Algebraic models for higher categories

Algebraic mo dels for higher ca tegories Thoma s Nikolaus ∗ F ac h b ereic h Mathemat ik, Universit¨ at Hamburg Sc h w erpunkt Algebr a und Zahlentheorie Bundesstraß e 55 , D – 20 146 Hamburg Abstract W e in tro duce the notion of algebraic fib ran t ob jects in a general mo del category and es- tablish a (combinato rial) mo del catego ry structure on algebraic fibrant ob jects. Based on this constr u ct ion we prop ose algebraic Kan complexes as an algebraic mo del for ∞ - group oids and algebraic quasi-categories as an algebraic m odel for ( ∞ , 1)-cat egories. W e furthermore giv e an explicit pro of of the homotop y hyp o thesis. Con ten ts 1 In tro duction 2 2 Algebraic fibran t ob jects 4 2.1 F ree a lg e braic fibrant ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Monadicit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 An aux iliary cons truction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Limits a nd Colimits in Alg C . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Mo del structure on Alg C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Algebraic Kan complexes 15 3.1 Kan complexes and ∞ -group oids . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Algebraic Kan complexes as ∞ -group oids . . . . . . . . . . . . . . . . . . . . 16 3.3 The homoto p y h yp othesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Algebraic quasi-categories 20 4.1 Quasi-categories as ( ∞ ,1)-catego rie s . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Algebraic q uasi-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Group oidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Outlo ok 24 ∗ Email address: Thoma s.Nik ola us@uni-ham burg.de 2000 Mathematics Sub ject C la ssification: Prima ry: 55U35; Secondar y: 18G30. 1 1 INTR ODUCTION 2 1 In tro duction Simplicial sets hav e b een in tro duced a s a com binatorial mo del for top ological spaces. It has b een kno wn for a long time that top ological spaces and certain simplicial sets called Kan complexes are ’the same’ from the viewp o in t of homotopy theory . T o mak e this statemen t precise Quillen [Qui67] in tro duced the concept of mo del categories and equiv alence o f mo del categories as a n abstract framew ork for homotop y theory . He endo w ed the c ategory T op of top ological spaces and the category sSet of simplic ial sets with mo del category structures and sho w ed that T o p and sSet are equiv alen t in his sense. He could ide n t ify Kan complexes as fi bran t ob jects in the mo del structure on sSe t. Later higher category theory came up. A 2 -category ha s not only ob jects and morphisms, like an ordinary (1-)catego r y , but also 2-morphisms, whic h are morphisms b et w een morphisms. A 3-category has also 3-mor phisms b et wee n 2-morphisms and so on. Finally an ∞ - catego ry has n -morphisms for a ll n ≥ 1. Unfortunately it is ve ry ha r d to giv e a tractable definition of ∞ -categories. See [Lei02] f or sev eral definitions of higher c ategories. An inte resting sub class of all ∞ - cat ego rie s are those ∞ -categories for whic h all n -mor phisms are inv ertible. They are called ∞ -group oids. A s tandard c onstruction from algebraic top ol- ogy is the fundamen tal group oid construction Π 1 ( X ) of a top ological space X . Allo wing higher paths in X (i.e. homotopies) extends this construction to a fundamental ∞ -gro u p oid Π ∞ ( X ). It is widely b eliev ed that ev ery ∞ - groupoid is, up to equiv alence, of this form. This b elief is called the homotop y h yp othesis [Bae07]. There is ano t h er imp ortan t sub c lass of ∞ -categories, called ( ∞ , 1)- cat ego rie s. These are ∞ -categories where all n -morphisms f or n ≥ 2 are in v ertible. Th us the only difference to ∞ -group oids is that there ma y b e non-inv ertible 1-morphisms. In particular the collection of all small ∞ -group oids forms a ( ∞ , 1)-category . Another example of a ( ∞ , 1)-category is the category of top ological spaces where the n -morphisms ar e give n by n -homotopies. In the language of ( ∞ , 1 )-categories a more refined v ersion of the homotopy h yp othesis is the assertion that the fundamen tal ∞ - groupoid construction prov ides an equiv alence of the re- sp e ctiv e ( ∞ , 1)-categories. F rom the p ersp e ctiv e of higher category theory Quillen mo del structures are really presen- tations of ( ∞ , 1)-categories, see e.g. [L ur 0 9] appendix A.2 and A.3. Hence w e think ab out a mo del category structure as a generators and relations description of a ( ∞ , 1)-category . A Quillen equiv alence the n b ecomes an adjoin t equiv alence of the presen t ed ( ∞ , 1) -categories. Th us the classical Quillen equiv a lence b et w een top ological spaces and simplicial sets really enco de s an equiv alence of ( ∞ , 1)-categories. Keeping this statement in mind it is reasonable to think of a simplicial set S a s a mo del for an ∞ -group oid. The n -morphisms a r e then the n simp lices S n . And in fact there has b een m uch prog r ess in higher category theory using simplicial sets as a mo del for ∞ - groupo ids. This mo del has certain disadv an tages. First of all a simplicial set do es not enco de how to comp ose n -morphisms. But suc h a comp osition is inevitable for higher categories. This problem is usually addres sed as follows: The mo del structure axioms on sSet imply that in the corresp onding ( ∞ , 1) category eac h simplicial set is equiv alent to a fibrant ob ject i.e . a Kan comple x. It is p ossible to in terpret the lifting prop erties o f a Kan complex S as the existence of comp ositions in the ∞ -g r oupoid, see section 3.1. Althoug h the lifting conditions ensure the existence of comp ositions fo r S , 1 INTR ODUCTION 3 these comp ositions a r e only unique up to homotopy . This makes it sometimes hard t o w ork with K a n complexes as a mo del for ∞ -group o ids . Another disadv an tage is that the sub cat- egory of Kan complexes is not v ery w ell b eha v ed, for example it do es not hav e colimits and is no t lo cally presen table. The idea of this pap er to solve these problems is to consider a more algebraic v ersion of Kan complexes as mo del for ∞ -groupo ids . More precisely w e will consider Kan complexes endo wed with the additiona l structure of distinguished fillers. W e call them algebraic Kan complexes. W e sho w that the category of a lg e braic Kan complexes has all colimits and limits and is lo cally presen t a ble (theorem 3.2.3). F urthermore we endo w it with a mo del structure and sho w that it is Quillen equiv alen t to simplicial sets (theorem 3.2.4-.5 ) . The name algebraic will b e j ustified b y iden tifying algebraic Kan complexes as algebras f or a certain monad on simplicial sets (theorem 3.2 .1 - 2). The fact that a lgebraic Kan complexes really mo del ∞ -gro upoids will b e justified b y a pro of of the appropria te v ersion o f the homoto p y h yp othesis (coro lla ry 3.6). W e will generalize this notion of algebraic K an complex to algebraic fibran t ob jects in a general mo del category C , whic h satisfies some tec hnical conditions, stated at the b eginning of section 2. In particular we sho w that C is Quillen equiv alen t t o the mo del category Alg C of algebraic fibran t o b jects in C (theorem 2.20). W e sho w that Alg C is mona dic ov er C (prop osition 2.2) and that all ob jects ar e fibran t. In addition we g iv e a form ula ho w to compute (co)limits in Alg C from (co)limits in C (section 2.4). Finally we apply the general construction to the Jo y al mo del structure on sSet. This is a simplicial mo del for ( ∞ , 1)-categories [Jo y08, Lur09]. The fibran t o b jects are called quasi- categories. W e prop ose the category AlgQuasi of algebraic quasi-categories as our mo del for ( ∞ , 1 )-categories (section 4.2). O n e of its ma jo r adv antages is that t he mo del structure on algebraic quasi-categories can b e describ ed v ery explicitly , in particular w e will give sets of generating cofibrations and trivial cofibrations (theorem 4.4). Suc h a generating set is not kno wn for the Jo y al structure. There ha v e b een other prop osals for algebraic definitions o f ∞ - groupo ids, see e.g. [Cis07]. But as p oin t ed out in the introduction of [Mal10] the issue to find a lo cally presen table algebraic mo del is still op en. Strictly sp eaking the problem describ ed there is solv ed b y the mo del of alg e braic Ka n complexes. Nev ertheless higher category theorists may feel that algebraic Kan complexes a r e after all not the kind of mo del that they w ere lo oking for. But since the mo del presen ted here is formally a lgebraic b y the usual definition, this at least indicates that the formal definition for what is supp ose d to coun t as an algebraic mo del for ∞ -group oids might need to b e refined. This pap e r is orga niz ed as fo llows. In Section 2 w e giv e the definition of the category Alg C for a general mo del catego r y C . W e pro v e t ha t Alg C enjoy s excellen t categor ic al prop erties and that it admits a mo del structure Quillen equiv alen t to C . In Section 3 w e in ve stigate algebraic Kan complexes a s a mo del for ∞ -groupo ids . In particular we pro v e the ho m otopy hypoth- esis. In section 4 w e inv estigate algebraic quasi-categories as a mo del for ( ∞ , 1)-categories. F urthermore w e compare algebraic Ka n complexes and algebraic quasi-categories. In Section 5 we sk etc h some further p ossible applications. Ac kno wledgemen ts. The a uthor is supp orted b y the Collab orativ e Researc h Centre 676 2 ALGEBRAIC FIBRANT OBJECTS 4 “P a rticle s, Strings and the Early Unive rse - the Structure of Matter and Space-Time”. The author thanks Urs Sc hreib er, Mik e Shulman, Emily Riehl, Ric ha rd G arner, Till Barmeier and Christoph Sc h weigert for helpful discussions and commen ts on the draft. 2 Algebraic fibr ant ob jects Let C b e a cofibrantly generated mo del category . F or the terminology of mo del categories w e refer to [Ho v07]. F urthermore w e make the assumption: All trivial cofibrations in C are monic. This is true in man y mo del catego r ies. F or example in simplicial sets with either the Quillen or the Joy al mo del structure. Cho ose a set of trivial cofibrations { A j → B j } j ∈ J in C suc h that an ob ject X ∈ C is fibrant iff for ev ery morphism A j → X with j ∈ J there exists a filler, that means a morphism B j → X rendering the diagram A j / /   X B j ? ?         (1) comm utat ive. W e could take J to b e a set of generating trivial cofibrations but in general J migh t b e smaller. W e assume that the domains A j are small o b jects and that C is co com- plete, so t ha t Quillen’s small ob ject argumen t yields a fibrant replacemen t . F or simplicity w e assume that the A j ’s are ω -small but ev erything is still v alid if they are o nly κ -small for an arbitrary (small) regular cardinal κ . In order t o hav e a more a lgebraic mo del fo r fibrant ob jects w e w an t to fix fillers f o r a ll diagrams. Definition 2.1. An algebr aic fi br ant obje ct (of C ) is an ob ject X ∈ C together with a distinguished filler for eac h morphism h : A j → X with j ∈ J . That means a morphism F ( h ) : B j → X rendering diagra m (1) commutativ e. A map of a lgebraic fibra nt ob jects is a map that sends distinguished fillers to distinguished fillers. The catego r y of algebraic fibrant ob jects is denoted by Alg C . In pa rticu lar fo r eac h algebraic fibrant ob ject the underlying ob ject X ∈ C is fibran t b ecaus e all fillers exist. No w w e ha v e the canonical forgetful functor U : Alg C → C whic h sends an algebraic fibran t ob ject to the underlyin g o b ject of C . The task of this section is to sho w that U induces an equiv alence b et w een mo del categories. Mor e precisely , w e wan t to endow Alg C with a mo del category structure and sho w that U is the righ t adjoint of a Quillen equiv alence. 2 ALGEBRAIC FIBRANT OBJECTS 5 2.1 F ree algebraic fibran t ob jects As a first step we give an explicit description of the left adjo in t F : C → Alg C called the free a lg e braic fibran t ob ject functor. W e wan t to use R ic hard Garner’s impro v ed ve rsion of Quillen’s small ob ject argument [Gar09]. The idea is to start with an ob ject X ∈ C and to successiv ely add fillers in a free wa y . So w e define X 1 to b e the pushout (in C ) of the diagram F A j / /   X   F B j / / X 1 where the disjoint unions are t a k en o v er all j ∈ J and morphisms A j → X . Not e tha t the inclusion X → X 1 is monic due to our a s sumptions that trivial cofibratio ns ar e monic. F or those morphisms h : A j → X 1 whic h factor through X the structure morphisms B j → X 1 pro vide fillers. These are w ell defined b ecaus e the factorization o f h through X is unique since X → X 1 is monic. Unfortunately there migh t b e mor ph isms h : A j → X 1 whic h do not factor thro ug h X . Th us in order to add additional fillers for these let X 2 b e the pushout F A j / /   X 1   F B j / / X 2 where the disjoint union is tak en ov er j ∈ J and those morphisms A j → X 1 whic h do no t factor through X . Note that this differs fro m the ordinary small ob ject argument where this colimit is t a k en ov er all morphisms A j → X 1 . W e again b o okmark the fillers B j → X 2 and pro ceed inductiv ely . Eve n tually we obtain a sequence X → X 1 → X 2 → X 3 . . . where all mor phisms are b y construction trivial cofibrations. Let X ∞ := lim − → ( X → X 1 → X 2 → X 3 . . . ) b e the colimit ov er this diagra m. Note that the inclusion X → X ∞ is b y construction a trivial cofibration, in particular a weak equiv alence. No w let h : A j → X ∞ b e a morphism. Because A j is ω -small this factors through a finite step and our construction implies that there is a unique smallest m suc h that h factors t hro ugh X m . Then we ha v e the filler F ( h ) : B j → X m +1 → X ∞ . This mak es X ∞ in t o an algebraic fibran t ob ject. 2 ALGEBRAIC FIBRANT OBJECTS 6 Prop osition 2.2. The assignment F : C → A l g C w h ich sends X to X ∞ is left adjoint to U : Alg C → C . F urthermor e the unit of this adjunc tion is the inclusion X → X ∞ , henc e a we ak e quivalenc e. Pr o of. Let Z b e an a lg e braic fibra n t ob ject. W e ha v e to sho w t ha t for a morphism ϕ : X → U ( Z ) in C there is a unique morphism ϕ ∞ : X ∞ → Z in Alg C rendering X ϕ / /   Z X ∞ ∃ ! ϕ ∞ = = | | | | comm utat ive. But this is trivial since all w e did in going fro m X to X ∞ w as gluing new distinguished fillers whic h hav e to b e sen t to the distinguished fillers in Z . Corollary 2.3. Each of the maps i : F A j → F B j admits a c anonic al r e tr act (le ft inverse). Pr o of. A retract is a map r : F B j → F A j suc h that the comp osition F A j i → F B j r → F A j is the iden t ity . Because F is left adjoin t to U , this is the same thing as a map r ′ : B j → U F A j suc h that the comp osition A j → B j r ′ → U F A j is the unit of the adjunction, i.e. the inclusion i ′ : A j → F A j . Hence r ′ is just a filler for the morphism i ′ , and suc h a filler exists canonically b ecause F A j is an algebraic fibrant ob ject. 2.2 M onadicit y In this section we wan t t o sho w that algebraic fibran t ob jects in C are algebras f o r a certain monad. Th is is a rat her direct justification to call them alg e braic. Let T b e the monad whic h is induced b y the adjunction F : C / / Alg C : U o o That means T = U ◦ F : C → C . Prop osition 2.4. The c a t e gory C T of T -algebr as in C is e quivalent to the c ate gory Alg C . Mor e pr e cisely the functor U induc es an e quivalenc e U T : A lg C → C T . In abstract langua g e the prop osition states that the adjunction ( F , U ) is monadic. By Bec k’s monadicit y theorem we ha v e to sho w that 1. a morphism f in Alg C is an isomorphism iff U ( f ) is an isomorphism in C ; 2. Alg C has co equalizers of U -split co equalizer pairs and U preserv es those co equalizers. 2 ALGEBRAIC FIBRANT OBJECTS 7 Let us first turn tow ards prop ert y 1. Assume f : X → Y is a morphism in Alg C suc h that U ( f ) is a n isomorphism in C with inv erse g : U ( Y ) → U ( X ). It suffices to sho w that g is a morphism in Alg C , i.e. it sends distinguished fillers to distinguished fillers. But this is satisfied since f and g induce isomorphisms b et w een sets hom C ( A j , X ) ∼ = hom C ( A j , Y ) and hom C ( B j , X ) ∼ = hom C ( B j , Y ) and th us g preserv es distinguished fillers since f do es. The second prop ert y is seemingly more inv olv ed. A parallel pair of arrows f , g : X → Y in Alg C is a U split co equalizer pair if the corresp onding co equalizer diagram in C U ( X ) U ( g ) / / U ( f ) / / U ( Y ) π / / Q allo ws sections s of π and t of U ( f ) such that U ( g ) ◦ t = s ◦ π . W e will endow Q with the structure of an algebraic fibrant ob j ect suc h that it is the co equalize r o f the initial pair f , g in Alg C . Therefore w e hav e to fix a filler F ( h ) : B j → Q for each morphism h : A j → Q . Since s is a section of π , the image of the morphism s ◦ h : A j → Y under s is h . Th us w e let F ( h ) b e the imag e of the distinguished filler f o r s ◦ h . Then t he follo wing lemma sho ws that prop ert y 2 and thus prop osition 2.4 holds. Lemma 2.5. The morph i s m π : Y → Q is a morp h ism in Alg C wh i c h is a c o e q uali z e r of the p air f , g : X → Y . Pr o of. Fir st w e che c k that π lies in Alg C . T ak e a mor phism h : A j → Y . By definition of fillers in Q w e ha v e to sho w that the fillers F ( h ) , F ( ˜ h ) : B j → Y for h : A j → Y and f or ˜ h := s ◦ π ◦ h : A j → Y are sen t to the same filler in Q . But we hav e h = U ( f ) ◦ t ◦ h and ˜ h = U ( g ) ◦ t ◦ h and therefore this f ollo ws from the fact that Q is the co equalizer. No w we w an t to v erify the univ ersal pr o perty . W e ha v e to c heck that for eac h morphism ϕ : Y → Z in Alg C , suc h that ϕ ◦ f = ϕ ◦ g : X → Z there is a unique ϕ Q : Q → Z in Alg C suc h that X g / / f / / Y π / / ϕ   Q ∃ ! ϕ Q       Z comm utes. F rom the fact that Q is the co equalizer in C we obtain a unique morphism ϕ Q and it o nly remains to show that it lies in Alg C , i.e. preserv es distinguished fillers. But this is automatic since ϕ preserv es distinguishe d fillers and fillers in Q ar e by definition images of distinguished fillers in Y . 2.3 A n auxiliary c on stru c t ion In this section w e w ant to prov e pr o positions 2.6 and 2 .10, whic h w e will use to in v estigate colimits in the next section. The impatien t reader can skip this section and just ta ke note of these pr o positions. 2 ALGEBRAIC FIBRANT OBJECTS 8 Prop osition 2.6. L et Y b e an algeb r aic fibr ant o b je ct, X b e an o bje ct in C and f : Y → X b e a morphism (that me ans a morph i s m f : U ( Y ) → X in C ). Then ther e is an algebr aic fibr ant obje ct X f ∞ to gether with a morphism X → X f ∞ such that the c omp osite Y → X → X f ∞ is a m ap of algebr aic fibr ant obje cts. F urthermor e X f ∞ is initial with this pr op erty. T hat me ans it sa tisfi e s the fol lowin g universal pr op erty: F or e ach mo rp hism ϕ : X → Z in C whe r e Z is an algeb r aic fibr ant obje ct, such that the c omp osite Y → X → Z is a m orphism of algebr aic fibr ant obje cts ther e exists a unique morphism ϕ ∞ : X f ∞ → Z r endering the diagr am Y f   alg. ! ! C C C C C C C C X   ϕ / / Z X f ∞ ∃ ! ϕ ∞ > > } } } } c ommutative. If f is monic in C , then X → X f ∞ is a trivial c ofibr ation. Before w e prov e this prop osition w e first draw some conclusions. Remark 2.7. • L et ∅ b e the initial algebr aic fibr ant obje ct. Assume that the underlying obje ct U ( ∅ ) i s initial in C . In p articular we ha v e a unique morp h ism f : U ( ∅ ) → X . Then the universal p r op erty of X f ∞ r e duc es to the universal pr op erty of X ∞ fr om pr op osition 2.2. We wil l use this observation to gi v e a v e ry s i m ilar c onstruction of X f ∞ for arbitr ary f . N ote that the assumption that U ( ∅ ) is initial is not always satisfie d, but it holds in the c ase of a l g ebr aic Kan c omplexes and algebr aic quasi-c ate gories. • In c ontr ast to the map X → X ∞ in the c as e Y = ∅ the morph ism X → X f ∞ is in gener al neither a c ofibr ation n or a we ak e quivalenc e. Nevertheless if f : Y → X is a monomorphism , the pr o p osition says that it is stil l a trivial c ofi br ation. W e will construct X f ∞ in tw o steps. First consider the images under f of distinguished filler diagrams of Y . These a re diagrams A j / /   Y / / X B j ? ?         7 7 o o o o o o o o o o o o o o o If w e w ant t o turn f into a morphism o f algebraic fibran t ob jects, these diagr a m s hav e to b e distinguished filler diagra m s of X . But then it might o ccur that a morphism A j → X factors in different wa ys through Y and provide s different fillers B j → X . In order to a v o id this a mbiguit y w e wan t to iden tify them. Note that this ambiguit y do es not o ccur if f is a 2 ALGEBRAIC FIBRANT OBJECTS 9 monomorphism, so in this case w e can skip this next step. Let us describe t his more technic ally: Let H b e the set of morphisms h : A j → X whic h factor through Y . F or eac h h ∈ H let F h b e the set of fillers ϕ : B j → X whic h are the images of distinguished fillers. F h has at least one eleme n t but it might of course b e an infinite set. Now fix an elemen t h 0 ∈ H . In order to iden tify the differen t fillers F h 0 w e tak e the co equ alizer X h 0 := CoEq  B j ϕ → X | ϕ ∈ F h 0  whic h comes with a morphism p h 0 : X → X h 0 . W e now w an t to rep eat t his pro cess in- ductiv ely in order to iden tify the fillers for all ho r n s H . Therefore w e endow H with a w ell-o r dering suc h that h 0 is the least elemen t. Assume that X h ′ is defined for all h ′ < h . Let X > > > > > > > a f @ @         c 3 ALGEBRAIC KAN COMPLEXES 16 No w a filler F ( h ) : ∆(2) → X is a 2 - s implex that fills this ho rn: b g   > > > > > > > >   a f @ @         t / / c The target t of this 2-cell should no w b e see n as a comp osition of f and g . Of course the filler F ( h ) is not unique a nd th us the comp osition o f 1-cells is also not unique. Nev ertheless, using the higher dimensional fillers one can sho w that comp osition defined in this w ay is unique up t w o 2- c ells (that means b et w een tw o comp osites there is a lw a ys a 2- c ell connecting t hem, whic h is also unique up to 3-cells...). But the lack of a fixed comp os itions is sometimes coun terintuitiv e or might lead to problems w orking with ∞ -group o ids . Thus the idea is to fix a filler for eac h pair of morphisms ( f , g ) and refer to this as the c omp osition of f and g . W e giv e a definition of Kan complexes with fixed fillers, called algebraic Ka n complexes, in the next section. But let us first r eturn to t h e inv estigation of lift ing prop erties. W e saw that the lif t ing against the inner horn Λ 1 (2) endows X with comp ositions o f 1-cells. Analo gously o ne can see that lif ting ag ains t higher inner horns Λ k ( n ) prov ides comp ositions of higher cells, whic h is a go o d exercise to do for n = 3. But we wan t to lo ok at the outer horns Λ 0 (2) and Λ 2 (2). A morphism Λ 0 (2) → X provides tw o morphisms f , g ∈ X 1 that fit together lik e this: b a f @ @         g / / c A filler for suc h a dia gram translates in to a diagram b t   > > > > > > > >   a f @ @         g / / c . This means that g could b e seen a s a comp o s ition t ◦ f o r equiv alen tly t = g ◦ f − 1 . In that wa y a Kan complex provides in verse s an th us mo dels ∞ -group oids rather than ∞ -categories. In our approac h to ∞ -g roupoids w e will also fix those fillers and thus hav e a c hoice of inv erses. Later in section 4 w e will consider quasi-categories where fillers are only r equired f or inner horns and th us there are no in v erses for 1-cells. 3.2 A lg ebraic Kan complexes as ∞ -group oids In t h is section w e will give the notion of a lg e braic Kan complex and use the general metho ds dev elop ed in section 2 to o b tain a mo del structure and to deduce pro perties for algebraic Kan complexes. As motiv a t ed in t he last section, an algebraic Kan complex should hav e fixed fillers for all horns, and th us fixed comp ositions a nd in v erses of cells. Definition 3.1. 1. An algebr ai c Kan c omplex is a simplicial set X together with a dis- tinguished filler for eac h horn in X . A map of a lgebraic Kan complexes is a map that sends distinguished fillers to distinguished fillers. The category o f alg e braic Kan complexes is denoted b y AlgKan 3 ALGEBRAIC KAN COMPLEXES 17 2. A morphism f : X → Y of algebraic Ka n complexes is a w eak equiv alence (fibration) if the underlying morphism U A ( f ) : U A ( X ) → U A ( Y ) is a w eak equiv alence (fibration) of simplicial sets. A morphism is a cofibration of alg e braic Kan complexes, if it has the LLP with resp e ct to trivial fibratio ns. The mo del category sSet is cofibran tly generated and the cofibrations are exactly the monomor- phisms. Th us from section 2 w e immediately hav e: Theorem 3.2. 1. The c anonic al for ge tful functor U A : A lgKan → sSet has a left adjoint F A : sSet → AlgKan which is c onstructe d by iter atively attaching n -simplic es as fil lers for al l horns. 2. Algebr aic K an c omplexes ar e pr e c isely a lgebr as fo r the mo n ad T := U ◦ F gener ate d by this adjunction. 3. AlgKan is smal l c omp le t e and c o c omplete. Limits and filter e d c o limits ar e c ompute d as limits r esp . c olimits of the underlying si mplicial sets. 4. AlgKan is a c ombinatorial mo del c ate gory with gener ating trivial c ofibr ations F A Λ k ( n ) → F A ∆( n ) and ge ner ating c ofibr ations F A ∂ ∆( n ) → F A ∆( n ) 5. The p a ir ( F , U ) is a Q u il len e quivalenc e. F urthermor e the functor U A pr eserves c ofi- br ations a n d trivial c ofibr ations. Pr o of. 1: prop. 2.2; 2: prop. 2.4; 3: prop 2.11 and corollary 2.14; 4: theorem 2.20 and prop osition 2.2 2; 5: theorem 2.20 and coro llary 2.18. Note that in con trast to sSet in this mo del structure on AlgKa n each ob ject is fibrant but not necessarily cofibrant. F o r example the p oint in AlgKan is not cofibran t. The cofibran t ob jects are exactly retracts of F A ∂ ∆( n ) → F A ∆( n )-cell complexes. W e will sa y some words ab out suc h cell complexes in o r der to g iv e a b etter understanding of the cofibrations. Let X b e an algebraic Kan complex. W e w a n t to glue a n- cell to X along its b oundary ∂ ∆( n ). F ormally sp eaking w e w an t to compute the pushout of a diagram F A ∂ ∆( n ) / /   X F A ∆( n ) where the upp er morphism comes fr om a morphism ∂ ∆( n ) → X of simplicial sets, whic h is just a combinatorial n -sphere in X . F rom prop 2.15 w e kno w tha t the pushout can now b e computed in tw o steps: first glue the n -cell a long its b oundary to X , i.e. form the pushout X ∪ ∂ ∆( n ) ∆( n ) in sSet. Intuitiv ely sp eaking we simply add a new n - c ell to o ur ∞ -group oid. But now some comp ositions are missing, namely those of the new n -cell with cells of the old ∞ -group oid X . Th us w e throw in freely all those comp ositions, i.e. form X f ∞ (see section 2.3). What w e finally obtain is the pushout in the category AlgKa n . 3 ALGEBRAIC KAN COMPLEXES 18 Note first that gluing a n -cell not a lo ng its b oundary , but along its horn w o rk s exactly the same. No w g e neral cell complexes are just an iterat ion of this gluing pro cess. The f act that filtered colimits are computed as colimits of the underlying simplicial sets means, that w e can do this iteration naive ly and finally obtain the right algebraic Kan complex. Hence w e hav e a v ery clear understanding of cofibrations and trivial cofibrations in AlgKan. This discussion also shows that the category AlgKan provides the righ t colimits, whereas colimits of (o rdinary) Kan complexes migh t no longer b e Kan complexes and thus are not the correct colimit of ∞ -group oids . 3.3 T he homotop y h yp othesis The ho mo t o p y h ypo the sis is informally sp eakin g t he idea that ∞ -g roupoids a r e the same as top ological spaces. Here w e prop ose algebraic Kan complexes as a mo del for ∞ -group oids. Therefore we should sho w that they are equiv a le n t to to pological spaces. More precisely we w ant to pro v e that the mo del categories ar e Quillen equiv a le n t. As mo del categories are a w ay to enco de the ( ∞ , 1)-category of ∞ -group oids, this could b e regarded as proving the homotop y hypothesis for our mo del of ∞ -groupo ids . First of a ll, it is a classical result of Quillen, that the (standard) mo del categories of top olog- ical spaces a nd simplicial sets are equiv alen t. The a djoin t functors whic h for m the Quillen pair are | ... | : sSet / / T o p : Sing o o where the left adjo int | ... | is the geometric realization functor and the righ t adj o in t Sing is the singular complex functor. W e could no w argue, using this result, that the catego ry of algebraic Kan complexes is Quillen equiv alen t to simplicial sets and th us is equiv alen t to top ological spaces. This is p erfectly fine on the lev el of ( ∞ , 1)-categories. But in this w ay w e will no t obtain a direct Quillen equiv alence b et w een algebraic Kan complexes and top ological spaces, b ecaus e the Quillen equiv alences cannot b e comp osed. Instead w e will giv e a direct Quillen equiv alence | ... | r : AlgKan / / T o p : Π ∞ o o where the left adjoint | ... | r is called reduced geometric realization and the right adjoin t Π ∞ is called fundamen tal ∞ -group oid. This will render the diagram AlgKan | ... | r   : : : : : : : : : : : : : : : U A                  sSet | ... | / / F A A A                T o p Sing o o Π ∞ ] ] : : : : : : : : : : : : : : : (4) comm utat ive (more precisely: the inner and the outer triang le). Let us start b y describing t h e fundamen tal ∞ -functor Π ∞ : T op → AlgKan. F or a top ological space M the or dina r y singular complex is b y definition the simplicial set Sing( M ) with Sing( M ) n = hom T op  | ∆( n ) | , M  3 ALGEBRAIC KAN COMPLEXES 19 where | ∆( n ) | denotes the top ological n - s implex | ∆( n ) | =  ( x 0 , . . . , x n ) ∈ R n +1 ≥ 0 | X x i = 1  . In order to mak e the diagram (4 ) comm utativ e, the underlying simplicial set of Π ∞ ( M ) has to b e the simplicial set Sing( M ). No w to endow Sing( M ) with the structure of an algebraic Kan complex, w e hav e to give distinguished fillers fo r a ll horns Λ k ( n ) → Sing( M ) . But due to the fact that Sing is righ t adjoint to | ... | suc h a horn is the same a s a mo r phism h : | Λ k ( n ) | → M of to pological spaces. It is easy to see that | Λ k ( n ) | is (homeomorphic to) the naiv e horn | Λ k ( n ) | = [ i 6 = k  ( x 0 , . . . , x n ) ∈ | ∆( n ) | | x i = 0  whic h is the union of all but o ne faces of the simplex | ∆( n ) | . F ro m the geometric p oin t of view, it is clear that there a re (linear) retractions R ( n, k ) : | ∆( n ) | → | Λ k ( n ) | . W e will not g iv e an explicit form ula f or the R ( n, k ) b ecause t ha t will not giv e mor e insigh ts, but in principle that can b e easily do ne. W e use these retractions to obtain morphisms | ∆( n ) | R ( n,k ) − → | Λ k ( n ) | h → M whic h by a dj o in tness are fillers ∆( n ) → Sing( M ) for horns in Sing( M ). W e denote the result- ing algebraic K an complex b y Π ∞ ( M ). F urthermore this assignmen t is obv iously functor ia l in M suc h tha t w e finally hav e defined the functor Π ∞ : T op → AlgKan Remark 3.3. The c onstruction o f the functor Π ∞ dep ends on the choic e o f r etr acts R ( n, k ) we have m a de. Every other choic e would le ad to a differ ent (b ut of c ourse we akly e quivalen t) algebr aic Kan c omplex Π ∞ ( M ) . This choic e p ar ame terize s the c omp osition of p aths or higher c el ls in the p ath ∞ -gr oup oid . No w let us turn tow ards t he reduced g e ometric realization functor | | r : AlgKan → T op . So let X b e a n algebraic Kan complex. First o f all, consider the geometric realization | U A ( X ) | of the underlying simplicial set. The distinguished fillers ∆( n ) → X prov ide n- s implices in | U A ( X ) | whic h are fillers for the horns | Λ k ( n ) | → | U A ( X ) | . But the comp osite | ∆( n ) | R ( n,k ) − → | Λ k ( n ) | → | U A ( X ) | pro vides another filler. Therefore w e define the reduced geometric realization | X | r as the space where those tw o differen t fillers for the same horn hav e b een identifie d. F ormally we ha v e | X | r := CoEq  G | ∆( n ) | ⇒ | U A ( X ) |  where the disjoin t unio n is t a k en ov er all horns Λ k ( n ) → X . With this definition w e ha v e: 4 ALGEBRAIC QUASI-CA TEGORIES 20 Prop osition 3.4. The functor | | r : A lgKan → T op is lef t ad j o int to Π ∞ . Pr o of. Let X b e an algebraic Kan complex, M a top ological space and f : | X | r → M b e a con t inuous map. By construction of | X | r as a co equalize r this is the same as a contin uous map ˜ f : | U A ( X ) | → M suc h that for each horn h : Λ k ( n ) → X with filler F ( h ) : ∆( n ) → X the t w o maps | ∆( n ) | R ( n,k ) − → | Λ k ( n ) | | h | − → | U A ( X ) | ˜ f → M and | ∆( n ) | | F ( h ) | − → | U A ( X ) | ˜ f → M agree. Using the adjunction ( | ... | , Sing) w e see that this is the same as a mo r p hism ˜ ˜ f : U A ( X ) → Sing( M ) suc h that the ima g e s of distinguished filler diagrams in X are sen t to the fillers in Sing( M ) obtained b y using the retractions R ( n, k ). That means ˜ ˜ f is a morphism of algebraic Kan complexes b et w een X and Π ∞ ( M ). Corollary 3.5. Diagr am (4) c ommutes (up to natur al iso morphism). Pr o of. The inner triangle comm utes b y construction o f Π ∞ . F or comm utativit y of the outer w e hav e to sho w tha t | ... | r ◦ F A ∼ = | ... | . F rom the fact that | ... | r is left adjoin t to Π ∞ and F A is left adjoint to U A w e deduce that | ... | r ◦ F A is left adjoint to U A ◦ Π ∞ . By commutativit y of the inner triangle the latter is equal to Sing. That means tha t | ... | r ◦ F A and | ... | are left adjoin t to Sing and t h us are naturally isomorphic. Corollary 3.6. The p air  | | r , Π ∞  is a Q uil len e quivalenc e. Pr o of. W e already know that ( | ... | , Sing) and ( F A , U A ) are Quillen equiv alences. By the 2- out- of-3 prop ert y for Quillen equiv alences it follows that ( | ... | r , Π ∞ ) is also a Quillen equiv alence. 4 Algebraic quasi- c ate gories In this section we w an t to apply the general principle t o the Jo y al mo del structure o n simplicial sets. Thereb y we are led to in tro duce the concept of alg e braic quasi-category as an alg ebraic mo del for ( ∞ , 1)-categories. F inally we will relate algebraic quasi-catego r ies to algebraic Kan complexes. 4.1 Qu a si-categories as ( ∞ ,1)-categories The category sSet carries another mo del structure b esides t he Quillen mo del structure ( see [Jo y08], [Lur09]). This second mo del structure is called t he Joy al mo del structure. Unfortu- nately it is more complicated than the Quillen structure, but it is a lso cofibrantly generated. The cofibrations are the same as in the Quillen structure and thu s the b oundary inclusions ∂ ∆( n ) → ∆( n ) . 4 ALGEBRAIC QUASI-CA TEGORIES 21 are a set of generating cofibrat io ns . But there is no kno wn description of a set of g e nerating trivial cofibra tions , although it is kno wn that suc h a set exists. The w eak equiv alences in this mo del structure are called categorical equiv alences or quasi-equiv alences. The fibran t o b jects in this mo del structure are called quasi-categories. These a re the sim- plicial sets X whic h hav e the righ t lifting prop ert y ag ains t all inner horns Λ k ( n ) → ∆( n ) for n ≥ 2 and 0 < k < n . W e describ ed in section 3.1 how these lift in g conditions could b e seen as providing comp ositions of cells. The fact that we only ha v e lifting conditions against inner horns thus means that w e do not hav e inv erses to 1- ce lls. A more precise t r e atmen t of these lifting prop erties sho ws that we still ha ve inv erses for n -cells with n ≥ 2. That means t h at quasi-categories are a mo del fo r ( ∞ , 1)-categories, i.e. ∞ -categories where all n -morphisms for n ≥ 2 are inv ertible. And in fact there ha s b een m uch work providing evidence that this is an appropria te mo del for ( ∞ , 1)-categories. See [Ber09] for a go o d in t r oduction. But as in the case o f Kan complexes it is desirable to ha v e a more alg e braic mo del where esp ecially comp ositions of morphisms are not only g uaran teed to exist but are sp ecified. W e will do this b y applying our general construction f rom section 2 to quasi-catego r ies, as we did for Kan complexes in section 3. 4.2 A lg ebraic quasi-categories Let J b e the set of inner ho r n inclusions Λ k ( n ) → ∆( n ) whic h are trivial cofibratio ns in the Joy al mo del structure. Using this set of morphisms w e follo w the general pattern from section 2: Definition 4.1. An algebr ai c quasi-c ate gory is a simplicial set X together with a distin- guished filler for eac h inner horn in X . A map o f quasi-categories is a map that sends distin- guished fillers to distinguished fillers. W e denote the category of algebraic quasi-categories b y AlgQuasi. Theorem 4.2. 1. The c anonic a l for g et ful functor U Q : AlgQuasi → sSet obtains a left adjoint F Q : sSet → A lgQuasi which is c ons t ructe d by iter atively attaching n -simplic e s as fil lers for al l in n er horns . 2. Algebr aic q uasi - c ate gories ar e algeb r as for the mo nad T Q := U Q ◦ F Q gener ate d by this adjunction. 3. AlgKan is smal l c omp le t e and c o c omplete. Limits and filter e d c o limits ar e c ompute d as limits r esp . c olimits of the underlying si mplicial sets. Pr o of. 1: theorem 2.2; 2: theorem 2.4 ; 3 : theorem 2.1 1 and corollary 2.1 4. Additionally w e ha v e the mo del structure on AlgQuasi: 4 ALGEBRAIC QUASI-CA TEGORIES 22 Definition 4.3. A morphism f : X → Y o f alg e braic quasi-catego r ies is a w eak equiv alence (fibration) if the underlying morphism U Q ( f ) : U ( X ) → ( Y ) is a catego r ical equiv a le nce (fibration) in the Jo y al mo del structure. A morphism is a cofibratio n of algebraic quasi- categories, if it has the LLP with resp ect to trivial fibra tions. No w according to theorem 2.20 this defines a cofibrantly generated mo del structure on AlgQuasi. One of the ma jor adv an tages of this new mo del structure is that we can ex- plicitly write down a set of generating trivial cofibrations. This follows from the fact that in the Jo y al mo del structure a morphism b et we en fibrant ob jects, i.e. quasi-categories, is a fibration iff it has the LLP with resp ect to inner horn inclusions Λ k ( n ) → ∆( n ) and the inclusion pt → I of an ob ject in the interv al group oid I . By definition I is the nerv e of the group oid with tw o ob jects and a n isomorphism b et w een them (see [Joy 08], Prop. 4.3 .2 ). Th us w e hav e: Theorem 4.4. A lgQuasi is a c ombina toria l mo del c ate gory with gener ating trivial c ofibr ations F Q Λ k ( n ) → F Q ∆( n ) for 1 < k < n F Q pt → F Q I and ge ner ating c ofibr ations F Q ∂ ∆( n ) → F Q ∆( n ) The p air ( F Q , U Q ) i s a Quil len e quivalenc e. F urthermor e the functor U Q pr eserves c ofibr a- tions and trivial c ofibr ations. Pr o of. Theorem 2.20 and prop osition 2.22 show that AlgQuasi is a com binat o rial mo del category with the giv en generating cofibrations and that the pair ( F Q , U Q ) is a Quillen equiv alence. F ro m corollary 2.18 w e kno w that U Q preserv es trivial cofibrations. It only remains to sho w that the giv en set o f morphisms is a set o f generating trivial cofibrations. W e show that f : X → Y is a fibration in AlgQuasi if it ha s the RLP with resp ect to the giv en morphisms. By definition f is a fibration iff U Q ( f ) is a fibratio n in the Jo y al mo del structure. Since U Q ( X ) and U Q ( Y ) are quasi-categories, this is the case if U Q ( f ) has the RLP with resp ect to inner horn inclusions and pt → I . Using t he fact that F Q is left adjoint to U Q w e see that f is a fibration in Alg Q u asi iff it has the RLP with resp ect to the given set of morphisms. 4.3 G roup oi dification In this section w e w an t to inv estigate how the (mo del) categories AlgKa n and AlgQuasi are related to eac h other. Remem b er that ob jects in b oth of them are simplicial sets with extra structure. In the case of AlgKan w e hav e fixed fillers for all horn inclusions and in t he case of AlgQuasi w e only hav e fixed fillers fo r inner horn inclusion. This sho ws that w e hav e a canonical forgetful functor V : AlgKan → AlgQuasi 4 ALGEBRAIC QUASI-CA TEGORIES 23 whic h forgets the fillers for the outer horns. W e will construct a left adjoint G : AlgQuasi → AlgKan called gr oup o idific ation and show that the pair ( G , V ) forms a Quillen adjunction (not a Quillen equiv alence!). This is the algebraic ana lo gue of the fact that the Quillen mo del structure on sSet is a left Bousfield lo calization of the Joy al mo del structure. More precisely w e ha v e a comm uting square AlgKan V / / U A   AlgQuasi G o o U Q   sSet Q I d / / F A O O sSet J I d o o F Q O O (5) of Quillen adjunctions, where sSet J denotes the category o f simplicial sets with the Jo y al mo del structure and sSet Q with the Quillen mo del structure. Here the inner and the outer squares commute (up to natura l isomorphism). No w let X b e an alg ebraic quasi-category . W e already ha v e fixed fillers for inner horns in X , i.e. morphisms Λ k ( n ) → X with 0 < k < n . In order to build an algebraic Kan complex out of X we will fr e ely add fillers for outer horns in X , i.e. morphisms Λ k ( n ) → X with k = 0 or k = n . The construction is muc h the same as the construction fro m section 2.1 a nd w e only sk etc h it. Let X 1 b e the pushout F Λ k ( n ) / /   X   F ∆( n ) / / X 1 where the colimit is taken o v er all outer horns in X . The next step X 2 is obtained b y gluing n -cells ∆( n ) alo ng outer horns Λ k ( n ) → X 1 that do not fa c tor through X . W e pro ceed lik e this a nd finally put G ( X ) := lim − → ( X → X 1 → X 2 → . . . ) . Prop osition 4.5. The functor G : AlgQuasi → AlgKan is left adj o int to V and the squar e (5) c ommutes. Pr o of. By definition of G it is clear that it is left adjoin t to V . In diagram (5) the comm uta- tivit y o f the outer square is just a trivial statemen t ab out the forgetful functors U Q , U A and V . Comm utativity of the inner square means that we ha v e t o sho w that G ◦ F Q and F A are naturally isomor phic. This follows from the fact that G ◦ F Q and F A are b oth left adjoin t to U Q ◦ V = U A . Prop osition 4.6. The p air ( G, V ) is a Quil len adj u nction. Pr o of. It is enough to sho w t ha t V preserv es fibra tions and trivial fibrations. Let f : X → Y b e a (trivial) fibration in AlgK a n. W e wan t to sho w that V ( f ) : V ( X ) → V ( Y ) is a (trivial) fibration in AlgQua s i. By definition 4.3 this is the case iff U Q ( V ( f )) is a Jo y al (trivial) fibration in sSet J . By definition 3.1 w e already kno w t ha t U A ( f ) = U Q ( V ( f )) is a Quillen 5 OUTLOOK 24 (trivial) fibration. Th us the claim follows f r om the f act that a Quillen (trivial) fibration is a Jo y al (trivial) fibration. This is equiv alen t to the statemen t that I d : sSet Q → sSet J is a righ t Quillen functor or to the statemen t that sSet Q is a left Bousfield lo calization of sSet J . 5 Outlo ok In this section w e w a nt to sk etc h furt her p ossib le a pp lications of the general theory of algebraic fibrant ob jects. They alw ays lead to an algebraic v ersion of the structure mo delled b y the mo del category . • There is a nice extension of the category of simplicial sets called dendroidal sets, in- tro duced in [MW07]. F urthermore there is a lso a mo del structure on dendroidal sets defined b y Cisinski and Mo erdijk, whic h extends the Joy al mo del structure on sim- plicial sets, see [CM09]. The fibran t ob jects are called inner Ka n complexes and a re a mo del for ( ∞ , 1)-op erads. The cofibrat io ns in the category a re (normal) monomor- phisms, hence w e can a pp ly our general result. This leads to a n algebraic mo del for ( ∞ , 1 )-operads. Similar to the case of a lgeb raic quasi-categories described in section 4.2 there is an explicit set of generating tr ivial cofibrations for algebraic dendroidal sets. D end roidal sets ha v e b een in tro duced to giv e recursiv e definitions of weak n - categories. It would b e in teresting to see whether algebraic dendroidal sets could b e used to pro duce algebraic analog u es o f t h ese constructions. • The model structure of Quillen and Jo yal b oth use simplicial sets to mo de l ∞ -group oids and ( ∞ , 1)-categories as explained in section 3.1 and 4.1. But simplicial sets can also b e used as a mo del for all w eak ∞ -categories. This go es back to ideas of Street. In order to do so, simplicial sets are equipp ed with the extra structure of t hin elemen ts whic h allow to k eep tr a c k of in v ertible higher cells. The category obtained in this w a y is called the category of stratified simplicial sets. On this category there is a mo del category structure constructed b y V erit y [V er08]. The fibran t ob jects are called w eak complicial sets. In this mo del structure the cofibrations are also mo no m orphisms, hence b y applying our g e neral construction this leads to an algebraic v ersion of w eak ∞ -categories. It w ould b e in teresting to in ves tigate this mo del structure in more detail, in particular to see how the nerv es of strict ∞ - categories lead t o algebraic w eak complicial sets. • Simplicial preshea v es o v er a site S are preshea v es with v alues in the category of simpli- cial sets. They also carry mo del structures whic h exhibits them as models for ∞ -stac ks. The tw o most imp ortant mo del structures are the lo cal pro j e ctiv e and the lo cal injec- tiv e mo del structure, see e.g. [Jar07] and [D ug 0 1]. In b oth of them the cofibrations a re monomorphisms and hence we obta in t w o categories of algebraic simplicial preshea v es. They thus form a mo del for algebraic ∞ -stack s. Using results of [DHI04] one can mak e explicit the descen t conditio n s in these algebraic ∞ -stack s. This might prov ide a fra m ew or k in whic h some gluing constructions can b e made direct a nd functorial. There are of course man y more p ossib le applications of the general construction. Let us finally note that the fibra nt replacemen t monad T : C → C inv estigated in Prop osition 2.2 3 REFERENCES 25 is v ery useful in some applications, eve n if w e do not w ant to deal with algebraic fibran t ob jects. It allo ws for example to replace diagrams ( e.g. homoto p y pushout diagra ms ) nicely in v ery general situations. References [AR94] J. Ad´ amek and J. R o s ic k ` y. L o c al ly pr esentable and ac c essible c ate gories . Cam bridge Univ Pr, 1994. [Bae07] J. Baez. The ho m otopy h yp othesis. 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