ar ranged for all m + k ≤ n + 2. Corollary 17. 3.6 Su pp ose A is a Se gal pr e c ate gory with A 0 = ∗ and A 1 / c onne cte d, such that A is ( m, k ) -arr ange d for al l m + k ≤ n . Then the natur al morphism |A 1 / | → Ω |A| induc es an isomorphism on π i for i < n − 1 . 17.3 The c alculus 369 Pr o of: Use Theo rem 17 .3.5 to obtain a morphis m A → A ′ with the prop erties sta ted there (which we refer to as (1)–(3)). W e hav e a diagram |A 1 / | → Ω |A| |A ′ 1 / | ↓ → Ω |A ′ | ↓ . By pr o pe rty (1) the vertical morphism on the right is a weak equiv a- lence. By prop erty (2) and Theore m 5.3.1 the mor phism on the b ottom is a weak eq uiv alence. By prop erty (3) the vertical mor phism o n the right induces is omorphisms on π i for i < n − 1. This gives the r equired statement.  Corollary 17.3.7 Fix n , and supp ose A is a Se gal pr e c ate gory with A 0 = ∗ and A 1 / c onne cte d. By applying the op er a tions A 7→ Ar r ( A , m ) for vario us m , a ﬁnite n u mb er of times (less tha n ( n + 2 ) 2 ) in a pr e deter- mine d way, we c an eﬀe ctively get to a morphism of S e gal pr e c ate gories A → B such that |A| → |B | is a we ak e quivalenc e of sp ac es, and such that B is ( m, k ) -arr ange d for al l m + k ≤ n + 2 . F urthermor e B 0 = ∗ and B 1 / is c onne cte d. Pr o of: By Cor ollary 17.3.2 any succes s ive applica tion of the op era - tions A 7→ Ar r ( A , m ) yields a mor phism |A| → |B | w hich is a weak equiv alence of spaces. By Pro po sition 17 .3.3 it s uﬃce s , fo r exa mple, to successively apply Ar r ( A , i ) for i = 2 , 3 , . . . , n + 2 a nd to r epe at this n + 2 times. These op era tio ns preserve connectedness of the pieces in degree 1, so B 1 / is c o nnected.  Corollary 17.3.8 Fix n , and supp ose A is a Se gal pr e c ate gory with A 0 = ∗ and A 1 / c onne cte d. L et B b e the r esult of the op er ations of Cor ol lary 17.3.7. Then the n -typ e of the simplicial set B 1 / is e quivalent to the n -typ e of Ω |A| . Pr o of: Apply Coro lla ries 17.3 .6 and 17.3 .7.  370 Gener ators and r elations for Se gal c ate gories 17.4 Computing the lo op space Suppo se X is a simplicial se t with X 0 = X 1 = ∗ , a nd with ﬁnitely many nondegenerate s implice s . Fix n . W e will obtain, by iterating an op eration closely related to the op eration gen of Chapter 16, a ﬁnite complex representing the n -type of Ω X . Let A be X considered as a Sega l precateg ory constant in the second v a riable, in o ther w ords A p,k := X p . Apply the arrang ement pr o cess, iterated a s in Co rollar y 1 7.3.7. Corollary 17.4.1 Fix n , and su pp ose X is a simplicial set with ﬁnitely many nonde gener ate simplic es, with X 0 = X 1 = ∗ . L et A b e X c o n- sider e d as a Se gal pr e c ate gory . L et B b e the r esult of the op er ations of Cor ol lary 17.3.7. Then the n -typ e of the simplicial set B 1 / is e quivalent to the n -typ e of Ω X . Pr o of: An immediate restatement of 17.3.8 .  R emark: Any ﬁnite region of the Segal precatego ry B is eﬀectiv ely computable. In fact it is just an iteration of o per ations pushout and mapping cone, a rranged in a wa y which dep ends on combinatorics of simplicial sets. Thus the n + 1-skeleton of the simplicial set B 1 / is eﬀec- tively calcula ble (in fact, o ne could b ound the num ber of simplices in B 1 / ). Corollary 17.4.2 Fix n , and su pp ose X is a simplicial set with ﬁnitely many nonde gener ate simplic es, with X 0 = X 1 = ∗ . Then we c an eﬀe c- tively c alculate H i (Ω | X | , Z ) for i ≤ n . Pr o of: Immediate from ab ov e.  In some sense this co r ollary is the “most e ﬀectiv e” part of the presen t argument, since we can get at the calcula tion after a b ounded n umber of ea sy steps o f the form A 7→ Arr ( A , m ). W e describ e how to use the ab ov e descr iption of Ω X inductively to obtain the π i ( X ). This seems to be a new algorithm, diﬀerent from those of E . Br own [52] and Kan–Curtis [131] [132] [79]. There is an unboundedness to the resulting algorithm, coming es sen- tially from a pr oblem with π 1 at each stage. Even though we know in adv ance that the π 1 is ab elian, we would need to k now “why” it is ab elian in a pr ecise w a y in order to sp ecify a strategy for making A 1 / connected 17.4 Computing the lo op sp ac e 371 at the a ppropriate place in the lo op. In the absence o f a particula r de- scription of the pro of we are forced to s ay “search over a ll pro o fs” at this sta ge. See Subsection 17.4.5 for further discussion. 17.4.1 Getting A 1 / to b e connected In the g eneral situatio n, we have to tackle the pro blem of computatio n of a fundamental g r oup using ge ner ators and re la tions, known to b e undecideable in general. Some sub-cases can still be treated eﬀectively . The ﬁr st question is how to arra nge A on the level of τ ≤ 1 ( A ). W e deﬁne op eratio ns Ar r 0 only ( A , m ) and Arr 1 only ( A , m ). These con- sist of do ing the op eratio n Ar r ( A , m ) but instead of using the e ntire mapping cone C , only adding on 0-cells to A m/ to get a surjection o n π 0 ; or only adding o n 1-cells to ge t an injection on π 0 . Note in the seco nd case that we do n ’t add extra 0-c ells. This is an impo rtant p oint, b eca use if w e added further 0-cells every time w e added some 1-cells , the pro cess would never stop. T o deﬁne Ar r 0 only ( A , m ), use the same constructio n as for Ar r ( A , M ) but instead o f setting C to b e the mapping cone, we put C ′ := A m/ ∪ sk 0 ( A 1 / × . . . × A 1 / . Here sk 0 denotes the 0-skeleton of the simplicial set, and im means the image under the Sega l map. Let C ⊂ C ′ be a subset where we choose only one p oint for e a ch connected comp onent of the pro duct. With this C the same construction as previously gives Ar r 0 only ( A , m ). With the subset C ⊂ C ′ chosen a s above (no te that this choice can eﬀectively b e made) the resulting simplicial set p 7→ π 0  Arr 0 only ( A , m ) p/  may be descr ib e d only in terms o f the simplicial set p 7→ π 0 ( A p/ ) . That is to say , this op eratio n Ar r 0 only ( A , m ) co mm utes with the op er- ation of co mpo nen twise applying π 0 . W e forma lize this as τ ≤ 1 Arr 0 only ( A , m ) = τ ≤ 1 Arr 0 only ( τ ≤ 1 A , m ) . T o deﬁne Ar r 1 only ( A , m ), let C b e the cone o f the map fr o m A m/ to im ( A m/ ) ∪ sk 1 ( A 1 / × . . . × A 1 / ) o where sk 1 ( A 1 / × . . . × A 1 / ) o denotes the union of connected comp onents 372 Gener ators and r elations for Se gal c ate gories of the 1-skeleton of the pro duct, whic h touch im ( A m/ ). In this case , note that the inclusio n A m/ ֒ → C is 0-co nnec ted (all connected co mpo nen ts of C contain elements of A m/ ). Using this C we obtain the op era tion Ar r 1 only ( A , m ). It do esn’t int ro- duce an y new connected compo nents in the new s implicial sets A ′ p/ , but may connect together some comp onents which were disjoint in A p/ . Again, the o per ation Arr 1 only ( A , m ) commutes w ith truncation: we hav e τ ≤ 1 Arr 1 only ( A , m ) = τ ≤ 1 Arr 1 only ( τ ≤ 1 A , m ) . Our goal in this section is to ﬁnd a seq uence of op e rations which makes τ ≤ 1 ( A ) 1 bec ome triv ial (equa l to ∗ ). In view of this, and the ab ov e commutations, we may henceforth work with simplicial sets (which we denote U = τ ≤ 1 A for example) and use the above op eratio ns follow ed by the trunca tio n τ ≤ 1 as mo diﬁcatio ns o f the simplicia l se t U . W e tr y to o btain U 1 = ∗ . This corr esp onds to making A 1 / connected. Our o per ations hav e the following interpretation. The o per ation U 7→ τ ≤ 1 Arr 0 only ( U, 2) has the eﬀect of for mally adding to U 1 all binary pro ducts of pairs o f elements in U 1 . (W e say that a binary pro duct of u, v ∈ U 1 is deﬁned if there is an element c ∈ U 2 with pr incipal edges u a nd v in U 1 ; the pro duct is then the image w o f the third edg e o f c ). The o per ation U 7→ τ ≤ 1 Arr 1 only ( U, 2) has the eﬀect o f identifying w and w ′ any time b oth w and w ′ are bina ry pro ducts of the sa me elements u, v . The o per ation U 7→ τ ≤ 1 Arr 0 only ( U, 3) has the eﬀect of introducing, for each triple ( u, v , w ), the v a rious binary pro ducts one c an ma ke (keeping the same order) and giving a formula ( uv ) w = u ( v w ) for certain of the bina ry pro ducts th us introduced. It is so mew ha t unclea r whether blindly applying the comp osed op er- ation U 7→ τ ≤ 1 Arr 1 only ( τ ≤ 1 Arr 0 only ( U, 3) , 2) 17.4 Computing the lo op sp ac e 373 many times m ust automa tically lead to U 1 = ∗ in case the actual fun- damental g roup is triv ial. This is b ecaus e in the pro cess of a dding the asso ciativity , w e also add in so me new bina ry pr o ducts; to which ass o- ciativity mig ht then ha v e t o be applied in order to get something trivial, and s o on. If the ab ov e do e sn’t work, then w e may need a slightly rev ised ver- sion of the op era tion Ar r 0 only ( U, 3) where w e add in only certain triples u, v , w . This can b e acco mplished b y c ho osing a subset of the or iginal C at each time. Similarly for the Arr 0 only ( U, 2) for binary pr o ducts. W e now o btain a situation wher e we have op era tio ns which eﬀect the appro - priate c hanges on U corresp onding to all of the v arious po ssible steps in an elemen tary pro of that the as so ciative unitary monoid gener ated by generator s U 1 with r e lations U 2 , is trivia l. Thus if w e have an elemen- tary pr o of that the asso c iative unitary monoid genera ted by U 1 with relations U 2 is trivia l, then we can read oﬀ from the steps in the pro of, the necessar y seq uence of op erations to apply to get U 1 = ∗ . On the level of A thes e same steps will result in a new A with A 1 / connected. In our case we ar e interested in the gr oup c omple tion of the monoid: we want to obtain the condition of be ing a Seg al gr oup oid not just a Segal catego r y . It is p ossible that the simplicia l set X w e star t with would yield a monoid which is not a gro up, when the a bove op erations are applied. T o ﬁx this, we take note of ano ther op era tion whic h can b e applied to A which do esn’t a ﬀect the weak type of the realization, a nd which guar antees that, when the monoid U is g enerated, it b eco mes a group. Let I b e the category with tw o o b jects and one morphism 0 → 1, and let I b e the category with tw o ob jects and a n isomorphism betw een them. Consider these as Segal catego ries (taking their nerve as bisimpli- cial sets constant in the second v a riable). Note that | I | and | I | are b oth contractible, so the obvious inclus ion I ֒ → I induces an eq uiv a le nce o f realizations . The bisimplicia l set I is just that which is represented b y (1 , 0 ) ∈ ∆ × ∆. Thus for a Sega l precategor y A , if f ∈ A 1 , 0 is an o b ject of A 1 / (a “ morphism” in A ) then it co rresp onds to a morphism I → A . Se t A f := A ∪ I I . Now the morphism f is strictly inv ertible in the precategor y τ ≤ 1 ( A f ) and in par ticular, when we apply the op erations desc rib ed a bove, the image of f becomes inv ertible in the res ulting ca tegory . If A 0 = ∗ (whence A f 0 = ∗ to o) then the ima ge of f b ecomes inv ertible in the resulting 374 Gener ators and r elations for Se gal c ate gories monoid. Note ﬁnally that |A| → |A f | = |A| ∪ | I | | I | is a weak equiv alence. In fact we wan t to inv ert all of the 1-morphisms. Let A ′ := A ∪ S f I   [ f I   where the union is taken over all f ∈ A 1 , 0 . Again |A| → |A ′ | is a weak equiv alence. Now, when we apply the previous pro cedure to τ ≤ 1 ( A ′ ) giving a categor y U (a monoid if A had only o ne ob ject), a ll morphisms coming fro m A 1 , 0 bec ome in v ertible. Note that the morphisms in A ′ , i.e. ob jects of A ′ 1 , 0 , are either m orphisms in A or their newly-added inverses. Thu s all o f the morphisms coming from A ′ 1 , 0 bec ome inv ertible in the category U . But it is clear fro m the o pe rations descr ibe d ab ov e that U is genera ted b y the morphisms in A ′ 1 , 0 . Therefo r e U is a gro upo id. In the ca se o f only one o b ject, U b ecomes a group. By Sega l’s theorem we then have U = π 1 ( |A| ). If we know for some reason that |A| is s imply connected, then U is the triv ia l gro up. More precisely , search for a pro of that π 1 = 1 , and when such a pro of is found, apply the co rresp onding series of op er ations to τ ≤ 1 ( A ′ ) to obtain U = ∗ . Applying the op erations to A ′ upstairs, we obtain a new A ′′ with |A ′′ | ∼ = |A ′ | ∼ = |A| and A ′′ 1 / connected. Another w ay of lo o king at this is to say that every time one nee ds to take the inv erse of an element in the pro of that the g roup is trivial, a dd on a co py of I over the corresp onding c o py of I . 17.4.2 The case of ﬁnite homotop y groups W e ﬁrst present our algor ithm for the case o f ﬁnite homotopy groups . Suppo se we want to calculate π n ( X ). W e assume known that the π i ( X ) are ﬁnite for i ≤ n . Start: Fix n and start with a simplicial s et X containing a ﬁnite n umber of no ndegenerate simplices. Supp ose we know that π 1 ( X, x ) is a given ﬁnite group; r e cord this gro up, a nd set Y equal to the cor resp onding cov ering s pa ce of X . Thus Y is simply connected. Now c o nt ract out a maximal tree to o btain Z with Z 0 = ∗ . Step 1. Let A p,k := Z p be the co r resp onding Segal prec ategory . It has only one o b ject. 17.4 Computing the lo op sp ac e 375 Step 2. Let A ′ be the copro duct of A with one copy o f the ner ve of the catego ry I (containing tw o isomor phic ob jects), for each mo rphism I → A (i.e. each p oint in A 1 , 0 ). Step 3. Apply the pro cedur e of Subsection 17.4.1 to obtain a mo rphism A ′ → A ′′ with A ′′ 1 / connected, and inducing a weak equiv alence o n realizations . (This s tep c an only be b ounded if we hav e a spe c iﬁc pro of that π 1 ( Y , y ) = 1). Step 4. Apply the pro ce dur e o f Corollar y 17.4.1 and Theorem 1 7.3.5 to obtain a morphism A ′′ → B inducing a weak equiv a lence on realiza tio ns, such that B is a Segal group oid. Note that the n − 1-type of B 1 / is eﬀec- tively calculable (the no n-eﬀective parts o f the pro o f of Theore m 17.3 .5 served only to prov e the prop erties in question). By Segal’s theorem, |B 1 / | ∼ Ω |B | ∼ Ω | Y | , which in turn is the connected comp onent of Ω | X | . Thus π n ( | X | ) = π n − 1 ( |B 1 / | ) . Step 5. Go back to the Start with the new n equal to the o ld n − 1, a nd the new X equal to the simplicia l set B 1 / ab ov e. The new fundamental group is known to b e ab elian (since it is π 2 of the previous X ). Th us we ca n calculate the new fundamental g roup as H 1 ( X ) and, under o ur hypothesis, it will b e ﬁnite. Keep re pea ting the pr o cedure until w e get down to n = 1 and hav e recorded the answer. 17.4.3 Ho w to get rid of free ab elian groups in π 2 In the case where the hig her ho motopy gr oups ar e inﬁnite (i.e. they contain factors of the form Z a ) we nee d to do something to get pa st these inﬁnite gr oups. If we go down to the case where π 1 is inﬁnite, then taking the universal cov ering no long er results in a ﬁnite co mplex. W e prefer to avoid this by tackling the problem a t the level of π 2 , with a geometrical argument. Namely , if H 2 ( X, Z ) is nonzero then we can take a clas s there as giving a line bundle, and take the total space o f the corres p o nding S 1 -bundle. This amounts to taking the ﬁb er of a map X → K ( Z , 2). This can b e done explicitly and eﬀectively , resulting again in a calculable ﬁnite complex. In the new complex w e will hav e reduced the r ank of H 2 ( X, Z ) = π 2 ( X ) (we are assuming that X is simply c onnected). The o riginal method of E. Bro wn [52] for eﬀectiv ely calculating the π i 376 Gener ators and r elations for Se gal c ate gories was basically to do this at all i . The technical problems in [5 2] a re ca used by the fact that one do esn’t hav e a ﬁnite complex repres ent ing K ( Z , n ). In the case n = 2 we don’t hav e these technical problems b eca use we can lo ok at cir c le ﬁbrations and the circle is a ﬁnite c o mplex. F o r this section, then, we are in some s e ns e r everting to an easy cas e of [5 2] and not us ing the Seifert- V an K amp en technique. Suppo se X is a simplicial set with ﬁnitely ma ny nondegenera te sim- plices, a nd supp os e X 0 = X 1 = ∗ . W e can calc ula te H 2 ( X, Z ) as the kernel of the diﬀerential d : Z X ′ 2 → Z X ′ 3 . Here X ′ i is the se t o f nondegener a te i - simplices. (Note that a basis of this kernel can eﬀectively be computed using Gaussian elimination). Pick an element β of this bas is, which is a collection of integers b t for each 2- simplex (i.e. triang le) t . F or each triangle t deﬁne an S 1 -bundle L t ov er t tog ether with trivialization L t | ∂ t ∼ = ∂ t × S 1 . T o do this, take L t = t × S 1 but change the triv ialization a long the bo undary by a bundle automorphism ∂ t × S 1 → ∂ t × S 1 obtained from a map ∂ t → S 1 with winding n umber b t . Let L (2) be the S 1 -bundle ov er the 2 -skeleton of X obtaine d by glueing together the L t along the tr ivializations over their boundaries. W e ca n do this eﬀectiv ely and obtain L (2) as a simplicial set with a ﬁnite n um ber of nondegenera te simplices. The fact that d ( β ) = 0 means tha t for a 3-simplex e , the restr iction of L (2) to ∂ e (which is top olog ically a n S 2 ) is a triv ia l S 1 -bundle. Thus L (2) extends to an S 1 -bundle L (3) on the 3-skeleton of X . F urthermore, it c a n be extended acros s any simplices of dimension ≥ 4 be c a use all S 1 -bundles on S k for k ≥ 3, are trivial ( H 2 ( S k , Z ) = 0). W e o btain an S 1 -bundle L on X . By sub dividing things appropr iately (including po ssibly sub dividing X ) we can assume that L is a simplicial set with a ﬁnite num ber o f nondegenerate simplices. It dep ends on the choice of basis element β , so ca ll it L ( β ). Let T = L ( β 1 ) × X . . . × X L ( β r ) where β 1 , . . . , β r are our basis elemen ts f ound above. It is a torus bundle 17.4 Computing the lo op sp ac e 377 with ﬁb er ( S 1 ) r . The lo ng exa ct homotopy seque nce fo r the map T → X gives π i ( T ) = π i ( X ) , i ≥ 3; and π 2 ( T ) = ker( π 2 ( X ) → Z r ) . Note that Z r is the dual of H 2 ( X, Z ) so the kernel π 2 ( T ) is ﬁnite. Finally , π 1 ( T ) = 0 since the map π 2 ( X ) → Z r is surjective. Note tha t we have a pro of that π 1 ( T ) = 0 . 17.4.4 The general algorithm Here is the g eneral situation. Fix n . Supp ose X is a simplicia l set with ﬁnitely many nondegenerate s implices, with X 0 = ∗ and with a pro of that π 1 ( X ) = { 1 } . W e will calcula te π i ( X ) for i ≤ n . Step 1. Ca lculate (b y Gaussian elimina tion) and r ecord π 2 ( X ) = H 2 ( X, Z ). Step 2. Apply the opera tion describ ed in the pr e vious subsection ab ove, to obtain a new T with π i ( T ) = π i ( X ) for i ≥ 3, with T 0 = ∗ , with π 1 ( T ) = 1, and with π 2 ( T ) is ﬁnite. Let A b e the Sega l pr ecategor y corresp onding to T . Step 3. Use the discussion of Sectio n 17.4.1 to obtain a morphism A → A ′ inducing a weak equiv a lence of r ealizations, such that A ′ 1 / is connected. F or this step we need a pro of that π 1 ( T ) = 1. In the absence of a sp eciﬁc (ﬁnite) pro of, search ov er all pro ofs. Step 4. Use Coro llary 17.3 .7 to r eplace A ′ by a Segal precateg ory B with |A ′ | → |B | a weak equiv alence, such that the n − 1-type of B 1 / is equiv alen t to Ω |B | which in turn is equiv alen t to Ω | X | . Let Y = B 1 / as a simplicial s et. Note tha t Y is c o nnected and π 1 ( Y ) is ﬁnite, b eing equal to π 2 ( T ). W e have π i ( X ) = π i − 1 ( Y ) for 3 ≤ i ≤ n . Step 5. Cho ose a universal cov er of Y , a nd mo d out by a maximal tree in the 1 -skeleton to obtain a simplicia l set Z , with ﬁnitely many nondegenera te simplices, with Z 0 = ∗ , and with a pro o f that π 1 ( Z ) = 1. W e have π i ( X ) = π i − 1 ( Z ) for 3 ≤ i ≤ n . Go back to the b e ginning of the algorithm and plug in ( n − 1) and Z . Keep doing this until, at the step wher e we calculate π 2 of the new ob ject, we end up having calc ula ted π i ( X ) as desir ed. 378 Gener ators and r elations for Se gal c ate gories 17.4.5 Pro ofs of Go demen t W e p ose the following questio n: how could o ne obtain, in the pro cess of applying the a bove algo rithm, a n explicit pro o f that at each s ta ge the fundamen tal g roup (of the universal c over Z in step 5) is trivial? This could then be plugged into the machinery a bove to obtain an explicit strategy , th us we would avoid having to try all poss ible str ategies. T o do this we would need an explicit pro of that π 1 ( Y ) is ﬁnite in step 4, a nd this in turn w ould b e based on a pro of that π 1 ( Y ) = π 2 ( T ) a s well a s a pro of of the Go dement prop erty that π 2 ( T ) is ab elian. 17.5 Example: π 3 ( S 2 ) The story behind the preprint [1 95] was that Ronnie Br own came b y T oulouse for Jea n P radines’ retirement pa rty , and we were discus sing Seifert-V an Kampen. He pointed out that the result o f [193] didn’t seem to lead to a ny actua l calculations. After that, I tr ie d to use that tech- nique (in its simpliﬁed Segal-ca tegoric version) to calculate π 3 ( S 2 ). It was appare nt fro m this calcula tion that the pro c e ss w as eﬀective in gen- eral. W e des c rib e here what happ ens for calculating π 3 ( S 2 ). W e take as simplicial mo del a simplicial set w ith the bas epo int as unique 0- cell ∗ and w ith o ne no ndegenerate simplex e in degree 2 . Note tha t this lea ds to ma n y degenerate simplices in degrees ≥ 2 (how ev er ther e is only one degenerate simplex which we denote ∗ in degree 1). W e follow out what happens in a lang uage of cell-additio n. Thus w e don’t feel required to take the whole cone C at eac h step of a n op er ation Arr ( A , m ); we take any addition of cells to A m/ lifting cells in A 1 / × . . . × A 1 / . W e keep the no tation A for the r esult of each op eration (since our discussion is linear, this shouldn’t cause to o muc h confusion). The ﬁrst step is to (2 , 0)-arra nge A . W e do this by adding a 1-cell joining the tw o 0-cells in A 2 / , in an op eration of t yp e Ar r ( A , 2). Note that b oth 0-cells ma p to the same po int A 1 / × A 1 / = ∗ . The ﬁrst result of this is to add on 1- cells in the A m/ connecting all of the v arious degeneracies of e , to the ba sep oint. Thus the A m/ bec ome co nnec ted. Additionally we get a new 1-cell added o nto to A 1 / corres p o nding to the third face (02). F urthermo re, we obtain all images of this ce ll by degeneracies m → 1. Thus we g et m circles attac hed to the pieces which 17.5 Ex ample: π 3 ( S 2 ) 379 bec ame connected in the ﬁrst pa rt of this ope ration. Now eac h A m/ is a wedge of m cir cles. In pa r ticular note that A is now ( m, 1 )-arra nged for all m . The next step is to (2 , 2)-ar range A . T o do this, note that the Sega l map is S 1 ∨ S 1 = A 2 / → A 1 / × A 1 / = S 1 × S 1 . T o arra nge this ma p we hav e to add a 2 -cell to S 1 ∨ S 1 with a ttaching map the commutator relation. Again, this has the result o f adding on 2-cells to all of the A m/ ov er the pairwise co mm utators of the lo ops. F urthermore, we o btain a n extra 2- cell added o n to A 1 / via the edge (02). The a tta ching map her e is the c o mm utator of the generator with itself, s o it is homoto pically trivial and we hav e added on a 2 - sphere. (Note in passing that this 2-sphere is what gives rise to the class of the Hopf map). Aga in, we obtain the imag es of this S 2 by all of the degeneracy ma ps m → 1. Now A 1 / = S 1 ∨ S 2 , A 2 / = ( S 1 × S 1 ) ∨ S 2 ∨ S 2 , and in general A is ( m, 2)-arr anged for all m . Lo oking for w ard to the next section, we see that adding 3-cells to A m/ for m ≥ 3 in the appropriate wa y as describ ed in the pro o f of 17.3.5, will end up re sulting in the addition of 4-c e lls (or higher) to A 1 / so this no long er aﬀects the 2-type of A 1 / . Thus (for the purpo ses of getting π 3 ( S 2 )) we may now ignore the A m/ for m ≥ 3. The remaining op era tion is to (2 , 3)-arr ange A . F or this, lo ok a t the Segal ma p A 2 / = ( S 1 × S 1 ) ∨ S 2 ∨ S 2 → A 1 / × A 1 / = ( S 1 ∨ S 2 ) × ( S 1 ∨ S 2 ) . Let C b e the mapping cone on this map. Then w e end up atta ching one copy of C to A 1 / along the third edg e map A 2 / → A 1 / . This gives the answer for the 2-type of Ω S 2 : τ ≤ 2 (Ω S 2 ) = τ ≤ 2  ( S 1 ∨ S 2 ) ∪ ( S 1 × S 1 ) ∨ S 2 ∨ S 2 C  . T o c a lculate π 2 (Ω S 2 ) we revert to a ho mological formulation (beca use it isn’t eas y to “see” the cone C ). In homolog y of degree ≤ 2, the ab ov e 380 Gener ators and r elations for Se gal c ate gories Segal ma p ( S 1 × S 1 ) ∨ S 2 ∨ S 2 → ( S 1 ∨ S 2 ) × ( S 1 ∨ S 2 ) is an iso mo rphism. Thus the map A 2 / → C is an isomor phism on ho- mology in degr ees ≤ 2, and adding in a copy of C along A 2 / do esn’t change the homology . Th us H 2 (Ω S 2 ) = H 2 ( S 1 ∨ S 2 ) = Z . Noting that (as we know from general principles) π 1 (Ω S 2 ) = Z acts trivially on π 2 (Ω S 2 ) and π 1 itself has no homo logy in degree 2, we get that π 2 (Ω S 2 ) = H 2 (Ω S 2 ) = Z . Exer cise: Calculate π 4 ( S 3 ) using the above method. R emark: our ab ov e r ecourse to homolog y calcula tions sug gests that it might b e interesting to do pusho uts and the o per ation C at in the context of simplicial chain complexes. 17.5.1 Seeing Kan’s simplicial free groups Using the ab ove pr o cedure, we c an a ctually see how K an’s simplicial free groups ar ise in the ca lculation for a n ar bitrary simplicial set X . They ar ise just from a ﬁrst stage wher e we add on 1- cells. Namely , if in doing the pro cedure Ar r ( A , m ) we r eplace C by a choice of 1-cell joining any tw o comp onents of A m/ which go to the same comp onent under the Seg al map, then applying this op eratio n for v arious m , we obtain a simplicia l space who se comp onents are connected and homo - topic to wedges of circles. (W e hav e to start with a n X having X 1 = ∗ ). The resulting simplicial spa ce has the sa me realiz ation as X . If X has only ﬁnitely many nondegenera te s implices then one can stop a fter a ﬁnite num ber o f applica tions o f this o per ation. T aking the fundamental groups of the compo nent spaces (based at the degene r acy of the unique basep oint) gives a simplicial free group. T a k ing the classifying simpli- cial sets o f these groups in each comp onent we obta in a bisimplicial set whose realization is equiv alent to X . This bisimplicial set actually satis- ﬁes A p, 0 = A 0 ,k = ∗ , in other words it satisﬁes the globular condition in bo th directions! W e ca n therefore vie w it as a Sega l precatego ry in tw o wa ys. The seco nd wa y , interc hanging the t wo v a riables, yields a Segal precatego r y where the Sega l maps are isomorphisms (bec ause a t ea ch stage it was the clas sifying simplicia l set for a gr oup). Thus, viewed in this wa y , it is a Sega l gro up oid a nd Sega l’s theorem implies that the 17.5 Ex ample: π 3 ( S 2 ) 381 simplicial set p 7→ A p, 1 , which is the under lying set of a simplicia l free group, has the ho motopy type of Ω X . P A R T IV THE MODEL STR UCTURE 18 Sequen tially free precategories In this chapter, we contin ue the s tudy of weakly enriched categ o ries by lo oking at some basic o b jects: these are the c a tegories with an ordered set of ob jects x 0 , . . . , x n and morphisms o ther than the identit y from x i to x j only when i < j . More pr ecisely we consider the fr e e catego ries of this type obtained by sp ecifying an ob ject B i ∈ M of mor phisms from x i − 1 to x i for 1 ≤ i ≤ n ; then the ob ject of morphis ms from x i to x j should b e the pr o duct of the B k for k = i + 1 , . . . , j . One of the main tasks is to lo o k a t a notion of pr ecategory which cor resp onds to this notion of ca tegory . At the end will b e our main calculation, which is w ha t happ ens when one takes the pro duct of t wo such catego ries. Throughout, the notion of weak equiv alence on PC ( X , M ) is the one given b y the mo del structure of Theorem 14.1.1 . 18.1 Imp osing the Segal condition on Υ Recall M -precatego ries Υ( B 1 , . . . , B k ) deﬁned in Sections 12.5 and 16 .1. These can b e strictly categoriﬁed, which is to say that w e can construct corres p o nding str ict M -categories which will b e weakly equiv alent (The- orem 1 8.1.2 b elow). Deﬁne a pr ecategory e Υ k ( B 1 , . . . , B k ) as follows: the set o f ob jects is the same a s for Υ k ( B 1 , . . . , B k ), tha t is [ k ] = { υ 0 , . . . , υ k } . F or any sequence υ i 0 , . . . , υ i n with i 0 ≤ . . . ≤ i n , we put e Υ k ( B 1 , . . . , B k )( υ i 0 , . . . , υ i n ) := B i 0 +1 × B i 0 +2 × · · · × B i n − 1 × B i n . (18.1.1) This includes the unitalit y co ndition e Υ k ( B 1 , . . . , B k )( υ i , . . . , υ i ) = ∗ . F or any other sequence, that is to say any sequence w hich is not increa sing, This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 386 Se quential ly fr e e pr e c ate gorie s the v alue is ∅ . Note in particular that e Υ k ( B 1 , . . . , B k )( υ i − 1 , υ i ) = B i = Υ k ( B 1 , . . . , B k )( υ i − 1 , υ i ) . (18.1.2) By the adjunction prop erty of Υ k ( B 1 , . . . , B k ) there is a unique map Υ k ( B 1 , . . . , B k ) → e Υ k ( B 1 , . . . , B k ) (18.1.3) inducing the identit y (18.1 .2) on a djacent pairs of ob jects. In fac t, w e can consider Υ k ( B 1 , . . . , B k ) as a subob ject of e Υ k ( B 1 , . . . , B k ) with this map a s the inclusion. Lemma 18.1.1 The pr e c ate gory e Υ k ( B 1 , . . . , B k ) is a stict M -c ate gory, in p articular it is a Se gal M -c ate gory. Pr o of If υ i 0 , . . . , υ i n is an incr easing sequence i.e. i 0 ≤ . . . ≤ i n , and if 1 ≤ j ≤ n − 1 then by the formula (18 .1.1) the natural maps obtained by splitting the sequence of ob jects at υ i j give an isomorphism e Υ k ( B 1 , . . . , B k )( υ i 0 , . . . , υ i n ) e Υ k ( B 1 , . . . , B k )( υ i 0 , . . . , υ i j ) × e Υ k ( B 1 , . . . , B k )( υ i j , . . . , υ i n ) . ∼ = ↓ By induction it follows that the Segal maps a re isomorphisms. One should think of e Υ k ( B 1 , . . . , B k ) as b eing the fr e e M -c ate gory gener ate d by morphism obje cts B i going fr om υ i − 1 to υ i . Here the o b jects are linear ly order ed, and the generating ob jects of M are placed b e t ween adjacent ob jects in the ordering. W e would like to make precis e this intuition by showing that it is the Seg al M -ca tegory genera ted by the preca tegory Υ k ( B 1 , . . . , B k ) as stated in the following theorem. Theorem 18 .1.2 F or any k and any se quenc e of obje cts B 1 , . . . , B k , the inclus ion (18.1 .3) is a we ak e qu ivalenc e in PC ( { υ 0 , . . . , υ k } , M ) . If e ach B i is c oﬁbr ant, then it is a trivial c oﬁbr ation in the inje ctive m o del structur e. 18.2 Sequen tially free precategories in general Before g etting to the pro o f of the previous theo rem, whic h will b e done at the end of the chapter o n pag e 395 , it is useful to ge ne r alize the 18.2 S e quential ly fr e e pr e c ate gorie s in gener al 387 ab ov e situation b y giving a criterio n for when a preca tegory will lead to a free Segal ca tegory with linea rly or dered ob ject set and ge ne r ators B i betw een adjacent ob jects. Deﬁnition 18.2 . 1 A sequentially free M -preca tegory c onsists of a ﬁnite line arly or der e d set X = { x 0 , . . . , x k } to gether with a s tructur e of M -pr e c ate gory A ∈ PC ( X , M ) , satisfying the fol lowing pr op erties: (SF1)—if x i 0 , . . . , x i n is a se quenc e of obje cts which ar e not incr e asing, i.e. ther e is some 0 < j ≤ n with i j − 1 > i j , t hen A ( x i 0 , . . . , x i n ) = ∅ ; and (SF2)—if x i 0 , . . . , x i n is a se quenc e of obje cts in incr e asi ng or der i.e. i j − 1 ≤ i j for al l 0 < j ≤ n , t hen the outer map for t he n -simplex pr ov ides a we a k e quivalenc e A ( x i 0 , . . . , x i n ) ∼ → A ( x i 0 , x i n ) . (18.2.1) Remark 18.2. 2 Note t hat c onditio n (SF2) for a se quenc e ( x i ) of length n = 0 says that the map ∗ = A ( x i ) → A ( x i , x i ) is a we a k e quiv- alenc e. W e often say that A is sequentially free with r esp e ct to a given or der on X if ( X, A ) is sequentially free for the ordering in question. Lemma 18.2.3 Both Υ k ( B 1 , . . . , B k ) and e Υ k ( B 1 , . . . , B k ) ar e se quen- tial ly fr e e M -pr e c ate go ries with r esp e ct to the or deri ng on [ k ] . Pr o of Both clear ly satisfy (SF1). F or e Υ k ( B 1 , . . . , B k ) the co ndition (SF2) holds by construction. F or Υ k ( B 1 , . . . , B k ), supp ose we hav e an increasing sequence of ob jects υ i 0 ≤ · · · ≤ υ i n . If i n = i 0 then the v a lues on b oth sides of (18.2.1 ) are ∗ . If i n = i 0 + 1 then the v alues on bo th sides are B i n , wherea s if i n > i 0 + 1 the v a lues on b oth sides ar e ∅ . In all three cases the ma p (18.2.1 ) is an isomorphis m, a fortiori a weak equiv alence. Lemma 18. 2.4 Supp ose X is a ﬁxe d line arly or der e d ﬁnite set, β is an or dinal, and { A ( b ) } b ∈ β is a tr ansﬁnite se quenc e of M -pr e c ate gories, that is to say a functor β → PC ( X, M ) . Supp ose that e ach ( X , A ( b )) is a se quential ly fr e e M -pr e c ate gory with r esp e ct to the given or der on X , and that for any b ≤ b ′ the tr ansition map A ( b ) → A ( b ′ ) is an inje ctive c oﬁbr ation in PC ( X , M ) . Then the c o limit ( X, colim b ∈ β A ( b )) is again a se quential ly fr e e M -pr e c ate gory with r esp e ct to t he same or dering. Pr o of The colimit is calculated lev elwise a s a ∆ o X -diagra m in M , since ﬁltered colimits pre serve the unit ality condition. A co limit of ob jects ∅ is 388 Se quential ly fr e e pr e c ate gorie s again ∅ , so the colimit satisﬁes (SF1). The left prop erness hyp o theses on M implies trans ﬁnite left prop erne s s (Prop os itio n 9.5.3), so the weak equiv alence (18.2.1) is pr eserved in the colimit whos e transitio n maps are co ﬁbrations, giving (SF2). W e now come to one of the main steps where we g ain some control ov er the pro cess of gener ators a nd r e lations. Rec a ll from Section 16.4 the op eration Gen consisting of a pplying one step of the calculus of generator s and relations. Lemma 18.2.5 S u pp ose ( X , A ) is a se quential ly fr e e M -pr e c ate gory with line arly or der e d obje ct set X = { x 0 , . . . , x m } , and x a , x a +1 , . . . , x b is a st r ictly incr e asi ng se quenc e of adjac ent obje cts with 0 ≤ a < b ≤ m . Then the new M -pr e c ate go ry G en (( X, A ); x a , . . . , x b ) is also s e quen- tial ly fr e e with the same or der e d set of obje cts X , and furt hermor e for any a < j ≤ b the map A ( x j − 1 , x j ) → Gen (( X , A ); x a , . . . , x b )( x j − 1 , x j ) is an isomorphism (henc e a we ak e quivalenc e) in M . Pr o of Fix a facto r ization E , e, p 1 , . . . , p b − a as used for the construction of G en (( X, A ); x a , . . . , x b ). Note that p : E ∼ → A ( x a , x a +1 ) × · · · × A ( x b − 1 , x b ) is a weak equiv a lence. The sequence of ob jects x a , . . . , x b is disjoint, so we can use the des cription of G en (( X, A ); x a , . . . , x b ) given in L e mma 16.4.2. Tha t says that for a ny sequence of the form x i 0 , . . . , x i p if a ≤ i 0 ≤ · · · ≤ i p ≤ b with i 0 + 2 ≤ i p then Gen ( A ; x a , . . . , x b )( x i 0 , . . . , x i p ) = A ( x i 0 , . . . , x i p ) ∪ A ( x a ,...,x b ) E . (18.2.2) but for any other sequence, Gen ( A ; x a , . . . , x b )( x i 0 , . . . , x i p ) = A ( x i 0 , . . . , x i p ) . (18.2.3) W e can now check the conditio ns (SF1) and (SF2). If the sequence of ob jects ( x i 0 , . . . , x i p ) is not increa sing, then it falls int o the second case (18.2.3), a nd A ( x i 0 , . . . , x i p ) = ∅ by (SF1) for A , which g ives (SF1) for Gen ( A ; x a , . . . , x b ). Supp ose that ( x i 0 , . . . , x i p ) is increas ing. W e need to chec k (SF2). If either i 0 < a or i p > b then we ag ain fall in to case (1 8.2.3) for b oth the full sequence ( x i 0 , . . . , x i p ) and also the pair o f endp oints 18.2 S e quential ly fr e e pr e c ate gorie s in gener al 389 ( x i 0 , x i p ). Th us, by (SF2) for A we hav e a weak equiv alence Gen ( A ; x a , . . . , x b )( x i 0 , . . . , x i p ) = A ( x i 0 , . . . , x i p ) A ( x i 0 , x i p ) = Gen ( A ; x a , . . . , x b )( x i 0 , x i p ) ∼ ↓ giving this case o f (SF2) for Gen ( A ; x a , . . . , x b ). Supp ose on the other hand that a ≤ i 0 ≤ i p ≤ b . If i p ≤ i 0 + 1 then we a gain are in case (18.2.3) for b oth the full sequence and the pair of endp oints, so we get the condition (SF2) as ab ove. Supp ose therefore that i 0 + 2 ≤ i p . In the diagram A ( x i 0 , . . . , x i p ) ← A ( x a , . . . , x b ) → E A ( x i 0 , x i p ) ↓ ← A ( x a , . . . , x b ) w w w w w w w w w → E w w w w w w w w w w the left vertical arr ow is an equiv alence b y (SF2) for A . By left pr o p e r - ness of M via Corollar y 9 .5.2, the vertical maps therefore induce a weak equiv alence from the pushout of the top row to the pushout o f the bo t- tom row. By eq uations (18.2.2) which apply b oth to the full sequence ( x i 0 , . . . , x i p ) and the pair of endp oints ( x i 0 , x i p ), these pushouts are r e- sp ectively Gen ( A ; x a , . . . , x b )( x i 0 , . . . , x i p ) and G en ( A ; x a , . . . , x b )( x i 0 , x i p ). Thu s, the map from the one to the other is a n equiv alence, which g ives condition (SF2) in this la st ca se. This proves that Gen (( X , A ); x a , . . . , x b ) is a gain a sequentially free M -precateg ory . F or the last statement, no te that the sequence ( x j − 1 , x j ) falls into cas e (18.2.3) beca use the space b etw een the endp oints is only 1. Hence the formula (18.2.3) says that the map A ( x j − 1 , x j ) → Gen (( X , A ); x a , . . . , x b )( x j − 1 , x j ) is a n is o morphism. Recall that the ma p A → Gen (( X, A ); x a , . . . , x b ) is a weak equiv a- lence in PC ( X ; M ). Lemma 18.2.6 S u pp ose ( X , A ) is a se quential ly fr e e M -pr e c ate gory with or der e d obje ct s et X = { x 0 , . . . , x m } . By iter ating a series of op er- ations of the form A 7→ Gen (( X , A ); x a , . . . , x b ) we c an obtain a we ak 390 Se quential ly fr e e pr e c ate gorie s e quiva lent se quential ly fr e e M -pr e c ate gory ( X, A ) → ( X, A ′ ) such that A ′ satisﬁes the S e ga l c onditio ns. Pr o of W e show how to obtain the Segal condition for strictly incr easing sequences of adjace n t ob jects. At the end of the pr o of we go fro m here to the Seg al condition for g eneral sequences. Fix a n integer n 0 and s uppo s e ( X , A ) satisﬁes the Segal condition for all a dja c en t sequence s ( x c , x c +1 , . . . , x d ) with d − c > n 0 , and a certain num b e r o f a dja c e n t s equences ( x c v , x c v +1 , . . . , x d v ) for ( c v , d v ) indexed by v ∈ V for some s et v , with d v − c v = n 0 . Supp ose 0 ≤ a < b ≤ m with b − a = n 0 . Then Gen (( X , A ); x a , . . . , x b ) satisﬁes the Segal co ndition for all adjacent sequences ( x c , x c +1 , . . . , x d ) with d − c > n 0 , for the given adjacent sequences ( x c v , x c v +1 , . . . , x d v ) with v ∈ V , and a lso for the sequence ( x a , . . . , x b ). Indeed, if ( x c , x c +1 , . . . , x d ) with d − c > n 0 then the terms entering into the Segal map for this sequence are a ll cov ered by the situation (18 .2.3) in the explicit descriptio n of Gen (( X, A ); x a , . . . , x b ) used in the pr evious pr o of (s e e Lemma 16.4 .2). Note that the ter ms in the pro duct on the rig h t hand s ide o f the Segal map are of the form Gen (( X, A ); x a , . . . , x b )( x j − 1 , x j ) = A ( x j − 1 , x j ). Here we use the condition tha t we ar e o nly lo oking at adjacent sequences . By the r ecurrence hypo thes is on A , the Seg al map for this s equence is an equiv a lence. Similarly , if ( x c v , x c v +1 , . . . , x d v ) is one of our given sequences with d v − c v = n 0 , and if it is diﬀer ent fro m the sequence ( x a , . . . , x b ), then either c v < a or d v > b so again everything en tering int o the Segal ma p for this sequence is the same a s for A , by (18 .2.3). Thus, again the inductive h yp othesis on A implies the Seg al conditio n for Gen (( X, A ); x a , . . . , x b )( x c v , x c v +1 , . . . , x d v ). A t the s equence ( x a , . . . , x b ) equation (18.2.2) gives Gen ( A ; x a , . . . , x b )( x a , . . . , x b ) = A ( x a , . . . , x b ) ∪ A ( x a ,...,x b ) E = E and the ma p E → A ( x a , x a +1 ) × · · · × A ( x b − 1 , x b ) is an equiv alence by h ypo thesis on the choice of E . Therefore, the Segal condition holds at the sequence ( x a , . . . , x b ) to o a nd we ca n a dd this to our co llection V of go o d sequences of lenght n 0 . So, the inductiv e pro cedur e is to start with the maximal sequence x 0 , . . . , x m of length m , imp ose the Segal condition here by changing A to Gen ( A ; x 0 , . . . , x m ); then successively impose the Seg a l condition on all 18.2 S e quential ly fr e e pr e c ate gorie s in gener al 391 sequences of le ngth m − 1, then m − 2 and so o n, using the ab ov e inductive observ ation. At eac h step A is replace d b y Gen ( A ; x a , . . . , x b ) and there is a ma p from the old to the new A whic h is a weak eq uiv a lence in PC ( X, M ). By Lemma 1 8.2.5, the new A is alw ays ag ain a sequen tially free M -pre c ategory . Combining these steps (there are a ﬁnite n umber) down to n 0 = 2 we arrive at a ma p ( X , A ) → ( X, A ′ ) which is a weak equiv alence in PC ( X , M ), and such that A ′ is a sequentially free M - precatego r y which sa tisﬁes the Segal condition for a ll sequences of the form ( x c , . . . , x d ). W e claim that this implies that A ′ satisﬁes the Segal condition for any sequence of the form x i 0 , . . . , x i p . Note that if p = 0 then the Segal co n- dition is a utomatic since A ′ ( X ) = ∗ . Supp ose p ≥ 1 and cons ide r our se- quence ( x i 0 , . . . , x i p − 1 , x i p ). If any i j − 1 > i j then A ′ ( x i 0 , . . . , x i p − 1 , x i p ) = ∅ and A ′ ( x i j − 1 , x i j ) = ∅ . T he seco nd statement, plus Lemma 10.0.1 0 on the dir e ct pro duct with ∅ , imply that the right hand side of the Segal map is ∅ , which is the same as the left s ide by the ﬁrst statement. So in this case , the Seg al co ndition is automatic. Hence we may assume that i j − 1 ≤ i j for all 1 ≤ j ≤ p . Let a := i 0 and b := i p , and denote by ( x a , . . . , x b ) the full sequence of adjacent elements (each counted once) going fro m x a to x b . There is a unique map σ : ( x i 0 , . . . , x i p − 1 , x i p ) → ( x a , . . . , x b ) in ∆ X , sending each x i j to the same ob ject at the unique place it o ccurs in ( x a , . . . , x b ). Note that we are using here the reductio n of the previous pa ragra ph that i j − 1 ≤ i j for all 1 ≤ j ≤ p , g uaranteeing that each x i j o ccurs in the list ( x a , . . . , x b ). This σ induces a map σ ∗ : A ′ ( x a , . . . , x b ) ∼ → A ′ ( x i 0 , . . . , x i p − 1 , x i p ) . (18.2.4) That this is a weak equiv a lence, can be seen by considering the diag ram with maps to the spa nning pa ir ( x a , x b ) = ( x i 0 , x i p ): A ′ ( x a , . . . , x b ) → A ′ ( x i 0 , . . . , x i p − 1 , x i p ) A ′ ( x a , x b ) ↓ = = = = = = = = A ′ ( x i 0 , x i p ) . ↓ The vertical ma ps a re weak equiv alences b y the sequentially free condi- tion, s o the to p map is a w eak equiv a lence by 3 for 2. On th e other hand, for ea ch pair in the origina l sequence, consider the 392 Se quential ly fr e e pr e c ate gorie s Segal ma p fo r it: σ i 0 ,i 1 : A ′ ( x i 0 , . . . , x i 1 ) → A ′ ( x i 0 , x i 0 +1 ) × · · · × A ′ ( x i 1 − 1 , x i 1 ) , . . . σ i p − 1 ,i p : A ′ ( x i p − 1 , . . . , x i p ) → A ′ ( x i p − 1 , x i p − 1 +1 ) × · · · × A ′ ( x i p − 1 , x i p ) . The mini-sequences app earing on the le ft hand sides, are the se quences of length i j − i j − 1 going from i j − 1 to i j by interv als of step 1. Whenever i j − 1 = i j , we have a sequence of length zero and b oth sides o f the map are equal to ∗ . The case of the Seg al condition which we already know, says that the σ i j − 1 ,i j are weak equiv alences. Putting these all together, we get a weak eq uiv a le nce who se ta rget is the full Segal pro duct for the sequence ( x a , . . . , x b ) considered ab ov e: A ′ ( x i 0 , . . . , x i 1 ) ×· · ·× A ′ ( x i p − 1 , . . . , x i p ) ∼ → A ′ ( x a , x a +1 ) ×· · ·× A ′ ( x b − 1 , x b ) . On the o ther hand, the map to the spanning interv al gives a map for each mini-sequence A ′ ( x i j − 1 , . . . , x i j ) ∼ → A ′ ( x i j − 1 , x i j ) (18.2.5) which is a w eak equiv alence, by the condition that A ′ is sequentially free. The mini-s e q uences ma p in to the full adjacent sequence ( x a , . . . , x b ), giving ma ps A ′ ( x a , . . . , x b ) → A ′ ( x i j − 1 , . . . , x i j ) . The Seg al map for ( x a , . . . , x b ) th us factors as A ′ ( x a , . . . , x b ) ∼ → A ′ ( x i 0 , . . . , x i 1 ) × · · · × A ′ ( x i p − 1 , . . . , x i p ) A ′ ( x a , x a +1 ) × · · · × A ′ ( x b − 1 , x b ) . ∼ ↓ (18.2.6) The seco nd arr ow is a weak equiv a lence as po int ed out pr e viously , a nd the comp os ition is a weak equiv alence b y the Seg a l co ndition for ( x a , . . . , x b ) which we a lready know, so the ﬁrs t map is a weak equiv a lence by 3 for 2. 18.2 S e quential ly fr e e pr e c ate gorie s in gener al 393 Now we hav e a commut ative diagr am A ′ ( x a , . . . , x b ) → A ′ ( x i 0 , . . . , x i p − 1 , x i p ) A ′ ( x i 0 , . . . , x i 1 ) × · · · × A ′ ( x i p − 1 , . . . , x i p ) ↓ → A ′ ( x i 0 , x i 1 ) × · · · × A ′ ( x i p − 1 , x i p ) ↓ where the top ar row is a weak eq uiv alence as s een ab ov e (18.2.4 ), the left vertical map is a weak equiv alences by (18 .2.6), and the bottom map is a w eak equiv a lence by combining to gether the equiv alences (18.2 .5). By 3 for 2, it follows that the right vertical map is a weak equiv alence, which is the Segal condition for the sequence ( x i 0 , . . . , x i p − 1 , x i p ). This completes the pro of of the lemma. Corollary 18. 2 .7 Supp ose ( X , A ) is a se quential ly fr e e or de r e d M - pr e c ate gory, and let r : ( X , A ) → ( X, A ′ ) b e a ﬁ br ant r eplac ement in either the pr oje ctive or inje ctive mo de l struct ur e on PC ( X ; M ) c on- structe d in The or em 14.1.1. Then ( X, A ′ ) is a se quential ly fr e e or der e d M -pr e c ate gory for the same or de r on X , and for any 0 < j ≤ n the map A ( x j − 1 , x j ) → A ′ ( x j − 1 , x j ) is a we ak e quivalenc e. The same c onclusions holds if, inste ad of a ﬁbr ant r eplac ement , r is a we ak e quiva lenc e to an obje ct A ′ which satisﬁes the Se gal c onditions. Pr o of Use either the pr o jective or the injective structure in what fol- lows. Note that the sequentially free condition is preser ved by levelwise weak equiv a lences of unital dia grams on ∆ o X . By Lemma 18.2 .6, there is a map A → A ′′ , weak equiv alence in PC ( X, M ), such that A ′′ satisﬁes the Segal conditions and is still se- quentially free. Consider furthermor e a map s : A ′′ → A 3 which is a trivial coﬁbra tion to a ﬁbrant ob ject in PC ( X, M ). Then A 3 also sat- isﬁes the Sega l conditions, so Le mma 1 4.4.1 says that s is a levelwise weak equiv a lence. It fo llows that A 3 is a gain sequentially free. Our diﬀerent ﬁbrant replacement r : A → A ′ is a trivia l coﬁbra tion, so there is a map g : A ′ → A 3 compatible with the maps fro m A . By 3 for 2 this map g is a weak equiv alence in PC ( X , M ), so ag ain by Lemma 14.4.1 it is a levelwise weak equiv alence, hence A ′ is sequentially free. F urthermore, A ( x j − 1 , x j ) → A ′′ ( x j − 1 , x j ) is a weak e q uiv a lence in M ; it follows fro m the ab ove levelwise weak equiv alences compatible with the maps from A , that the same is true fo r A 3 , then A ′ . F or the la s t para graph, supp ose r : A → A ′ is a weak equiv a lence in 394 Se quential ly fr e e pr e c ate gorie s PC ( X, M ) and A ′ satisﬁes the Segal conditions. Then c ho osing ﬁr st a ﬁbrant replacement A 3 of A ′ , then a factorization, we can get a square A → A ′ A ′′ ↓ → A 3 ↓ such that the v ertical arrows are ﬁbrant repla cement s a nd all ar rows are weak equiv alences. By Lemma 14.4 .1, the r ight vertical and b o tto m maps are le velwise weak eq uiv alences since A ′ and A ′′ satisfy the Segal conditions. By the ﬁrst part of the pr esent lemma, A ′′ is sequentially free, it follows that A 3 and then A ′ are sequentially free. O n a djacent pairs of o b jects the left v ertical map induces an equiv a le nce, so b y 3 for 2 the top ma p do es to o. This shows that the require d statements hold for A ′ . In the situation of the previous coro llary , s upp os e ( x i 0 , . . . , x i n ) is an increasing sequence of o b jects. Then we have weak equiv a lences A ′ ( x i 0 , x i n ) ∼ → A ′ ( x i 0 , . . . , x i n ) ∼ → A ′ ( x i 0 , x i 1 ) × · · · × A ′ ( x i n − 1 , x i n ) . On the other hand if the s e quence is not increasing , the co rresp onding morphism space is ∅ . Suppo se that our sequence of ob jects is a s equence of adjacent ob jects, that is to s ay look at a sequence of the fo r m ( x a , x a +1 , . . . , x b ) for a ≤ b . The condition of the corolla ry says furthermor e that we hav e weak equiv - alences A ( x j − 1 , x j ) ∼ → A ′ ( x j − 1 , x j ) for a < j ≤ b . These eq uiv a le nces go together to give an equiv alence on the level of the direct pro duct, b y the pro duct conditions (PROD) and (DCL) for M as in Lemma 1 0.0.11. Thu s, the pair of weak equiv alences in the pr evious paragr aph extends to a chain o f equiv a lences A ′ ( x a , . . . , x b ) ∼ → A ′ ( x a , x a +1 ) × · · · × A ′ ( x b − 1 , x b ) A ′ ( x a , x b ) ∼ ↓ A ( x a , x a +1 ) × · · · × A ( x b − 1 , x b ) . ∼ ↑ (18.2.7) All in all, this says that ( X , A ′ ) lo oks up to homotopy very muc h like the e Υ k ( B 1 , . . . , B k ) deﬁned at the s tart, with B i = A ( x i − 1 , x i ). 18.2 S e quential ly fr e e pr e c ate gorie s in gener al 395 Corollary 1 8 .2.8 Supp ose f : ( X, A ) → ( X, B ) is a morphism in PC ( X ; M ) b etwe en M -pr e c ate gori es which ar e b oth se quential ly fr e e for the same or de ring o f the un derlying set of obje cts X . Supp ose that for any adjac ent obje cts x j − 1 , x j in the or de ring, f induc es a we ak e quivale nc e A ( x j − 1 , x j ) ∼ → B ( x j − 1 , x j ) . Then f is a we ak e quivalenc e in P C ( X ; M ) . Pr o of View the ob ject set a s num bered X = { x 0 , . . . , x k } . Let ( X , A ′ ) and ( X , B ′ ) be ﬁbrant replacements for ( X , A ) and ( X , B ) resp ectively , and w e may a ssume that f extends to a map f ′ : ( X , A ′ ) → ( X , B ′ ). F or any 0 ≤ a ≤ b ≤ k , f ′ and f induce a mor phism b etw een the chains of equiv alences considered in (18.2.7), for A, A ′ and B , B ′ . The hypothesis of the pres e nt c o rollar y says that the induced map on the bo ttom rig ht corner is a weak eq uiv a lence; thus the induced map f ′ : A ′ ( x a , x b ) → B ′ ( x a , x b ) is a weak e q uiv a lence. Note on the other hand that if a > b then A ′ ( x a , x b ) = ∅ a nd B ′ ( x a , x b ) = ∅ . Thus f ′ induces a weak eq uiv alence on the mor phis m space for any pa ir of ob jects. By the Segal condition, it fo llows that f ′ is an ob jectwise weak eq uiv a le nce in the ca tegory of ∆ o X -diagra ms in M , so it is a weak e quiv a lence in the mo del structures. Since f ′ is a ﬁbrant replac emen t for f , this implies that f was a weak equiv alence. Pr o of of The or em 18.1.2 : the map f : Υ k ( B 1 , . . . , B k ) → e Υ k ( B 1 , . . . , B k ) satisﬁes the hyptheses of the pre vious co rollary , so we conclude that it is a weak equiv alence. If the B i are c o ﬁbrant, then the map in question is a coﬁbration in the injective mo del structure, b ecause is is an o b ject wise coﬁbration when viewed in the injective mo del structure: the induced maps ar e either iden tities, or maps from ∅ in to a pro duct of copies of B i . A pro duct of copies of B i is coﬁbrant, b eca us e of the car tesian conditions on M (see Lemma 10.0.12 ).  19 Pro ducts In this chapter, we consider the direct pro duct of tw o M -enriched pr e- categorie s. On the one hand, we would like to maintain the cartesia n or pro duct c o ndition (P ROD) for the new mo de l categ ory we ar e con- structing. On the other hand, compatibilit y with direct pro duct pr o- vides the main technical to ol we need in o rder to s tudy pushouts by weak eq uiv alences in PC ( M ). Recall that we already ha ve the injectiv e and pro jective model structures of Theorem 14.1.1 on PC ( X, M ) a t o ur disp o sal. Here is the place where we rea lly use th e full structure o f the catego ry ∆, as well as the unitality co ndition. At the end o f the chapter, we’ll discuss some c ounterexamples sho wing why these asp ects are necess ary . 19.1 Pro ducts of sequen tially free precategor ies Suppo se X = { x 0 , . . . , x m } and Y = { y 0 , . . . , y n } are ﬁnite linearly o r- dered sets, and ( X, A ) and ( Y , B ) are sequen tially f ree M -preca tegories, with resp ect to the orderings. The pro duct of thes e ob jects co nsidered in the ca tegory PC ( M ), has the for m ( X × Y , A ⊠ B ) where ( A ⊠ B )(( x i 0 , y j 0 ) , . . . , ( x i p , y j p ) := A ( x i 0 , . . . , x i p ) × B ( y j 0 , . . . , y j p ) . Let A → A ′ and B → B ′ be weak eq uiv a lences tow ards ob jects satis- fying the Sega l condition and which are therefor e also sequentially free (Corollar y 1 8.2.7). W e obtain a map of M -pre c ategories A ⊠ B → A ′ ⊠ B ′ on the ob ject set X × Y . W e would like to sho w that this is a w eak eq uiv- alence in PC ( X × Y ; M ). F or an y subset S ⊂ X × Y , let i ( S ) : S ֒ → X × Y deno te the inclusion This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 19.1 Pr o ducts of se quential ly fr e e pr e c ate go ries 397 map, a nd i ( S ) ∗ : PC ( X × Y , M ) → PC ( S, M ) the pullback map on M -prec a tegory structures. This can b e pushed ba ck to a M -pre c ategory str ucture with ob ject set X × Y , and the resulting functor will be deno ted a s i ( S ) ∗ ! for brevity: i ( S ) ∗ ! := i ( S ) ! i ( S ) ∗ : PC ( X × Y , M ) → PC ( X × Y , M ) . This may b e describ ed explicitly as follows. Suppos e C ∈ PC ( X × Y , M ). Recall from the ﬁrst paragr a ph of Section 12.3, that i ( S ) ! i ( S ) ∗ ( C ) is the same as the pre category i ( S ) ∗ ( C ) ov er set of ob jects S , extended by adding o n the discrete M -enriched categ ory on the complementary set X × Y − S . Thus, for any sequence of ob jects (( x i 0 , y i 0 ) , . . . , ( x i p , y i p )) we hav e: i ( S ) ∗ ! ( C )(( x i 0 , y i 0 ) , . . . , ( x i p , y i p )) = C (( x i 0 , y i 0 ) , . . . , ( x i p , y i p )) if a ll the ( x i j , y i j ) are in S ; if any o f the pairs is not in S and the sequence is no t constant then i ( S ) ∗ ! ( C )(( x i 0 , y i 0 ) , . . . , ( x i p , y i p )) = ∅ , while the unitality condition requires that i ( S ) ∗ ! ( C )(( x i 0 , y i 0 ) , . . . , ( x i 0 , y i 0 )) = ∗ , for a c onstant sequence at any po int ( x i 0 , y i 0 ) ∈ X × Y . Lemma 19.1.1 If C satisﬁes the Se gal c ondition, then so do es i ( S ) ∗ ( C ) (on obje ct set S ) and i ( S ) ∗ ! ( C ) (on obje ct set X × Y ). Pr o of This follows immediately from Lemma 12.3.1 . The ab ov e discussion is us ua lly applied to an ob ject of the form C = A ⊠ B ∈ PC ( X × Y ; M ). Here are the sp ecia l t yp es of subsets S we will consider. A b ox will b e a subset o f the form B a,b := { x 0 , . . . , x a } × { y 0 , . . . , y b } ⊂ X × Y , 398 Pr o ducts which can b e pictur ed for example with ( a, b ) = (5 , 3) as r r r r r r r r r r r r r r r r r r r r r r r r ( x 0 , y 0 ) ( x 0 , y b ) ( x a , y 0 ) ( x a , y b ) A notche d b ox is a subset B a,b ⊃ B ν a,b := B a,b − 1 ∪ { ( x ν , y b ) , . . . , ( x a , y b ) } , deﬁned when b ≥ 1 . Thus, a no tch ed b ox B ν a,b is a b ox B a,b min us a part of the top ro w { ( x 0 , y b ) , . . . , ( x ν − 1 , y b ) } . F or example a notc hed b ox with ( a, b ) = (5 , 3) a nd ν = 3 lo oks like r r r r r r r r r r r r r r r r r r r r r ( x 0 , y 0 ) ( x 0 , y b − 1 ) ( x a , y 0 ) ( x ν , y b ) ( x ν , y b − 1 ) ( x a , y b ) The b oxes and no tc hed boxes ar e deﬁned for 0 ≤ a ≤ m and 0 ≤ b ≤ n , and 0 ≤ ν ≤ a . If a = m, b = n then B a,b = X × Y , and if ν = 0 then B ν a,b = B a,b . F or any ν , the upp er right corner of B ν a,b is equa l to the p oint ( x a , y b ). The low er left corner is ( x 0 , y 0 ). The c orner of the notch of B ν a,b is the po int ( x ν , y b ). Note that B ν a,b − { ( x ν , y b ) } = B ν +1 a,b 19.1 Pr o ducts of se quential ly fr e e pr e c ate go ries 399 whenever 0 ≤ ν < a . F or ν = a we hav e B ν a,b − { ( x ν , y b ) } = B a,b − 1 . A tail is a subset o f the form T i 0 ,...,i p j 0 ,...,j p := { ( x i 0 , y j 0 ) , . . . , ( x i p , y j p ) } which will be considered whenever i 0 ≤ i 1 ≤ · · · ≤ i p , i u ≤ i u − 1 + 1 , j 0 ≤ j 1 ≤ · · · ≤ j p , j u ≤ j u − 1 + 1 , and ( i u , j u ) 6 = ( i u − 1 , j u − 1 ). Put another w a y , at eac h place in the pair of sequences, ther e ar e thr e e p oss ibilities for ( i u , j u ) in terms of ( i u − 1 , j u − 1 ): ( i u , j u ) = ( i u − 1 + 1 , j u − 1 ) or ( i u − 1 , j u − 1 + 1 ) o r ( i u − 1 + 1 , j u − 1 + 1 ) . Thu s, the sequence mov es in single horizontal, vertical or dia gonal steps. W e say that the ta il go es fr om ( x i 0 , y j 0 ) to ( x i p , y j p ). A ful l tail is o ne which go es from ( x 0 , y 0 ) to ( x m , y n ). A dipp er is a subset S which is the unio n of a notched box B ν a,b , with a tail T i 0 ,...,i p j 0 ,...,j p going fro m ( x i 0 , y j 0 ) = ( x a , y b ) to ( x i p , y j p ) = ( x m , y n ). These overlap a t the point ( x a , y b ). F or example, a dipp e r sta rting with the previous notched b ox, a nd g oing out to ( m, n ) = (9 , 7 ) could lo ok like this: q q q q q q q q q q q q q q q q q q q q q ( x a , y b ) ❫ ( x ν , y b ) ⑦ q q   q q q   q ( x m , y n ) F or b ≥ 1 , the union o f B b a,b with a tail T going from ( x a , y b ) to ( x m , y n ), is equal to the union of B 0 a,b − 1 with a tail T ′ = { ( x a , x b − 1 ) } going from ( x a , x b − 1 ) to ( x m , y n ), with the union o verlapping at the po int ( x a , x b − 1 ). F or exa mple, in the previous picture if we set ν = b it 400 Pr o ducts bec omes q q q q q q q q q q q q q q q q q q q ( x a , y b ) ❫ ( x a , y b − 1 ) ✛ q q   q q q   q ( x m , y n ) In this situation if furthermo re b − 1 = 0, t hen the s ubset S b eco mes a full tail go ing from ( x 0 , y 0 ) to ( x m , y n ). Note also that a b ox with a = 0 plus a tail, is again equal to a full tail. In view of this, w e consider the v ariables a, b, ν for the notc hed box in a dipp er S , only in the range 0 ≤ ν < a and 0 < b . The co rner of the notch ( x ν , y b ) is w ell-deﬁned, a nd S − { ( x ν , y b ) } is again either a dipper , or a full tail. The pro duct of the linea r o rders on X and Y is a pa rtial or de r on X × Y where ( x a , y b ) < ( x a ′ , y b ′ ) whenever a ≤ a ′ , b ≤ b ′ and ( a, b ) 6 = ( a ′ , b ′ ). Given a dipper S , let ( x ν , y b ) b e the cor ne r of the notch and deﬁne new subsets S > ( x ν ,y b ) < := { ( x ′ , y ′ ) ∈ S, either ( x ′ , y ′ ) < ( x ν , y b ) or ( x ν , y b ) < ( x ′ , y ′ ) } and S ≥ ( x ν ,y b ) ≤ := { ( x ′ , y ′ ) ∈ S, either ( x ′ , y ′ ) ≤ ( x ν , y b ) or ( x ν , y b ) ≥ ( x ′ , y ′ ) } . Note tha t S ≥ ( x ν ,y b ) ≤ = S > ( x ν ,y b ) < ∪ { ( x ν , y b ) } . 19.1 Pr o ducts of se quential ly fr e e pr e c ate go ries 401 F or example, for the dipper pic tur ed previously with a = 5 , b = 3 , ν = 3 , m = 9 , n = 7, we have the pic tur e for S > ( x ν ,y b ) < q q q q q q q q q q q q q q ( x a , y b ) ❫ ( x ν +1 , y b ) q ( x ν , y b − 1 ) ✛   q q   q q q   q ( x m , y n ) and the pic tur e for S ≥ ( x ν ,y b ) ≤ q q q q q q q q q q q q q q q ( x a , y b ) ❫ ( x ν , y b ) ⑦ q q   q q q   q ( x m , y n ) Lemma 19.1 .2 If S is a dipp er with notche d b ox B ν a,b for 0 ≤ ν < a and 0 < b , then either ν > 0 and b > 1 in which c ase S > ( x ν ,y b ) < and S ≥ ( x ν ,y b ) ≤ ar e dipp ers with notche d b ox B 0 ν,b − 1 and tails going fr om ( x ν , y b − 1 ) to ( x m , y n ) ; or else either ν = 0 or b = 1 in which c ase S > ( x ν ,y b ) < and S ≥ ( x ν ,y b ) ≤ ar e ful l tails. Pr o of Lo ok at the ab ov e pictures. The idea of the pro of of the pro duct prop erty is to consider subsets S which ar e dipp ers o r full tails, and prov e that i ( S ) ∗ ( A ⊠ B ) → i ( S ) ∗ ( A ′ ⊠ B ′ ) is a weak equiv a lence in PC ( S ; M ). W e ﬁrst disc us s the ca s e of a full tail. Then the case o f dipp ers will b e treated b y inductio n on a, b , ν , even tually g e tting to the ca se a = m, b = n which gives the desir ed theor em. T he ca se of full tails, treated in the following prop osition, encloses the cas e of all increas ing single-step paths go ing fr om (0 , 0) to ( m, n ). This for malizes the intuit ion 402 Pr o ducts that to understand the pr o duct w e should unders tand what happ ens on each pa th. This is similar to wha t is going on in the decomp osition of a pro duct of simplices : the pro duct has a deco mpo sition into simplices indexed b y the same collection of paths—althoug h the simplices of max- imal dimens ion corre spo nd to paths without diagonal steps. Prop ositio n 19.1. 3 Supp ose ( X , A ) and ( Y , B ) ar e se quential ly fr e e M -pr e c ate gories. Supp ose T = T i 0 ,...,i p j 0 ,...,j p is a ful l tail. The induc e d or- dering on T is a line ar or der, and i ( T ) ∗ ( A ⊠ B ) is a se quential ly fr e e M -pr e c ate gory on the line arly or der e d obje ct set T . F urthermor e, sup- p o se A → A ′ and B → B ′ ar e we ak e quivalenc es towar ds se quential ly fr e e pr e c ate go ries s atisfying the Se gal c ondition, in t he mo del c ate gories of The or em 14.1 .1. Then i ( T ) ∗ ( A ⊠ B ) → i ( T ) ∗ ( A ′ ⊠ B ′ ) is a we ak e quivalenc e whose tar get satisﬁes the Se gal c ondi tion. The same is also true of i ( T ) ∗ ! ( A ⊠ B ) whose obje ct set is X × Y . Pr o of An incr easing sequence of ob jects of T has the form ( z 0 , . . . , z r ) where z k = ( x u ( k ) , y v ( k ) ) with u ( k ) = i a ( k ) and v ( k ) = j a ( k ) for 0 ≤ a (0) ≤ . . . ≤ a ( r ) ≤ p an increasing sequence in th e set of indices for the ob jects of T . The tail condition means that for any such increas ing se- quence, the sequences x u (0) , . . . , x u ( r ) and y u (0) , . . . , y u ( r ) are incr easing sequences in X and Y respectively . Now i ( T ) ∗ ( A ⊠ B )( z 0 , . . . , z r ) = A ( x u (0) , . . . , x u ( r ) ) × B ( y u (0) , . . . , y u ( r ) ) whereas i ( T ) ∗ ( A ⊠ B )( z 0 , z r ) = A ( x u (0) , x u ( r ) ) × B ( y u (0) , y u ( r ) ) so the fact that A and B are sequentially free implies, via the car tesian condition for M , tha t the ma p i ( T ) ∗ ( A ⊠ B )( z 0 , . . . , z r ) → i ( T ) ∗ ( A ⊠ B )( z 0 , z r ) is a weak equiv a lence. Suppo se A → A ′ and B → B ′ are weak equiv alences tow ards sequen- tially free precateg o ries satisfying the Seg al co ndition. It follows from Corollar y 18 .2.7 that for a ny adjace nt pair o f o b jects x i − 1 , x i ∈ X , the map A ( x i − 1 , x i ) → A ′ ( x i − 1 , x i ) is a weak equiv alence. The sequentially fr e e co nditio n implies that A ( x i , x i ) 19.1 Pr o ducts of se quential ly fr e e pr e c ate go ries 403 and A ′ ( x i , x i ) are cont ractible (see Remark 18.2.2 ). In particular for an y ob ject x i the map A ( x i , x i ) → A ′ ( x i , x i ) is a weak equiv alence. The same tw o statements hold for B . But now the tail pr o p e r ty of T says that an adjacent pair of ob jects in T is of the form ( x i , y j ) , ( x k , y l ) where x i , x k are either adjacent ob jects or the sa me ob ject in X , and y j , y l are either adjacent ob jects or the same ob ject in Y . It follows that the map A ( x i , x i ) × B ( y j , y l ) → A ′ ( x i , x i ) × B ′ ( y j , y l ) is a weak equiv a lence. By Co rollary 1 8.2.8, the map i ( T ) ∗ ( A ⊠ B ) → i ( T ) ∗ ( A ′ ⊠ B ′ ) is a weak equiv alence. The set of ob jects of T injects into X × Y , so upo n pushing forward to precategor ies on ob ject set X × Y , ag ain i ( T ) ∗ ! ( A ⊠ B ) → i ( T ) ∗ ! ( A ′ ⊠ B ′ ) is a weak equiv a lence. The nex t step is to note that when we r emov e the corner of the notch from a dipper S we have a pusho ut diagram. In order to give the statement with some g enerality , say that C ∈ PC ( X × Y , M ) is or- der e d if C (( x i 0 , y j 0 ) , . . . , ( x i p , y j p )) = ∅ whenever we hav e a no nincreas- ing sequence ( x i 0 , y j 0 ) , . . . , ( x i p , y j p ) in the pro duct order , that is to say whenever there is some k such that either i k − 1 > i k or j k − 1 > j k . Let PC ord ( X × Y ; M ) denote the catego ry of or dered M -enriched precate- gories over X × Y . Prop ositio n 19.1. 4 Supp ose S = B ν a,b ∪ T is a dipp er and ( x ν , y b ) the c orner of t he notch. As s u me 0 ≤ ν < a and 0 < b . Then the pushout expr ession for su bsets of X × Y S = ( S − { ( x ν , y b ) } ) ∪ S > ( x ν ,y b ) < S ≥ ( x ν ,y b ) ≤ extends to a pushout ex pr essio n for any or der e d C ∈ PC ord ( X × Y ; M ) : the squar e i ( S > ( x ν ,y b ) < ) ∗ ! ( C ) → i ( S ≥ ( x ν ,y b ) ≤ ) ∗ ! ( C ) i ( S − { ( x ν , y b ) } ) ∗ ! ( C ) ↓ → i ( S ) ∗ ! ( C ) ↓ 404 Pr o ducts is a pushout squar e in the c ate gory PC ( X × Y ; M ) . Pr o of It suﬃces to v erify this levelwise on ∆ X × Y , in other words we hav e to verify it for every sequence of ob jects ( x i 0 , y i 0 ) , . . . , ( x i p , y i p ). If either o f x · or y · is not incre a sing then it is trivia lly true, so we may assume that both are increasing . If none of the elements in the sequenc e are ( x ν , y b ) then it is again trivially true. So we may as sume that ther e is a j with ( x i j , y i j ) = ( x ν , y b ). Then the full sequence is contained in the r egion S ≥ ( x ν ,y b ) ≤ . It follows tha t b oth vertical maps in the ab ov e diagram, at the level of the sequence ( x · , y · ), are isomo rphisms. This implies tha t the dia gram is a pushout. Lo oking at this b efor e putting everything back o n to the same ob ject set X × Y , one can als o say that the squar e i ( S > ( x ν ,y b ) < ) ∗ ( C ) → i ( S ≥ ( x ν ,y b ) ≤ ) ∗ ( C ) i ( S − { ( x ν , y b ) } ) ∗ ( C ) ↓ → i ( S ) ∗ ( C ) ↓ is a pusho ut square in PC ( M ). This version of the sta tement p erha ps explains better wha t is going on, but is le s s useful to us since we are currently working in the mo del category structure on PC ( X × Y ; M ) for a ﬁx ed se t o f ob jects X × Y . Corollary 19.1.5 Supp ose g : C → C ′ is a map of or der e d M -enriche d pr e c ate gories over obje ct set X × Y , such that b oth C and C ′ ar e levelw ise c oﬁbr ant, that is c oﬁbr ant in PC inj ( X × Y , M ) . Supp ose that for any ful l tail T going fr om (0 , 0) to ( m, n ) the map i ( T ) ∗ ! ( C ) → i ( T ) ∗ ! ( C ′ ) is a we ak e quivalenc e. Then g is a we ak e quivalenc e. Pr o of W e show by induction that for an y dipper of the form S = B ν a,b ∪ T , the map i ( S ) ∗ ! ( C ) → i ( S ) ∗ ! ( C ′ ) is a weak equiv a lence. The induction is by ( b, a − ν ) in lexico graphic order. In the initial case b = 1 and ν = a , S is a tail so the inductive statement is one of the hypotheses . Suppos e the s tatemen t is known for all dipp ers S ′ corres p o nding to ( a ′ , b ′ , ν ′ ) with b ′ < b or b ′ = b and a ′ − ν ′ < a − ν . Use the expre s sion o f Pro po sition 1 9 .1.4: i ( S ) ∗ ! ( C ) a nd 19.1 Pr o ducts of se quential ly fr e e pr e c ate go ries 405 i ( S ) ∗ ! ( C ′ ) are r esp ectively the pushouts of the top and b ottom rows in the diag ram i ( S − { ( x ν , y b ) } ) ∗ ! ( C ) ← i ( S > ( x ν ,y b ) < ) ∗ ! ( C ) → i ( S ≥ ( x ν ,y b ) ≤ ) ∗ ! ( C ) i ( S − { ( x ν , y b ) } ) ∗ ! ( C ′ ) ↓ ← i ( S > ( x ν ,y b ) < ) ∗ ! ( C ′ ) ↓ → i ( S ≥ ( x ν ,y b ) ≤ ) ∗ ! ( C ′ ) . ↓ The vertical maps in the dia gram induce the given map i ( S ) ∗ ! ( C ) → i ( S ) ∗ ! ( C ′ ). The ho rizontal maps are co ﬁbrations in the injective mo del structure PC inj ( X × Y , M ). Indeed, on a n y seq uence ( x i 0 , y j 0 ) , . . . , ( x i p , y j p ) of po int s in X × Y , eac h of t he horizon tal maps is either the ident ity , or the inclusion fro m ∅ to C ( x i 0 , y j 0 ) , . . . , ( x i p , y j p ) or C ′ ( x i 0 , y j 0 ) , . . . , ( x i p , y j p ). By hypothesis, C and C ′ are levelwise coﬁbrant , so the inclusio ns from ∅ are co ﬁbrations. This shows that the horizontal maps are coﬁbratio ns. The inductive hypothesis applies to each of the vertical maps. This is seen by noting that the inv ar ia nt s ( b ′ , a ′ − ν ′ ) for the dipp ers S − { ( x ν , y b ) } , S > ( x ν ,y b ) < and S ≥ ( x ν ,y b ) ≤ are strictly smaller than ( b, a − ν ) in lexic ographic order: —for S − { ( x ν , y b ) } we hav e a ′ = a , ν ′ = ν + 1 and b ′ = b , unless ν = a − 1 in which case b ′ = b − 1; —for S > ( x ν ,y b ) < and S ≥ ( x ν ,y b ) ≤ we hav e b ′ = b − 1. Hence, by the inductiv e h yp othesis, ea ch o f the vertical ma ps is a weak equiv alence. Now PC inj ( X × Y , M ) is left pro p er b ecause it is a left Bous ﬁeld lo calizatio n c f Theor em 14.1 .1, which implies by Coro l- lary 9.5.2 that coﬁbrant pushouts ar e pre s erved by w eak equiv alences (see alternatively Lemma 1 6.3.4). This shows that the map on pushouts i ( S ) ∗ ! ( C ) → i ( S ) ∗ ! ( C ′ ) is a weak equiv alence. This completes the inductive pro of. A t the last case S = X × Y we obtain the conclusion of the cor ollary , that g : C → C ′ is a weak equiv a lence. This g ives the ﬁrst main result of this ch apter. Theorem 19 .1.6 Supp ose ( X , A ) and ( Y , B ) ar e s e quential ly fr e e M - pr e c ate gories, c oﬁbr ant in PC inj ( X × Y , M ) . Supp ose A → A ′ and B → B ′ ar e trivial c oﬁbr ations in PC inj ( X, M ) and PC inj ( Y , M ) r e- sp e ctively, towar ds se quential ly fr e e M -pr e c ate gories. Then the map ( X × Y , A ⊠ B ) → ( X × Y , A ′ ⊠ B ′ ) is a trivial c oﬁbr ation in PC inj ( X × Y , M ) . 406 Pr o ducts Pr o of Supp o se ﬁrst that A ′ and B ′ satisfy the Segal condition. In this case, Corollar y 19.1 .5 applies with C = A ⊠ B and C ′ = A ′ ⊠ B ′ . The h y- po thesis on full tails used in Co rollary 19.1.5 is provided b y Prop osition 19.1.3. If A ′ and B ′ do not themselves satisfy the Segal condition, we can choose further morphisms A ′ → A ′′ and B ′ → B ′′ which are triv ial coﬁbrations in PC inj ( X, M ) a nd PC inj ( Y , M ) resp ectively , such that A ′′ and B ′′ satisfy the Segal condition. The ﬁrst case of this pro of then applies to the maps fr om A and B , and also to the maps from A ′ and B ′ . These show that ( X × Y , A ⊠ B ) → ( X × Y , A ′′ ⊠ B ′′ ) and ( X × Y , A ′ ⊠ B ′ ) → ( X × Y , A ′′ ⊠ B ′′ ) are tr ivial coﬁbrations in PC inj ( X × Y , M ). By 3 for 2 it follows that ( X × Y , A ⊠ B ) → ( X × Y , A ′ ⊠ B ′ ) is a w eak equiv alence in PC inj ( X × Y , M ), and it is a triv ial coﬁbration by the cartesian pro per ty o f M (conditions (PROD) and (DCL), applied as in Lemma 10.0.12 ). 19.2 Pro ducts of general precategories The next step is to extend the result of Theorem 19.1.6 from the sequen- tially free ca se, to the pro duct of arbitrar y M -enriched precatego ries. Recall that we hav e deﬁned morphisms Υ k ( B 1 , . . . , B k ) → e Υ k ( B 1 , . . . , B k ) of sequentially free M -precateg o ries, the targ et satisﬁes the Seg al con- ditions, and the morphism is a w eak equiv alence in PC ([ k ] , M ) by The- orem 1 8.1.2. Us e the notation Σ([ k ]; B ) := Υ k ( B , . . . , B ) where the same ob ject B o ccurs k times. Lemma 19.2.1 In the c ase wher e B 1 = . . . = B k = B ∈ M , t he map Υ → e Υ factors as Σ([ k ]; B ) → h ([ k ]; B ) → e Υ k ( B , . . . , B ) , 19.2 Pr o ducts of gener al pr e c ate gori es 407 and b oth maps ar e glob a l we ak e quivalenc es b etwe en se quential ly fr e e M - enriche d pr e c ate gories. Pr o of All three are s e quent ially free, and Coro lla ry 18.2 .8 applies. F or Σ([ k ]; B ) and e Υ k ( B , . . . , B ) this is exa ctly what we s aid in the pro of of Theorem 18.1.2 ; for h ([ k ]; B ) see the ex plicit descriptio n in Sectio n 12.5. Prop ositio n 19 .2.2 Supp ose A ∈ PC ( M ) . Then we c an obtain a glob al we ak e quivalenc e A → A ′ such t hat A ′ satisﬁes t he Se gal c ondi- tions i.e. A ′ ∈ R , by taking a tr ansﬁnite c omp osition of pus hout s along morphisms of the form Σ([ k ]; V ) ∪ Σ([ k ]; U ) h ([ k ]; U ) → h ([ k ]; V ) (19.2.1) for gener ating c oﬁbr ations U f → V in M . Pr o of The maps (19.2.1 ) are the same as the Ψ([ k ] , f ) co nsidered in Corollar y 16.2.5 a nd which ma ke up the new pieces in K Reedy . Note that this co llection is missing the piece of K Reedy consisting of the gen- erators for levelwise trivial Reedy c oﬁbrations. How ev er, for the present statement that pie c e is not needed: if we apply the small ob ject argument to the present c o llection of morphis ms we can o btain a map A → A ′ which is a tr ansﬁnite comp ositio n of pushouts alo ng morphis ms of the form (19.2.1), such that A ′ satisﬁes the left lifting prop erty with resp ect to this co llection. The pus houts in question pre serve weak equiv alences, indeed the maps a r e a part o f K Reedy so that follows fro m the construc- tion of the mo del str uctur e of Theorem 1 4.3.2 by direct left Bo us ﬁeld lo calization; or else one could a pply Theorem 16.3.3 and L e mma 16.3.5 which is really saying pretty muc h the sa me thing. Now, an o b ject which satisﬁes the left lifting pr op erty with res pec t to the Ψ([ k ] , f ), satisﬁes the Segal conditions b ecause the pro duct maps satisfy lifting along any generating coﬁbra tio n U → V for M , thus they are tr ivial ﬁbrations in M , which s hows that A ′ satisﬁes the Segal condition. Notice that in the c o nstruction of the pro p os ition, the r esulting A ′ will not in general b e even levelwise ﬁbrant, one would hav e to include pushouts alo ng morphisms of the for m h ([ k ] , f ) for f a g enerating trivia l coﬁbration of M . Theorem 19.2. 3 Supp ose A ∈ PC ( M ) is Re e dy c oﬁbr ant, k ∈ N and B ∈ M is a c oﬁbr ant obje ct. Then the map A × Σ([ k ]; B ) → A × h ([ k ]; B ) 408 Pr o ducts is a glob al we ak e quivalenc e. Pr o of The pro of go es in several steps. (i) Supp ose A is a sequentially free M -enriched prec a tegory . Note that Σ([ k ]; B ) → h ([ k ]; B ) is a triv ial co ﬁbration o f sequentially fr e e M - enriched prec ategories , inducing an order -preserv ing isomorphism on ob jects. Apply Theorem 19 .1.6 to this map and the identit y of A , to conclude that A × Σ([ k ]; B ) → A × h ([ k ]; B ) is a glo bal weak equiv alence. This completes the pro of when A is se- quentially free. This a pplies in particula r to the h ([ k ]; B ) which are se- quentially free. (ii) Recall fr om Theo r em 16 .3.3, that we a lready know that any pushout along a g lobal triv ial coﬁbra tion inducing an isomorphism on sets of ob jects, is ag ain a g lo bal trivia l co ﬁbration. (iii) Supp ose we know the statement of the theo rem for A , A ′ and A ′′ and supp ose given a diagram in whic h one of the ar rows is a t least a n injectiv e (i.e. levelwise) coﬁbratio n A ′ ← A → A ′′ , then we claim that the statement of the theo r em is tr ue for Q := A ′ ∪ A A ′′ . Indeed, Q × Σ([ k ]; B ) = ( A ′ × Σ([ k ]; B )) ∪ A× Σ([ k ]; B ) ( A ′′ × Σ([ k ]; B )) by c o mm utation of c olimits and direct pr o ducts in PC ( M ) (which is part (DCL) of the car tesian condition 1 0.0.9). The same is true for the pro duct with h ([ k ]; B ). The map Q × Σ([ k ]; B ) → Q × h ([ k ]; B ) (19.2.2 ) 19.2 Pr o ducts of gener al pr e c ate gori es 409 is ther efore obtained b y functoriality of the pushout of the columns in A ′ × Σ([ k ]; B ) → A ′ × h ([ k ]; B ) A × Σ([ k ]; B ) ↑ → A × h ([ k ]; B ) ↑ A ′′ × Σ([ k ]; B ) ↓ → A ′′ × h ([ k ]; B ) . ↓ W e are suppo sing that we k now that each of the horizontal ma ps is a global weak equiv alence, also they induce isomorphis ms on ob jects. By the cartesian prop erty for M applied levelwise, the sa me o ne o f the vertical maps is a levelwise coﬁbratio n. By Lemma 16.3.4, the induced map on pushouts (19.2.2) is a globa l weak equiv a lence. T his prov es the claim for step (iii). (iv) Supp ose given a sequence A i indexed by an or dina l β , with injective coﬁbrant tr ansition maps. Suppos e the statement of the theorem is true for each A i , then it is true for Q := colim i ∈ β A i . Indeed, just as in the previous par t Q × h ([ k ]; B ) ca n b e expressed as a tra nsﬁnite comp osition of pushouts of Q × Σ([ k ]; B ) along maps which are by hypothesis globa l trivial coﬁbrations which induce isomorphisms on ob jects. By Theorem 16.3.3 and Lemma 1 6.3.5, the co mpo sition is a global weak equiv a lence. (v) W e show by induction on m ∈ N that if sk m ( A ) ∼ = A then the statement of the theor em holds for A . It is eas y to s ee in case m = 0 bec ause then A is just a discr e te set. Supp ose this is known for any m ≤ n , a nd supp os e A = sk n ( A ). B y P rop osition 15.5.1 we can express A as a trans ﬁnite comp os ition o f pushouts of sk n − 1 ( A ) along maps of the for m h ([ n ] , ∂ [ n ]; U → V ) → h ([ n ]; V ). O n the other hand, h ([ n ] , ∂ [ n ]; U → V ) = h ([ n ]; U ) ∪ h ( ∂ [ n ]; U ) ∂ h ( ∂ [ n ]; V ) , and h ( ∂ [ n ]; U ) = sk n − 1 h ([ n ]; U ). By the inductive h yp othesis the state- men t of the theorem is kno wn for h ( ∂ [ n ]; U ) a nd similarly for h ( ∂ [ n ]; V ). It is known for h ([ n ]; U ) b y (i). So by (iii) the s tatement of the theorem is known for h ([ n ] , ∂ [ n ]; U → V ). F urthermor e it is known for sk n − 1 ( A ) by the inductiv e hypo thesis. Again b y (iii) and (iv) we co nclude the statement for A . (vi) An y Reedy coﬁbr ant A ca n b e expressed as a trans ﬁnite co mpo - 410 Pr o ducts sition o f the maps sk m ( A ) → sk m − 1 ( A ), so by (iv) a nd (v) we get the statement of the theorem for a ny Ree dy co ﬁbrant A . This completes the pro of. Recall from Coro llary 16.2.5 and the r emark at the b eginning of the pro of o f P rop osition 1 9.2.2 ab ove, for a n y c o ﬁbration f : U → V we hav e the notation srcΨ([ k ] , f ) = Σ([ k ]; V ) ∪ Σ([ k ]; U ) h ([ k ]; U ) and the ma p Ψ ([ k ] , f ) g o es from her e to h ([ k ]; V ). Corollary 19. 2 .4 Su pp ose A ∈ PC ( M ) is Re e dy c oﬁbr ant, k ∈ N , and f : U → V is a c oﬁbr ation in M . Then t he map A × srcΨ([ k ] , f ) → A × h ([ k ]; v ) is a glob al we ak e quivalenc e. Pr o of In the co c artesian diagram A × Σ([ k ]; U ) → A × h ([ k ]; U ) A × Σ([ k ]; V ) ↓ → A × srcΨ([ k ] , f ) ↓ the upp er a rrow is a glo bal weak eq uiv alence inducing a n isomorphism on the set o f o b jects, by the pr e vious theorem. F urthermor e it is a Ree dy coﬁbration b y Propo s ition 15.6 .12, s o it is a Reedy isotrivial coﬁbration. By Theorem 16.3.3 , the b ottom map is a global weak equiv alence. The statement o f the cor ollary no w follows b y aga in using the previous The- orem 1 9.2.3 a s w ell as 3 for 2. Theorem 19.2. 5 Assume M is a tr actab le left pr op er c artesia n mo del c ate gory. F or any A , B ∈ PC ( M ) , the map A × B → Seg ( A ) × Seg ( B ) is a glob al we ak e quivalenc e. Pr o of W e supp os e ﬁr st that A and B are Reedy coﬁbr ant. There is a map A → A ′ to an ob ject satisfying the Segal conditio ns , which is a trans ﬁnite co mpo sition of pushouts along morphisms of the form srcΨ([ k ] , f ) Ψ([ k ] ,f ) → h ([ k ]; V ). Each of these pushouts is a global tr iv ial 19.2 Pr o ducts of gener al pr e c ate gori es 411 Reedy coﬁbra tion inducing an isomo r phism on the set of o b jects, by Theorem 1 6.3.3. The map A × B → A ′ × B is the corr esp onding transﬁnite comp os ition of pushouts a long mor - phisms o f the form srcΨ([ k ] , f ) × B → h ([ k ]; U ) × B . This is b ecause pa rt of the car tesian hypothes is for M is (DCL) com- m utation o f direct pro ducts and colimits. B y Co rollar y 19.2 .4, the mo r- phisms srcΨ ([ k ] , f ) × B → h ([ k ]; U ) × B a re glo bal weak equiv alences; they are also Reedy co ﬁbr ations by Pr op osition 15 .6.12 b ecause we as - sumed tha t B is Reedy coﬁbr a nt . These maps a re again isomorphisms on ob jects, so we can apply Theorem 16.3.3 which sa ys that global triv- ial coﬁbrations which induce isomor phis ms on the set of o b jects are preserved under pushout (and see Lemma 16 .3.5 for the tr a nsﬁnite com- po sition). T he r efore A × B → A ′ × B is a glo bal w eak e q uiv a lence. Arguing in the s ame wa y for the pro duct of a map B → B ′ with A ′ , then comp osing the tw o equiv alences we conclude tha t the map A × B → A ′ × B ′ is a global weak equiv a lence. O n the other hand, A ′ → Seg ( A ′ ) a nd B ′ → Seg ( B ′ ) are levelwise weak equiv alences, s o A ′ × B ′ → Seg ( A ′ ) × Seg ( B ′ ) is a levelwise weak equiv alence. Similar ly , the fact that A → A ′ is a global weak equiv alence inducing an isomorphis m on ob jects, implies that Seg ( A ) → Seg ( A ′ ) is a levelwise weak equiv alence, a nd by the same remark for B then taking the pr o duct, we ge t that Seg ( A ) × Seg ( B ) → Seg ( A ′ ) × Seg ( B ′ ) is a levelwise w eak equiv alence. Thus we obtain a dia gram A × B → A ′ × B ′ Seg ( A ) × Seg ( B ) ↓ → Seg ( A ′ ) × Seg ( B ′ ) ↓ 412 Pr o ducts where the top map is a global weak equiv alence, and the right vertical and bo ttom maps a re levelwise hence global weak equiv alence s (Le mma 14.5.3). By 3 for 2 it follows that the left vertical map is a global weak equiv alence as requir ed for the theore m. This co mpletes the pro of for the ca se o f Reedy c o ﬁbrant ob jects. Now supp ose A and B are general ob jects of PC ( M ). Co nsider Reedy coﬁbrant repla cements A ′ → A and B ′ → B ; these ma y be chosen a s levelwise equiv alences of diagra ms, which are then g lo bal weak equiv a- lences by Lemma 14 .5 .3. In particular, S eg ( A ′ ) → Seg ( A ) is a levelwise weak equiv a lence a nd the same for B ′ so Seg ( A ′ ) × Seg ( B ′ ) → Seg ( A ) × Seg ( B ) is a levelwise w eak equiv alence. Note also that A ′ × B ′ → A × B is a levelwise weak equiv alence of diagr ams over ∆ o Ob( A ) × Ob( B ) , so it is a global weak eq uiv a le nce by Lemma 14.5 .3 . The ﬁrst part of the pr o of treating the Reedy coﬁbrant case shows that A ′ × B ′ → Seg ( A ′ ) × Seg ( B ′ ) is a g lo bal w eak equiv a lence. In the s quare dia gram A ′ × B ′ → A × B Seg ( A ′ ) × Seg ( B ′ ) ↓ → Seg ( A ) × Seg ( B ) ↓ the top, b ottom and left vertical arrows ar e global weak equiv alences, so b y 3 for 2 the r ight vertical arrow is a global weak eq uiv alence. This completes the pro of. Corollary 19.2.6 Supp ose A → B and C → D ar e glob al we ak e quiv- alenc es. Then the map A × C → B × D is a glob al we ak e quivalenc e. Pr o of Supp o se ﬁrst of all that A , B , C and D are ob jects sa tisfying the Seg a l co nditions. Then the pro ducts also satisfy the Segal condi- tions. T runcatio n of these is compatible with dir ect pro ducts, b y Lemma 14.5.2, so the map in question is essentially surjective. By looking at the 19.3 The ro le of u nitality, de gener acies and higher c oher enc es 4 13 morphism ob jects we see tha t it is fully faithful, so it is a global weak equiv alence by following the deﬁnition. Next supp ose that A , B , C and D are any o b jects, and lo ok a t the diagram A × C → B × D Seg ( A ) × Seg ( C ) ↓ → Seg ( B ) × Seg ( D ) . ↓ The vertical maps are g lobal w eak eq uiv a le nces by Theor em 19.2.5 , while the b otto m map is a global weak eq uiv a le nc e by the ﬁrst para graph o f the pro of. B y 3 for 2, the top map is a globa l weak equiv alence. 19.3 The role of unitalit y , degeneracies and higher coherences In this section, w e point out wh y w e need to impose the unitality condi- tion A ( x 0 ) = ∗ , to include the degeneracy maps in ∆ (which als o cor re- sp ond to s ome sor t of unit co ndition), and why w e can’t tr uncate ∆ by , say , dropping the ob jects [ n ] for n ≥ 4 . These all hav e to do with the arguments of this chapter abo ut pro ducts. In some sense it go e s back to the Eilenberg -Zilb er theorem; our pr o duct condition can be viewed as a generaliza tion to the pr esent con text where the informa tion of direction of a rrows is retained. 19.3.1 The unitality condition Suppo se we tried to use no n- unital precatego ries. These would b e pairs ( X, A ) where A : ∆ o X → M is an arbitrar y functor. The Seg al condition would include, for sequences of length n = 0, the fact that A ( x 0 ) → ∗ should be a weak equiv alence, in other words A ( x 0 ) is weakly con- tractible. So , this would c onstitute a weak version of the unitality co n- dition. W e would pro ceed muc h as ab ove, imp osing the Sega l co ndition by the small o b ject arg ument in a n op eration denoted A 7→ Seg n ( A ). It see ms likely that this would lea d to a mo del categor y , conjecturally Quillen equiv alent to the model categor ies on unital precategories whic h we are co nstructing here. How ev er, even if the mo del structure existed, it could not b e car tesian. 414 Pr o ducts The reason for this o ccurs at so me v ery degenerate o b jects: co ns ider the non-unital pre c ategory with ob ject set Ob( B ) = { y } a s ingleton, but with functor the constant functor with v alues the initial ob ject: B ( y , . . . , y ) := ∅ . This includes the case o f se q uences o f length 0: B ( y ) = ∅ , so B do es n’t satisfy the unitality condition. No w Seg n ( B ) w ould b e some kind of M - enriched ca tegory with a single ob ject; it seems clear that it would b e the co initial ∗ but in any case has to c ontain ∗ as a r etract. Suppo se A is another non-unital precategor y (whic h might in fact b e unital). C o nsider A × B . The o b ject set is Ob( A ) × { y } ∼ = Ob( A ). But for any sequence (( x 0 , y ) , . . . , ( x n , y )) we hav e ( A × B )(( x 0 , y ) , . . . , ( x n , y )) = A ( x 0 , . . . , x n ) × B ( y , . . . , y ) = A ( x 0 , . . . , x n ) × ∅ = ∅ . In particular, the s tructure of A × B dep ends only on Ob( A ) and not on A itself. This would b e incompatible with the ca rtesian c ondition (for any reasonable choice of M ), b eca use Seg n ( A ) × ∗ → Seg n ( A ) × Seg n ( B ) is c o ntained as a retract, but Seg n ( A × B ) is e s sentially trivial. T o make the last step of the ab ov e argument precis e we would need to inv estigate Seg n explicitly . In the cas e M = Set , the same discussion as in Section 16 .8 applies, expres sing S eg n ( A ) as the categ ory generated by A considered as a s y stem of generato r s a nd relations; the ﬁrst step would b e to imp os e the Segal c o ndition at n = 0 which, for M = Set , is exactly the unitalit y condition; from there the rest is the s ame. In this case we see that S eg n ( B ) is really just ∗ , Seg n ( A × B ) is the discr ete category o n ob ject se t Ob( A ) × { y } , and it cannot contain Seg n ( A ) as a retract in g eneral. 19.3.2 Degeneracies Let Φ ⊂ ∆ denote the categor y consis ting o nly o f face maps, in o ther words the ob jects of Φ are the no nempty ﬁnite linearly ordered sets [ k ] for k ∈ N , but the maps a re the injective order-pr eserving maps. W e could try to create a theory o f weak catego ries based on Φ rather than ∆. It w ould be a ppropriate to call these “w eak semicategories”, because the deg eneracy maps in ∆ cor resp ond to inserting identit y morphisms 19.3 The ro le of u nitality, de gener acies and higher c oher enc es 4 15 int o a comp osa ble sequence. This theory is undoubtedly in teresting and impo rtant, and has no t be e n fully w orked out as far as I know. This would undoubtedly b e r elated to the work of J. Ko ck on w eakly unital hig her categor ie s [140] [1 41], a s o ne could s tart by consider ing weak semicategorie s , then imp ose a w eak unitality conditio n. Unfortunately the theory of pro ducts again do esn’t work if ∆ is re- placed by Φ. Indeed, a slig ht mo diﬁca tion of the example o f the pre- vious subsection aga in pr ovides a counterexample. Let B b e the pre- category with a single ob ject y , with B ( y ) = ∗ so it is unital, but with B ( y , . . . , y ) = ∅ for any sequence of length n ≥ 1 (that is to say , w ith n + 1 elements). This will still b e a v a lid functor from Φ { y } to M , how ever, taking the pr o duct A × B will destroy the structure of A . As in the pr evious subsectio n, this can b e made precise in the case M = Set . The non- unita l precateg ories may then b e consider e d a s systems of genera tors a nd relations for a categor y , but the system do esn’t contain the de g eneracies. It is interesting to lo o k more clo sely at how this works in the ca se of systems of g enerator s for a monoid, that is to say for a category with a single ob ject. The 1-cells of a preca tegory A cor resp ond to g e nerators of the monoid, and the 2-cells corr esp ond to r elations of the form f = g h among the genera to rs. In this case, the system of genera to rs and rela tio ns is just g iven by t wo sets A ( x, x ) and A ( x, x, x ) with three maps A ( x, x, x ) → → → A ( x, x ) . The pro cess o f genera tors and relatio ns would hav e to include the addi- tion o f iden tities. In this case, for example, if A has a sing le generator and no relations , then its pro duct with itse lf A × A will a g ain b e a s y stem with a single generator and no relatio ns; but A generates the monoid N and N × N is diﬀerent from N , so the pro duct of systems of ge ne r ators a nd r elations do esn’t g enerate the pro duct of the corres po nding monoids. One can see in this simple example how t he degeneracies come to the rescue. A sys tem of ge nerators and rela tio ns with unitality and deg en- eracies corresp onds to a diagram of the form A ( x, x, x ) → ← → ← → A ( x, x ) → ← → A ( x ) = ∗ . This means that there is an explicit element 1 a mong the gener ators, with the relations 1 · f = f and f · 1 = f for any other genera tor f . Now let’s lo ok again at N gener a ted by a sy stem A consisting of a 416 Pr o ducts single genera tor. W e have A ( x, x ) = { f , 1 } with r elations corr esp onding to the le ft a nd right identities for b oth f and 1 itself. The pr o duct now has generator s A × A ( x, x ) = { f × f , f × 1 , 1 × f , 1 × 1 } where we hav e noted the single o b ject o f A × A a s x again r a ther than ( x, x ). The unit generato r is 1 × 1. The rela tions include the left and right identities with the unit gener a tor 1 × 1, plus tw o new relations of the for m ( f × 1 ) · (1 × f ) = f × f and (1 × f ) · ( f × 1) = f × f . The ﬁr st o f these t wo relations serves to eliminate the generator f × f so we get to a monoid with tw o generators f × 1 a nd 1 × f , then the second relation gives the commutativit y ( f × 1) · (1 × f ) = (1 × f ) · ( f × 1). So, the mono id generated b y the system A × A is indeed N × N . W o rking out this ex ample demonstra tes how the degener acies of ∆ ent er into the cartesia n condition in an imp ortant wa y . 19.4 Wh y w e can’t truncate ∆ The ab ove examples could all be do ne in the ca se of M = Set , where the passage fr o m precatego ries to categor ies is the pro cess of gener ating a category by generato rs and r elations. F or that, we didn’t need to consider the part of ∆ involving [ n ] for n ≥ 4 (the cas e n = 3 b eing nee de d for the as so ciativity condition). On the other hand, for weak enrichmen t in a g eneral mo del ca tegory M , we ca n’t replace ∆ by a ny ﬁnite t runcation, that is by a subcatego ry ∆ ≤ m of ﬁnite ordered sets of size ≤ m . This can b e seen by the req uirement tha t there s ho uld be a hig her Poincar´ e g roup oid constructio n; in the case when M = K is the mo del category of simplicial sets, the K -enriched precatego r ies should be real- izable int o arbitrar y homotopy t yp es; and in pa rticular the Segal group oids should be eq iv a lent to homo to p y t yp es by a pair of functors including Poincar´ e-Seg a l catego ry and realiza tion. These should b e compatible with homo topy groups in a wa y similar to that des crib ed in Cha pters 3 and 4 . 19.4 W hy we c an ’t trunc ate ∆ 417 If we imp ose these conditions, it b ecomes ea sy to see that the Segal group oids deﬁned using only ∆ ≤ m (say for m ≥ 3) don’t mo del a ll homotopy types . This means that, for the pro g ram w e ar e pursuing here, the category ∆ ≤ m cannot be s uﬃcient . It s hould also be p os sible to show that ∆ ≤ m can’t be used to mo del homotopy n -type s for n > m , in any wa y at all; how ever, it doesn’t seem completely clear how to formulate a go o d statement of this kind. If for 1-ca tegories it suﬃces to lo ok at ∆ ≤ 3 , we could exp ect mor e generally that in order to consider n -ca tegories it would s uﬃce to lo ok at ∆ ≤ n +2 , indeed this show ed up in the explicit example of Chapter 17 and will show up a gain in our discuss ion of stabiliza tion in a future version (see [196]). 20 In terv al s Given our trac ta ble, left prop er and cartesian mo del categ ory M , the main r e maining problem in order to construct the globa l mo del structure on PC ( M ) is to consider the notion of interval which should b e an M -precategor y (to be called Ξ( N | N ′ ) in our notations b elow), weak equiv alen t to the usual category E with tw o isomorphic ob jects υ 0 , υ 1 ∈ E , and with a single morphism b etw een any pair of o b jects. If A ∈ PC ( M ) is a w eakly M -enriched categ ory , an int ernal e quiv- alenc e b etw een x 0 , x 1 ∈ Ob( A ) is a “ morphism from x 0 to x 1 ” (s e e (20.2.1) b elow), whic h pro jects to an is omorphism in the truncated cat- egory τ ≤ 1 ( A ). This ter minology was in tro duced by T amsa mani in [206]. It plays a vital role in the study o f global weak eq uiv alences. Essen- tial s urjectivity of a morphism f : A → B means (ass uming that B is levelwise ﬁbrant) that fo r any ob ject y ∈ O b( B ), there is an ob ject x ∈ Ob( A ) and a n internal equiv a lence b et ween f ( x ) and y . Unfortunately , an internal equiv alence b etw een x 0 and x 1 in A do esn’t necessarily tra nslate into the exis tence of a morphism E → A . This will work after we hav e esta blished the mo del structure on PC ( M ) if we a ssume that A is a ﬁbrant ob ject. How ever, in order to ﬁnish the construction of the mo del structure , we should s tart with the weak er hypothesis 1 that A satisﬁes the Segal conditions and is levelwise ﬁbra n t. The “interv al ob ject” Ξ( N | N ′ ) should b e contractible, a nd hav e the versalit y prop erty that whenever x 0 and x 1 are internally equiv alent, there is a morphism Ξ( N | N ′ ) → A relating them. The co nstruction o f s uch a versal interv al was the sub ject of a n er ror in [1 93], found a nd co rrected by Pelissier in [1 7 1]. This was so mewhat similar to a mistak e in Dwyer-Hirsc hhorn-K an’s origina l construction of 1 As o bserved by Bergner [36] this hypothesis will b e equiv alent t o ﬁbrancy i n the global pro jectiv e model structure, once we know that it exists. This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 20.1 Contra ctible obje ct s and intervals in M 419 the mo del catego ry structure for simplicial catego ries ([87], now [88]), po int ed out by T o e n and subsequently ﬁxed for simplicia l ca tegories by Bergner [3 3]. P elissier ﬁxed this problem for the mo del category o f Segal categorie s by constructing an explicit interv a l ob ject and verifying its top ological pr op erties using the compar ison b etw een Segal 1-gro upo ids and spaces. Drinfeld has co nstructed an interv al ob ject for diﬀerential graded categories [8 3]. Pelissier’s corre ction as written cov ered only the case of K -enric hed weak c ategories , and one of o ur purpo ses here is to po int out that his argument serves to co nstruct the re q uired in terv als in general, by func- toriality w ith r e spe c t to a left Q uillen functor K → M . F or the main result which is contractibility of the Ξ( N | N ′ ) we pro ce ed therefor e in t wo s teps : ﬁrst considering the problem for the case of Segal categ ories i.e. K -enriched weak catego ries a s was done in [171]; then going to the case of M -enriched weak categ o ries by tra ns fer a long K → M . Sec- tions 11.8, 14.7, a nd 16.7 ab out tr ansfer along a left Quillen functor w ere motiv a ted by this mov emen t. The pos sibility of do ing that is one o f the adv an tages of the fully iterative p oint o f view orig inally sug gested by Andr´ e Hirschowitz in Pelissier’s thesis to pic, in which M is a genera l input into the c onstruction. It should also be p os sible to ada pt Pelissier’s correctio n dire c tly to the original n -nerves cons idered in [2 06] [193], by using T amsamani’s theorems o n the topo logical realization of weak n - group oids which in tur n applied Segal’s origina l results in a pa rtially iterative w ay . That would be mor e g eometrically motiv ated, but for the present treatment the fully iter ative approach is b oth more g e neral and more dir ect. I w ould like to thank Regis Pelissier for ﬁnding and cor recting this problem. 20.1 Con tractible ob jects and in terv als in M An ob ject A ∈ M is c ontr actib le if the unique morphism A → ∗ is a weak equiv a lence. An interval obje ct is a triple ( B , i 0 , i 1 ) wher e B ∈ M and i 0 , i 1 : ∗ → B such tha t B is contractible and i 0 ⊔ i 1 : ∗ ∪ ∅ ∗ → B is a co ﬁbration. Assumption (AST) in the cartesia n co ndition 10.0 .9 says that ∗ is a coﬁbrant ob ject, so an interv al ob ject is itself coﬁbrant. A morphism b etwe en intervals from ( B , i 0 , i 1 ) to ( B ′ , i ′ 0 , i ′ 1 ) means a 420 Intervals morphism f : B → B ′ such tha t f ◦ i 0 = i ′ 0 and f ◦ i 1 = i ′ 1 . Since B a nd B ′ are co nt ractible, a morphism f is a utomatically a weak equiv alence. Lemma 20.1.1 Supp ose ( B , i 0 , i 1 ) and ( B ′ , i ′ 0 , i ′ 1 ) ar e two interval ob- je ct s. Then ther e is a thir d one ( B ′′ , i ′′ 0 , i ′′ 1 ) and morphisms of intervals f : B → B ′′ and f ′ : B ′ → B ′′ . These may b e assu m e d to b e trivial c oﬁbr ations. Pr o of Put A := B ∪ ∗∪ ∅ ∗ B ′ and choose a factorization A f → B ′′ → ∗ where the ﬁrs t morphism is a c oﬁbration and the second morphism is a weak eq uiv alence. Now i 0 and i ′ 0 are the sa me when considered a s ma ps ∗ → A b ecause of the copro duct in the deﬁnition of A . Thus f ◦ i 0 = f ◦ i ′ 0 gives a map i ′′ 0 : ∗ → B ′′ . Similar ly i ′′ 1 := f ◦ i 1 = f ◦ i ′ 1 . The map ∗ ∪ ∅ ∗ → A is co ﬁbrant, and since f is coﬁbrant the comp osition in to B ′′ is coﬁbra nt. Note that the maps B → A and B ′ → A are coﬁbr ations, so the same is true of the maps to B ′′ , a nd since the in terv als are weakly equiv alen t to ∗ these maps are trivial coﬁbra tions. Recall that we deﬁned in Chapter 14 a functor τ ≤ 0 : M → Set by τ ≤ 0 ( A ) := Hom ho ( M ) ( ∗ , A ′ ) where A → A ′ is a ﬁbra nt replacement. Lemma 20.1.2 Supp ose A is a ﬁbr ant obje ct and a, b : ∗ → A . The fol lowing c onditions ar e e quivale nt: (a)—The classes of a and b in τ ≤ 0 ( A ) c oi ncide; (b)—for a ny interval ob je ct ( B , i 0 , i 1 ) ther e exists a map B → A sending i 0 to a and i 1 to b ; (c)—ther e exists an interval obje ct ( B, i 0 , i 1 ) and a map B → A sending i 0 to a and i 1 to b . Pr o of This is an exe r cise in Quillen’s theor y of the homo topy category of a mo del ca tegory , which we do for the reader’s conv enience. Note that (b) ⇒ (c) ⇒ (a ) ea sily . Assume that A is also coﬁbrant. T o prov e that (a) ⇒ (c), supp ose that the classes of a a nd b coincide in τ ≤ 0 ( A ). This is equiv alen t to saying that the tw o maps a, b : ∗ → A pro ject to the same map in ho( M ). Rec a ll fro m Quillen [175] that ho( M ) is also the categor y of ﬁbrant and co ﬁbr ant ob jects of M , with homo- topy classes of maps. As ∗ is automatica lly ﬁbrant, and coﬁbrant by hypothesis; and we are ass uming that A is co ﬁbrant a nd ﬁbrant, then condition (a) sa ys t hat the t w o maps a and b are homotopic in the sense of Quillen [17 5], which says exactly condition (c). F or the implication 20.2 I n tervals for M -enriche d pr e c ate gories 421 (a) ⇒ (c), but with A ass umed o nly to b e ﬁbrant, choose a trivial ﬁ- bration from a c o ﬁbrant o b ject A ′ → A . Lift to maps a ′ , b ′ : ∗ → A ′ . Since A ′ → A pro jects to an is omorphism in ho( M ), the maps a ′ and b ′ are equiv alen t in τ ≤ 0 ( A ′ ) so b y condition (c) pro ven for A ′ previously , there exists an interv al ob ject ( B , i 0 , i 1 ) a nd a n ex tension of a ′ ⊔ b ′ to B → A ′ . Co mpo sing gives the required map B → A . T o ﬁnis h the pr o of it suﬃces to show (c) ⇒ (b). Suppo se ( B , i 0 , i 1 ) and ( B ′ , i ′ 0 , i ′ 1 ) are t wo interv al o b jects. Ap plying Lemma 20 .1 .1 there is an in terv al ob ject ( B ′′ , i ′′ 0 , i ′′ 1 ) with trivial coﬁbra tions fro m b oth B and B ′ . If a ⊔ b : ∗ ∪ ∗ ∗ → A extends to a map B → A , and if A is ﬁbrant, then the lifting prop erty for A g ives the extension to B ′′ → A which then restricts to a map B ′ → A as required to show (c) ⇒ (b). Using the as sumption that M is cartesian, we can make a simila r statement ex pla ining the rela tion o f ho motopy between mor phisms using an in terv al, if the targe t is a ﬁbrant ob ject of M . Lemma 20.1.3 Supp ose M is a c artesia n m o del c ate gory. Supp ose A is a c oﬁbr ant obje ct and C is a ﬁbr ant obje ct. Then, for t wo morphisms f , g : A → B t he fol lowing statements ar e e quiva lent: (a)— f and g ar e homotopic in Qu il len ’s sense, me aning that the classes of f and g in Ho m ho ( M ) ( A, C ) c oincide; (b)—for any interval obje ct ( B , i 0 , i 1 ) ther e exists a map h : A × B → C such that h ◦ (1 A × i 0 ) = f and h ◦ (1 A × i 1 ) = g ; (c)—ther e exists an interval obje ct ( B , i 0 , i 1 ) and a map h : A × B → C such that h ◦ (1 A × i 0 ) = f and h ◦ (1 A × i 1 ) = g . Pr o of The cartesian prop erty of M implies that for any interv al ob ject B , the diagr am A × ( ∗ ∪ ∅ ∗ ) = A ∪ ∅ A → A × B → A is an A × I -ob ject in Quillen’s s ense, and so ca n b e used to measur e homotopy betw een our t wo maps. 20.2 In terv als for M -enric hed prec at egor ies Let E := Υ( ∗ ) denote the categor y w ith tw o ob jects υ 0 , υ 1 and a single morphism b etw een them. Thus, E ( υ 0 , . . . , υ 0 ) = ∗ , E ( υ 1 , . . . , υ 1 ) = ∗ , E ( υ 0 , . . . , υ 0 , υ 1 , . . . , υ 1 ) = ∗ and the remaining v alues a re ∅ . This is the image of the usual categ ory [0 → 1] under the map PC ( Set ) → PC ( M ). 422 Intervals An alterna tiv e description of E in terms of the re pr esentable ob ject notation o f Section 12 .5 is E = h ([1] , ∗ ). If A is any M -enriched pre- category , a ma p E → A is the same thing as a triple ( x 0 , x 1 , a ) where x 0 , x 1 ∈ Ob( A ) a nd a : ∗ → A ( x 0 , x 1 ) is an element o f the “ set of morphisms from x 0 to x 1 ”. This “set of morphisms” may be denoted by Mor 1 A ( x 0 , x 1 ) := Hom M ( ∗ , A ( x 0 , x 1 )) = Hom x 0 ,x 1 PC ( M ) ( E , A ) (20.2 .1) where the sup erscript on the righ t designates the subset of maps E → A sending υ 0 to x 0 and υ 1 to x 1 . Let E denote the image of the category with tw o isomorphic ob jects under the map PC ( Set ) → PC ( M ). W e think of E as containing E as a sub catego ry . Thus E aga in ha s ob jects υ 0 , υ 1 , but E ( x 0 , . . . , x p ) = ∗ for any sequence of ob jects. One can also view it as the co discrete precatego r y with tw o ob jects, E = co dsc ([1]) = co dsc ( { υ 0 , υ 1 } ) in the notation of Sec tio n 12.5. If A ∈ PC ( M ), a map E → A is sure to co rresp ond to an internal equiv alence b etw een the images of the tw o endp o int s υ 0 , υ 1 . Say that a ma p E → A which ex tends to E → A , is s tr ongly invertible . An impo rtant little observ ation is that the identit y morphis ms (i.e. thos e given b y the imag es o f the degener acies ∗ = A ( x 0 ) → A ( x 0 , x 0 )) are strongly in vertible. Unfortunately , given a g e ne r al morphism from x 0 to x 1 in A , the corres p o nding map E → A will not in genera l extend to E → A . That is to say , not a ll int ernal eq uiv alences will b e stro ngly in vertible. This is why we need to do some further work to construct versal interv al ob jects. Assuming that A satisﬁes the Se g al condition and is levelwise ﬁbran t, suppo se x 0 , x 1 ∈ Ob( A ) and supp ose a : ∗ → A ( x 0 , x 1 ) is a morphism from x 0 to x 1 . The c ondition of a b eing an internal eq uiv a lence means that there should b e morphisms b and c from x 1 to x 0 , such that ba is homotopic to the identit y of x 0 and ac is homotopic to the ident ity of x 1 . In turn, these ho mo topies can b e repres e nted b y maps from int erv al ob jects in M which we shall denote by N and N ′ resp ectively . W e will build up a big co pro duct r epresenting this collection of data. It turns out to be conv enien t to r elax slig h tly the conditio ns that the homotopies go b etw een a b and the identit y (r e sp. ca and the identit y). Instead, we say that the homo to pies go b etw een ab or ca a nd strong ly inv ertible mor phisms. In pa rticular, the source of c could be an o b ject x ′ 1 diﬀerent from x 1 and the targ et of b could be an ob ject x ′ 0 diﬀerent from x 0 . 20.2 I n tervals for M -enriche d pr e c ate gories 423 This s ituation c an b e represented diagra matically by x ′ 1 x 1 x 0 x ′ 0 ⇔ N ′ ⇔ N c a b ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❯ ☛ ❑ ☛ ❯ ✕ . The g oal in this section is to construct a pr ecategor y Ξ( N | N ′ ) suc h that a map Ξ( N | N ′ ) → A is the same thing as s uch a diagra m. Notice that the diagram may be div ided into tw o triangles which are indep endent except for the fact that they shar e the sa me edge lab eled a . Our Ξ( N | N ′ ) will b e a copro duct of tw o preca tegories Ξ( N ) and Ξ( N ′ ) alo ng E , where each of the pieces r epresents a triangular diagram. So to sta rt, suppo se given an interv al ob ject ( N , i 0 , i 1 ) with N ∈ M and i 0 , i 1 : ∗ → N . W e will co ns truct a precategor y Ξ( N ) such that a map fr om Ξ( N ) to A is the same thing as a diagram of the form y 1 y 0 y 2 ⇔ N ◗ ◗ ◗ ◗ ◗ ◗ s ✑ ✑ ✑ ✑ ✑ ✑ ✰ ☛ ❯ ✕ in A . Ther e are three pieces. The part inv olving N is a map to A from an M -enriched precategory o f the for m Υ( N ) (see Section 16.1 a nd Chap- ter 18), which comes with t wo maps Υ( i 0 ) and Υ( i 1 ) fr o m E to Υ ( N ). The commutativ e tr iangle c orresp onds to a map fro m a representable precatego r y of the for m h ([2 ] , ∗ ) to A . The strongly inv ertible mo rphism on the left cor resp onds to an ex tens io n of one o f the E → Υ( N ) → A to a map E → A . 424 Intervals Motiv ated by this picture, deﬁne the M -enriched precateg ory Ξ( N ) to b e the co pro duct of three terms corresp onding to these three pieces: Ξ( N ) := E ∪ Υ( i 0 )( E ) Υ( N ) ∪ Υ( i 1 )( E ) h ([2] , ∗ ) . The map at the end of the copro duct nota tio n is E = h ([1] , ∗ ) → h ([2] , ∗ ) corres p o nding to the edge [1] → [2] sending 0 to 0 a nd 1 to 2. The ob jects of Ξ( N ) will be denoted ξ 0 , ξ 1 , ξ 2 . These corresp ond to the three o b jects of h ([2] , ∗ ). In case of a map Ξ( N ) → A corr esp onding to a diagra m as ab ov e, the images of ξ i are the o b jects lab eled y i ab ov e. Thus the tw o ob jects υ 0 , υ 1 of b oth copies of E as well as Υ( N ) corr esp ond to ξ 0 and ξ 2 resp ectively . Lemma 20.2. 1 Supp ose A ∈ PC ( M ) . Then a map Ξ( N ) → A c or- r esp onds to giving thr e e obje cts x 0 , x 1 , x 2 ∈ Ob( A ) , to giving an element t : ∗ → A ( x 0 , x 1 , x 2 ) , to giving a map b : N → A ( x 0 , x 2 ) and to giving a map g : E → A such that b ◦ Υ ( i 1 ) = ∂ 02 ( t ) , and b ◦ Υ( i 0 ) = g ( e 01 ) wher e e 01 : ∗ → E ( υ 0 , υ 1 ) is the unique map. Pr o of This comes from the copro duct description for Ξ( N ). W e think of t : ∗ → A ( x 0 , x 1 , x 2 ) as cor resp onding to a commutativ e triangle with ma ps ∂ 01 ( t ) and ∂ 12 ( t ) whose “comp osition” is ∂ 12 ( t ) ◦ ∂ 01 ( t ) = ∂ 02 ( t ) . Then N can be a homoto p y from ∂ 02 ( t ) to the map g ( e 01 ) (in our ap- plication N will b e cont ractible). Then the extensio n o f this map to g deﬁned o n E says that g ( e 01 ) is s trictly in vertible. So, roughly sp eaking when we lo o k at a map Ξ( N ) → A we are lo oking at tw o mo r phisms whose co mpo s ition ∂ 12 ( t ) ◦ ∂ 01 ( t ) is equiv alen t to a n inv ertible map. The tw o diﬀerent maps in question corr esp ond to maps ζ 01 , ζ 12 : E → Ξ( N ) with ζ 01 corres p o nding to ∂ 01 ( t ) and ζ 12 to ∂ 12 ( t ). The construction Ξ also works for the other half of Ξ( N | N ′ ). W e distinguish the tw o interv al ob jects which are used here, for cla r ity of notation. Obviously one could c ho ose the same on bo th sides. Given t wo interv al ob jects N and N ′ , we can form Ξ( N | N ′ ) := Ξ( N ) ∪ E Ξ( N ′ ) where the map E → Ξ( N ) is ζ 01 and the ma p E → Ξ( N ′ ) is ζ 12 . Thes e bec ome the same map denoted η : E → Ξ( N | N ′ ). Denote the fo ur ob jects o f Ξ( N | N ′ ) by ξ | 0 , ξ 0 | 1 , ξ 1 | 2 and ξ 2 | , these co rresp onding with the ob jects of Ξ( N ) or Ξ( N ′ ) b y saying that ξ i | j corres p o nds to ξ i in the 20.2 I n tervals for M -enriche d pr e c ate gories 425 left piece Ξ( N ) and to ξ j in the r ight piece Ξ( N ′ ) to give the following picture of Ξ( N | N ′ ): ξ | 0 ξ 1 | 2 ξ 0 | 1 ξ 2 | Υ( N ′ ) Υ( N ) E E h ([2] , ∗ ) h ([2] , ∗ ) η ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❯ ☛ ❑ ☛ ❯ ✕ displaying η as a mo rphism from ξ 0 | 1 to ξ 1 | 2 i.e. an element o f the set Mor 1 Ξ( N | N ′ ) ( ξ 0 | 1 , ξ 1 | 2 ) deﬁned in (20.2.1). Lemma 20. 2.2 Supp ose ( N , i 0 , i 1 ) and ( N ′ , i ′ 0 , i ′ 1 ) ar e interval obje cts of M . If A is an M -enriche d pr e c ate gory, t hen a map Ξ( N | N ′ ) → A c orr esp onds to the data of a morphism ( x 0 , x 1 , a ) in A , of two other obje cts x ′ 0 and x ′ 1 , t o gether with c ommutative t riangles s : ∗ → A ( x 0 , x 1 , x ′ 0 ) , t : ∗ → A ( x ′ 1 , x 0 , x 1 ) , with maps h : Υ( N ) → A and h ′ : Υ( N ′ ) → A and two maps u, v : E → A su ch that various maps E → A induc e d by t hese data c oincide (se e the diagr am in t he pr o of b elow). Pr o of This comes fro m the copr o duct description of Ξ( N | N ′ ) and the corres p o nding pro pe r ties for Ξ( N ) and Ξ( N ′ ). The o b jects x 0 and x 1 are the images of ξ 0 | 1 and ξ 1 | 2 while x ′ 0 is the image of ξ 2 | and x ′ 1 is the image o f ξ | 0 . The ma ps which are suppos ed to coincide may b e rea d o ﬀ fro m the 426 Intervals diagram x ′ 1 x 1 x 0 x ′ 0 ⇔ h ′ ⇔ h t  a s  u v ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❯ ☛ ❑ ☛ ❯ ✕ in whic h the 2-cells represent maps fro m Υ ( N ) or Υ( N ′ ), the thic k lines represent maps from E , a nd the triangles represent maps from h ([2] , ∗ ). F or example the b oundary ∂ 02 ◦ t : ∗ → A ( x ′ 1 , x 1 ) corr esp onds to a map E → A sending υ 0 to x ′ 1 and υ 1 to x 1 . This should b e the same as the map E = Υ( ∗ ) Υ( i ′ 0 ) → Υ( N ′ ) h ′ → A . The o ther iden tiﬁcations are similar. Lemma 20 .2.3 F or any two interval obje cts N and N ′ , t he M -pr e c ate gory Ξ( N | N ′ ) is Re e dy c oﬁbr ant, and inde e d the inclusion disc { ξ 0 | 1 , ξ 1 | 2 } → Ξ( N | N ′ ) is Re e dy c oﬁbr ant henc e inje ctively c oﬁbr ant. Pr o of Use Corollar ies 15.6.5 and 15.6 .6 , and Lemma 16.1.3. How ev er, Ξ( N | N ′ ) is not pro jectively coﬁbrant, b ecause the inclusions of edges E → h ([2 ] , N ) ar e Reedy but not pro jective c o ﬁbrations. This issue will be addres s ed further in the comments after Remark 20 .3 .2 below. Record here what happ e ns when we change the interv als used in the construction. Lemma 20 . 2.4 Supp ose f : N → P is a morphism b et we en interval obje cts ( N , i 0 , i 1 ) and ( P, j 0 , j 1 ) , that is f ◦ i 0 = j 0 , f ◦ i 1 = j 1 . Supp ose similarly f ′ : N ′ → P ′ is a morphism b et we en interval obje cts ( N ′ , i ′ 0 , i ′ 1 ) and ( P ′ , j ′ 0 , j ′ 1 ) . Then t hese induc e a glob al we ak e quivalenc e Ξ( N | N ′ ) → Ξ( P | P ′ ) . 20.3 The versality pr op ert y 42 7 Pr o of It is a levelwise w eak equiv alnce, be ing a pushout of maps whic h are levelwise a w eak equiv alences. 20.3 The v ersalit y prop erty F rom the universal pr op erty of Lemma 20 .2.2, we obtain the versality prop erty of Ξ( N | N ′ ). Theorem 20. 3.1 Supp ose A ∈ PC ( X , M ) satisﬁes the Se gal c ondi- tion and is ﬁbr ant in the Re e dy diagr am mo del st ructur e Func Reedy (∆ o X /X , M ) . Supp ose that x, y ∈ X = O b( A ) and a : ∗ → A ( x, y ) is an element of Mor 1 A ( x, y ) . Su pp ose that a is an inner e quivalenc e, in other wor ds the image of a in the t runc ate d c ate gory τ ≤ 1 ( A ) ∈ Ca t , is invertible. Then for any interval obje cts N and N ′ in M ther e exist s a morphism Ξ( N | N ′ ) → A sending ξ 0 | 1 and ξ 2 | to x and ξ 1 | 2 and ξ | 0 to y , sending the taut olo gic al morphism η to a , and sending t he two c opies of E to the identities of x and y r esp e ctively. Pr o of Since A satisﬁes the Segal condition, the truncation τ ≤ 1 ( A ) ma y be deﬁned using A itself, that is to s ay that the trunca tion is the 1- category with x + O b( A ) as set of ob jects, and whose ner ve rela tive to this se t is the functor ∆ o x → Set , ( x 0 , . . . , x n ) 7→ τ ≤ 0 A ( x 0 , . . . , x n ) . The Reedy ﬁbra n t co nditio n for A implies that it is levelwise ﬁbrant, which means that for any sequenc e ( x 0 , . . . , x n ) ∈ ∆ o X the image A ( x 0 , . . . , x n ) is a ﬁbra nt ob ject of M , so τ ≤ 0 A ( x 0 , . . . , x n ) = Hom M ( ∗ , A ( x 0 , . . . , x n )) / ∼ where ∼ is the r e la tion of homo topy o ccuring in L emma 20.1.2 . The fact that a maps to an isomo rphism in τ ≤ 1 A therefore mea ns that there is an inv erse b ∈ τ ≤ 0 A ( y , x ); and b y the levelwise ﬁbr ant condition it can be r epresented b y b : ∗ → A ( y , x ). By the Segal condition the morphism A ( x, y , x ) → A ( x, y ) × A ( y , x ) is a weak equiv alence. On the o ther hand, the Reedy ﬁbra nt condition in the diagram categor y means that the matching ma p at ( x, y , z ) is a ﬁbration in M , which in turn implies that the Segal map ab ov e is a 428 Intervals ﬁbration. Hence it is a trivial ﬁbratio n, in particular the element ( a, b ) : ∗ → A ( x, y ) × A ( y , x ) lifts to a map ∗ → A ( x, y , x ). This g ives a dia gram s : h ([2] , ∗ ) → A representing “the comp osition b ◦ a ”, ﬁtting into the lower left tr iangle in the picture o n page 4 26. The image o f s in the nerve of τ ≤ 1 A is the commutativ e triangle for the co mpo s ition of the images o f b a nd a . W e chose b as repr esenting an inv erse to a in the truncated categor y , so the 02 edge of s is homotopic to the identit y of x ; by Lemma 20 .1 .2 there exists a map N → A ( x , x ) r epresenting this homotopy , or b y adjunction h : Υ( N ) → A . This gives the low er or Ξ( N ) part of the required diagra m. A similar discussion using the fact that a ◦ b is homotopic to the iden tit y of y , giv es the upper or Ξ( N ′ ) part, and these g lue together to g ive the req uired map Ξ( N | N ′ ) → A . Remark 20.3. 2 L et Ξ pro j ( N | N ′ ) denote a c oﬁbr ant r eplac ement for Ξ( N | N ′ ) in the pr oj e ctive diagr am c ate gory Func Reedy (∆ o X /X , M ) . Then it has the same versali ty pr op erty with r esp e ct to any A which is levelwise ﬁbr ant and s atisﬁes t he Se gal c ondition. The pro jectively coﬁbra n t version Ξ pro j ( N | N ′ ) could b e constructed explicitly by inserting o b jects of the for m Υ( L ) and Υ( L ′ ) in betw een ζ and the tw o triangles, for in terv als L and L ′ , according to the picture ξ | 0 ξ 1 | 2 ξ 0 | 1 ξ 2 | N ′ N L L ′ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✟ ✙ ❯ ☛ ❑ ☛ ❯ ✕ . This cor resp onds to the step in the pro o f of the theo r em where we used the Reedy ﬁbrant prop er t y to lift ( a, b ) to an element s : ∗ → A ( x, y , x ). If A is assumed only ﬁbrant in the pro jective structure (i.e. levelwise ﬁbrant) then ( a, b ) only lifts up to a homotopy giv en by L → A ( x, y ) × 20.4 Contra ctibility of intervals for K -pr e c ate gori es 429 A ( y , x ). The second ter m may b e neglected since we don’t care ab o ut the bo ttom ar row of the big diagra m, and the pie c e L → A ( x, y ) corresp onds to a map Υ( L ) → A . W e presented the Reedy version in our ma in discussion a bove b ecause the diagra ms are easier to picture. Let e Ξ( N | N ′ ) ⊂ Seg (Ξ( N | N ′ )) b e the full sub categor y co n taining o nly the tw o ob jects ξ | 0 and ξ 2 | . In the situa tio n of the theor em, we get by restriction plus functoriality of Seg a map e Ξ( N | N ′ ) → Seg ( A ) . Similarly if e Ξ pro j ( N | N ′ ) ⊂ Seg (Ξ pro j ( N | N ′ )) is the full sub catego ry containing only ξ | 0 and ξ 2 | , then in the s ituation of the remark we get a map a s stated in the following corolla r y . Corollary 20.3. 3 Supp ose A is levelwise ﬁbr ant and s at isﬁes t he Se- gal c onditions, and supp o se x, y ∈ Ob( A ) ar e two internal ly e quiva lent obje cts. Then ther e is a map e Ξ pro j ( N | N ′ ) → Seg ( A ) sending t he two obje cts of e Ξ pro j ( N | N ′ ) t o x and y r esp e ctively. Pr o of As in the ab ov e r emark we get a map Ξ pro j ( N | N ′ ) → A , hence by functoriality of Seg and comp osition with the inclusion, e Ξ pro j ( N | N ′ ) ⊂ Seg (Ξ pro j ( N | N ′ )) → Seg ( A ) as re quired. It remains to b e seen that Ξ( N | N ′ )) and thus e Ξ( N | N ′ ) ar e con- tractible. 20.4 Con tractibility of in terv als for K -p recategories Given the ab ov e cons truction, the main pro blem is to prove that Ξ( N | N ′ ) is contractible. In this section we do tha t for enrichment ov er the Kan- Quillen model category K of s implicial sets. Theorem 20 .4.1 Supp ose N , i 0 , i 1 and N ′ , i ′ 0 , i ′ 1 ar e two interval ob- je ct s in the Kan-Quil len mo del c ate gory of simplicial sets K . Then Ξ( N | N ′ ) is c ontr actib le in PC ( K ) , t hat is Ξ( N | N ′ ) → ∗ is a glob al we ak e quivalenc e. We have a map Ξ( N | N ′ ) → E × E which is a glob al 430 Intervals we ak e qu ivalenc e and an isomorphism on the sets of obj e cts. In p articular the map Seg (Ξ( N | N ′ )) → E × E induc es an obje ctwise we ak e quivalenc e, which is to say that Seg (Ξ( N | N ′ ))( x 0 , . . . , x p ) is c ontr actible in M for any se quenc e of obje cts x 0 , . . . , x p ∈ { ξ | 0 , ξ 0 | 1 , ξ 1 | 2 , ξ 2 | } . Pr o of This was treated in the last c hapter o f [171] and our prese nt version is only slightly diﬀerent in that we hav e expanded so mewhat Ξ as s omething with 4 ob jects. Our present picture is p erhaps close r to Drinfeld’s interv als fo r DG-categories [83]. Elements of PC ( K ) may b e considere d as ce r tain kinds of bisimpli- cial sets (see Section 12 .7 and Chapter 17), and this comm utes with copro ducts. Similarly the diagonal realization from bisimplicial sets to simplicial sets commutes with co pro ducts and takes Reedy o r injec- tive coﬁbrations 2 in PC ( K ) to coﬁbr ations in K (which ar e just the monomorphisms). Call the co mpo sition of these tw o op erations | · | : PC ( K ) → K . Note that |E | , |E | , and | h ([2] , ∗ ) | ar e contractible sim- plicial sets, and if N is an interv al ob ject in K then | Υ( N ) | is con- tractible. Th us, | Ξ( N ) | is a successive coﬁbra n t co pro duct of contractible ob jects ov er contractible ob jects, so it is contractible. Similarly the co - pro duct of tw o of these ov er the cont ractible |E | (mapping into b oth sides by coﬁbra tions) is contractible, s o | Ξ( N | N ′ ) | is contractible. In general a map A → Seg ( A ) induces a weak equiv alence of simplicial sets | A | ∼ → | Seg ( A ) | . Thus in our case, | Seg (Ξ( N | N ′ ) | is co n tractible. On the other ha nd, all of the 1-mor phisms in Ξ( N | N ′ ) go to in vert- ible morphisms in Seg (Ξ( N | N ′ ), in eﬀect the ma in middle mor phism η has by constr uction a left and a right inv erse up to equiv alence; so it g o es to a n equiv alence, and its inv erses go to equiv alences to o. Thus Seg (Ξ( N | N ′ ) is a Seg al group oid. Now, a Segal gr oup oid whose re- alization is contractible, is co n tractible (see Pr op osition 17.2.3). Thus Seg (Ξ( N | N ′ ) is contractible, which pr ov es the theor em in the cas e of K . 2 The Ree dy and injective c oﬁbrations are the same in PC ( K ) by Pr oposition 15.7.2 as was p ointe d out i n Coroll ary 17.0.2. 20.5 Const ruction of a left Quil len functor K → M 431 20.5 Construction of a left Quillen functor K → M In order to transfer the ab ove contractibilit y r esult for Ξ( N | N ′ ) in the K -enriched ca s e, to the general case, we e x plain in this section the essentially well-known c o nstruction o f a left Q uillen functor K → M . In Hovey [1 20] was expla ine d the int uition that every monoidal mo del category is a mo dule ov er K , and even without the mo noidal structure there is a left Quillen fun ctor from K into M . The constr uction is based on a choice of contractible cosimplicial ob ject in M , or more pre c isely a c osimplici al r esolution in the sense of Hir s chhorn [116]. Tha t means a functor R : ∆ → M which is coﬁbrant in the Reedy mo del s tr ucture Func Reedy (∆ , M ). Recall that an ob ject A ∈ M is c ontr ac tible if the unique mor phism A → ∗ is a weak equiv alence. W e say that a cosimplicial ob ject R : ∆ → M is levelwise c ontr actible if R ([ n ]) is contractible for each o b ject [ n ] ∈ Ob(∆). Lemma 20. 5.1 Ther e exists a choic e of R e e dy-c oﬁbr ant levelwise c on- tr actible c osimp licial obje ct R : ∆ → M . Pr o of See [116]. W e ﬁx o ne such choice, from now on. The ob jects R ([ n ]) may b e thought of as the “standard n -simplices” in M . If A ∈ M , deﬁne R ∗ ( A ) : ∆ o → Set to b e the functor R ∗ ( A ) : [ n ] 7→ Hom M ( R ([ n ]) , A ) . Theorem 2 0 .5.2 If R is a Re e dy-c oﬁbr ant levelwise c ontr actible c osim- plicial obje ct, then R ∗ is a right Q u il len functor fr om M to K = Func (∆ o , Set ) . It s left adjoint R ! : K → M is a left Quil len functor given by the u s ual formula for t he top o lo gic al r e alization of a simplicial set, but with t he st andar d n -simplex r eplac e d by R ([ n ]) ∈ M . Pr o of See [116]. Corollary 20.5.3 The r e a lization funct or induc es, fo r every obje ct set X , a fu n ctor PC ( X ; R ! ) : PC c ( X, K ) → PC c ( X, M ) . This is a left Quil len functor for c denoting either the pr oje ctive, inje ctive 432 Intervals or R e e dy mo del str u ctur es on K and M -enriche d pr e c ate gories over X . It is c omp atible with change of set X , and gives a fu n ctor PC ( R ! ) : PC ( K ) → PC ( M ) fr om the Se gal pr e c ate gories t o the M -enriche d pr e c ate gories. Pr o of Combine Theo r em 2 0.5.2, with the discussion of Prop osition 14.7.2. The co rresp onding rig h t Q uillen functor PC ( R ∗ ) : PC ( M ) → PC ( K ) should be a pplied to A ∈ PC ( M ) only after taking a ﬁbra nt r eplace- men t A → A ′ . Deﬁne Int R ( A ) := PC ( R ∗ )( A ′ ). W e call this the R - interior of A , since it is obtained b y lo o king at maps from the s ta ndard simplices R ([ n ]) int o A ( x 0 , . . . , x n ) so it measures A “from the inside”. This c o nstruction is compatible with truncation: Lemma 20.5 .4 F or A ∈ PC ( M ) we have an isomorphism of c ate- gories τ ≤ 1 ( A ) ∼ = τ ≤ 1 ( Int R ( A )) . Pr o of This follows from the deﬁnition of τ ≤ 1 in Section 14.5. By its construction as a colimit, PC ( R ! ) preser ves copro ducts, pr e- serves constructions Υ and h , preserves the v a rious no tio ns of co ﬁbr ancy . Since M is ca rtesian, Prop ositio n 16.7 .3 says that PC ( R ! ) ta kes weak equiv alences in PC ( X, K ) to weak equiv a lenes in PC ( X , M ). F urther- more since it preserves trunca tions, PC ( R ! ) prese r ves global weak equiv - alences, a nd preserves the notion of in terv al ob jects. 20.6 Con tractibilit y in general W e can now us e the functor PC ( X ; R ! ) to tra nsfer the the contractibilit y result for K -enriched precategor ies, to PC ( M ). This yie lds the ﬁrst main theo r em of the present chapter saying that Ξ( N | N ′ ) is a go o d int erv al ob ject in PC ( M ). This was the last step miss ing in Pelissier’s [171] co rrection of [193], but which is actually str aightforw ard from a Quillen-functorial po in t of view. The contractibilit y statement is made b efore we hav e completely ﬁn- ished the co nstruction of the mo del structure, a ltho ugh it is the penult i- mate s tep. Some c are is still therefore necessar y in using only the par ts of the mo del s tructure which are already known. 20.6 Contra ctibility in gener al 433 Theorem 20 .6.1 Supp ose N , i 0 , i 1 and N ′ , i ′ 0 , i ′ 1 ar e two interval ob- je ct s. Then Ξ( N | N ′ ) is c ontr actible in PC ( M ) , that is Ξ( N | N ′ ) → ∗ is a glob al we ak e quival enc e. We have a map Ξ( N | N ′ ) → E × E which is a glob al we ak e quivalenc e and an isomorphism on t he sets of obje cts. In p articular the m ap Seg (Ξ( N | N ′ )) → E × E induc es an obje ctwise we ak e quivalenc e, which is to say that Seg (Ξ( N | N ′ ))( x 0 , . . . , x p ) is c ontr actible in M for any se quenc e of obje cts x 0 , . . . , x p ∈ { ξ | 0 , ξ 0 | 1 , ξ 1 | 2 , ξ 2 | } . Pr o of First notice that the statement of the theorem is indep e ndent of the choice of interv al ob ject: if N → P and N ′ → P ′ are maps of in terv al ob jects then the statement of the theo rem for ( N , N ′ ) is equiv alent to the statement for ( P , P ′ ). See Lemma 20.2 .4. In particular it suﬃces to prov e the statement for one pair of interv als. Theorem 20.4.1 gives the sa me statement for precategor ies enriched ov er the Kan-Q uillen model categor y K of s implicial sets. Then choose a left Quillen functor R ! : K → M . This gives a functor PC ( R ! ) : PC ( K ) → PC ( M ) which preser ves copr o ducts. Suppo se ( B , i 0 , i 1 ) is an in terv al ob ject in K . Then R ! ( B ) is a n interv a l o b ject in M a nd PC ( R ! )(Ξ( B | B ) = Ξ( R ! ( B ) | R ! ( B )) . Now since PC ( R ! ) pre s erves global weak equiv alences, we o btain the statement of the theorem for the pair o f interv a l ob jects R ! ( B ) | R ! ( B ). By the inv aria nce discussed in the ﬁrst pa ragra ph of the pro of, this implies the s tatement of the theorem for all N , N ′ . Recall that e Ξ( N | N ′ ) ⊂ Seg (Ξ( N | N ′ )) is the full sub categor y contain- ing only the tw o ob jects ξ | 0 and ξ 2 | . Since all o b jects of Seg (Ξ( N | N ′ )) are eq uiv a len t, the inclusion e Ξ( N | N ′ ) ֒ → Seg (Ξ( N | N ′ )) is a globa l weak equiv alence (it is by deﬁnition fully faithful a nd b oth sides satisfy the Sega l co nditions). It follows from Theorem 2 0.6.1 and the 3 for 2 pr op erty of global weak equiv alences, that the functor p N ,N ′ : e Ξ( N | N ′ ) → E is a g lo bal w eak equiv alence; furthermo re this induces isomorphis ms on the sets of ob jects (there are e x actly tw o o b jects on e a ch s ide), and 434 Intervals bo th sides satisfy the Segal conditions, s o p N ,N ′ is an ob jectwise w eak equiv alence of diagra ms . 20.7 Pushout of trivial coﬁbrations These interv a l ob jects allow us to analyze pusho uts a long triv ia l coﬁbra - tions which ar e no t isomorphisms on ob jects. In this discussion, we us e injectiv e c o ﬁbrations since this encompas ses the Reedy and pro jective coﬁbrations to o. W e start by considering the pushout along the standard interv al E . Lemma 2 0 .7.1 Supp ose A ∈ PC ( M ) , and y ∈ Ob( A ) . Then the pushout morphism a : A → A ∪ { y } E obtaine d by identifying υ 0 and y , is a glob al we ak e quivalenc e. Pr o of By Coro llary 19.2.6 applied to the identit y of A and the map p : E → ∗ , the map 1 A × p : A × E → A is a global weak equiv alence. Let i 0 , i 1 : ∗ → E b e the tw o inclusions of ob jects υ 0 and υ 1 . The tw o maps 1 A × i 0 , 1 A × i 1 : A → A × E are g lobal weak equiv alences, a s c an b e seen b y comp osing with 1 A × p and us ing 3 for 2. Now, consider the morphism g : E × E → E × E equal to the identit y on E × { υ 0 } and sending E × { υ 1 } to the single ob ject ( υ 0 , υ 1 ). Set B := A ∪ { y } E . Let q : B → A denote the pro jection obtained by se nding all of E to the single ob ject y ∈ Ob( A ) . Then B × E =  A × E  ∪ { υ 0 }× E  E × E  . Apply the map g to the second factor o f this pushout, to obtain a map f : B × E → B × E such tha t f restr icts to the identit y o n B × { υ 0 } , while f | B×{ υ 1 } is the 20.7 Pu shout of trivial c oﬁbr ations 435 pro jection q : B → A . By the ﬁrst para g raph o f the pro o f, the ma ps induced by f , B × { υ 0 } → B × E and B × { υ 1 } → B × E are globa l w eak equiv alences. The comp osition of 1 B × p : B × E → B with the morphism f consider ed a bove, is a morphism (1 B × p ) ◦ f : B × E → B such that the comp ositio n (1 B × p ) ◦ f ◦ (1 B × i 0 ) is the identit y of B , and the c omp o sition (1 B × p ) ◦ f ◦ (1 B × i 0 ) is the comp osition B q → A a → B . The facts that (1 B × p ) ◦ f ◦ (1 B × i 0 ) is the identit y of B , and that (1 B × i 0 ) is a g lobal weak equiv alence, imply by 3 for 2 that (1 B × p ) ◦ f is a glo bal weak equiv alence. But then, comp osing with the glo bal weak equiv alence (1 B × i 1 ) w e see that (1 B × p ) ◦ f ◦ (1 B × i 0 ) is a globa l w eak equiv alence, in other words that the comp osition aq is a global weak equiv alence. In the other direction, the comp ositio n A a → B q → A is the identit y of A . Thus, we conclude from the last sentence of The o rem 14.6.4 that b oth q a nd the inclusio n A → B are global weak equiv alences. This la st s tatement is what we are supp osed to prov e. Corollary 20.7.2 Supp ose B is an M -enriche d pr e c ate gory with two obje cts b 0 , b 1 . Supp ose B satisﬁes the Se gal c onditions, and is c ontr actible, that is the m ap B → ∗ is a glob al we ak e quivalenc e. Then for any A ∈ PC ( M ) and any obje ct y ∈ Ob( A ) the map A → A ∪ { b 0 } B obtaine d by identifying b 0 to y , is a glob al we ak e quivalenc e. Pr o of There is a unique map f : B → E sending b 0 to υ 0 and b 1 to υ 1 . This map is a globa l weak equiv alence, as seen by applying 3 for 2 to the co mpo s ition B → E → ∗ . But s ince it induces a n is omorphism on ob jects, and b oth s ide s satisfy 436 Intervals the Segal co nditio ns, it is an ob jectwise weak equiv alence of diagrams. Applying f to the seco nd piece of the g iven pushout, we get a map g : A ∪ { b 0 } B → A ∪ { y } E to the pushout co nsidered in the previous coro llary . How ev er, g is an ob ject wise weak equiv alence of diagra ms, so it is a glo bal weak equiv a- lence. No te that g commutes with the maps fro m A , so b y the prev ious corolla r y and 3 for 2 we conclude that the map of the present s tatement is a g lo bal w eak equiv a lence. Suppo se f : A → B is an injective trivia l coﬁbration, a nd s uppo se B is levelwise ﬁbrant and satisﬁes the Sega l conditions. Le t Z := Ob( B ) − f (Ob( A )). F or each z ∈ Z choose e ( z ) ∈ Ob( A ) and a ( z ) ∈ B ( f ( e ( z )) , z ) such that the imag e of a ( z ) is inv ertible in the truncated categ ory . This is p os sible b y the deﬁnition of essential surjectivity of A → B . Applying Theor e m 2 0.3.1 There exis t collections o f interv al ob jects N z , N ′ z indexed by z ∈ Z , and functors t i : Ξ( N z | N ′ x ) → B sending ξ | 0 to e ( z ), ξ 2 | to z , and sending the tautological mor phism η to a ( z ). By functoriality of the construction Seg we get Seg (Ξ( N z | N ′ x )) → Seg ( B ) , and restricting this gives ˜ t i : e Ξ( N z | N ′ z ) → Seg ( B ). Now, ˜ t i sends the ﬁrst ob ject to e ( z ) ∈ A and the second ob ject to z . Putting these a ll together we g et a morphism in PC ( M ), A ∪ ` z ∈ Z { ˜ t i ξ ( | 0) } a z ∈ Z e Ξ( N z | N ′ z ) T → Seg ( B ) , and now T induces an isomorphis m o n sets o f ob jects. It is no longer necessarily a co ﬁbration. W e would like to show that T is a global weak equiv alence. W e star t by considering pushouts alo ng the inter- v a l 1-ca tegory E . F or this pro of we make essential use o f the cartesia n prop erty of M and the discussio n of pro ducts in Chapter 19. Corollary 20.7.3 Su pp ose A ∈ PC ( M ) , and y ∈ Ob( A ) . S upp ose N , N ′ ar e interval obje ct s in M . Then the pushout morphism A → A ∪ { y } e Ξ( N z | N ′ x ) obtaine d by identifying ξ ( | 0) and y , is a glob al we ak e quival enc e. Pr o of Apply Corolla ry 20.7.2 with B = Seg (Ξ( N z | N ′ x )). 20.7 Pu shout of trivial c oﬁbr ations 437 Corollary 20.7. 4 In the situation describ e d ab ove the pr e c e ding c o r ol- lary, t he morphism A → A ∪ ` z ∈ Z { ˜ t i ξ ( | 0) } a z ∈ Z e Ξ( N z | N ′ z ) is a glob al we ak e quivalenc e. Given that A → B was a glob al we ak e quiv- alenc e, the functor T : A ∪ ` z ∈ Z { ˜ t i ξ ( | 0) } a z ∈ Z e Ξ( N z | N ′ z ) → Seg ( B ) is a glob al we ak e quivalenc e inducing an isomorphism on s et s of obje ct s. Pr o of Cho ose a well-ordering of Z , giving an exhaustion o f Z by s ubsets Z i indexed by an ordina l i ∈ β . Let B i ⊂ B b e the full sub ob ject whose ob ject set is f (Ob( A )) ∪ Z i . By tr ansﬁnite induction we obtain that the functors T i : A ∪ ` z ∈ Z i { ˜ t i ξ ( | 0) } a z ∈ Z i e Ξ( N z | N ′ z ) → Seg ( B i ) are global weak equiv alences, using the previous co rollary at ea ch step. A t the e nd o f the induction we obtain the required statement. W e ar e now ready to show the preserv ation of global trivia l coﬁbra - tions under pusho uts. Theorem 20. 7.5 Supp ose A → B is an inje ctive trivial c o ﬁbr ation. Supp ose A → C is any morphism in PC ( M ) . Then the pushout mor- phism C → C ∪ A B is a glob al we ak e quivalenc e. Pr o of W e ﬁrst show this statement assuming that all three ob jects A , B and C satisfy the Seg al co nditions. Noting that B → Seg ( B ) is an isomorphism on sets of ob jects and a pplying Lemma 16.3.4 , it suﬃces to s how that the map C → C ∪ A Seg ( B ) is a g lo bal w eak equiv a lence. Deﬁne F := A ∪ ` z ∈ Z { ˜ t i ξ ( | 0) } a z ∈ Z e Ξ( N z | N ′ z ) , and consider the map T : F → B deﬁned above. By Corollar y 2 0.7.4, T 438 Intervals is a glo bal weak eq uiv a lence inducing an isomorphism on se ts of ob jects. By Lemma 1 6.3.4 it fo llows that the map C ∪ A F → C ∪ A Seg ( B ) is a g lo bal w eak equiv a lence, s o b y 3 for 2 it suﬃces to show that C → C ∪ A F is a global weak eq uiv a le nce. But the map A → F is obtained as a trans- ﬁnite comp os itio n of pusho uts a lo ng things of the form { ξ ( | 0) } → e Ξ( N z | N ′ z ), and by Coro llary 20 .7 .3 these pushouts are global weak equiv a lences. Thu s, the map C → C ∪ A F is a global weak equiv alence, which ﬁnishes this pa r t of the pr o of. Starting with C ← A → B in genera l, let A ′ := Seg ( A ), then B ′ := Seg ( A ′ ∪ A B ) , C ′ := Seg ( A ′ ∪ A C ) . W e get a diag ram C ← A → B C ′ ↓ ← A ′ ↓ → B ′ ↓ such that the b o ttom row sa tisﬁes the hypothesis for the ﬁrs t part of the pro of (all ob jects satisfy the Segal condition and the seco nd map is a global trivial coﬁbration), and suc h that the v ertical arrows ar e global weak equiv alences inducing iso morphisms on s ets of ob jecs. By Lemma 16.3.4, the b ottom ma p in the diagram C → C ′ C ∪ A B ↓ → C ′ ∪ A ′ B ′ ↓ is a g lobal w eak equiv alence. The to p vertical map is a g lobal w eak equiv alence by co nstruction of C ′ and the right vertical map is one to o, by the ﬁrst part of the pro of ab ov e. By 3 for 2 we conclude that the left vertical map is a global weak equiv alence, as requir ed. 20.8 A versality pr op erty 439 20.8 A v ersalit y prop ert y The v ersality prop erties for the interv a ls constructed ab ov e, yield a sim- ilar versalit y pr op erty for any coﬁbra nt replacement of E if the target A is ﬁbr ant in the diagram structure Func c (∆ i Ob( A ) / Ob( A ) , M ). Theorem 20.8. 1 Su pp ose A ∈ PC ( M ) , and supp o se A is ﬁbr ant as an obje ct of PC c (Ob( A ) , M ) wher e c indic ates either the pr oje ctive, the R e e dy or the inje ctive stru ct ur es. L et B → E b e a c oﬁbr ant r eplac ement in PC c ([1] , M ) , so Ob( B ) is stil l [1] = { υ 0 , υ 1 } . Then if x, y ∈ Ob( A ) ar e two obje cts, they pr oj e ct to e quivalent obje cts in τ ≤ 1 ( A ) if and only if ther e ex ists a morphism B → A sending υ 0 to x and υ 1 to y . Pr o of Since τ ≤ 1 B = E is the ca tegory with tw o isomorphic ob jects, existence of a map B → A sending υ 0 to x and υ 1 to y implies that x and y are in ternally equiv alen t in A . Suppo se x and y are in ternally equiv a lent . If A is a ﬁbrant ob ject for the Reedy or injective mo del str uctures rela tive to Ob( A ), there is a morphism Ξ( N | N ) → A g iven by Theo r em 20.3.1 . F or the pr o jective structure use Ξ pro j ( N | N ′ ) given by Remark 20.3.2 instea d. Denote either of these maps by C → A . Let e C ⊂ Seg ( C ) be the full sub category consisting of only the tw o main ob jects, but iden tify Ob( e C ) with the t w o element set [1] = { υ 0 , υ 1 } = Ob( E ). The map C → S eg ( C ) is a isotr iv ial coﬁbration so A → A ∪ C Seg ( C ) is an isotr iv ial coﬁbration by Theor em 1 6.3.3, it follows that our map extends to Seg ( C ) → A . This now res tricts to a map e C → A sending υ 0 to x a nd υ 1 to y . By contractibility , Theor em 2 0.6.1, e C → E is a weak equiv alence inducing an isomor phism on sets of ob jects. Cho ose a factorization e C i → e C ′ p → E where i is a trivial coﬁbration and p is a trivial ﬁbration in PC ([1] , M ). Again our map ex tends to e C ′ → A , but now since B is coﬁbrant and p is a trivial ﬁbration there is a lifting B → e C ′ inducing the iden tit y on the set o f ob jects. W e get the requir ed map B → A . 440 Intervals The imp orta nc e of this versality prop er t y is that it allows us to r eplace a global weak equiv a lence b y one which is surjective on sets of ob jects. Corollary 20. 8.2 L et B → E b e a c oﬁbr ant re plac emen t in PC c ([1] , M ) Supp ose f : A → C is a glob al we ak e quivalenc e, and supp ose C is a ﬁ- br ant obje ct in PC c (Ob( C ) , M ) . Then ther e exists a pushout A → A ′ by a c ol le ction of c opies of { υ 0 } ֒ → B , and a map A ′ → C which is a glob al we ak e quivalenc e and a surje ction on sets of obje cts. Pr o of F or each ob ject y ∈ Ob( C ) choose x ∈ Ob( A ) such that f ( x ) is in ternally equiv alen t to y . F o r each such pair we get a map B → C sending υ 0 to f ( x ) and υ 1 to y ; a ttaching a cop y of B to A b y s ending υ 0 to x and then doing this for a ll o b jects y w e obta in the required pushout A ′ and e xtension of the map. 21 The mo del category of M -en ric hed precategories In this chapter, we ﬁnish the pr o of that the ca tegory PC ( M ) of M - enriched precateg o ries, with v ar iable set of ob jects, has natural injectiv e and pro jectiv e model structur es. Given the pr o duct theorem of Chapter 19 and the discus sion o f in terv als in Chapter 20, the pro o f pr esents no further obstacles. W e also show that the Reedy structure PC Reedy ( M ) is again tra ctable, left prop er and c artesian, allowing us to iterate the op eration. 21.1 A standard factorization F ollow up on Co r ollary 20.8 .2 of the previous chapter, by analyzing further the case of maps which are surjective on the set of ob jects. Lemma 21 . 1.1 Supp ose f : A → B is a morphism in PC ( M ) such that Ob( f ) is surje ctive. L et Ob( f ) ∗ ( B ) ∈ PC (Ob( A ); M ) b e the pr e- c ate gory obtaine d by pul ling b ack along O b( f ) : Ob( A ) → Ob( B ) . Then f factors as A → Ob( f ) ∗ ( B ) → B , wher e the ﬁrst map is an isomorph ism on sets of obje cts, and the se c ond map Ob( f ) ∗ ( B ) → B s atisﬁ es the right lifting pr op erty with r esp e ct to any morphism g : U → V su ch that Ob( g ) is inje ctive. This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 442 The mo del c ate go ry of M -enriche d pr e c ate gories Pr o of Given a diagram U → Ob( f ) ∗ ( B ) V ↓ → B ↓ in or der to get a lifting it suﬃces to c ho ose a lifting on the level o f ob jects Ob( V ) → Ob(Ob( f ) ∗ ( B )) = Ob( A ) . This is po ssible since Ob( q ) is injective and Ob( f ) sur jective. Corollary 21.1.2 In the situation of t he lemma, if I is a set of mor- phisms in PC ( M ) which ar e al l inje ctive on the level of obje cts, and if the ﬁrst map A → Ob( f ) ∗ ( B ) is in inj ( I ) then f ∈ inj ( I ) . Pr o of Combine the lifting pro pe rty for inj ( I ) with the one of the pr e- vious lemma. 21.2 The mo del str uctures W e will be applying Theor em 9.9.7 of Chapter 9 to construct the mo del structure on PC ( M ). W e ﬁx a class o f coﬁbratio ns denoted generica lly by c , with c = pro j or c = Reedy . This choice determines the c orresp onding no tions of coﬁ- brations in PC c ( M ) or PC c ( X ; M ). Let I b e a set of generato r s for the c -coﬁbratio ns in PC c ( M ), as discussed in Cha pter 15. W e can choos e I to consist o f maps with c -coﬁbrant domains, for the Reedy and pro jective structures, see Chapter 15. Let K lo c denote a set of mo rphisms which are pseudo- generator s for the lo ca l w eak equiv alences in PC c ([ k ] , M ), as from The o rem 14 .1.1 or Theor em 14.3.2. W e may a s sume that they are c - coﬁbrations, with c -coﬁbrant domains if c is Reedy or pro jective. Recall that E = Υ ( ∗ ) is the c a tegory with a s ing le non- identit y mor - phism υ 0 → υ 1 , and E is the categor y obtained by in verting this map, that is with a s ingle isomorphism υ 0 ∼ = υ 1 . Co nsider the morphism { υ 0 } ֒ → E . Cho ose a c -co ﬁbrant replacement { υ 0 , υ 1 } → P → E 21.2 The mo del structu r es 44 3 in the mo del ca tegory PC c ([1] , M ), and le t { υ 0 } i 0 → P denote the inclusion mor phis m of a s ing le ob ject. This is s till a c - coﬁbration in PC c ( M ) (b ecause of Co nditio n (AST) in Deﬁnition 10 .0.9). Let K glob := K lo c ∪ { i 0 } . Note tha t the domain of i 0 is the sing le ob ject pre c ategory { υ 0 } whic h is c -coﬁbrant for any of the c . Theorem 21.2 .1 The class of glob al we ak e quivalenc es is pseudo- gener ate d by K glob in the s ense of c onst ruction (PG) of Se ctio n 9.9. F urthermor e, I and J s atisfy axioms (PGM1)–(PGM6), so they deﬁne tr actable left pr op er mo d el structu r es by The or em 9.9.7. F or these mo del structur es, the we ak e quivalenc es ar e the glob al we ak e quivalenc es; the c oﬁbr ations ar e the pr oj e ctive (r esp. inje ct ive r esp. R e e dy) c oﬁbr ations, and the ﬁbr ations ar e the pr oj e ctive (r esp. inje ctive r esp. R e e dy) glob al ﬁbr ations. Pr o of W e hav e to show that K glob leads to the class o f global weak equiv alences via prescription (PG). This a mo un ts to showing tha t a map f : X → Y is a global weak equiv alence if a nd o nly if ther e exists a diagram X → A Y ↓ → B ↓ with the horizo ntal ma ps in cell ( K glob ) a nd the right v ertical ma p in inj ( I ). The maps in K glob are trivial co ﬁbrations in the pro jective struc- ture, and the global trivial coﬁbra tions ar e pres erved b y pushout (The- orem 20.7 .5) and transﬁnite comp osition (Lemma 16.3.5), so the maps in cell ( K glob ) are glo bal trivial coﬁbra tions. By 3 for 2 for global trivial coﬁbrations (Pro p o s ition 14.6.4) it follows that if there exis ts a squa r e diagram as ab ov e then f is a glo bal weak eq uiv alence. Suppo se f is a globa l weak equiv alence, we w ould like to constr uct a square as ab ov e. Let r : Y → B b e the map given by applying the small ob ject argument to Y with resp ect to K lo c . Thus B is K lo c -injective. It follows that it satisﬁes the Segal conditions, and is a ﬁbrant ob ject in PC c (Ob( B ) , M ) (see Theo rems 14.1.1 and 1 4.3.2). By Co rollar y 20.8.2 of the pr eceding Chapter 20, there exists a pushout 444 The mo del c ate go ry of M -enriche d pr e c ate gories X → X ′ by a c o llection o f copies of the map i 0 : { υ 0 } → P and an extension of r f to a glo bal weak eq uiv a le nce g : X ′ → B which is surjective on the set of ob jects. Note that X → X ′ is in cell ( K glob ). Consider the fa ctorization X ′ → Ob( g ) ∗ B → B of Lemma 21.1.1 ab ov e. The ﬁr st ma p is an isomo rphism on the set of ob jects, so it ca n b e consider ed as a ma p in PC (Ob( X ′ ) , M ). Apply the small ob ject arg ument for the set K lo c , to the ﬁr st map to yield a factorization X ′ → A → O b( g ) ∗ B such that X ′ → A is in cell ( K lo c ) and A → Ob( g ) ∗ B is a ﬁbratio n in PC (Ob( X ′ ) , M ). Ho wev er, the comp ose d map X ′ → Ob( g ) ∗ B is a lo cal weak e q uiv a lence, s o b y 3 for 2 in the lo cal mo del structure, the map A → Ob( g ) ∗ B is a trivia l ﬁbratio n; hence it is in inj ( I ). Apply now Corolla ry 21.1.2 . Note that the factor ization o f Lemma 21.1.1 for the map A → B is just A → Ob( g ) ∗ B → B where the ﬁrst map is the same as prev iously; thus the ﬁrst map is in inj ( I ) and by Coro llary 21.1.2 the full map A → B is in inj ( I ). The comp osition X → X ′ → A of tw o maps in cell ( K glob ) is again in cell ( K glob ). This completes the veriﬁcation that our g lobal weak equiv alence f satisﬁes the condition (PG). W e now verify axioms (PGM1)–(PGM6 ) needed to apply Theorem 9.9.7 of Chapter 9. (PGM1)—by hypothesis M is lo ca lly prese n table, and I and K glob are chosen to be small sets o f morphisms; (PGM2)—we have chosen the domains of arrows in I a nd K glob to b e coﬁbrant, and K glob consists of c -coﬁbrations in other words it is con- tained in cof ( I ); (PGM3)—the class of g lobal weak equiv alences is closed under retracts by Prop ositio n 14.6.4 of C ha pter 14; (PGM4)—the class of global weak equiv a lences satisﬁes 3 for 2 again by Prop ositio n 14.6.4; (PGM5)—the cla ss of g lobal trivia l c -c o ﬁbrations is clo sed under pusho uts 21.3 The c artesian pr op erty 4 4 5 by Theorem 20.7.5 ; (PGM6)—the class of global trivial c -coﬁbratio ns is closed under trans- ﬁnite comp osition, indeed the coﬁbratio ns ar e closed unter transﬁnite comp osition since they hav e generating sets, see Cha pter 15, a nd the weak equiv a lences are to o b y Lemma 16.3.5 . Theorem 9.9 .7 now a pplies to show that PC ( M ) with the given classes o f c -coﬁbr ations, globa l weak equiv alences, hence globa l trivial c -coﬁbratio ns whence global c -ﬁbratio ns, is a tractable left prop er clo sed mo del ca tegory . 21.3 The cartesian prop erty Lemma 21.3. 1 Supp ose A → B and C → D ar e glob al R e e dy c oﬁbr a- tions, with the ﬁrst one b eing a we ak e quivalenc e. Then the map A × D ∪ A×C B × C → B × D is a glob al trivial R e e dy c oﬁbr ation. Pr o of It is a Reedy coﬁbr ation by Co rollary 15.6 .13. W e just hav e to show that it is a globa l weak equiv alence. By Co rollary 19.2.6 , the ver- tical maps in the dia gram A × C → A × D B × C ↓ → B × D ↓ are global weak equiv alences. Applying Theor em 20.7.5 to pushout along the left vertical glo bal trivia l coﬁbra tion, then using 3 for 2 , it follows that the map A × D → A × D ∪ A×C B × C is a g lo bal weak eq uiv a lence. Then b y 3 for 2 using the rig h t vertical global weak equiv alence, the map in the statement of the lemma is a global w eak equiv alence. Theorem 21.3.2 Supp ose M is a tr a ctable lef t pr op er c artesian mo del c ate gory. Then t he mo del c ate gory PC Reedy ( M ) of M -enriche d pr e c at- e gori es with R e e dy c oﬁbr atio ns is again a tr actable left pr o p er c artesian mo de l c ate gory. 446 The mo del c ate go ry of M -enriche d pr e c ate gories Pr o of Observe ﬁrst of a ll tha t direct pro duct commutes with co limits in PC ( M ), as ca n be seen from the explicit description of products and colimits and using the corres po nding condition for M . Next, note that the map ∅ → ∗ is a Reedy coﬁbration, from the deﬁnition. Prop ositio n 1 5.6.12 gives the coﬁbr ant prop erty of the pushout-pr o duct map; and the pr evious Lemma 2 1.3.1 gives the trivial coﬁbr ation prop- erty . This shows that PC Reedy ( M ) is cartesian. T o ﬁnish the pro of w e need to no te that it is tractable. This can b e seen by inspec tion of the genera ting coﬁbratio ns for the Reedy structure, given in Prop osition 15.6.11 . Of c o urse, the pro jective mo del structure is deﬁnitely not ca rtesian. On the o ther hand, one can hop e to tr e at the injective mo del structure. There is alr e a dy a problem with tra ctability: Lurie a nd Bar wick don’t men tion if their constructions of injective mo del catego ries preserve the tractability condition, at least unt il a most rece n t version of a pap er in which Bar wick states this prop erty . That would clearly b e an im- po rtant res ult, giving tractability for PC inj ( M ) in general. In the cas e of preshea f categ ories w ith monomorphisms as coﬁbr ations, of cour se this co ndition b ecomes automatic. Similarly , it do es n’t seem immedi- ately clear whether PC inj ( M ) will satisfy condition (PR OD) in general, although a gain this is r elatively easy to see fo r presheaf categories w ith monomorphisms as coﬁbrations. 21.4 Prop erties of ﬁbran t ob jects Julie Bergner made the very interesting observ ation [36] that one could give an explicit character iz ation for the ﬁbrant ob jects in the case of Segal ca tegories M = K . W e get the same kind of prop erty in ge neral. Prop ositio n 21.4.1 L et c = pr o j or c = Reedy . In the mo del c ate gory PC c ( M ) c onstructe d ab ove, an obje ct A with Ob( A ) = X is ﬁ br ant if and only if it is ﬁbr ant when c onsider e d as an obje ct of t he mo del c ate gory PC c ( X, M ) of Th e or em 14.1.1 or 14.3. 2. In turn this c ondition is e quivalent to saying that A satisﬁes the Se gal c onditions, and i s ﬁbr ant as an obje ct of the un ital diagr am mo del c ate gory Func c (∆ o X /X , M ) . Pr o of Left to the reader in the current version. F or c = pro j then, an M -precateg ory A is ﬁbrant if and only if it 21.5 The mo del c ate gory of strict M -en riche d c ate gories 447 satisﬁes the Segal conditions, and is lev elwise ﬁbran t. F or c = Reedy the ﬁbrancy conditio n is also pretty easy to chec k: it just means that the standart ma tc hing maps are ﬁbrations in M . 21.5 The mo del categor y of st rict M -enrich ed categories Dwyer and Kan prop osed, in a s eries of pa p er s, a mo del c a tegory struc- ture on the category o f str ict simplicia l ca tegories. Their progr am was ﬁnished by B ergner [3 3]. Lur ie then gener alized this to construc t a mo del category of strict M -enriched categorie s in [153, App endix], then used that to construct the mo del ca tegory of weakly M -enriched pr ecategor ies as we hav e done ab ove. Theorem 21.5.1 Supp ose M is a tr a ctable lef t pr op er c artesian mo del c ate gory. L et Ca t ( M ) denote the c ate gory of strict M -enriche d c ate- gories. Deﬁne the notion of we ak e quiva lenc e in the u sual way (se e Se c- tion 14.5). Then Ca t ( M ) has a t ra ctable left pr op er mo del stru ctur e in which the gener a ting c oﬁbr atio ns ar e obtaine d by fr e e additions of gener- ating c oﬁbr ations of M in t he m orphism sp ac e b etwe en any t wo obje ct s. Ther e is a Quil len adjunction Ca t ( M ) ← → PC pro j ( M ) and inde e d the mo del stru ct ur e on Ca t ( M ) c an b e use d to gener ate the mo de l structure on PC pro j ( M ) . However, Ca t ( M ) is not in gener al c artesia n. It fol lows that any obje ct of PC pro j ( M ) is e quivalent to a strict M -enriche d c ate gory. Pr o of See Bergner [33] fo r M = K and Lurie [153] for arbitra ry M . The strictiﬁca tion result, for the case of T amsamani n - g roup oids, was prov en by Paoli [170] using techniques of C at n -groups. This theo rem oﬀers a n a lter native r oute to the construction o f the mo del s tr ucture on PC pro j ( M ), who se proof is somewhat diﬀerent fro m ours. The adv antage is that it also gives the model structure on Ca t ( M ) and hence the strictiﬁca ton r esult; the disadv an tage is that it do e s n’t give the cartesian property . The cartesian questio n has been treated b y Rezk [179] for the case of iterated Rezk ca tegories. W e leave it to the reader to e x plore these diﬀerent p oints of view. 22 Iterated higher categories The conclusion of Theor em 2 1.3.2 matches the hypotheses we imp os ed that M b e tracta ble , left prop er a nd c a rtesian. Therefor e, we ca n it- erate the constr uction to obtain v arious versions o f mo del categ ories for n -c a tegories a nd simila r o b jects. This pro cess is inherent in the def- initions of T a msamani [206] a nd Pelissier [1 71]. Rezk co nsidered the corres p o nding iter ation of his deﬁnition in [179] following Barwick, and T rimble’s deﬁnition is also iterative. Such a n itera tion is also related to Dunn’s iter ation o f the Segal delo oping machine [85], and go es back to the well-kno wn iter a tive prese ntation of the notion o f s tr ict n -catego r y , see Bourn [46 ] for ex a mple. In what follows unless other wise indicated, the mo del categ ory PC ( M ) will mea n b y deﬁnition the Reedy structure PC Reedy ( M ). F or any n ≥ 0 deﬁne by induction PC 0 ( M ) := M and for n ≥ 1 PC n ( M ) := PC ( PC n − 1 ( M )) . This is the model category of M -enriche d n -pr e c ate gories . Notations for ob jects ther ein will b e discus sed b elow. In the iter a ted situation, we can in tro duce the following deﬁnition. Deﬁnition 22. 0.2 An M -enriche d n - pr e c ate gory A ∈ PC n ( M ) sat- isﬁes the full Segal condition if it satisﬁes the S e ga l c ondition as an PC n − 1 ( M ) -pr e c ate gory, and furthermor e inductively for any se quenc e of obje cts x 0 , . . . , x m ∈ Ob ( A ) the M -enriche d ( n − 1) -pr e c ate gory A ( x 0 , . . . , x m ) ∈ PC n − 1 ( M ) satisﬁes the ful l Se gal c onditio n. Lemma 22 . 0.3 If A is a ﬁbr ant obje ct in the (iter ate d R e e dy) mo del structur e on PC n ( M ) , then A satisﬁes the ful l Se gal c ondition. This is draft material from a forthcoming b o ok to b e published by Cambridge Uni- v ersity Press in the New Mathematical Monographs ser ies. This publication is in cop yright . c  Carlos T. Simpson 2010 . 22.1 I n itialization 449 Pr o of The Segal condition for the PC n − 1 ( M )-precategor y comes from Prop ositio n 2 1.4.1 (see Theo rem 14.3.2). Ho wev er, if A is ﬁbrant then the A ( x 0 , . . . , x m ) are ﬁbra nt in PC n − 1 ( M ) so by induction they also satisfy the full Segal condition. 22.1 Initialization Here are a few p oss ible choices for M to start with. If M = Set is the mo del catego ry of sets, with coﬁbra tions and ﬁbrations b eing arbitra ry morphisms and weak equiv alences b eing iso- morphisms, then PC n ( Set ) is the mo del c ate gory of n -pr e c ats which was considered in [193] a nd for which w e hav e now ﬁxed up the pro of. Let ∗ denote the mo del c a tegory with a single ob ject and a s ing le morphism. Then PC ( ∗ ) is Quillen equiv alen t (by a pro duct-preser ving map) to the mo del categor y {∅ , ∗} co nsisting of the empt yset and the one-element s et, where weak e quiv a lences ar e isomor phisms. Itera ting again, PC 2 ( ∗ ) is Quillen-eq uiv a le n t to PC ( {∅ , ∗} ). Thes e are b oth mo del categorie s of gr aphs , the ﬁrst allowing multiple edges b etw een no des and the second allo wing only zer o or one edges b et ween t wo no des. The weak equiv alences are deﬁned by requiring isomorphism on the level o f π 0 i.e. the set of connected co mpo nen ts o f a graph. These mo del catego ries of graphs are Q uillen-equiv alen t to Set but hav e the adv a nt age that the coﬁbrations ar e monomorphis ms. They a re rela ted to the notion of setoid in co nstructive type theory . In particular , PC n +2 ( ∗ ) is Quillen-equiv alen t to PC ( Set ) a nd should per haps be thought of a s the “ true” mo del categor y of n -catego ries. If we start with M = K the Kan-Quillen mo del categor y of simpli- cial se ts, then P C n ( K ) is the mo del ca tegory o f Segal n -preca teg ories int ro duced in [117]. One can imagine further constructions starting with M as a category of diagrams or other s uch things. Starting with Z / 2-equiv ariant s ets should be us e ful for co nsidering n - categor ie s with duals. 22.2 Notations By Lemma 12.7.4, once we sta rt iterating , the hypothesis (DISJ) o f 12.7 will b e in vigour. F urthermore , most of our exa mples of starting categorie s (even M = ∗ ) satisfy (DISJ), see Lemma 12.7.5. Whenever 450 Iter ate d higher c ate gories such is the cas e, it is rea sonable to in tro duce an iteration of the notation A n/ of Sec tio n 12.7. One can note, on the other ha nd, that even ba sed on this notatio n as the general framework, most of wha t was done in [193] and [1 94] really used the notation A ( x 0 , . . . , x n ) a t the crucial places. So, in a certain sense, the notations we introduce here a re not r eally the fundamental ob jects, no netheless it is co n venien t to hav e them for co mparison. In PC n ( M ) for a n y k ≤ m and any multi-index m 1 , . . . , m k we can introduce the notatio n A m 1 ,...,m k / ∈ PC n − k ( M ) deﬁned by in- duction on k . At the initial k = 1 (whenever n ≥ 1), by noting that A ∈ PC ( PC n − 1 ( M )) we can use the notation A m/ ∈ PC n − 1 ( M ) considered in Section 12.7. Then for k ≥ 2 deﬁne inductively A m 1 ,...,m k / := ( A m 1 ,...,m k − 1 / ) m k / ∈ PC n − k ( M ) . F or k < n , deﬁne on the other hand A m 1 ,...,m k := Ob ( A m 1 ,...,m k / ) ∈ Set . One can remark that the notations A m 1 ,...,m k / ∈ PC n − k ( M ) a nd A m 1 ,...,m k ∈ Set make sense for k < n b ecause we hav e seen that PC ( M ) sa tisﬁes Condition (DISJ), even if M itself do es not. In a related direction, notice that if M is a preshea f catego ry then PC ( M ) is also a presheaf ca tegory , b y the discussion o f Section 12.7 . In- ductively the sa me is tr ue of PC n ( M ). If M = Presh(Φ) then PC n ( M ) = Presh( C n (Φ)) in the notations of P rop osition 12.7.6. 22.3 The case of n -nerv es Start with M = Set with the trivial mo del str uctur e (as Lurie calls it [1 53]), where the weak eq uiv a le nc e s are isomorphisms and the c o ﬁ- brations and ﬁbrations are arbitrary maps. Iter ating the construction of Theorem 2 1.3.2 w e obtain the iterated Reedy mo de l category structure PC n ( Set ) . The underlying ca tegory is the c a tegory of pres heav es of sets on an iterated version of the construction of Section 12.7, PC n ( Set ) = Presh( C n ( ∗ )) . 22.3 The c ase of n -nerves 451 The under ly ing categor y C n ( ∗ ) may be se e n as a q uotient o f ∆ n = ∆ × · · · × ∆, indeed it is the same as the c a tegory which was denoted Θ n in [193] a nd [1 94]. If A ∈ PC n ( Set ) then the notation discussed in the previous section a pplies , a nd for a n y multi-index ( m 1 , . . . , m k ) with k ≤ n we g et a set A m 1 ,...,m k ∈ Set . F or k = n , the no tation A m 1 ,...,m n := A m 1 ,...,m n / may be used s ince the mo del categ o ry M is equal to S et . That yields a s ystem o f nota tio ns coinciding with that o f [193], [194], [196] etc. A slight diﬀerence is that fo r A ∈ PC n ( Set ), and for any sequence of o b jects x 0 , . . . , x m , what w e w ould b e denoting here by A ( x 0 , . . . , x m ) ∈ PC n − 1 ( Set ) was deno ted in thos e pr eprints by A m/ ( x 0 , . . . , x m ). W e hav e dropp ed the subsc r ipt ( ) m/ for brevity . The iterated injective a nd Reedy mo del str uc tur es coincide in the case of PC n ( Set ), by a pplying Prop osition 15.7.2 inductiv ely . A map A → B is a coﬁbr ation if and o nly if, for any multiindex m 1 , . . . , m k with k < n the map A m 1 ,...,m k → B m 1 ,...,m k is an injection of sets. The monomorphism co ndition is not impo sed at multiindices of le ngth k = n , indeed at the top le vel, all maps of Set are co ﬁbrations for its trivial mo del str ucture. The reader may refer to [193] for a fuller discussio n o f the notio n of co ﬁbrations. W e call PC n ( Set ) the categ ory of n - pr enerves 1 ; a nd the o b jects sa t- isfying the full Sega l condition a re the n - nerves of T amsamani [20 6]. A ﬁbrant o b ject of PC n ( Set ) is a n n - nerve, indee d it satis ﬁe s the Sega l conditions at the las t iteration (corresp onding to the ﬁrst elemen t m 1 of a m ultiindex), and furthermore the n − 1-prenerves A ( x 0 , . . . , x m ) are themselves ﬁbrant in PC n − 1 ( Set ) so by induction they also satisfy the Segal conditions at a ll of their levels. T aking the disjoint union ov er all sequences x 0 , . . . , x m yields A m/ which is an n − 1-nerve. A t n = 1, the category of 1- pr enerves is the category of simplicia l sets, and the 1-nerves are the simplicial sets which are nerv es of a 1- category , that is to sa y the category of 1-nerves is equiv a lent to Ca t . The pro cess A 7→ Seg ( A ) is th e generation of a categ ory b y g enerators and relations discussed in Section 16.8. 1 The obj ects of PC n ( Set ) w ere also called n -pr e c ats in [193] [194] 452 Iter ate d higher c ate gories 22.4 T runcation and equiv alences The deﬁnition of w eak equiv alence w e hav e a dopted for PC ( M ) in gen- eral is des ig ned for enrichm ent ov er a general mo del catego ry M . In T amsamani’s original deﬁnition of n -ne r ves, the notion o f eq uiv a lence and the trunca tion op era tions τ ≤ k were de ﬁned inductively a long the wa y . So , in ca se of PC n ( Set ) there r emains the question of e q uating these tw o deﬁnitions of equiv alences. F or any tra ctable left pr op er ca rtesian mo del ca tegory M , deﬁne the pr etrunc ation τ p ≤ n : PC n ( M ) → PC n ( Set ) as the functor induced by τ ≤ 0 : M → Set . Applied to PC n − k ( M ) for any 0 ≤ k ≤ n , this gives a pr etruncation functor τ p ≤ k : PC n ( M ) → PC k ( Set ) . If M = Set a nd n = k then it is the identit y . Recall that for A ∈ PC ( M ) the truncation op era tion was deﬁned by τ ≤ 1 ( A ) := τ p ≤ 1 ( Seg ( A )). Remark 22 .4.1 It do esn ’t se em to b e true in gener al that the trun - c atio n functor fr om PC ( M ) to Ca t use d starting in Chapter 12 c ould b e expr esse d in t erms of the gener ators and r elations op er ation fr om 1 - pr enerves to 1 -nerves as τ ≤ 1 ( A ) ∼ = Seg ( τ p ≤ 1 ( A )) . Inde e d, t he op er atio n Seg ( A ) might alter t hings in a way which isn ’t se en on t he level of 1 -tru nc ation. One should imp os e the full Sega l condition in orde r to b e able to use the pre truncation. Prop ositio n 22 .4.2 If A i s an M - enriche d n -pr e c ate gory wh ich satis- ﬁes t he f ul l Se gal c ondition, then fo r any k ≤ n the p r etrunc ation τ p ≤ k ( A ) is a k -n erve, and these trunc ations may b e c omp ose d le a ding at k = 1 t o the usu al tru nc ation τ ≤ 1 . They ar e c omp atible with dir e ct pr o ducts. Supp ose A f → B is a we ak e quivalenc e in PC n ( M ) , and A and B b oth satisfy the ful l Se gal c ondition. Then for any 0 ≤ k ≤ n , the t runc ation τ p ≤ k ( f ) is an e quivalenc e of k -nerves in the sense of [206]. F or M = Set , a morphism A f → B in PC n ( Set ) b etwe en n -nerves, is a we ak e quival enc e if and only if it is an e quivalenc e of n -nerves in the sense of [206]. 22.4 T runc ation and e quivalenc es 453 Pr o of F or n = 1, s ee Le mma 14.5.1. W e still should show the compat- ibilit y with comp o sing truncation op erations. If P := PC ( M ) then we hav e deﬁned the tr uncation τ ≤ 0 : P → Set us ing the mo del structure of P : τ ≤ 0 ( A ) it is the s et of morphis ms from ∗ to A in ho( P ). O n the other hand, we have deﬁned the trunca tio n denoted als o τ ≤ 0 : PC ( M ) → Set as sending A to the set of isomor phism classe s of τ ≤ 1 ( A ). T o show that they are the same, note ﬁrst that b oth are inv ariant un der weak equiv a - lences so we ma y assume that A is ﬁbrant. Then Hom ho ( P ) ( ∗ , A ) is the set of morphisms fro m ∗ to A , up to the relation of homo topy . The s et of morphisms is just Ob( A ), and the relation of homo topy says that x is equiv alen t to y if and only if there exists a map fro m an interv a l ob ject to A sending the endp oints to x and y resp ectively (see Lemma 2 0.1.3). If this co ndition holds then lo oking at the image of the interv al in τ ≤ 1 ( A ) we conclude that the p oints x and y go to the same isomorphism class. In the other direction, if x and y g o to isomorphic o b jects in τ ≤ 1 ( A ) then b y the versality pro p e rty Theorem 20.3.1 plus the contractibilit y of the in terv als in ques tion, Theo rem 20.6.1, the cor resp onding maps x, y : ∗ → A ar e ho motopic. This shows that the tw o v ersions of τ ≤ 0 ( A ) coincide. Assume tha t n ≥ 2 and the prop os ition is known for PC n − 1 ( M ). F or k = 1 , the truncatio n τ p ≤ 1 ( A ) is the op er ation of Lemma 14.5.1 which corres po nds to the r ight truncation when a pplied to A satisfy- ing the Segal conditions. F or k ≥ 2, we have the truncation functor τ p ≤ k − 1 : PC n − 1 ( M ) → PC k − 1 ( Set ), taking weak equiv alences b etw een ob jects which sa tisfy the full Segal condition, to weak equiv alences. It follows that if A ∈ PC n ( M ) satisﬁes the full Segal conditions, then applying τ p ≤ k − 1 levelwise to A consider ed a s a dia gram from ∆ o Ob( A ) to PC n − 1 ( M ), it yie lds a diag ram fro m ∆ o Ob( A ) to PC k − 1 ( Set ) whic h again satisifes the Seg al conditions, as well as the full Segal conditions levelwise. But τ p ≤ k − 1 applied levelwise is b y deﬁnition τ p ≤ k . This shows that τ p ≤ k ( A ) is a k -nerve. The compos ition of tw o truncation op era tions is a gain a truncation: τ p ≤ r ( τ p ≤ k ( A )) = τ p ≤ r ( A ) whenever r ≤ k . By induct ion they are co mpatible with direct pro ducts. Suppo se A f → B is a weak equiv alence in PC n ( M ), and A and B bo th satisfy the full Seg al condition. Since they sa tisfy the regular Segal condition, f is es sentially surjective, meaning that τ ≤ 0 ( A ) → τ ≤ 0 ( B ) is surjective, and induces equiv a lences A ( x, y ) → B ( f ( x ) , f ( y )) fo r a ll 454 Iter ate d higher c ate gories pairs of ob jects x, y ∈ Ob( A ). But A ( x, y ) a nd B ( f ( x ) , f ( y )) als o sa t- isfy the full Sega l condition, so by the inductive s ta temen t known for PC n − 1 ( M ), f induces equiv alences of k − 1 -nerves ( τ ≤ k A )( x, y ) = τ ≤ k − 1 A ( x, y ) ∼ → τ ≤ k − 1 ( B ( f ( x ) , f ( y ))) = ( τ ≤ k B )( f ( x ) , f ( y )) . This now implies tha t f induces an equiv alence of k - nerves fr om τ ≤ k A to τ ≤ k ( B ). In the case M = Set , the above argument works in the other direction to show that if f is an equiv alence of n -nerves in the sense of [20 6], it is essentially surjective and, b y applying the inductive sta temen t for n − 1, it is a lso fully faithful. 22.5 The ( n + 1) -category nC AT The cartesian mo del categor y structure on PC n ( M ) allows us to de- ﬁne a structure of M -enriched n + 1-ca teg ory denoted C AT ( n ; M ). In par ticular, starting with M = Set we obtain the n + 1-ca tegory nC AT = C AT ( n ; Set ) which was orig inally discus s ed in Cha pter 3 . In Section 10.2, starting with a tractable cartesian model categor y P we get the strict P -enriched category Enr ( P ). Recall that the ob jects of Enr ( P ) ar e the coﬁbrant and ﬁbra n t ob jects of P , and if X , Y are tw o suc h o b jects then the mor phism ob ject is given by the internal Hom , En r ( P )( X , Y ) = H om P ( X, Y ). The ident ity and comp os itio n op erations are the o b vious ones. This g eneral discus sion now applies to the mo del categ ory P = PC n ( M ). Deﬁne nC AT ( M ) := Enr ( PC n ( M )) . Note tha t nC AT ( M ) is a PC n ( M )-enriched category , so (with the pre- viously mentioned confusion of notation) nC AT ( M ) ∈ PC ( PC n ( M )) = PC n +1 ( M ) . As nC AT ( M ) is a strict PC n ( M )-enriched category , its Segal maps are isomo r phisms. Notice that for any ﬁbrant coﬁbrant ob jects A , B ∈ PC n ( M ), nC AT ( M )( A , B ) = Hom PC n ( M ) ( A , B ) is also ﬁbr ant. By the Segal isomor phisms and the fact that ﬁbra nt and 22.5 The ( n + 1 ) -c ate gory nC AT 455 coﬁbrant ob jects are preser ved by dir ect pro duct, it follows that fo r any sequence A 0 , . . . , A m the ob ject nC AT ( M )( A 0 , . . . , A m ) ∈ PC n ( M ) is ﬁbra n t. In particular, nC AT ( M ) is a pro jectively ﬁbrant PC n ( M )- enriched precateg o ry . One unfortunate consequence of the strictness on the ﬁrst lev el is that nC AT ( M ) is not Reedy-ﬁbr ant. Therefore it is n’t quite corre ct to write “ nC AT ( M ) ∈ Ob(( n + 1) C AT ( M ))” since nC AT ( M ) is not a ﬁbrant ob ject of PC n +1 ( M ). Le t n C AT ( M ) → nC AT ′ ( M ) deno te its ﬁbrant (and a utomatically coﬁbrant) replacement. Then nC AT ′ ( M ) ∈ Ob(( n + 1 ) C AT ( M )) . The diﬀerence betw een nC AT ( M ) and nC AT ′ ( M ) was one o f the main obstacles whic h needed to b e ov ercome in the treatment of limits [19 4]. If A , B are coﬁbr ant a nd ﬁbrant M -enriched n -precatego ries, then a morphism f : A → B corr e spo nds to an ob ject of then M -enriched n - precatego r y H om ( A , B ), or equiv alently to a 1-morphis m in nC AT ( M ). The morphism f is a weak equiv alence in PC n ( M ) if and only if it is an internal equiv alence in nC AT ( M ) i.e. it pro jects to an isomor phism in τ ≤ 1 ( nC AT ( M )), and we have an equiv alence of categories τ ≤ 1 ( nC AT ( M )) ∼ = ho( PC n ( M )) . 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