Homotopy fiber products of homotopy theories

HOMOTOPY FIBER PR ODUCTS OF HOMOTOPY THEORIES JULIA E. BER GNER Abstract. Giv en an appropriate diagram of l eft Quillen functors b et ween model categories, one can define a notion of homotop y fib er pro duct, but one might ask if it is really the correct one. Here, we show t hat this homotop y pullbac k is we ll- behav ed wi th respect to translating i t into the setting of more general homotop y theories, given by complete Segal spaces, where we hav e we ll- defined homotop y pullbac ks. 1. Introduction Homotopy theory originates with the study of topo logical spa ces up to weak homotopy e q uiv alence, where tw o spaces a r e w eakly homoto p y equiv alent if there is a map betw een them inducing isomor phisms on all the homotopy groups. The question of whether the nice prop erties of topolog ic a l spaces held in other s ettings led to the development of the notion o f a mo del ca tegory by Quillen [22]. The study of mo del categor ie s is then a more abstract form of homotopy theor y , one which has b een in vestigated extensively . One could then co nsider model categories themselves as ob jects of a catego r y and consider Quillen equiv alences as weak equiv alences betw een them. In this fra me- work, one c o uld a sk ques tio ns a bo ut relationships betw een mo del categor ies; for example, wha t would a homotopy limit or homotopy colimit of a diag ram of mo del categorie s be? Unfortunately , there are no immediate a nswers to these questions bec ause at present there is no known model s tructure on the categ ory of mo del categorie s. In one particular case of a homo topy limit, we hav e a plausible con- struction, that of the ho motopy fib er pro duct. This cons truction was first explained to the author by Smith; it appears in the literature in a pap er of T o¨ en [27], where it is useful in his development of derived Hall algebras. One mig ht ask whether, without a mo del catego ry of mo del categories, w e can really accept this definition as a gen uine homoto p y pullback construction. How ev er, one c a n a ls o c onsider a homo to p y theo r y to b e something more general than a mo del category , such as a category with a s pecified class of weak equiv a- lences, or maps one would lik e to co ns ider as equiv alences but whic h are not nec- essarily isomorphisms. While there may or may not be a mo del structure o n suc h a categor y , one ca n heur istically think of forma lly inverting the weak equiv a lences, set-theoretic problems notwithstanding. F urthermore, such “ homotopy theories” can b e regarded as the ob jects of a model category . Date : Septe mber 1, 2018. 2000 Mathematics Subje ct Class ific ation. Pri mary: 55U40; Secondary: 55U35, 18G55 , 18G30, 18D20. Key wor ds and phr ases. model categories, simplicial cat egories, complete Segal spaces, homo- top y theories, homotop y fib er pro ducts. The author wa s partially supp orted b y NSF grant DM S-0805951. 1 2 J.E. BER GNER In a series o f papers [7], [9], [10], Dwyer and K an develop the theor y of s implicial lo calizations, which ar e simplicial c ategories cor resp onding to mo del catego ries, or, more genera lly , categorie s with sp ecified w eak equiv alences. Since ev ery s uch cate- gory has a simplicial lo calization, and since, up to a natural notion of equiv alence of simplicial categories , every simplicial ca tegory arises a s the simplicial lo caliza- tion of some category with w eak equiv alenc e s [8, 2.5], simplicial categories can be considered to b e models for ho motopy theories. With these weak equiv alences, often called Dwyer-Kan equiv alences, the categ ory of (small) simplicial categories can itself b e considered a s a homotop y theo ry , and in fact it has a mo del structure [3]. Thus, within this fra mework one ca n address questions ab out ho motopy theo - ries that are natur a l to a sk in a given mo del category , such a s character izations o f homotopy limits and colimits. Unfortunately , the mo del structure o n the category of simplicia l catego ries is techn ically difficult to work with. The weak equiv alences are particularly challeng- ing to identify , for e xample. F or tuna tely , we hav e a choice of several other equiv alent mo del catego ries in whic h to address thes e ques tions. W ork of the author and of Joy al and T ie r ney has prov ed that the co mplete Segal space mo del structure on the category of simplicial spaces, tw o different Segal categ o ry mo del structures on the catego r y of Seg a l precategor ies, and the quasi-category mo de l structure on the category of simplicial sets a r e all equiv alent as mo del categories , and thus each is a mo del for the homotopy theory o f homotop y theor ies [4], [17], [19]. In this pape r, we address the c onstruction of the homotopy fib er pro duct of mo del categorie s and its a nalogue within the co mplete Segal space mo del structure. Of the v arious models mentioned above, the c o mplete Segal space mo del structure is the bes t setting in whic h to answ er this ques tion due to the pa r ticularly nice des c ription of the relev ant weak equiv alences. Some clarification of ter minology is needed here, as there are actually tw o notions of wha t is mea n t by a homotopy fib er pro duct of mo del categor ies. One is more restrictive than the other; the stronger is the one we would e xpec t to corre s po nd to a homotopy pullback, but the weak er is the one whic h can b e given the s tructure of a mo del categ ory . W e show that at least in some cases this mo del str ucture ca n b e lo calized so that the fibra n t-cofibra nt o b jects satisfy the more restrictive condition. The main res ult of this pap er is tha t the stricter definition do es in fact corresp ond to a homo topy pullback when we work in the co mplete Sega l space setting. W e also give a c hara cterization of the complete Seg al spaces aris ing from the less restrictive description. W e should p o int out that this constructio n has b een use d in sp ecial cas es, for example by H ¨ uttemann, K lein, V ogell, W aldhausen, and Williams in [15 ], and as an example of more gener al co nstructions, for ex ample the twisted diagra ms of H¨ uttemann and R¨ ondig s [16], and the mo del categ ories of comma categories as given by Stanculescu [26]. It is also imp ortant to no te that transla ting this question into the setting of more general homoto p y theories is not mer ely a temp o rary solution until one finds a mo del catego ry o f mo del categories. In practice, mo del structures ar e often hard to establish, and furthermore, the condition of having a Quillen pair b etw een t wo such mo del structures is a rigid one. Being able to consider ho motopy theories as more flexible kinds of o b jects, and ha ving mor phisms b etw een them less structured, makes it more likely that w e can actually implement such a c onstruction. W e HOMOTOPY FIBER PR ODUCTS 3 consider suc h a c ase in Ex a mple 3 .3. Y et, with the relatio nship b etw een the tw o settings esta blished, we can use the additional structure when we do indeed have it. In fact, par t of our motiv ation for making the compariso n in this pa per is to generalize T o¨ en’s development o f derived Hall algebras. Where he defines an asso- ciative algebra corresp onding to stable mo del categor ies given by mo dules ov er a dg category , we w ould like to define such an algebra using a mor e gener al stable homotopy theory , namely , one given by a stable complete Sega l space. The main result of this paper a llows us to use a homotop y pullback o f complete Seg a l spaces in the setting where T o¨ en uses a homotopy fiber pro duct of model categories. The dual notion o f homo topy pushouts o f model ca tegories, as well as more general homo topy limits and homotopy co limits, will be considered in later work. 2. Model ca tegories and more general homotopy theories In this section we giv e a brief review of mo del categories and th eir relations hip with the complete Segal space mo del for more general homotop y theories. Recall that a mo del c ate gory M is a categor y with thr ee distinguished classes of morphisms: weak equiv alences , fibrations, and c ofibrations, sa tis fying five axioms [11, 3.3]. Given a mo del catego ry structur e, one can pass to the homotopy cate- gory Ho( M ), in which the w eak eq uiv a lences from M b ecome isomorphisms. In particular, the weak eq uiv a lences, as the morphisms that we wish to inv ert, make up the most impo rtant part of a mo del c a tegory . An ob ject x in M is fi bra nt if the unique ma p x → ∗ to the terminal ob ject is a fibration. Dually , an o b ject x in M is c ofibr ant if the unique map φ → x from the initial ob ject is a cofibratio n. Given a mo del categor y M , there is also a model structure on the category M [1] , often called the morphis m catego r y of M . The ob jects of M [1] are mor phisms of M , and the mo rphisms of M [1] are given by pairs of morphis ms making the appropria te square diagram comm ute. A morphism in M [1] is a w eak equiv alence (or cofibra tion) if its comp onent maps ar e weak eq uiv a lences (or cofibrations) in M . More g enerally , M [ n ] is the catego ry with ob jects string s of n comp osable morphisms in M ; the model structure can b e defined analo gously . One co uld, mo re genera lly , consider categor ies with weak e q uiv a lences and no additional structure, and then forma lly in vert the weak equiv alences. This pr o cess do es give a homotopy ca teg ory , but it fre quent ly has the disadv a n tage of having a prop er class of mo rphisms b e t ween any tw o given ob jects. If we ar e willing to acce pt such set-theoretic problems, then we can work in this situation; the adv an tage of a mo del structur e is that it provides enough additio na l structur e so that we can ta ke homotopy clas ses of maps a nd hence a void these difficulties. In this paper, w e will use b oth notions of a “ homotopy theor y ,” dep e nding o n the circumstances. W e would, how ever, like to work with nice ob jects modeling categories with w eak equiv alences. While there are sev eral options, the mo del that w e use in this paper is that of complete Seg al spac es. Recall that the s implicial ca tegory ∆ op is defined to b e the category with ob jects finite order ed sets [ n ] = { 0 → 1 → · · · → n } and mor phisms the opp os ites of the order-pr eserving ma ps b etw een them. A simplicial set is then a functor K : ∆ op → S ets. 4 J.E. BER GNER W e denote by S S ets the catego ry of simplicial sets, and this ca tegory has a natural mo del category structure equiv alent to the standard model structure on topo logical spaces [12, I.10]. One can just a s ea sily consider mor e gener al simplicia l ob jects; in this pap er we consider simplicial sp ac es (also called bisimplicial sets ), or functors X : ∆ op → S S ets. There a re several model catego ry structures on the category of bisimplicia l sets. An impor tant one is the Reedy mo del str uc tur e [23], which is equiv alen t to the injectiv e mo del structure, where the w eak equiv alences ar e given by levelwise weak equiv alences of simplicial s ets, a nd the cofibr ations are given likewise [13, 15 .8.7]. Given a simplicial set K , we also denote b y K the s implicial space which has the simplicial s et K at every level. W e denote by K t , or “ K -trans p os ed”, the constant simplicial space in the other direc tio n, where ( K t ) n = K n , where on the r ight-hand side K n is r egarded as a discrete s implicial set. W e use the idea, or iginating with Dwyer and Kan, that a simplicial ca tegory , or category enric hed o ver simplicial sets, models a homotop y theory , in the following wa y . Using either of their tw o no tio ns of simplicial lo ca liz ation, one can obtain from a category with w eak equiv alences a s implicial ca tegory [9], [10], and there is a mo del structure S C on the categor y of all sma ll simplicial categor ies [3]; thus, we hav e a so-called ho motopy theor y of homo topy theories . One useful co nsequence of taking the simplicial categor y corresp onding to a model ca tegory is that w e can use it to describ e homotop y function c omplexes , or homotopy-inv ariant mapping spaces Map h ( x, y ) be t ween ob jects of a mo del c ategory which is not necessar ily equipp ed with the additional structure o f a simplicia l mo de l ca tegory . W e use, in particula r, the simplicia l set Aut h ( x ) o f homo to p y inv ertible self-ma ps of a n ob ject x . Here w e a lso conside r simplicial spaces sa tisfying co nditions imp os ing a notion of comp osition up to homotopy . These Segal spaces and complete Segal spaces were first introduced b y Rezk [24], and the name is meant to be sugg estive of simila r ideas firs t presented by Sega l [25]. Definition 2.1. [24, 4.1 ] A S e gal sp ac e is a Reedy fibrant simplicial space W s uch that the Segal maps ϕ n : W n → W 1 × W 0 · · · × W 0 W 1 | {z } n are weak equiv alences of simplicial sets for all n ≥ 2. Given a Segal space W , we can consider its “ob jects” ob( W ) = W 0 , 0 , a nd, betw een any tw o ob jects x and y , the “mapping spac e” map W ( x, y ), given by the homotopy fibe r o f the map W 1 → W 0 × W 0 given by the tw o face ma ps W 1 → W 0 . The Sega l co ndition given her e tells us that a Segal space has a notion of n -fold compositio n of mapping spaces, at least up to ho motopy . Mor e pr ecise constructions are given b y Rezk [24, § 5]. Using this comp osition, we can define “homotopy equiv alences” in a natural w ay , a nd then speak of the subspace of W 1 whose components con tain homotopy equiv alences, denoted W ho equiv . Notice that the degenera cy map s 0 : W 0 → W 1 factors throug h W ho equiv ; hence we may make the following definition. Definition 2.2. [24, § 6] A c omplete Se gal sp ac e is a Seg al spac e W such that the map W 0 → W ho equiv is a w eak equiv alence of simplicia l sets . HOMOTOPY FIBER PR ODUCTS 5 Given this definition, we ca n des crib e the mo del structur e on the category of simplicial spa c es that is used thro ughout this paper. Theorem 2. 3 . [24, § 7] Ther e is a mo del c ate gory stru ctur e C S S on the c ate gory of simplicia l sp ac es, o btaine d as a lo c alization of the R e e dy mo del st r u ctur e such that: (1) the fibr ant obje cts ar e the c omplete Se gal sp ac es, (2) al l obje cts ar e c ofibr ant, and (3) the we ak e quivalenc es b etwe en c omplete S e gal s p ac es ar e levelwise we ak e quivale nc es of simplicial sets. Now we retur n to the idea that a complete Segal space models a homoto py theory . Theorem 2. 4 . [4] The mo del c ate gories S C and C S S ar e Quil len e quivale nt. F urthermore, due to work o f Rezk [24] which w as contin ued by the author [2], w e can actually characterize, up to weak equiv alence, the co mplete Sega l space a rising from a simplicial categor y , or more sp ecifically , from a mo del category . Rez k defines a functor whic h we deno te L C from the category of mo del categories to the ca tegory of simplicial spaces; given a mo del category M , w e hav e that L C ( M ) n = nerve (we ( M [ n ] )) . Here, M [ n ] is defined as ab ov e, and w e( M [ n ] ) denotes the sub categ o ry of M [ n ] whose morphisms a r e the weak equiv a lences. While the resulting simplicial s pace is not in general Reedy fibrant, and he nc e no t a complete Segal space, Rezk proves that taking a Reedy fibrant replacement is sufficient to o btain a complete Segal space [24, 8 .3]. Hence, for the rest of this pap er we a ssume that the functor L C includes comp osition with this Reedy fibra nt r eplacement and ther e fore gives a complete Sega l space. One o ther difficulty that a r ises in this definitio n is the fa c t that it is only a well- defined functor on the catego ry whose ob jects are mo de l categ ories a nd whose mor- phisms preserve weak equiv a lences. This restrictio n on the morphisms is stronger than we would like; it w ould b e preferable to hav e such a functor defined on the category of mo del categor ies with mor phisms Quillen functors, where weak equiv a- lences are only preserved betw een either fibrant or cofibrant ob jects. T o obtain this more s uita ble functor , we c o nsider M c , the full subca teg ory o f M whose ob jects are co fibrant. While M c may no longer ha ve the structure of a model category , it is still a categ ory with w eak equiv alences, and therefore Rezk’s definition can still be applied to it, so our new definition is L C ( M ) n = nerv e(w e(( M c ) [ n ] )) . Each spa ce in this new diagra m is weakly equiv alent to the one given by the previous definition, and now the construc tio n is functorial on the ca tegory o f mo del c a tegories with mor phisms the left Quillen functors. If one wan ted to consider right Quillen functors instead, we could tak e the full sub c ategory o f fibrant o b jects, M f , r ather than M c . Before stating the theor em giving the c hara cterization, w e give some facts abo ut simplicial mo noids, o r functors from ∆ op to the categ o ry of monoids. Given a sim- plicial monoid M (or , more co mmonly , a simplicial group), w e can find a classifying complex of M , a simplicial set whose geometric realization is the classifying space B M . A precise construction can be made for this classifying space by the W M 6 J.E. BER GNER construction [12, V.4.4 ], [21]. As we ar e no t so co nc e rned here with the precise construction as with the fact that such a c la ssifying space exists, we will simply write B M for the classifying co mplex of M . Theorem 2.5. [2, 7.3] L et M b e a mo del c ate gory. F or x an obje ct of M denote by h x i the we ak e quivalenc e class of x in M , and denote by Aut h ( x ) the simplici al monoid of self we ak e quivalenc es of x . Up t o we ak e quivalenc e in the m o del c ate gory C S S , the c omplete Se gal sp ac e L C ( M ) lo oks like a h x i B Aut h ( x ) ⇐ a h α : x → y i B Aut h ( α ) ⇚ · · · . (W e should point out that the reference (Theorem 7 .3 of [2 ]) giv es a character - ization o f the complete Segal spa ce arising fro m a simplicial catego ry , not from a mo del catego r y . How ever, the results o f § 6 o f that same pap er allow one to transla te it to the theorem a s stated here.) This characterization, together with the fact tha t weak equiv a lences b e tw een complete Segal spaces are le velwise weak equiv alences of simplicial se ts, enables us to compar e complete Segal spaces aris ing from diff erent mo del c ategories . 3. Homotopy fiber pr oducts of model ca tegories W e beg in with the definition of ho motopy fib er pro duct a s given by T o ¨ e n in [27]. First, supp ose that M 1 F 1 / / M 3 M 2 F 2 o o is a diagr am of left Q uille n functors of mo del catego ries. Define their homotop y fib er pr o duct to be the mo del catego ry M = M 1 × h M 3 M 2 whose ob jects are given b y 5-tuples ( x 1 , x 2 , x 3 ; u, v ) such that each x i is an ob ject of M i fitting int o a diagram F 1 ( x 1 ) u / / x 3 F 2 ( x 2 ) . v o o A morphism of M , say f : ( x 1 , x 2 , x 3 ; u, v ) → ( y 1 , y 2 , y 3 ; z , w ), is given b y a triple of maps f i : x i → y i for i = 1 , 2 , 3, such that the following diag ram commutes: F 1 ( x 1 ) u / / F 1 ( f 1 )   x 3 f 3   F 2 ( x 2 ) v o o F 2 ( f 2 )   F 1 ( y 1 ) z / / y 3 F 2 ( y 2 ) . w o o This category M can b e given the structur e of a mo del catego r y , where the weak equiv ale nc e s and cofibr ations are g iven levelwise. In other words, f is a weak equiv alence (or cofibratio n) if each map f i is a weak eq uiv a lence (or cofibra tion) in M i . A more res tricted definition of this construction requires that the maps u and v be weak equiv a lences in M 3 . Unfortunately , if we impose this additiona l condition, the resulting category cannot b e given the structure of a mo del catego ry bec ause it is no t closed under limits and colimits. Howev er, intuition suggests tha t w e really wan t to require u a nd v to b e weak equiv alences in or der to get an appropr ia te homotopy pullbac k. W e w ould like to hav e a lo caliza tion of the mo del structure on M described ab ov e such that the fibran t-cofibrant ob jects have the maps u and v weak equiv alences. In at least some situations, we can find suc h a lo calization. HOMOTOPY FIBER PR ODUCTS 7 Recall that a model ca tegory is c ombi natorial if it is co fibrantly g e ne r ated and lo cally pr e sent able as a categor y [5, 2.1]. Theorem 3. 1. L et M b e the homotopy fib er pr o duct of a diagr am of left Quil len functors M 1 F 1 / / M 3 M 2 F 2 o o wher e e ach of the c ate gories M i is c ombi natorial. F urther assume that M is righ t pr op er. Then ther e exists a right Bousfield lo c alizatio n of M whose fi br ant and c ofibr ant obje cts ( x 1 , x 2 , y 2 ; u, v ) have b oth u and v we ak e quivalenc es in M 3 . Pr o of. Since the categ o ries M 1 , M 2 , and M 3 are combinatorial, a nd hence lo cally presentable, we can find, for each i = 1 , 2 , 3, a set A i of ob jects of M i which generates all of M i by filter ed colimits [5 , 2.2]. F urthermor e, w e can assume tha t the ob jects o f A i are all c ofibrant. (An explicit such s et can b e found, for example, using Dugger’s notion of a presentation of a combinatorial mo del categ ory [6].) Given a 1 ∈ A 1 and a 2 ∈ A 2 , consider the class of all ob jects x 3 such that there a re pairs of w eak equiv alences F 1 ( a 1 ) ≃ / / x 3 F 2 ( a 2 ) . ≃ o o Since A 1 and A 2 are sets, we can cho ose one repres e ntative of x 3 for each pair a 1 and a 2 with F 1 ( a 1 ) w eakly equiv alent to F 2 ( a 2 ). T aking the unio n of this set together with the gener ating set A 3 for M 3 , w e obtain a set which w e denote B 3 . F or i = 1 , 2 , let B i = A i . In M , consider the following set of ob jects: { ( x 1 , x 2 , x 3 ; u, v ) | x i ∈ B i , u, v weak equiv alences in M 3 } . By taking filtered colimits, w e ca n obta in from this set all ob jects ( x 1 , x 2 , x 3 ; u, v ) of M for which the maps u and v ar e weak equiv alences; while arbitra ry colimits do not necessar ily preserve these weak equiv ale nces, filtered colimits do [5, 7.3]. Thus, we can tak e a right Bo usfield lo calizatio n of M with res pect to this set of ob jects; if M is rig h t prop er, then this loc alization has a model structure [13, 5.1.1 ], [1].  Unfortunately , it s e ems to be difficult to descr ibe co nditions on the mo del cat- egories M 1 , M 2 , and M 3 guaranteeing that M is right pro p er . W e can weak en this co ndition s o mewhat, using a remark o f Hirschhorn [1 3, 5.1.2]. Alternativ ely , Barwick dis c us ses the structure which is retained after tak ing a right Bous field lo- calization of a mo del ca tegory which is not necessa rily right pro per [1]. Nonetheless, when the conditions of th is theore m are not satisfied, we can still use the origina l levelwise mo del s tr ucture on M and simply r estrict to the appr opriate subca teg ory when we wan t to r equire u and v to be w eak equiv alences. In order to determine whether this c o nstruction really gives a homotopy fib er pro duct of homotop y theories, we need to translate it in to the complete Seg al space mo del structure via the functor L C . When we req uire the maps u and v to be weak equiv alences, we ca n still tak e the asso c ia ted complete Segal spa ce e ven without a mo del structure, and we do g et a homotop y pullback in the mo del ca tegory C S S . The pro of of this s ta temen t is given in the next section. How ever, the more general construction also has a precis e description as w ell, which we give in the following section. W e co nc lude this section with a few examples. 8 J.E. BER GNER Example 3.2. W e b egin with some comments on the use of homotopy fib er pro d- ucts o f mo del categor ies as used by T o¨ en to pro ve asso ciativity o f his derived Hall algebras [2 7]. In this situation, we hav e a stable mo del ca tegory; this extra a ssump- tion tha t the homotopy categor y is triangulated implies that our mo del category has a “zero ob ject” so that the initial and terminal ob jects coincide. W e denote this ob ject 0 . Let T b e a dg category , or ca tegory enric hed over chain complexes o ver a finite field k . Then a dg mo dule ov er T is a dg functor T → C ( k ), w he r e C ( k ) denotes the category of chain complexes of mo dules ov er k . There is a mo del str ucture M ( T ) on the category of such mo dules over a fixed T , where the weak eq uiv a lences and fibrations ar e giv en levelwise [28, § 3]. Given an ob ject of M ( T ) [1] , namely a map f : x → y , let F : M ( T ) [1] → M ( T ) be the targ et map, s o that F ( f : x → y ) = y . Let C : M ( T ) [1] → M ( T ) b e the cone map, so that C ( f : x → y ) = y ∐ x 0. Using these functors, w e get a diagram M ( T ) [1] C   M ( T ) [1] F / / M ( T ) . T o understand the homo topy fib er product M of this diagr a m, T o¨ en uses the mo del structure on the ho motopy fib er pro duct giv en by lev elwise maps; even tually in the pro of he adds the a dditional assumption tha t the maps u and v in the definition b e weak equiv alences [27, § 4]. The homotopy fib er product M giv en b y this diagra m is equiv alent to the mo del categor y M ( T ) [2] whose ob jects are pairs o f co mpo sable morphisms in M ( T ). Example 3.3. Here we co nsider the following special case of a ho motopy pullback, the homotopy fib er of a map. Therefor e, this definition of homo topy fib er product of mo del categories leads to the following definition. Definition 3.4. Le t F : M → N b e a left Quillen functor of mo del categories. Then the ho motopy fib er of F is the homotop y fib er product of the dia gram M F   ∗ / / N where the map ∗ → N is necessar ily the map from the trivial model ca tegory to the initial ob ject φ of N . Using o ur definition, the ob jects of this homoto p y fib er are triples ( ∗ , m, n ; u , v ), where ∗ denotes the single ob ject of the trivial mo del category ∗ , m is an o b ject of M , n is an ob ject of N , u : φ → n is the unique such map, and v : F ( m ) → n . Impo sing our condition that u and v b e weak equiv alences, w e get that n m ust b e weakly equiv alen t to the initial ob ject of N , and m is any ob ject of M whose image under F is weakly equiv alen t to the initial ob ject of N . While this definition follows naturally fro m the usual notio ns, it is unsatisfactory for many purp oses. The requirement that the functors in the pullback diagra m b e left Quillen is a very rigid o ne. One might p erhaps prefer to look a t the ho motopy fiber o ver s ome other ob ject, but here one c a nnot. HOMOTOPY FIBER PR ODUCTS 9 Example 3.5. A further sp ecia liz ation of this definition illustrates its particula rly o dd nature. I f we ta ke the analog ue of a lo op space and define the “lo op mo del category ” as the homotopy pullback of the diagr am ∗   ∗ / / M for a n y mo del catego ry M , we simply get the subcategor y of M whose ob jects a re weakly equiv alent to the initial ob ject. 4. Homotopy pull ba cks o f complete S egal sp aces Consider the functor L C which takes a mo de l category (or simplicial categor y) to a complete Seg a l space. Given a homotopy fiber squa r e of mo del catego ries as defined in the previo us s ection (na mely , where w e require the maps u and v to be weak equiv alences), w e can apply this functor to obtain a homotopy co mm utative square L C M / /   L C M 2   L C M 1 / / L C M 3 . Alternatively , we could apply the functor L C only to the origina l diagram and take the homotopy pullback, whic h w e denote P , and obtain the following diag r am: P / /   L C M 2   L C M 1 / / L C M 3 . Since P is a ho motopy pullba c k, ther e exists a natural map L C M → P . Theorem 4. 1 . Th e map L C M → P = L C M 1 × h L C M 3 L C M 2 is a we ak e quivalenc e of c omplete Se gal sp ac es. T o prov e this theorem, we would like to b e able to use Theorem 2.5 whic h characterizes the complete Segal spaces that result fr o m applying the functor L C to a mo del c ategory . How ever, this theorem only gives the homotopy type of each space in the simplicial diagr am, not an explicit description o f the pre c ise space s we obtain. T hus, we b egin by unpacking this characterization in order to obtain actual maps b etw een these “nice” versions o f the co mplete Segal space s L C M i . More details can b e fo und in [2, § 7], where Theo r em 2.5 is prov ed. W e beg in with the descriptio n of the space at level zero. Given a mo del cate- gory M , we c an ta ke its cor resp onding simplicia l ca tegory L M given by Dwyer- Kan simplicial localizatio n. Denote by C the s ub-simplicial categ ory of L M who s e morphisms are all inv ertible up to homo topy . Then there is a w eak equiv alence F ( C ) → C , wher e F ( C ) denotes the fre e simplicial catego ry on C [10, § 2]. Then tak- ing a group oid completion of F ( C ) gives a simplicia l g r oup oid F ( C ) − 1 F ( C ). The characterization o f the corresp onding complete Segal space us e s the fact that this 10 J.E. BER GNER simplicial gro upo id is equiv alent to one which is the dis join t unio n of simplicial groups, say G . Ther e is a functor F ( C ) − 1 F ( C ) → G c ollapsing each compo nen t down to one with a single ob ject. Thus, we obtain a zig-z a g of w eak equiv a le nces of simplicial categories G ← F ( C ) − 1 F ( C ) ← F ( C ) → C . Since all these constructions a re functorial, b y using Theorem 2.5, w e are es sent ially passing from w orking with C to w orking with G . W e can apply these same constr uctions to the morphism category M [1] to under - stand the spa ce at level one, and, more genera lly , to M [ n ] to obtain the description of the space at level n . Pr o of of The or em 4.1. Since all the ob jects in q ue s tion a re complete Segal spa c e s, i.e., lo ca l ob jects in the mo del structure C S S , it suffices to show that the map L C M → P is a levelwise weak equiv alence of simplicial sets . L e t us b egin by comparing the space at le vel zero for each. The space P 0 lo oks like ( L C M 1 ) 0 × h ( L C M 3 ) 0 ( L C M 2 ) 0 = a h x 1 i B Aut h ( x 1 ) × h a h x 3 i B Aut h ( x 3 ) a h x 2 i B Aut h ( x 2 ) . On the other hand, ( L C M ) 0 lo oks like a h ( x 1 ,x 2 ,x 3 ; u,v ) i B Aut h (( x 1 , x 2 , x 3 ; u, v )) . How ev er, since the classifying space functor B commutes with taking the disjoin t union, this space is equiv alent to B   a h ( x 1 ,x 2 ,x 3 ; u,v ) i Aut h (( x 1 , x 2 , x 3 ; u, v ))   . Thu s, ( L C M ) 0 lo oks like the nerve of the category w ho se ob jects a r e dia grams of the for m F 1 ( x 1 ) u / / F 1 ( a 1 )   x 3 a 3   F 2 ( x 2 ) v o o F 2 ( a 2 )   F 1 ( x 1 ) u / / x 3 F 2 ( x 2 ) v o o where ea c h a i ∈ Aut h ( x i ). In other words, Aut h (( x 1 , x 2 , x 3 ; u, v )) consists of triples ( a 1 , a 2 , a 3 ) such that the ab ov e diagram commutes. F or the momen t, let us supp ose tha t we have no homotop y inv ar ia nce problems and that P 0 can be giv en b y a pullback, rather than a homotopy pullback; further explanation on this p oint w ill be given sho rtly . HOMOTOPY FIBER PR ODUCTS 11 Since B is a right a djoint functor (see [12, I I I.1] for deta ils), it commutes with pullbacks, and we hav e that P 0 ≃ a h x 1 i B Aut h ( x 1 ) × a h x 3 i B Aut h ( x 3 ) a h x 2 i B Aut h ( x 2 ) ≃ B     a h x 1 i Aut h ( x 1 ) × a h x 3 i Aut h ( x 3 ) a h x 2 i Aut h ( x 2 )     ≃ B   a h x 1 i , h x 2 i , h x 3 i Aut h ( x 1 ) × Aut h ( x 3 ) Aut h ( x 2 )   . Thu s, P 0 also lo oks like the nerve of the category whose ob jects a re diagrams of the form giv en above, since the le ftmos t and rightmost vertical a rrows are indexed b y maps in Aut( x 1 ) and Aut ( x 2 ), not by their imag es in M 3 . So, if F 1 , for example, ident ifies tw o maps of Aut( x 1 ), we still count t w o different diagrams. Howev er, if we are taking a s trict pullbac k, the horizontal maps m ust be equa lities. W e claim that taking the ho motopy pullback, rather than the str ict pullback, gives precisely all the diagrams as giv en ab ov e, without this restriction, a s follo ws. Since our diag ram consists of fibr a nt ob jects in C S S , we can apply [13, 19.9.4] and obta in a homotop y pullback by r eplacing one of the maps in the dia gram with a fibration. In do ing so, an ob ject x 1 , for example, is replaced by a pair g iven by x 1 together w ith a map F 1 ( x 1 ) → x 3 . Doing the same for the other map (since there is no harm in replacing b oth of them by fibrations) w e obtain all diagra ms of the form given a bove. So, we hav e shown that we hav e the desired w eak equiv a lence on level zero. Now, it remains to show that w e also get a weak equiv a lence of spaces at level one. The argument here is essentially the same but with larger diag rams. Again, we take ordinary pullbac ks to reduce notation, but this iss ue can be r esolved just as in the level zer o cas e . The s pace P 1 = ( L C M 1 × L C M 3 L C M 2 ) 1 can b e written as follo ws: 0 @ a h f 1 : x 1 → y 1 i B Aut h ( f 1 ) 1 A × 0 @ a h f 3 : x 3 → y 3 i B Aut h ( f 3 ) 1 A 0 @ a h f 2 : x 2 → x 2 i B Aut h ( f 2 ) 1 A ≃ B 0 @ a h f i : x i → y i i “ Aut h ( f 1 ) ” × Aut h ( f 3 ) Aut h ( f 2 ) 1 A . Note that when we tak e h f i : x i → y i i , the notation is meant to signify that w e are v arying x i and y i as ob jects, as well as maps b etw een them, and then taking distinct weak equiv alence cla sses. On th e other hand, if w e let f = ( f 1 , f 2 , f 3 ) : ( x 1 , x 2 , x 3 ; u, v ) → ( y 1 , y 2 , y 3 ; w , z ) , 12 J.E. BER GNER the spa ce ( L C M ) 1 can b e written as a h f i B Aut h ( f ) ≃ B   a h f i Aut h ( f )   . As ab ov e, let a i denote a ho motopy automorphism of x i , and le t b i denote a homotopy automorphism of y i . Then, b oth of the ab ov e spaces are given by the nerve of the catego ry who se ob jects ar e diagr ams of the form F 1 ( x 1 ) u / / F 1 ( f 1 )   F 1 ( a 1 ) $ $ I I I I I I I I I x 3 a 3   ? ? ? ? ? ? ? ? ? f 3   F 2 ( x 2 ) F 2 ( a 2 ) $ $ I I I I I I I I I F 2 ( f 2 )   v o o F 1 ( x 1 ) u / / F 1 ( f 1 )   x 3 f 3   F 2 ( x 2 ) v o o F 2 ( f 2 )   F 1 ( y 1 ) F 1 ( b 1 ) $ $ I I I I I I I I I w / / y 3 b 3   ? ? ? ? ? ? ? ? F 2 ( y 2 ) z o o F 2 ( b 2 ) $ $ I I I I I I I I I F 1 ( y 1 ) w / / y 3 F 2 ( y 2 ) z o o One could show that the higher-deg ree spa ces of each of thes e complete Segal spaces ar e a lso weakly equiv alent, but since these s paces are deter mined b y these t wo, the above a rguments are sufficien t.  5. The more general construction on complete Segal sp aces In this section, we dro p the co ndition that the maps u a nd v in the definition o f the homotopy fib er pro duct a re weak equiv alences in M 3 and give a characterizatio n of the resulting complete Segal s pace. Again, let M 2 F 2   M 1 F 1 / / M 3 be a diag ram of mo del categ ories and left Quillen functors. Let N b e the category whose ob jects are given b y 5 -tuples ( x 1 , x 2 , x 3 ; u, v ), where x i is an ob ject of M i for ea ch i , a nd the maps u a nd v fit in to a diagram F 1 ( x 1 ) u / / x 3 F 2 ( x 2 ) . v o o The 0 -space of the complete Seg al space L C N has the homotopy t yp e a h ( x 1 ,x 2 ,x 3 ; u,v ) i B Aut h (( x 1 , x 2 , x 3 ; u, v )) . An element of the group Aut h (( x 1 , x 2 , x 3 ; u, v )) lo oks lik e a diagram F 1 ( x 1 ) u / / ≃   x 3 ≃   F 2 ( x 2 ) ≃   v o o F 1 ( x 1 ) u / / x 3 F 2 ( x 2 ) . v o o HOMOTOPY FIBER PR ODUCTS 13 Using this diagram as a guide, we can for mu late a co ncise descr iption of ( L C N ) 0 . Prop ositio n 5. 1. L et N 0 denote the nerve of the c ate gory given by ( · → · ← · ) . The sp ac e ( L C N ) 0 has the ho motopy t yp e of the pul lb ack of the diagr am Map ( N t 0 , L C M 3 )   Map (∆[0] t , L C M 1 ) × Map (∆[0] t , L C M 2 ) / / Map (∆[0] t , L C M 3 ) 2 wher e t he horizo ntal map is given by Map (∆[0] t , L C F 1 ) × Map (∆[0] t , L C F 2 ) and the vertic al arr ow is induc e d t he p air of sour c e m aps ( s 1 , s 2 ) : N 0 → ∆[0] ∐ ∆[0] . Pr o of. F or i = 1 , 2 , 3, let a i denote a homotopy automor phism of x i in M i . The collection o f diagra ms of the form F 1 ( x 1 ) u / / F 1 ( a 1 )   x 3 a 3   F 2 ( x 2 ) F 2 ( a 2 )   v o o F 1 ( x 1 ) u / / x 3 F 2 ( x 2 ) . v o o can b e written as the pullback Aut h ( u ) × Aut h ( x 3 ) Aut h ( v ) . T aking classifying spaces and copro ducts ov er all isomorphism classes o f ob jects, we obta in the pullback (5.2) a h u : F 1 ( x 1 ) → x 3 i B Aut h ( u ) × a h x 3 i B Aut h ( x 3 ) a h v : F 2 ( x 2 ) → x 3 i B Aut h ( v ) . How ev er, notice that the s pace a h u : F 1 ( x 1 ) → x 3 i B Aut h ( u ) is equiv alent to the pullback a h x 1 i B Aut h ( x 1 ) × a h x 3 i B Aut h ( x 3 ) a h f 3 : y 3 → x 3 i B Aut h ( f 3 ) , since an elemen t of Aut h ( u ) lo o ks like a diag ram y 3 f 3 / / ≃   x 3 ≃   y 3 f 3 / / x 3 where y 3 = F 1 ( x 1 ). Analogously , the space a h v : F 2 ( x 2 ) → x 3 i B Aut h ( v ) 14 J.E. BER GNER is equiv alent to the pullback a h x 2 i B Aut h ( x 2 ) × a h x 3 i B Aut h ( x 3 ) a h f 3 : y 3 → x 3 i B Aut h ( f 3 ) . Putting these t wo equiv alences together, w e get that the pullback (5.2) can b e written as 0 B B @ a h x 1 i B Aut h ( x 1 ) × a h x 3 i B Aut h ( x 3 ) a h f 3 i B Aut h ( f 3 ) 1 C C A × a h x 3 i B Aut h ( x 3 ) 0 B B @ a h x 2 i B Aut h ( x 2 ) × a h x 3 i B Aut h ( x 3 ) a h f 3 i B Aut h ( f 3 ) 1 C C A . How ev er, this pullback can be written in a muc h more managea ble wa y using our characterization of the complete Seg al spaces co rresp onding to a mo del category . Thu s, w e get a pullback ( L C M 1 ) 0 × ( L C M 3 ) 0 ( L C M [1] 3 ) 0 × ( L C M 3 ) 0 ( L C M 2 ) 0 × ( L C M 2 ) 0 ( L C M [1] 3 ) 0 . Rearra ng ing terms in the pullbac k gives an eq uiv a lent formulation of this space as (( L C M 1 ) 0 × ( L C M 2 ) 0 ) × ( L C M 3 ) 2 0 (( L C M [1] 3 ) 0 × ( L C M 3 ) 0 ( L C M [1] 3 ) 0 ) . How ev er, this spa ce is precisely the pullback of the diagr am given in the statement of the propos ition, since Ma p(∆[0] t , L C M 1 ) = ( L C M 1 ) 0 and analo gously for M 2 , and the pullbac k on the right a grees with the space Map( N t 0 , L C M 3 ).  Now we give a characteriza tio n of the space ( L C N ) 1 . Prop ositio n 5.3. If N 1 denotes the nerve of the c ate gory given by · / /   ·   ·   o o · / / · · o o then the sp ac e ( L C N ) 1 is we akly e quivalent to the homotopy pul lb ack of the diagr am Map ( N t 1 , L C M 3 )   Map (∆[1] t , L C M 1 ) × Map (∆[1] t , L C M 2 ) / / Map (∆[1] t , L C M 3 ) 2 wher e t he maps ar e analo gous to the ones in the pr evious pr op osition. Pr o of. Again, let f = ( f 1 , f 2 , f 3 ) : ( x 1 , x 2 , x 3 ; u, v ) → ( y 1 , y 2 , y 3 ; w , z ) . HOMOTOPY FIBER PR ODUCTS 15 Notice that, b y definition, the homotopy type of the space ( L C N ) 1 is g iven by a h f i B Aut h ( f ) . An element of the group Aut h ( f ) is given by a diagram F 1 ( x 1 ) u / / F 1 ( f 1 )   F 1 ( a 1 ) $ $ I I I I I I I I I x 3 a 3   ? ? ? ? ? ? ? ? ? f 3   F 2 ( x 2 ) F 2 ( a 2 ) $ $ J J J J J J J J J F 2 ( f 2 )   v o o F 1 ( x 1 ) u / / F 1 ( f 1 )   x 3 f 3   F 2 ( x 2 ) v o o F 2 ( f 2 )   F 1 ( y 1 ) F 1 ( b 1 ) $ $ I I I I I I I I I w / / y 3 b 3   ? ? ? ? ? ? ? ? F 2 ( y 2 ) z o o F 2 ( b 2 ) $ $ J J J J J J J J J F 1 ( y 1 ) w / / y 3 F 2 ( y 2 ) . z o o If we let α 1 : u → w and α 2 : v → z be maps in M [1] , such a diagr am can also be regarde d as an element o f the homoto py pullback Aut h ( f 1 ) × Aut h ( f 3 ) Aut h ( α 1 ) × Aut h ( f 3 ) Aut h ( α 2 ) × Aut h ( f 2 ) Aut h ( f 3 ) . T aking classifying spaces and copro ducts o ver all p ossible cla sses of ob jects a nd morphisms, we obtain a pullba c k 0 B B @ a h f 1 i B Aut h ( f 1 ) × a h f 3 i B Aut h ( f 3 ) a h α 1 i B Aut h ( α 1 ) 1 C C A × a h f 3 i B Aut h ( f 3 ) 0 B B @ a h f 2 i B Aut h ( f 2 ) × a h f 3 i B Aut h ( f 3 ) a h α 2 i B Aut h ( α 2 ) 1 C C A . This pullback can b e rewr itten in terms of the corres po nding complete Segal spaces as  ( L C M 3 ) 1 × ( L C M 3 ) 1 ( L C M [1] 3 ) 1  × ( L C M 3 ) 1  ( L C M 2 ) 1 × ( L C M 3 ) 1 ( L C M [1] 3 ) 1  . A t this p o int , notice that this space is also the pullba ck of the diagr am ( L C M [1] 3 ) 1 × ( L C M 3 ) 1 ( L C M [1] 3 ) 1   ( L C M 1 ) 1 × ( L C M 3 ) 1 ( L C M 2 ) 1 / / ( L C M 3 ) 2 1 . How ev er, since the upper space is equiv alent to Ma p( N t 1 , L C M 3 ), we hav e c o m- pleted the pro of.  16 J.E. 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