Adding inverses to diagrams II: Invertible homotopy theories are spaces

ADDING INVERSES TO DIA GRAMS I I: INVER TIBLE HOMOTOPY THEORIES ARE SP A CES JULIA E. BER GNER Abstract. In previous work, we sho w ed that there are appropriate mo del category structures on t he category of simplicial cat egories and on the ca tegory of Segal pr ecategories, a nd that they are Quillen equiv alen t to one another and to Rezk’s complete Segal space mo del structure on the category of s implicial spaces. Here, we sho w that these results still hold if we instead use groupoid or “inv ertible” cases. Namely , we sho w that m odel structures on the categories of simplicial groupoids, Segal pregroupoids, and inv ertible sim plicial spaces are all Quillen equiv alen t to one another and to the standard mo del structure on the cate gory of spaces. W e prov e this r esul t using t wo different approac hes to inv ertible complete Segal spaces and Segal group oids. 1. Introduction The notio n of homotopy theor ies as mathematical ob jects is b ecoming a useful to ol in top ology as mor e mathematica l structures are b eing viewed fr om a homo- topical or higher -categorica l viewp oint. Cur rently , there are four known models for homotopy theor ies: simplicial categor ies, Segal categ ories, complete Seg al spaces, and quasi- categories. There ar e corr e sponding mo del catego ry structures for each; the fir st thr e e were shown to b e Quillen equiv alent to ea c h other b y the author [7], and the fourth was shown to be eq uiv a len t to the first by Joy al [19]; explicit equiv- alences b et ween the fourth a nd the other tw o a re a lso g iv en by Joyal a nd Tier ney [21]. Each of these mo dels has pr o ved to b e useful in different co n texts. Simplicial categorie s are na tur ally mo de ls for homoto py theories, in that they arise naturally from mo del categ o ries, o r mor e gener ally from catego ries with weak equiv alences, via Dwy er and Kan’s s implicial lo calizatio n techniques [11], [13]. F o r this rea son, one imp ortant motiv ation for studying any of these mo dels is to understa nd sp ecific homotopy theories and relations hips b etw een them. Quasi- categories, on the o ther hand, are more clearly a g e neralization o f ca tegories and mor e suited to construc- tions that lo ok like those appe aring in categ o ry theory . In fact, b oth Joyal [20] and Lurie [22] hav e written extensively on extending categ ory theo ry to quasi- category theory . The o b jects in all four models ar e often called ( ∞ , 1)-catego r ies, to indicate that they can b e reg arded as categor ies with n -morphisms for a ny n ≥ 1, but for which these n -mor phisms a re all invertible whene ver n > 1. In this current pa per , we would lik e to show that the first thre e mo dels can be restricted to the gr oupo id ca se without muc h difficulty . Such structur es could 2000 Mathematics Subject Classific a tion. 55U35, 18G30, 18E35. Key wor ds and phr ases. homot op y theories, si mplicial categories, simpli cial group oids, com- plete Segal spaces, Segal gr oupoids, mo del catego ries, ( ∞ , 1)-categ ories and group oids. Support f rom the Fields Institute for a visit in M ay 2007, when muc h of the work for this paper was completed, i s gratefully ackno wledged. 1 2 JULIA E. BERG NER be called ( ∞ , 1)-group oids, but they a re r eally ( ∞ , 0)-catego ries, since even the 1- morphisms are inv ertible in this ca se. In fact, we go on to prove that these mo del structures are Quillen equiv alent to the standard mo del structure on the catego r y of simplicial sets (and therefore to the standard mo del structure on the ca tegory of top ological spa ces). It has b een pr opo sed by a num b e r of p eople, b eginning with Grothendieck [16], that ∞ -group oids, or ( ∞ , 0)-ca teg ories, should b e mode ls for homotopy t yp es of spaces, so this r esult can b e se en a s further e v idence for this “homotopy hypothesis.” Many authors hav e prov e d results in this area, including T amsama ni [29], Ber ger [3], Cisinski [10], Paoli [24], Biedermann [8], a nd B arwick [2], and a nice ov erview is g iv en by Baez [1 ]. W e should further note her e that this compariso n actually encompa sses an in- vertible version of the fourth mo del, that of qua si-categor ies, since a quasi-ca tegory with inverses is just a Ka n co mplex , and the fibrant ob jects in the standard mo del category structure on the ca tegory of simplicial se ts ar e precisely the K an com- plexes. Organization of the P ap er. In section 2, w e g iv e a new pro of o f the existence of a mo del str ucture on the category of simplicial group oids. In sections 3 and 4 , we define invertible versions o f complete Sega l space s and Sega l categories using the ca tegory I∆ op rather than ∆ op as a means of enco ding in verses. W e pro ve the existence of appropr iate mo del category structures a s well. In s ection 5, we prov e that these s implicial gro upoid, Segal g roup oid, and inv ertible complete Segal space mo del structur e s are Quillen equiv alent to one another and to the standa rd mo del ca tegory structure on the categ ory of simplicial sets. In section 6, we g ive an alternate approach to inv ertible versions of Segal catego ries and complete Segal spaces b y changing the pro jection maps in the categor y ∆ op , and we again sho w that we have a zig -zag of Quillen equiv a lences b et ween the r esulting mo del c a tegories. W e refer the re a der to the previo us pap er [4] for our notations and conven tions regar ding simplicial ob jects and mo del catego ries. Ac knowledgmen ts. I would lik e to tha nk Andr´ e Jo yal and Simona Paoli for discussions on the material in this pap er, as well as the re fer ee for suggestio ns for its improv ement. 2. A model ca tegor y structure on the ca tegor y of simplicial gr oupoids A simplicial c ate gory is a categor y C e nric hed over simplicial sets, or a categ o ry such that, for ob jects x and y of C , there is a simplicial set of mor phisms Map C ( x, y ) betw een them. Recall that the c ate gory of c omp onent s π 0 C o f a simplicial categor y C is the category with the sa me ob jects as C and s uc h that Hom π 0 C ( x, y ) = π 0 Map C ( x, y ) . W e use the following notion of equiv alence of simplicial ca teg ories. Definition 2.1. [11, 2.4] A functor f : C → D b etw een tw o simplicial categor ies is a Dwyer-Kan e quivalenc e if it satisfies the following tw o conditions: • (W1) for any ob jects x a nd y o f C , the induced map Map C ( x, y ) → Map D ( f x, f y ) is a weak equiv alence of simplicial sets, and INVER TIBLE HOMOTOPY THEORIES 3 • (W2) the induced ma p of ca tegories of comp onents π 0 f : π 0 C → π 0 D is a n equiv alence of ca tegories. Dwyer and K an pr o ved in [12, 2.5] that there is a mo del structure on the cate- gory S G pd of small simplicial g roup oids with the Dwyer-Kan eq uiv alences as weak equiv alences. In fact, they wen t on to show that this mo del s tructure has the a ddi- tional structur e of a simplicial mo del categ ory , and tha t it is Q uillen equiv alent to the usual mo del structure o n the category S S ets of simplicial sets. In this section, we give an alter nate pro of of the existence of this mo del structure, following the pro of for the mo del ca tegory of small simplicia l categor ies [5]. Consider the category S G pd O of simplicial g roup oids with a fixed set O of ob- jects. In particular, the morphisms in this category are r equired to be the identit y map on this set of ob jects. There is a mo del categor y str uc tur e on S G p d O in which the weak equiv a lences are defined to b e those s implicial functors f : C → D suc h that for any o b jects x and y o f C , the ma p Map C ( x, y ) → Map D ( f x, f y ) is a weak equiv alence of simplicial sets. The fibratio ns are defined a nalogously [4]. This model categ o ry can also b e shown to ha ve the a dditional s tructure a simplicial mo del category , just as Dwyer and Ka n s ho w for the analog ous mo del categ ory S C O of small simplicial categorie s with a fixed ob ject set [13, 7 .1]. Theorem 2.2. The c ate gory S G pd of smal l simplicial gr oup oids has a c ofibr antly gener ate d mo del c ate gory structur e given by the fol lowing thr e e classes of mor- phisms: (1) The we ak e quivalenc es ar e the Dwyer-Kan e qu ivalenc es of simplicial gr oup oids. (2) The fibr ations ar e the maps f : C → D su ch that • (F1) for any obje cts x and y of C , the map Map C ( x, y ) → Map D ( f x, f y ) is a fi br ation of simplicial sets, and • (F2) for any obje ct x in C , z in D , and morphism g : f x → z in D 0 , ther e is an obje ct y in C and morphism d : x → y in C 0 such t hat f d = g . (3) The c ofibr ations ar e the maps with the left lifting pr op erty with r esp e ct to the maps which ar e b oth fi br ations and we ak e quivalenc es. T o pr o ve this theorem, we need to define candidates for our sets of generating cofibrations and generating acy clic cofibrations. T o do s o , we begin by defining the functor U G : S S ets → S G p d taking a s implicial set X to the simplicial group oid with ob jects x and y and mapping spac e s Map( x, y ) = Map( y , x ) = X and Map( x, x ) = Map( y , y ) = ∆[0] . In other w o rds, the mapping spaces Map( x, y ) and Map( y , x ) a re in verse to o ne another. Using this functor, we define the set of generating cofibr ations to consist of the maps 4 JULIA E. BERG NER • (C1) U G ˙ ∆[ n ] → U G ∆[ n ] for all n ≥ 1 , and • (C2) ∅ → { x } . Similarly , we define the set of g enerating acyclic cofibr ations to c onsist of the maps • (A1) U G V [ n, k ] → U G ∆[ n ] for all n ≥ 1, 0 ≤ k ≤ n , and • (A2) { x } → F , where F is the gr oupo id with t wo ob jects x and y and with all the mapping spaces given by ∆[0]. Notice in par ticular that the set of generating acyclic cofibrations is substantially smaller than the analo gous set for the mo del structure on the ca tegory of simplicial categorie s. Prop osition 2.3. A map f : C → D of simplicia l gr oup oids has the rig ht lif ting pr op erty with r esp e ct to the maps in (A1) and (A 2) if and only if it satisfies (F1) and (F2). Pr o of. Using the standar d model structur e S S e ts on the categor y of simplicial sets, it is not ha r d to show that condition (F1) is equiv alent to having the right lifting prop erty with resp ect to the ma ps in (A1). Thu s, let us supp o se that f has the right lifting pro p erty with resp ect to (A1) and (A2) and show that f satisfie s (F2). In other w ords, given a n ob ject x of C and ob ject z of D , we need to show that a map g : f x → z in D lifts to a map d : x → y for s ome ob ject y o f C suc h that f y = z and f d = g . Let us cons ider the ob jects w = f x a nd z in C . First suppos e that w 6 = z . Define E to b e the full s implicial sub category o f D with ob jects w and z . Let F b e the simplicial category with t wo ob jects w and z and a single isomorphism h : w → z . Consider some map i : F → E which preser v es the ob jects. Note that the map { x } → F is precisely the map in (A2). Cons ider the co mposite inclusion map F → E → D . These maps fit into a diagra m { x } / /   C f   F / / = = ④ ④ ④ ④ ④ D . The dotted a rrow lift exis ts b ecause we assume that f : C → D has the rig h t lifting prop erty w ith r espect to the map in (A2). But, the exis tence of this lift implies that any mor phism in D , s inc e it must b e an iso morphism, lifts to a morphism in C . Then, suppose that w = z . Define E ′ to b e the simplicial group o id with tw o ob jects w and w ′ such that ea c h function complex of E ′ is the simplicial set Map D ( w, w ) and compos ites are defined a s they a re in D . W e then define the map E ′ → D which sends both ob jects of E ′ to w in D and is the ide ntit y map on all the function co mplexes. Using this simplicial group oid E ′ for the simplicia l group oid E use d ab ov e, the a r gument pro ceeds as b efore. Now supp ose that f sa tisfies (F1) and (F2). W e s eek to show that f has the right lifting prop erty with res pect to the map (A2). In other words, we need to show that the do tted arr o w lift exis ts in the diagr am { x } / /   C f   F / / = = ④ ④ ④ ④ ④ D . INVER TIBLE HOMOTOPY THEORIES 5 How ever, the existence of such a lift follows fro m the fact tha t the ma p f satisfies prop erty (F2).  The pr oo f of the following prop osition follows just as the analogo us statement for simplicia l categor ies [5, 3.1]. Prop osition 2.4. The c ate gory S G pd has al l smal l limits and c olimits and its class of we ak e quivalenc es is close d under r etr acts and satisfies the “two out of thr e e” pr op erty. Notice that, as for the case of S C , a map f : C → D satisfies b oth (F1) and (W1) if and o nly if it has the right lifting pro per t y with resp ect to the maps in (C1). How ever, we need the following stro nger statement. Prop osition 2 .5. A map in S G pd is a fibr ation and a we ak e quivalenc e if and only if it has the right lifting pr op erty with r esp e ct to the maps in (C1) and (C2). Pr o of. Firs t, we supp ose that f : C → D is a fibration and a weak equiv alence . As noted ab ov e, it follows from the definitions that f has the right lifting pro perty with resp ect to the maps in (C1). Thus, it r e mains to show that f has the r ight lifting pro per t y with res p ect to the map (C2), i.e., with resp ect to the ma p ∅ → { x } . Ho wev er , satisfying suc h a lifting prop erty is equiv a len t to being surjectiv e on o b jects. The fact that f is essentially surjective, or surjective on isomo rphism classes of ob jects, follows from co ndition (W2). Given this pro perty , the fact that f satisfies condition (F2) guar ant ees that it is actually s urjectiv e. Conv er sely , supp ose that f : C → D has the rig h t lifting prop erty with resp ect to the maps in (C1) and (C2). W e need to show that f is a fibration and a weak equiv alence. As noted ab ov e, it follows immediately from the definitions that f satisfies conditions (F1) a nd (W1). The fac t that f satisfies (W1) implies that the ma p Hom π 0 C ( x, y ) → Hom π 0 D ( f x, f y ) is a n isomorphism of sets. F urthermore, the fact that f has the right lifting prop- erty with r espe ct to the map in (C2) guarantees that f is s urjectiv e on o b jects. Therefore, the ma p π 0 f : π 0 C → π 0 D is an equiv alence of catego ries; i.e., condition (W2) is satisfied. It r emains to s ho w that f satisfies condition (F2). Since we have already prov ed that f satisfies (F1) if and only if it ha s the r ig h t lifting pr oper t y with resp ect to (A1), and that if f has the right lifting prope rt y with resp ect to (A1) a nd (A2), then it satisfies (F2), it suffices to show that f has the right lifting prop erty with r espect to (A2). How ever, the map { x } → F in (A2) can b e wr itten as a comp osite of a pusho ut along ∅ → { x } followed by pushouts along ma ps of the form U G ˙ ∆[ n ] → U G ∆[ n ] for n = 0 , 1. But, f has the right lifting prop erty with resp ect to a ll such maps since they a re in (C1) a nd (C2).  Prop osition 2.6. A map in S G pd is an an acyclic c ofibr ation if and only if it has the left lifting pr op erty with r esp e ct to t he fibr ations. The pro of of this propo s ition can be proved formally , just as in the cas e of simplicial categor ie s [5, 3 .3 ]. It do es, howev er , r equire the following lemma, whose pro of essentially follows the one for S C O in [1 3, 7.3]. Lemma 2.7. The mo del c ate gory S G pd O is pr op er. 6 JULIA E. BERG NER Pr o of. W e fir st pro ve that S G p d O is right pr ope r , namely , that a pullbac k of a weak e q uiv alence along a fibr ation is a weak equiv alence. How ever, since fibrations and weak equiv alences are defined in terms o f fibrations and weak eq uiv alences of mapping spac e s, this fact follows from the right prop erness of S S ets [1 7, 13.1 .1 3]. T o prov e that S G pd O is left prop er, we need to s ho w that the pushout o f a weak equiv alence along a c o fibration is a weak equiv alence. Supp ose that the following diagram A i / / g   X f   B j / / Y is a pushout diagram with the map g a c o fibration a nd i a w eak equiv alence. T o prov e that the map j is a weak equiv alence , we can as sume that the map g : A → B is a free map, since cofibrations are retra cts of free maps [12, 2.4]. F urther more, a ny free map can b e written as the co limit of a sequenc e of free maps for which all of the nondeg enerate genera to rs are in the same dimension. So, suppo se that C is an ordinary group oid, r egarded a s a simplicial g roup oid, a nd let C ⊗ ∆[ n ] denote the simplicial gro upoid given by the simplicial structur e on S G pd O . Then, it suffices to show that j is a weak equiv alence in the diagra m (ma de up o f tw o pusho ut squar es) C ⊗ ˙ ∆[ n ]   / / A i / / g   X f   C ⊗ ∆[ n ] / / B j / / Y . This result follows from technical res ults on pushouts in [1 3, § 8].  Pr o of of The or em 2.2. W e need to verify the co nditions of Theorem 2.1 o f the pre- vious pap er [4]. The fact tha t M has small limits and colimits and that its weak equiv alences satisfy the ne c e ssary co nditions was prov ed in Prop osition 2.4. Con- dition 1 follows from the same a rguments as used for simplicial categories [5 , 1.1]. Condition 2 was proved in Pro positio n 2.3. Condition 3 was pr ov ed in Pro pos itio n 2.5. Condition 4 was proved in Prop osition 2.6.  The following prop osition is prov ed similarly to the analo gous result for simplicial categorie s [5, 3 .5]. Prop osition 2.8. The mo del c ate gory S G pd is right pr op er. Pr o of. Supp ose tha t A f / /   B g   C h / / D is a pullback diagram of simplicial group oids, where g : B → D is a fibration and h : C → D is a Dwyer-Kan equiv alence. W e would like to show that f : A → B is a Dwyer-Kan equiv a lence as well. INVER TIBLE HOMOTOPY THEORIES 7 W e first need to show that Ma p A ( x, y ) → Map B ( f x, f y ) is a weak equiv alence of simplicial sets for any ob jects x and y of A . How ever, this fact follows s ince the mo del categor y s tr ucture on simplicia l sets is r ight prop er [17, 13.1 .4]. It remains to prove that π 0 A → π 0 B is an equiv alenc e of categ ories. Given what we have prov ed th us far, it suffices to show that A → B is essentially surjective on ob jects. Consider an o b ject b of B a nd its image g ( b ) in D . Since C → D is a Dwyer-Kan equiv alence, there exists a n ob ject c of C together with an isomor phism g ( b ) ∼ = h ( c ) in D . Since B → D is a fibra tion, there exists an o b ject b ′ and is omorphism b ∼ = b ′ in B such that g ( b ′ ) = h ( c ). Using the fact that A is a pullback, w e hav e an isomorphism b ∼ = f ( b ′ , c ), completing the pro of.  3. Inver tible Segal sp aces In this section, we define “gr oupo id versions” of Rezk’s Sega l spaces and c omplete Segal spa c es. T o do so, we first summarize a few general fa c ts using the categ ory I∆ op ; further details can b e found in the previous pap er [4 , § 4]. W e no te that Rezk’s original definition of (complete) Seg al s paces can b e recovered from the definitions in this section by r eplacing the categor y I∆ op with the categ ory ∆ op throughout. The same ho lds for our definitions of Segal group oids in the next se c tion. W e give fewer details in this section a nd the following ones, compared to the previous section, due to the fact that most of the pro ofs are not only s imilar but actually follow just as in the original ca ses. In the ca tegory I∆ , there ar e maps β i : I [1 ] → I [ k ] giv en b y 0 7→ i a nd 1 7→ i + 1. W e denote the corresp onding map in I∆ op by β i : I [ k ] → I [1]. Each map β i induces a n inclusion o f inv ertible simplicial sets I ∆[1] → I ∆[ n ]. W e thus define the inv ertible simplicial space I G ( k ) t = k − 1 [ i =0 β i I ∆[1] t and the inclusion map ξ k : I G ( k ) t → I ∆[ n ] t F urthermor e, this inclusion map induces, for any inv e r tible simplicial space X , a map Map( I ∆[ k ] t , X ) → Map( I G ( k ) t , X ) which can b e wr itten as the Seg al map ξ k : X k → X 1 × X 0 · · · × X 0 X 1 | {z } k . (This treatment should be compa r ed to Rezk’s in the non-invertible case [27, § 4].) Using I∆ op rather than ∆ op , we still hav e the injective and pro jective mo del structures o n the ca tegory S S ets I∆ op . These mo del structures ca n b e defined via the inclusion maps I∆ op → ∆ op . F or the injective structure, then, the generating cofibrations a re the maps ˙ ∆[ m ] × I ∆[ n ] t ∪ ∆[ m ] × I ˙ ∆[ n ] t → ∆[ m ] × I ∆[ n ] t for all n, m ≥ 0. F or the pro jectiv e struc tur e, the genera ting cofibr a tions ar e the maps ˙ ∆[ m ] × I ∆[ n ] t → ∆[ m ] × I ∆[ n ] t 8 JULIA E. BERG NER for all m, n ≥ 0. Definition 3. 1. An invertible Se gal sp ac e is an injective fibrant in vertible simplicial space W such that the Seg al maps ξ k are weak equiv a lences of simplicial s e ts for k ≥ 2. Define the map ξ = a k ≥ 1 ( ξ k : I G ( k ) t → I ∆[ k ] t ) . Theorem 3.2. L o c alizing the inje ctive m o del c ate gory structur e on the c ate gory S S ets I∆ op of invertible simplic ial sp ac es with r esp e ct to the map ξ r esults in a mo del structu r e I S e S p c in which the fibr ant obje cts ar e the invertible Se gal sp ac es. W e ca n a lso define a model structure I S e S p f on the categor y of simplicial spa ces by lo calizing the pro jectiv e, rather than the injective, mo del s tructure with res pect to the map ξ . In this case the fibrant ob jects are in vertible s implicia l spa ces fibr ant in the pro jectiv e mo del structure for which the Sega l maps ar e isomorphisms . Like a Segal space, an inv er tible Sega l space has a set of “ob jects,” given by the set W 0 , 0 , and, for any pair ( x, y ) of ob jects, a “mapping space” map W ( x, y ). This mapping spac e is defined to b e the fib er ov er ( x, y ) of the map W 1 d 1 × d 0 / / W 0 × W 0 . How ever, the fa ct that we ar e considering an invertibl e Seg al space implies that these mapping spaces have inv erses, so that for any x and y ob jects in W , the simpli- cial se ts map W ( x, y ) and map W ( y , x ) are isomorphic. Thus, all maps are homotopy equiv alences and the space W ho equiv ⊆ W 1 consisting of “ho motopy e q uiv alences” (as defined by Rezk for Segal s paces [2 7, § 5]) is in fact all of W 1 for a n inv ertible Segal spa c e W . (It is helpful here to rega r d an invertible Segal space simply as a Segal spac e via the inclusion functor ∆ op → I∆ op , in which case Rezk’s definitions can b e used as is.) Definition 3. 3. An invertible c omplete Se gal sp ac e is an inv ertible Seg al s pace W such that the degeneracy map s 0 : W 0 → W 1 is a weak equiv alence of simplicial sets. Notice that in so me sense this definition is silly , b ecause an invertible complete Segal space is one for which W 0 and W 1 are w eakly equiv alent, and therefore, using the definition of Segal space, each equiv alent to W n for ea c h n ≥ 0. Thus, inv ertible complete Sega l spa ces are just, up to weak eq uiv a lence, constant simplicia l spaces, and equiv a le nt to simplicial sets. W e prove this fact more explicitly in sec tio n 5. How ever, we can take the sa me map ψ : ∆[0] t → E t that Rezk used to obtain a complete Segal space mo del structure on the catego ry of simplicial spa ces to obta in an inv ertible co mplete Segal s pace mo del structure on the category of in vertible simplicial spaces. Here, E denotes the nerve o f the catego r y F with tw o ob jects and a single isomorphism b et ween these tw o o b jects a nd no other nonidentit y mor - phisms. The map ψ is just the inclusion of one of these ob jects. The following theorem can then b e prov ed just as Rezk proves the analo gous theorem for the complete Sega l space mo del structure [27, 7.2] Theorem 3.4. Ther e is a mo del struct ur e I C S S on the c ate gory of invertible sim- plicial sp ac es such that the fibr ant obje cts ar e the invertible c omplete Se gal sp ac es. INVER TIBLE HOMOTOPY THEORIES 9 4. Segal gr oupoids T o define the notio n of a Segal group oid, we ag ain use Segal maps in the context of inv er tible s implicial spaces. W e first us ed this appro ach to them in [4], but the idea of a Segal gro upoid ha s also been studied fro m a sligh tly different angle by Simpson [28] and Pellissier [25]. Definition 4.1 . A Se gal pr e gr oup oid is an inv er tible simplicial space X suc h that the simplicia l set X 0 is dis crete. W e denote the categ ory o f Segal pregro upoids by S e G pd . Recall from the pr e vious sectio n that, given an in vertible simplicial s pace X , the maps β k for k ≥ 2 induce the Seg al maps ξ k : X k → X 1 × X 0 · · · × X 0 X 1 | {z } k . Definition 4.2. A Se gal gr oup oid is a Seg al gr oupo id X such that the Segal maps ξ k are weak equiv a lences of s implicia l sets for all k ≥ 2. In the prev ious pa p er, we proved that there mo del category s tructures on the category of Segal group oids with a fixed ob ject set O . Prop osition 4.3. [4, 4.1] Th er e is a m o del c ate gory st ructur e I S S p O ,f on the c ate gory of Se gal pr e gr oup oids with a fi x e d set O in de gr e e zer o in which the we ak e quivalenc es and fibr ations ar e given levelwise. Similarly, ther e is a mo del c ate- gory structur e I S S p O ,c on the same un derlying c ate gory in which the we ak e quiv- alenc es and c ofibr ations ar e given levelwise. F urthermor e, we c an lo c alize e ach of these mo del c ate gory structu res with r esp e ct to a map to obtai n mo del structur es I LS S p O ,f and I LS S p O ,c whose fibr ant obje cts ar e Se gal gr oup oids. (Recall here tha t the subscripts c and f ar e meant to suggest the injective and pro jectiv e mo del structures, with the r espective letter indicating whether cofibra- tions or fibrations a r e given levelwise.) F urthermor e, the following rigidification result holds in this fixed ob ject s et case. Prop osition 4.4. [4, 4.2] Ther e is a Qu il len e quivalenc e b etwe en t he m o del c ate- gories I LS S p O ,f and S G pd O . W e conjectured in [4] that this result should s till hold when we gener alize to the category o f all small simplicial g roup oids and all Segal gr o upoids . Be fo re proving this r esult, w e nee d to establish that w e have the necess a ry mo del structures for Segal group oids, i.e., mo del structures on the catego ry of Segal pregroup oids in which the fibrant ob jects are Sega l group oids. The fir st step in finding such mo de l struc tur es is mo difying the generating cofi- brations and gener ating acyclic cofibrations of S S e ts I∆ op c and S S e ts I∆ op f so that they are maps b etw een Sega l pregroup oids rather than maps betw een ar bitrary inv ertible simplicial spa ces. Recall that for S S ets I∆ op c , a set of gener ating cofibra tions is given by { ˙ ∆[ n ] × I ∆[ n ] t ∪ ∆[ m ] × I ˙ ∆[ n ] t → ∆[ m ] × I ∆[ n ] t | m, n ≥ 0 } . T o find the set that w e nee d, we apply a reduction functor ( − ) r which makes the space in degre e 0 discr ete and then c heck p otentially pr oblematic v alues of n a nd m , as in [7, § 4]. Thus, we obtain the s et I c given by { ( ˙ ∆[ m ] × I ∆[ n ] t ∪ ∆[ m ] × I ˙ ∆[ n ] t ) r → (∆[ m ] × I ∆[ n ] t ) r | m ≥ 0 when n ≥ 1 , n = m = 0 } 10 JULIA E. BERG NER as a p oten tial set of g enerating co fibr ations. F or the mo dification o f the gener ating cofibrations of S S ets I∆ op f , we need to take a different appro ach . T o do so, we ma k e the following definitions. Let I P m,n be the pusho ut in the diagr am ˙ ∆[ m ] × ( I ∆[ n ] t ) 0   / / ˙ ∆[ m ] × I ∆[ n ] t   ( I ∆[ n ] t ) 0 / / I P m,n . If we repla ce ˙ ∆[ m ] with ∆[ m ] in the ab ov e diagram, we denote the pushout I Q m,n . W e then define a s et of g enerating cofibratio ns for S e G pd f to b e I f = { I m,n : I P m,n → I Q m,n | m, n ≥ 0 } . Now, to g iv e the definitions o f the desired weak equiv alences for our mo del struc- tures, we establish a n appro priate “lo calization” functor . Since our mo del struc- tures ar e not actually o btained by lo calizing another mo del structur e, this functor is not technically a lo calization, but it is analogo us to the fibrant repla c emen t func- tor in I S e S p f , which is obtained fr o m lo calizatio n, and it do es turn out to b e a fibrant r eplacement functor in S e G pd f . F or the inv ertible Segal space mo del str uc tur e I S e S p c , a choice o f g enerating acyclic cofibra tions is the set { V [ m, k ] × I ∆[ n ] t ∪ ∆[ m ] × I G ( n ) t → ∆[ m ] × I ∆[ n ] t | n ≥ 0 , m ≥ 1 , 0 ≤ k ≤ m } . T o have thes e maps defined betw een Seg al pregr oupo ids rather than b etw een ar- bitrary invertible simplicial spaces, we restrict to the ca se wher e n ≥ 1 . As in [7, § 5] it can b e shown that taking an co limit of iterated pushouts along a ll such maps results in a Seg al g r oupo id which is also an inv ertible Sega l space. W e will denote this functor L c . There is an a nalogous functor in the mo del category I S e S p f which we denote L f . Thu s, using the fa ct that an in vertible Segal space X ha s “ob jects,” “ma pping spaces,” and a “ homotopy ca teg ory” Ho ( X ) (aga in, as given by Rezk in [27, § 5]), we can define the cla sses o f maps we need for the mo del structure S e G p d c . First, we define a Dwyer-Kan e quivalenc e of inv ertible Sega l spaces to b e a map f : W → Z such that • the map map W ( x, y ) → map Z ( f x, f y ) is a weak equiv alence of simplicial sets for any pair of ob jects x a nd y of W , a nd • the map Ho( W ) → Ho( Z ) is an equiv a lence of categ ories. Theorem 4.5. Th er e is a mo del c ate gory stru ctur e S e G pd c on the c ate gory of Se gal pr e gr oup oids su ch that (1) a we ak e quivalenc e is a map f : X → Y such that the induc e d map L c X → L c Y is a Dwyer-Kan e quivalenc e of Se gal sp ac es, (2) a c ofibr ation is a monomorphi sm (so every Se gal pr e gr oup oid is c ofibr ant), and (3) a fibr ation is a map with the right lifting pr op erty with r esp e ct to the m aps which ar e c ofibr ations and we ak e quivalenc es. INVER TIBLE HOMOTOPY THEORIES 11 This theorem can be proved using a s gener ating cofibratio ns the set I c defined ab ov e and as genera ting acy clic cofibr ations the set J c = { i : A → B } o f r epresen- tatives o f isomorphism classes of maps in S e G pd sa tisfying (1) for all n ≥ 0 , the spa ces A n and B n hav e countably many simplices , and (2) the map i : A → B is a monomo rphism and a weak equiv alence. Given these definitions, the pro of of the existence of the mo del structure S e G pd c , while technical, follows just as the pr oo f for the mo del str ucture S e C at c [7, 5.1]. The other mo del structure, S e G pd f , has the same w eak equiv alences a s S e G pd c , but not all mono morphisms are cofibrations. Instead, w e take the maps o f Segal pregro up oids which are cofibra tions in the pro jective mo del structure S S ets I∆ op f . Theorem 4 .6. Ther e is a c ofibr antly gener at e d mo del structur e S e G pd f on the c ate gory of Se gal pr e gr oup oids given by the fol lowing classes of morphisms: (1) the we ak e quivalenc es ar e the same as those of S e G pd c , (2) the c ofibr ations ar e t he maps which c an b e obtaine d by taking iter ate d pushouts along the maps in the set I f , and (3) the fibr ations ar e the maps with t he right lifting pr op erty with r esp e ct to the maps which ar e b oth c ofibr ations and we ak e quivalenc es. Here, the set of g enerating cofibrations is I f , a nd the set of generating acyclic cofibrations is given by the set J f which is defined a nalogously to J c but using the new definition of cofibration. Again, the pro of follows just as in [7 , 7.1 ]. 5. Quillen equiv alences Here, we show that w e still ha ve Quillen equiv alences b etw e en these v a rious mo del categor ies. In fact, the pro ofs that w e giv e in [7] con tinue to ho ld. Here we give a sketc h of what the v ar ious functor s are co nnecting these catego ries. W e beg in with the simplest example. Throughout, the topmost arrow indicates the left Quillen functor. Prop osition 5.1. The identity functor id : S e G pd f ⇄ S e G pd c : id is a Q uil len e quivalenc e of mo del c ate gories. T o c ompare the model categories S e G pd c and I C S S , first notice that we can take the inclusion functor I : S e G p d c → I C S S . This functor has a rig ht adjoint R given as follows. Recall tha t, given any s implicial space X , we can consider its 0-c oskeleton, de- noted cosk 0 ( X ) [26, § 1]. Let W be an inv er tible simplicia l space, and reg ard W 0 and W 0 , 0 as constant simplicial spaces . Consider the in vertible simplicial spaces U = c o sk 0 ( W 0 ) and V = cosk 0 ( W 0 , 0 ) and the natural maps W → U ← V . Define RW to b e the pullback of the diagr am RW / /   V   W / / U. As in [7, § 6 ], we can s e e that R is in fact a right adjoint and that the following result holds: 12 JULIA E. BERG NER Prop osition 5.2. The adjoint p air I : S e G p d c ⇄ I C S S : R is a Q uil len e quivalenc e of mo del c ate gories. T o prov e the existence of a Quillen equiv alence b etw ee n S G pd and S e G pd f , w e first notice that via the nerve functor a simplicial gro upoid ca n be reg a rded as a strictly lo cal ob ject in the categor y of I∆ op diagrams of simplicial sets with resp ect to the ma p ξ used to define the invertible Sega l s pace mo del structure. Thus, the nerve functor R : S G pd → S e G pd f can b e shown to ha ve a left adjoin t via the following lemma. Lemma 5.3. [6, 5.6] Consider t wo c ate gories, the c ate gory of al l diagr ams X : D → S S ets and the c ate gory of strictly lo c al diagr ams with r esp e ct t o some set of maps S = { f : A → B } . The for getful functor fr om t he c ate gory of st rictly lo c al diagr ams to the c ate gory of al l diagr ams has a left adjoint. Denoting this left adjoint F , the following result o nce a gain follows from the same ar gumen ts given in [7, 8.6]. Theorem 5.4. The adjoint p air F : S e G pd f ⇄ S G pd : R is a Q uil len e quivalenc e of mo del c ate gories. How ever, in this case w e a lso hav e the following additional result, emphasizing the idea that an ( ∞ , 0)-catego ry should just b e a space (or simplicial set). Theorem 5.5 . L et T : I C S S → S S ets b e the functor taking an invertible c omplete Se gal sp ac e W to its 0-sp ac e W 0 . This fun ct or has a left adjoi nt given by the fu n ctor C : S S ets → I C S S taking a simpl icial set K to t he c onstant invertible simplicial sp ac e C K . Thi s adjoint p air gives a Quil len e quivalenc e of mo del c ate gories. Pr o of. It is not hard to show that C is left adjoint to T . T o prove that this a djoin t pair is a Quillen pair , w e observe that the left adjoin t functor C pres erves co fi- brations b ecause they a re just monomorphisms in ea c h ca tegory . It also preserves acyclic cofibratio ns b ecause the ima ge o f a weak equiv alence of simplicial sets is a Reedy weak equiv alence of simplicial spa ces, and therefore also a weak equiv alence in I C S S . By the same reaso ning , C also re flects weak equiv ale nce s b etw een co fibrant ob- jects. Thus, to prove that w e hav e a Q uillen e quiv alence it remains to show that the map C ( T W ) → W is a weak equiv alence for any in vertible complete Segal spa c e W . How ever, C ( T W ) = C ( W 0 ), the co ns tan t in vertible simplicial set which has W 0 in each degree. How ever, if W is a n inv e r tible complete Seg al space, then the spaces in each degree a r e all weakly eq uiv alent. Thus, the desired map is a w eak equiv alence.  Again, we note that this result is not surpr ising, in tha t Dwyer and Kan prov ed that S G p d is Quillen equiv alent to S S ets in [12], but this particula r Quillen equiv- alence with I C S S makes the relationship esp ecially clear . W e co nclude this section by noting some r elationships b et ween mo dels for inv ert- ible ho mo top y theories as g iv en in this pa p er and mo dels for homotopy theor ies. Recall that the inclusion map ∆ op → I∆ op induces a map S S ets I∆ op → S S ets ∆ op INVER TIBLE HOMOTOPY THEORIES 13 which has a n “ in vert” map a s a left adjoint. This a djoin t pair is in fac t a Quillen pair when we consider the res p ective injectiv e model categ ory structures on the t wo categorie s. F urthermore, we still hav e a Quillen pair I C S S ⇆ C S S betw een the lo calized mo del catego ries. This adjoint relationship b et ween C S S and I C S S (where the “inv ert” map is the left a djoin t) can b e contrasted with a n Quillen pair b e tween C S S and S S e ts for which the analogo us map is the right adjoint. The map C S S → S S ets given by ta k ing a simplicial space X to the simplicia l set X 0 can b e sho wn to ha ve a left adjoint (giving, essen tially , a constant simplicial space). It can be sho wn to be a rig h t Q uillen functor b ecause if X is a fibrant ob ject (i.e., a complete Segal space and therefore Reedy fibrant), then X 0 is fibra n t in S S ets as well, and becaus e weak equiv alences betw een co mplete Seg al spaces give weak equiv alences betw een 0-spaces . 6. A Bousfield appr oa ch to Segal groupoids and inver tible co mplete Segal sp aces As in [4 ], we can a lso use a different appro ach to defining Segal group oids a nd inv ertible (complete) Seg al spaces. The idea b ehind this metho d is , when mo ving from the ordinary Segal case to the inv er tible o ne, to ch ange the pro jections used to define the Seg al maps, a n idea used by Bousfield [9]. In the categor y ∆ , consider the maps γ k : [1] → [ n ] given b y 0 7→ 0 and 1 7→ k + 1 for any 0 ≤ k < n . Just as the maps α k are used to define the or dina ry Segal maps ϕ k , we can use the maps γ k to define the Bousfi eld-Se gal m aps χ n : X n → X 1 × X 0 · · · × X 0 X 1 | {z } n for each n ≥ 2 . Definition 6.1. A Bousfield-Se gal sp ac e is a Reedy fibran t simplicial spa ce W satisfying the condition that the Bousfield-Segal maps χ n are weak equiv alences for n ≥ 2. Notice that Bousfield-Segal spaces are really just another wa y to think ab out inv ertible Sega l s paces. W e hav e given them a different name here to distinguish them from our previous definition; when we co nsider model ca tegory structures the distinction is more impor tan t, since the underlying category here is that of simplicial spac es, rather than that of inv ertible simplicial spa c es. T o define a mo del categor y s tructure o n the category o f simplicial spaces in whic h the fibrant ob jects a re Bo usfield-Segal s pa ces, we need to define an a ppropriate map with which to lo calize the Reedy mo del structure. F or γ i defined a s ab ov e , define H ( k ) t = k − 1 [ i =1 γ i ∆[1] t and the inclusion map χ k : H ( k ) t → ∆[ k ] t . 14 JULIA E. BERG NER Combining these maps for a ll v alues o f k , we obtain a map χ = a k ≥ 1 ( χ k : H ( k ) t → ∆[ k ] t ) . Theorem 6.2. L o c alizing the R e e dy mo del structur e with r esp e ct to the map χ r esults in a mo del st ructur e B S e S p c in which the fibr ant obje cts ar e the Bousfield- Se gal sp ac es. As usual, we can also lo calize the pro jective mo del s tructure with resp ect to the map χ to obta in a mo del categor y B S e S p f . The prop erties of inv er tible Seg al spaces discuss ed previously co ntin ue to hold for Bo us field-Segal spaces. In particular , we have the fo llowing definition. Definition 6 .3. A c omplete Bousfield-Se gal sp ac e is a Bousfield-Seg a l space W for which the degenera cy ma p s 0 : W 0 → W 1 is a w eak equiv alence. Recall the map ψ : ∆[0] t → E t used to define the (in vertible) complete Seg al space mo del structure. W e use it again here to establish wha t we call the complete Bousfield-Sega l space mo del structure on the catego ry of s implicial spaces. Theorem 6.4. L o c alizing the m o del structur e B S e S p c with r esp e ct to the map ψ r esults in a mo del structur e C B S S in which the fi br ant obje cts ar e t he c omplete Bousfield-Se gal sp ac es. W e now turn to the Bousfield a pproach to Segal gr oupo ids . W e gav e the following definition in [4, § 6]. Definition 6.5. A Bousfield-Se gal c ate gory is a Segal precategor y for which the Bousfield-Sega l maps χ n are weak equiv alences for e a c h n ≥ 2 . In [4], we consider the fixed ob ject set ca s e, in which we lo ok at Bous field-Segal categorie s with a given set O in deg r ee zero. In particular , we defined a fixed-ob ject version o f the map χ a s follows: χ O = a k ≥ 0  χ k O : H ( k ) t O → ∆[ k ] t O .  . Prop osition 6.6. [4, 6.1] L o c alizing the c ate gory S S p O ,f with r esp e ct to the map χ O r esults in a mo del st ructur e L B S S p O ,f whose fibr ant obje cts ar e Bousfield- Se gal c ate gories. Similarly, lo c alizing S S p O ,c with r esp e ct to χ O r esults in a mo del c ate gory we denote L B S S p O ,c . Using these model structures, w e can pro ceed to the more gene r al case, in which we have mo del str uctures B S e C at c and B S e C at f on the categor y o f Segal precate- gories in which the fibrant ob jects a re Bous fie ld- Segal catego ries. W e first consider B S e C a t c , which is analogo us to S e C at c . Considering the a cyclic cofibrations given by the set { V [ m, k ] × ∆[ n ] t ∪ ∆[ m ] × H ( n ) t → ∆[ m ] × ∆[ n ] t } for n ≥ 1 , m ≥ 1, and 0 ≤ k ≤ m enables us to define a “ loc alization” functor L B ,c taking a Sega l preca tegory to a Bousfield-Segal categ ory by taking a c o limit of pushouts along the maps of this set. Theorem 6.7. Ther e is a mo del c ate gory stru ctur e B S e C at c on the c ate gory of Se gal pr e c ate gories su ch that INVER TIBLE HOMOTOPY THEORIES 15 (1) a we ak e quivalenc e is a map f : X → Y such that the induc e d map L B ,c X → L B ,c Y is a Dwyer-Kan e quivalenc e of Bousfield-Se gal sp ac es, (2) a c ofibr ation is a monomorphism, and (3) a fibr ation is a map which has t he right lifting pr op erty with r esp e ct to the maps which ar e c ofibr ations and we ak e quivalenc es. In particular, in this mo del structure, the co fibrations sho uld b e monomo rphisms so that all ob jects ar e cofibr an t. Therefo re, we ca n use the s ame set of g e ne r ating cofibrations, I c = { ( ˙ ∆[ m ] × ∆[ n ] t ∪ ∆[ m ] × ˙ ∆[ n ] t ) r → (∆[ m ] × ∆[ n ] t ) r } , where m ≥ 0 when n ≥ 1, and when n = m = 0 , that we used for S e C at c in [7, § 5]. W e can define a set of generating acyc lic co fibrations v ery similar ly , namely , by a set of r epresentativ es of isomorphis m classes of maps i : A → B w hich a re cofibrations a nd w eak equiv alences such that for a ll n ≥ 0 the simplicial sets A n and B n hav e only countably many simplices . Notice that this definition only differs from the one in [7] in that we weak equiv alences are defined here in terms of the functor L B ,c rather than b y the functor L c taking a Segal precateg o ry to a Segal category . Given these genera ting sets, the pro of of the existence of this mode l structure follows just as in [7, 5.1]. Once aga in, we also have a co mpanion mo del structure B S e C a t f . F or this mo del str ucture, w e use a functor L B ,f taking a Segal preca tegory to a Bousfield-Sega l categ o ry which is fibrant in the pro jective, r ather than the Reedy mo del structure, and then define the weak equiv alences in terms o f this functor. How ever, as befo r e, it turns o ut that the t wo functors define the same class o f w eak equiv alences [7, § 7]. Just a s in S e C at f , we make use of the set I f = { P m,n → Q m,n | m, n ≥ 0 } . Theorem 6.8. Ther e is a mo del c ate gory structur e B S e C at f on the c ate gory of Se gal pr e c ate gories su ch that (1) we ak e quivalenc es ar e the same as those in B S e C at c , (2) a c ofibr ation is a map which c an b e obtaine d by taking iter ate d pushouts along the maps in the set I f , and (3) a fibr ation is a map which has t he right lifting pr op erty with r esp e ct to the maps which ar e c ofibr ations and we ak e quivalenc es. Now, we have the fo llowing r esults, analog ous to those of the previous section. Prop osition 6.9. The identity functor induc es a Quil len e quivalenc e id : B S e C at f ⇄ B S e C at c : i d. Theorem 6.10. The inclusion functor I : B S e C at c → C B S S has a right adjoint, and this adjoint p air is a Qu il len e quivalenc e. As descr ibed in the previous section, the rig h t adjoint here, applied to an ob ject W of C B S S , is given by a pullback o f the diagram W → cosk 0 ( W 0 ) ← cosk 0 ( W 0 , 0 ) . 16 JULIA E. BERG NER The pro of that this map is adjoint to the inclus ion map a nd tha t the adjoint pa ir is in fact a Quillen equiv alence of mo del categ ories follows just a s the o ne given in [ § 6]thesis. W e can again make use o f Lemma 5.3 and a n a rgument like the one in [7, 8.6] to prove the fo llo wing r esult. Theorem 6.11. The n erve funct or R : S G pd → B S e C at f has a left adjoi nt, and this adjoint p air is a Q uil len e quivalenc e. Lastly , w e can co mpare the mo del structure C B S S to the mo del structur e on simplicial sets just as we did in the pro of o f Theorem 5 .5. Theorem 6.12. L et T : C B S S → S S ets b e t he funct or t aking an c omplete Bousfield- Se gal sp ac e W to its 0-sp ac e W 0 . Th is functor has a left adjoint C : S S ets → C B S S taking a simplici al set K to the c onstant invertible simplicial sp ac e C K . This ad- joint p air gives a Quil len e quivalenc e of mo del c ate gories. References [1] John Baez, The homotop y h yp othesis, talk given at the Fi elds Institute, January 2007, slides a v ailable at http ://www.math.ucr.edu/home/baez/ homotop y/ . [2] Clar k Barwick, pr iv ate comm unication. [3] Clemens B er ger, A cell ular nerv e for higher categories, A dv. M ath. 169 (2002), no. 1, 118-175. [4] J.E. Bergner, Adding inv erses to diagrams encoding algebraic structures, prepri n t av ailable at math.A T/0610291. [5] J.E. Bergner, A model category structure on the category of simpl icial categories, T r ans. Am er. Math. So c. 359 (2007), 2043-2058. [6] J.E. Bergner, Rigidification of algebras ov er mu lti-sorted theories, A lgebr. Ge om. T op ol. 6(2006) 1925-1955. [7] J.E. Bergner, Three mo dels for the homotop y theo ry of homotopy theories, T op olo gy 46 (2007), 397-436. [8] G. Biedermann, On the homotopy theory of n -t ypes, preprint av ail able at math.A T/0604514. [9] A.K . Bousfield, The simplicial homotop y theory of i terated lo op spaces, unpublished man u- script. [10] Denis-Charl es Cisinski, Batanin higher group oids and homotop y ty p es, preprint a v ailable at math.A T/0604442. [11] W.G. Dwyer and D. M . Kan, F unction complexes in homotopical algebra, T op olo gy 19 (1980), 427-440. [12] W.G. Dw yer and D.M. Kan, Homotop y theory and simpl i cial group oids, Pr o c. Kon. A c ad. van Wetensch. , A87(4) (1984), 379-385. [13] W.G. Dwyer and D.M. Kan, Simpli cial l ocalizations of categories, J. Pur e Appl. Algebr a 17 (1980), no. 3, 267–284. [14] W.G. Dwyer and J. Spalinski, Homotop y theories and mo del categories, in Handb o ok of Algebr aic T op olo gy , Elsevier, 1995. [15] P .G. Go erss and J.F. Jardine, Simplicial Homotopy The ory, Pr o gr ess in Mathematics, 174 , Birkh¨ auser V erlag, Basel, 1999. [16] A. Grothendiec k, Pursuing stac ks, lett er to D. Quillen, a v ailable at www.grothendiec kcircle.org. [17] Phili p S. Hirs chhorn, Mo del Cate gories and Their L o c alizations , Mathematical Surveys and Monographs, 99. A merican Mathematical Society , Providence , RI, 2003. [18] Mark Hov ey , Mo del Cate gories , M athemat ical Surveys and Monographs 63. American Math- ematical Society , Providence, R I, 1999. [19] A. Joy al, Simplicial categories vs quasi-categories, i n preparation. [20] A. Joy al, The theory of quasi- categ ories I, in preparation. INVER TIBLE HOMOTOPY THEORIES 17 [21] Andr´ e Jo yal and Myles Tierney , Quasi- categ ories vs Segal spaces, Contemp. M ath. 431 (2 007) 277-326. [22] Jacob Lurie, Higher topos theory , prepri n t av ailable at math.CT/0608040. [23] Saunders M ac Lane, Catego ries for the working mathematician. Se c ond ed ition. Gr aduate T exts in Mathematics, 5. Springer-V erlag, New Y or k, 1998. [24] Simona P aoli, Semistrict T amsamani n -gr oupoids and connect ed n - t ypes, preprint a v ailable at math.A T/0701655. [25] Regis Pellissier, Cat´ egories enrichies f aibles, preprint a v ailable at math.A T/0308246 . [26] C.L. Reedy , Homotop y theo ry of model categories, unpublished man uscript, av ailable at h ttp://www-math.mit.edu/ ∼ psh. [27] Charles Rezk, A mo del for the homotop y theory of homotop y theory , T ra ns. Amer. Math. So c. , 353 (2001), no. 3, 973-1007. [28] Carlos Simpson, Effective generalized Seifer t-V an Kamp en: ho w to calculate Ω X , preprint a v ailable at q-alg/9710011. [29] Zouhair T amsamani, Equiv alence de la th´ eorie homotopique des n - groupo ¨ ıdes et celles des espaces top ologiques n - tronqu ´ es, preprint av ail able at alg-geom/9607010 . E-mail addr ess : bergner j@member.a ms.org Dep ar tment of Ma thema tics, Kansas St a te Univ ersity, 138 Cardwell Hall, Ma nha t- t an, KS 66506 Curr ent addr ess : Uni versity of California, Riverside, CA 92521 ERRA TUM TO “ADDING INVERSES TO DIA GRAMS ENCODING ALGEBRAIC STR UCTURES” AND “ADDING INVERSES TO DIA GRAMS II: INVER TIBLE HOMO TOPY THEORIES ARE SP A CE S” JULIA E. BER GNER 1. St a tement o f previous err or In previous w ork, we studied v arious kinds of functor s X : ∆ op → S S et s satis- fying a Segal co ndition, so that the maps X n → X 1 × X 0 · · · × X 0 X 1 | {z } n are weak equiv a lences of simplicial sets for n ≥ 2. When we impo sed the additional condition that X 0 = ∆[0], such ob jects, called re duc e d Se gal c ate gories o r Se gal monoids were shown to b e equiv a len t to simplicial monoids [4]. When instead X 0 = ∐ ∆[0], some discrete simplicia l set, then X is a Se gal c ate gory can more generally b e rega r ded as an up-to-homotopy mo del for a s implicial categor y with this same ob ject set [4]. These results were used in the comparison b etw e e n the model str uc tur e for all Segal ca tegories (no t just with a fixe d set in deg ree zero ) and the mo del structure for simplicial categories [6]. F urthermore, the former mo del structure was shown to be Q uillen equiv alent to the mo del s tructure for complete Segal s paces, o r functors ∆ op → S S e ts wher e the discr ete level zer o condition is replaced with a “complete- ness” condition [6]. A natural ques tion w as then whether these results c ould be generalized to an “inv er tible” and t wo metho ds were prop osed in both of the pap ers [2], [3]. The first was to repla ce ∆ op in the ab ov e definitions with a categor y I∆ op in which the ob jects had an inv olution map. How ever, these results were in fact incorrec t, in that this involution do es not adequately enco de an inverse map. In this note, we clarify that this diagra m should e nc o de the str ucture of a mono id with inv o lutio n rather than a group, o r categ ory with inv olution rather than a g r oupo id, in the ca se of multiple ob jects. The second a pproach g iv en in these pap ers, g iv en by using differe n t pro jection maps as first used by Bousfield [7 ], is s till corr ect. Ac knowledgmen ts. The author would like to thank Philip Hackney fo r conv er- sations ab out this work. 2010 Mathematics Subject Classific ation. 55U35, 18G30 , 18E35. The author wa s partially supp or ted by NSF gran t DMS-0805951. 1 2 JULIA E. BERG NER 2. Repla cing groups with monoids with inv olution In the case of monoids, we consider functors ∆ op → S S ets , where the categor y ∆ op has as ob jects finite ordered sets [ n ] = (0 → 1 → · · · → n ) for each n ≥ 0 and as morphisms the o ppos ites of the o rder-prese rving maps b etw een them. Notice that each [ n ] can b e rega rded as a category with n + 1 ob jects and a sing le morphism i → j whenever i ≤ j . In [2] we defined a category I∆ op , whose ob jects are given by sma ll g roup oids I [ n ] = (0 ⇄ 1 ⇆ · · · ⇄ n ) for n ≥ 0 . In other words, each I [ n ] is a category with n + 1 ob jects and a single isomo r phism b etw een a n y t wo ob jects. The morphisms of I∆ op are g enerated b y tw o sets o f ma ps: the o pp osite of the order- preserving maps from ∆ op , and a n inv olution morphism on each I [ n ] whic h s e nds each i to n − i . The ho p e was that functor s I∆ op → S S e ts satisfying a Segal conditio n enco ded a group structure. Unfortunately , in verses a re not a dequately given, so such functors actually give the structure of a monoid with inv o lutio n. Thu s, we change the terminology g iven in [2 ] as follows. In the cas e of ∆ , the simplicial set ∆[ n ] is giv e n b y the representable functor Hom ∆ ( − , [ n ]). Similarly , we can define an o b ject I ∆[ n ] which is given b y the representable functor Hom I∆ ( − , I [ n ]). T he s e n - simplic es with involution are the standard building blo c ks o f the spaces we consider here. In pa rticular, every simplex should b e regar ded a s having a corresp onding “ reverse” simplex. As with simplicial sets, we can consider the bo undary of I ∆[ n ], denoted ∂ I ∆[ n ], which consists o f the nondegenera te simplices of I ∆[ n ] of degree less tha n n . Thu s, we can define a n simplicia l set with involution to b e a functor I∆ op → S ets and, mor e generally , an simplicial obje ct with involution in a catego r y C to be a functor I∆ op → C . W e denote the catego ry of simplicial sets with inv olution by I S S ets . W e further cons ider the cas e of simplicial s pa ces with in volution, or functors I∆ op → S S ets . Since there is a forgetful functor U : I S S ets → S S ets (resp ectively , U : S S ets I ∆ op → S S ets ∆ op ), w e define a map f of s implicial sets (resp ectively , spa ces) with inv olutio n to be a weak equiv a le nce if U ( f ) is a w eak equiv alence of simplicial sets (r espectively , spa ces). In par ticular, we define a Se gal pr e c ate gory with involution to b e an simplicial space with in volution X such that the simplicial s et X 0 is discr ete. If X 0 = ∆[0], then we call it a Se gal pr emonoid with involution . T o define a Sega l categor y with inv olution, use the maps ξ n : X n → X 1 × X 0 · · · × X 0 X 1 | {z } n defined in [2, § 4]. Thus, a Se gal c ate gory with involution is a Segal precategory with inv olution X suc h that for each n ≥ 2 the map ξ n is a weak equiv alence of simplicial sets. Obtaining an appro pr iate mo del structure requires lo calization with resp ect to the following map: ξ O = a n ≥ 1 ( ξ n : a x ∈O n +1 ( I G ( n ) t x → I ∆[ n ] t x )) . The pro ofs of the following tw o pro pos itions contin ue to ho ld, with the necessary changes in terminolo g y . ERRA TUM TO “ADDING INVERSES” 3 Prop osition 2.1. [2, 4.1] Ther e is a mo del c ate gory structur e LS S ets I∆ op O ,f on the c ate gory of Se gal pr e c ate gories with involution with a fixe d set O in de gr e e zer o in which the we ak e quivalenc es and fibr ations ar e given levelwise. Similarly, t her e is a mo del c ate gory struct ur e LS S ets I∆ op O ,c on the same u nderlying c ate gory in which the we ak e qu ivale nc es and c ofibr ations ar e give n levelwise. F urt hermor e, we c an lo c alize e ach of these mo del c ate gory structur es with the m ap ξ O to obtain mo del structur es LS S ets I∆ op O ,f and LS S ets I∆ op O ,c whose fibr ant obje cts ar e Se gal c ate gories with involution. Prop osition 2.2. [2, 4 .2] The adjoint p air given by the identity functor induc es a Quil len e quivalenc e of mo del c ate gories LS S ets I∆ op O ,f / / LS S ets I∆ op O ,c . o o In [2 ], we claimed that there w as a Quillen equiv alence LS S ets T G ∗ ⇆ LS S e ts I∆ op ∗ ,f . Unfortunately , we did no t adequately esta blish that we obtained g roup s tructures using this catego ry I∆ op . W e s eek to establish that our pr e vious pro of instead gav e a Quillen equiv alence LS S ets T M I ∗ ⇆ LS S e ts I∆ op ∗ ,f where T M I is the the ory of monoids with involution . This theory T M I has as ob jects T k which a re giv e n b y the free monoid with inv olution on k generator s. In other words, T k is free on g enerators x 1 , . . . , x k and x 1 , . . . , x k with inv o lutio n I ( x k ) = x k . T her e is a monoid map τ : T k → T k given by x j 7→ x k − j +1 . This map will corresp ond to the flip map of each ob ject I [ k ] in I∆ op . Using Ba dzioch’s theor em from [1], such a Quillen e q uiv alence will complete the pro of of the following theorem. Theorem 2. 3. The mo del c ate gory structure A l g T M I is Qu il len e quivalent t o the mo del c ate gory structur e LS S ets I∆ op ∗ ,f . As in [4], we prove this theorem using several lemma s . Note that in the mo del structure LS S ets I∆ op ∗ ,c , we denote b y L 1 the lo calization, o r fibr ant replacement functor. Analogously , we denote by L 2 the lo caliza tion functor in LS S ets T M I ∗ . The first s tep in the pr o o f o f the theore m is to show w ha t the lo caliza tion functor L 1 do es to the n -simplex with inv olution I ∆[ n ] t . By I ner v e( − ) t , we denote the r ep- resentable functor Hom( I [ n ] , − ), viewed as a transp osed cons tan t simplicial space. It is her e , Lemma [2, 4 .5], tha t the ma jor error o ccurred. The correct statement is as follows. Prop osition 2.4 . L et F n denote the fr e e monoid with involution on n gener ators. Then in LS S ets I∆ op ∗ ,c , L 1 I ∆[ n ] t ∗ is we akly e quivalent t o I nerve ( F n ) t for e ach n ≥ 0 . In the pro of, we defined a filtratio n of I nerve ( F 1 ) t as follows: Ψ k ( I nerve( F 1 ) t ) j = ( ( x n 1 | · · · | x n j ) | j X ℓ =1 | n ℓ | ≤ k ) where x and its “ in verse” x − 1 denote the tw o nondegenera te 1-simplices of I ∆[1] t ∗ = Ψ 1 . The problem is that we a ssumed her e that x could b e canceled with x − 1 , when 4 JULIA E. BERG NER there is no structure built in to I∆ op to make this cancelation p ossible. In short, we do not really hav e the str ucture of a g roup, but o nly of a monoid with inv olutio n. Therefore, we instead define the filtratio n of I ner ve( F 1 ) t by Ψ k ( I nerve( F 1 ) t ) j = { ( w 1 | · · · | w j ) | ℓ ( w 1 · · · w j ) ≤ k } where the w i are words in x and x , and ℓ denotes word length. With this mo dification, the pre v ious pro of g oes thr ough as befo re. T o obtain the appropria te a djoin t pair for the Quillen eq uiv a lence, we firs t define a functor J : I∆ op → T M I . On o b jects, this functor is defined by I n 7→ T n . On face and degener a cy maps coming from ∆ op , this functor b ehav es the same as the functor ∆ op → T M as defined in [4]. Specifically , the coface a nd co degeneracy maps maps in I∆ ar e given b y d i ( x k ) =      x k i > k x k x k +1 i = k x k +1 i < k and s i ( x k ) =      x k i ≤ k e i = k − 1 x k − 1 i < k − 1 . The inv o lution map is sent to the map τ : T ℓ → T ℓ defined by x k 7→ y ℓ − k +1 . This functor J induces a map J ∗ : LS S ets T M I ∗ ,f → LS S e ts I∆ op ∗ ,f for which we have a left a djoin t J ∗ via left Kan e x tension. As in [2], define I M [ k ] to b e the functor T M I → S S e ts given b y F n 7→ Ho m T M I ( F k , F n ) = ( F k ) n , and let H = I nerve( F k ) t . The following results contin ue to hold, replacing T G with T M I . Lemma 2.5. [2, 4.9 ] In LS S ets T M I ∗ , L 2 J ∗ ( H ) is we akly e quivalent to I M [ k ] . Prop osition 2. 6. [2, 4.10 ] F or any obje ct X in LS S ets I∆ op ∗ ,c , L 1 X is we akly e quiv- alent to J ∗ L 2 J ∗ X . Prop osition 2.7. [2, 4.1 1, 4.3] The adjoi nt p air J ∗ : LS S ets I∆ op ∗ ,f / / S S ets T M I ∗ : J ∗ o o is a Q uil len e quivalenc e. The more g eneral fixed-o b ject ca se, as w ell as the more general c a se a s devel- op ed in [3 ], can be co rrected by co nsidering categor ies with inv olution ra ther tha n group oids whenever I∆ op is the indexing categor y . References [1] Bernard Badzio c h, A lgebraic theories in homotopy theory , Ann. of Math. (2) 155 (2002) , no. 3, 895–913. [2] J.E. Bergner, Adding i n v erses to diagrams encoding algebraic structures, Homolo gy , Homotopy Appl. 10(2), 2008, 149174. [3] J.E. Bergner, Adding i n v erses to diagrams I I: Inv ertible homotopy theories are spaces, Homol- o gy, Homotopy Appl. 10(2), 2008, 175193. [4] J.E. Bergner, Simplicial monoids and Segal categ ories, Contemp. M ath. 431 (2007) 59-83. [5] J.E. B er gner, Correction to “Simplicial monoids and Segal categories,” a v ailable at math.A T/0806.1767. [6] J.E. Bergner, Three models for the homot op y theory of homotop y theories, T op olo gy 46 (2007) , 397-436. ERRA TUM TO “ADDING INVERSES” 5 [7] A.K. Bousfield, The simplicial homotopy theory of i terated lo op spaces, unpublished manu- script. E-mail addr ess : bergner j@member.a ms.org University of California, Riverside, CA 9252 1