HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES

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1 HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES JULIA E. BERGNER Abstract. Gien an appropriate diagram of left Qillen fnctors between model categories, one can define a notion of homotopy fiber prodct, bt one might ask if it is really the correct one. Here, we show that this homotopy pllback is well-behaed with respect to translating it into the setting of more general homotopy theories, gien by complete Segal spaces, where we hae well-defined homotopy pllbacks. 1. Introdction Homotopy theory originates with the stdy of topological spaces p to weak homotopy eqialence, where two spaces are weakly homotopy eqialent if there is a map between them indcing isomorphisms on all the homotopy grops. The qestion of whether the nice properties of topological spaces held in other settings led to the deelopment of the notion of a model category by Qillen [22]. The stdy of model categories is then a more abstract form of homotopy theory, one which has been inestigated extensiely. One cold then consider model categories themseles as objects of a category and consider Qillen eqialences as weak eqialences between them. In this framework, one cold ask qestions abot relationships between model categories; for example, what wold a homotopy limit or homotopy colimit of a diagram of model categories be? Unfortnately, there are no immediate answers to these qestions becase at present there is no known model strctre on the category of model categories. In one particlar case of a homotopy limit, we hae a plasible constrction, that of the homotopy fiber prodct. This constrction was first explained to the athor by Smith; it appears in the literatre in a paper of Toën [27], where it is sefl in his deelopment of deried Hall algebras. One might ask whether, withot a model category of model categories, we can really accept this definition as a genine homotopy pllback constrction. Howeer, one can also consider a homotopy theory to be something more general than a model category, sch as a category with a specified class of weak eqialences, or maps one wold like to consider as eqialences bt which are not necessarily isomorphisms. While there may or may not be a model strctre on sch a category, one can heristically think of formally inerting the weak eqialences, set-theoretic problems notwithstanding. Frthermore, sch homotopy theories can be regarded as the objects of a model category. Date: October 9, Mathematics Sbject Classification. Primary: 55U40; Secondary: 55U35, 18G55, 18G30, 18D20. Key words and phrases. model categories, simplicial categories, complete Segal spaces, homotopy theories, homotopy fiber prodcts. The athor was partially spported by NSF grant DMS

2 2 J.E. BERGNER In a series of papers [7], [9], [10], Dwyer and Kan deelop the theory of simplicial localizations, which are simplicial categories corresponding to model categories, or, more generally, categories with specified weak eqialences. Since eery sch category has a simplicial localization, and since, p to a natral notion of eqialence of simplicial categories, eery simplicial category arises as the simplicial localization of some category with weak eqialences [8, 2.5], simplicial categories can be considered to be models for homotopy theories. With these weak eqialences, often called Dwyer-Kan eqialences, the category of (small) simplicial categories can itself be considered as a homotopy theory, and in fact it has a model strctre [3]. Ths, within this framework one can address qestions abot homotopy theories that are natral to ask in a gien model category, sch as characterizations of homotopy limits and colimits. Unfortnately, the model strctre on the category of simplicial categories is technically difficlt to work with. The weak eqialences are particlarly challenging to identify, for example. Fortnately, we hae a choice of seeral other eqialent model categories in which to address these qestions. Work of the athor and of Joyal and Tierney has proed that the complete Segal space model strctre on the category of simplicial spaces, two different Segal category model strctres on the category of Segal precategories, and the qasi-category model strctre on the category of simplicial sets are all eqialent as model categories, and ths each is a model for the homotopy theory of homotopy theories [4], [17], [19]. In this paper, we address the constrction of the homotopy fiber prodct of model categories and its analoge within the complete Segal space model strctre. Of the arios models mentioned aboe, the complete Segal space model strctre is the best setting in which to answer this qestion de to the particlarly nice description of the releant weak eqialences. Some clarification of terminology is needed here, as there are actally two notions of what is meant by a homotopy fiber prodct of model categories. One is more restrictie than the other; the stronger is the one we wold expect to correspond to a homotopy pllback, bt the weaker is the one which can be gien the strctre of a model category. We show that at least in some cases this model strctre can be localized so that the fibrant-cofibrant objects satisfy the more restrictie condition. The main reslt of this paper is that the stricter definition does in fact correspond to a homotopy pllback when we work in the complete Segal space setting. We also gie a characterization of the complete Segal spaces arising from the less restrictie description. We shold point ot that this constrction has been sed in special cases, for example by Hüttemann, Klein, Vogell, Waldhasen, and Williams in [15], and as an example of more general constrctions, for example the twisted diagrams of Hüttemann and Röndigs [16], and the model categories of comma categories as gien by Stanclesc [26]. It is also important to note that translating this qestion into the setting of more general homotopy theories is not merely a temporary soltion ntil one finds a model category of model categories. In practice, model strctres are often hard to establish, and frthermore, the condition of haing a Qillen pair between two sch model strctres is a rigid one. Being able to consider homotopy theories as more flexible kinds of objects, and haing morphisms between them less strctred, makes it more likely that we can actally implement sch a constrction. We

3 HOMOTOPY FIBER PRODUCTS 3 consider sch a case in Example 3.3. Yet, with the relationship between the two settings established, we can se the additional strctre when we do indeed hae it. In fact, part of or motiation for making the comparison in this paper is to generalize Toën s deelopment of deried Hall algebras. Where he defines an associatie algebra corresponding to stable model categories gien by modles oer a dg category, we wold like to define sch an algebra sing a more general stable homotopy theory, namely, one gien by a stable complete Segal space. The main reslt of this paper allows s to se a homotopy pllback of complete Segal spaces in the setting where Toën ses a homotopy fiber prodct of model categories. The dal notion of homotopy pshots of model categories, as well as more general homotopy limits and homotopy colimits, will be considered in later work. 2. Model categories and more general homotopy theories In this section we gie a brief reiew of model categories and their relationship with the complete Segal space model for more general homotopy theories. Recall that a model category M is a category with three distingished classes of morphisms: weak eqialences, fibrations, and cofibrations, satisfying fie axioms [11, 3.3]. Gien a model category strctre, one can pass to the homotopy category Ho(M), in which the weak eqialences from M become isomorphisms. In particlar, the weak eqialences, as the morphisms that we wish to inert, make p the most important part of a model category. An object x in M is fibrant if the niqe map x to the terminal object is a fibration. Dally, an object x in M is cofibrant if the niqe map φ x from the initial object is a cofibration. Gien a model category M, there is also a model strctre on the category M [1], often called the morphism category of M. The objects of M [1] are morphisms of M, and the morphisms of M [1] are gien by pairs of morphisms making the appropriate sqare diagram commte. A morphism in M [1] is a weak eqialence (or cofibration) if its component maps are weak eqialences (or cofibrations) in M. More generally, M [n] is the category with objects strings of n composable morphisms in M; the model strctre can be defined analogosly. One cold, more generally, consider categories with weak eqialences and no additional strctre, and then formally inert the weak eqialences. This process does gie a homotopy category, bt it freqently has the disadantage of haing a proper class of morphisms between any two gien objects. If we are willing to accept sch set-theoretic problems, then we can work in this sitation; the adantage of a model strctre is that it proides enogh additional strctre so that we can take homotopy classes of maps and hence aoid these difficlties. In this paper, we will se both notions of a homotopy theory, depending on the circmstances. We wold, howeer, like to work with nice objects modeling categories with weak eqialences. While there are seeral options, the model that we se in this paper is that of complete Segal spaces. Recall that the simplicial category op is defined to be the category with objects finite ordered sets [n] = {0 1 n} and morphisms the opposites of the order-presering maps between them. A simplicial set is then a fnctor K : op Sets.

4 4 J.E. BERGNER We denote by SSets the category of simplicial sets, and this category has a natral model category strctre eqialent to the standard model strctre on topological spaces [12, I.10]. One can jst as easily consider more general simplicial objects; in this paper we consider simplicial spaces (also called bisimplicial sets), or fnctors X : op SSets. There are seeral model category strctres on the category of bisimplicial sets. An important one is the Reedy model strctre [23], which is eqialent to the injectie model strctre, where the weak eqialences are gien by leelwise weak eqialences of simplicial sets, and the cofibrations are gien likewise [13, ]. Gien a simplicial set K, we also denote by K the simplicial space which has the simplicial set K at eery leel. We denote by K t, or K-transposed, the constant simplicial space in the other direction, where (K t ) n = K n, where on the right-hand side K n is regarded as a discrete simplicial set. We se the idea, originating with Dwyer and Kan, that a simplicial category, or category enriched oer simplicial sets, models a homotopy theory, in the following way. Using either of their two notions of simplicial localization, one can obtain from a category with weak eqialences a simplicial category [9], [10], and there is a model strctre SC on the category of all small simplicial categories [3]; ths, we hae a so-called homotopy theory of homotopy theories. One sefl conseqence of taking the simplicial category corresponding to a model category is that we can se it to describe homotopy fnction complexes, or homotopy-inariant mapping spaces Map h (x, y) between objects of a model category which is not necessarily eqipped with the additional strctre of a simplicial model category. We se, in particlar, the simplicial set At h (x) of homotopy inertible self-maps of an object x. Here we also consider simplicial spaces satisfying conditions imposing a notion of composition p to homotopy. These Segal spaces and complete Segal spaces were first introdced by Rezk [24], and the name is meant to be sggestie of similar ideas first presented by Segal [25]. Definition 2.1. [24, 4.1] A Segal space is a Reedy fibrant simplicial space W sch that the Segal maps ϕ n : W n W 1 W0 W0 W 1 } {{ } n are weak eqialences of simplicial sets for all n 2. Gien a Segal space W, we can consider its objects ob(w ) = W 0,0, and, between any two objects x and y, the mapping space map W (x, y), gien by the homotopy fiber of the map W 1 W 0 W 0 gien by the two face maps W 1 W 0. The Segal condition gien here tells s that a Segal space has a notion of n-fold composition of mapping spaces, at least p to homotopy. More precise constrctions are gien by Rezk [24, 5]. Using this composition, we can define homotopy eqialences in a natral way, and then speak of the sbspace of W 1 whose components contain homotopy eqialences, denoted W hoeqi. Notice that the degeneracy map s 0 : W 0 W 1 factors throgh W hoeqi ; hence we may make the following definition. Definition 2.2. [24, 6] A complete Segal space is a Segal space W sch that the map W 0 W hoeqi is a weak eqialence of simplicial sets.

5 HOMOTOPY FIBER PRODUCTS 5 Gien this definition, we can describe the model strctre on the category of simplicial spaces that is sed throghot this paper. Theorem 2.3. [24, 7] There is a model category strctre CSS on the category of simplicial spaces, obtained as a localization of the Reedy model strctre sch that: (1) the fibrant objects are the complete Segal spaces, (2) all objects are cofibrant, and (3) the weak eqialences between complete Segal spaces are leelwise weak eqialences of simplicial sets. Now we retrn to the idea that a complete Segal space models a homotopy theory. Theorem 2.4. [4] The model categories SC and CSS are Qillen eqialent. Frthermore, de to work of Rezk [24] which was contined by the athor [2], we can actally characterize, p to weak eqialence, the complete Segal space arising from a simplicial category, or more specifically, from a model category. Rezk defines a fnctor which we denote L C from the category of model categories to the category of simplicial spaces; gien a model category M, we hae that L C (M) n = nere(we(m [n] )). Here, M [n] is defined as aboe, and we(m [n] ) denotes the sbcategory of M [n] whose morphisms are the weak eqialences. While the reslting simplicial space is not in general Reedy fibrant, and hence not a complete Segal space, Rezk proes that taking a Reedy fibrant replacement is sfficient to obtain a complete Segal space [24, 8.3]. Hence, for the rest of this paper we assme that the fnctor L C incldes composition with this Reedy fibrant replacement and therefore gies a complete Segal space. One other difficlty that arises in this definition is the fact that it is only a welldefined fnctor on the category whose objects are model categories and whose morphisms presere weak eqialences. This restriction on the morphisms is stronger than we wold like; it wold be preferable to hae sch a fnctor defined on the category of model categories with morphisms Qillen fnctors, where weak eqialences are only presered between either fibrant or cofibrant objects. To obtain this more sitable fnctor, we consider M c, the fll sbcategory of M whose objects are cofibrant. While M c may no longer hae the strctre of a model category, it is still a category with weak eqialences, and therefore Rezk s definition can still be applied to it, so or new definition is L C (M) n = nere(we((m c ) [n] )). Each space in this new diagram is weakly eqialent to the one gien by the preios definition, and now the constrction is fnctorial on the category of model categories with morphisms the left Qillen fnctors. If one wanted to consider right Qillen fnctors instead, we cold take the fll sbcategory of fibrant objects, M f, rather than M c. Before stating the theorem giing the characterization, we gie some facts abot simplicial monoids, or fnctors from op to the category of monoids. Gien a simplicial monoid M (or, more commonly, a simplicial grop), we can find a classifying complex of M, a simplicial set whose geometric realization is the classifying space BM. A precise constrction can be made for this classifying space by the W M

6 6 J.E. BERGNER constrction [12, V.4.4], [21]. As we are not so concerned here with the precise constrction as with the fact that sch a classifying space exists, we will simply write BM for the classifying complex of M. Theorem 2.5. [2, 7.3] Let M be a model category. For x an object of M denote by x the weak eqialence class of x in M, and denote by At h (x) the simplicial monoid of self weak eqialences of x. Up to weak eqialence in the model category CSS, the complete Segal space L C (M) looks like BAt h (x) BAt h (α). x α : x y (We shold point ot that the reference (Theorem 7.3 of [2]) gies a characterization of the complete Segal space arising from a simplicial category, not from a model category. Howeer, the reslts of 6 of that same paper allow one to translate it to the theorem as stated here.) This characterization, together with the fact that weak eqialences between complete Segal spaces are leelwise weak eqialences of simplicial sets, enables s to compare complete Segal spaces arising from different model categories. 3. Homotopy fiber prodcts of model categories We begin with the definition of homotopy fiber prodct as gien by Toën in [27]. First, sppose that F 1 F M 1 M 3 2 M 2 is a diagram of left Qillen fnctors of model categories. Define their homotopy fiber prodct to be the model category M = M 1 h M 3 M 2 whose objects are gien by 5-tples (x 1, x 2, ;, ) sch that each x i is an object of M i fitting into a diagram F 2 (x 2 ). A morphism of M, say f : (x 1, x 2, ;, ) (y 1, y 2, y 3 ; z, w), is gien by a triple of maps f i : x i y i for i = 1, 2, 3, sch that the following diagram commtes: F 1 (y 1 ) F 1(f 1) f 3 w F 2 (x 2 ) F 2(f 2) z y 3 F 2 (y 2 ). This category M can be gien the strctre of a model category, where the weak eqialences and cofibrations are gien leelwise. In other words, f is a weak eqialence (or cofibration) if each map f i is a weak eqialence (or cofibration) in M i. A more restricted definition of this constrction reqires that the maps and be weak eqialences in M 3. Unfortnately, if we impose this additional condition, the reslting category cannot be gien the strctre of a model category becase it is not closed nder limits and colimits. Howeer, intition sggests that we really want to reqire and to be weak eqialences in order to get an appropriate homotopy pllback. We wold like to hae a localization of the model strctre on M described aboe sch that the fibrant-cofibrant objects hae the maps and weak eqialences. In at least some sitations, we can find sch a localization.

7 HOMOTOPY FIBER PRODUCTS 7 Recall that a model category is combinatorial if it is cofibrantly generated and locally presentable as a category [5, 2.1]. Theorem 3.1. Let M be the homotopy fiber prodct of a diagram of left Qillen fnctors F 1 F M 1 M 3 2 M 2 where each of the categories M i is combinatorial. Frther assme that M is right proper. Then there exists a right Bosfield localization of M whose fibrant and cofibrant objects (x 1, x 2, y 2 ;, ) hae both and weak eqialences in M 3. Proof. Since the categories M 1, M 2, and M 3 are combinatorial, and hence locally presentable, we can find, for each i = 1, 2, 3, a set A i of objects of M i which generates all of M i by filtered colimits [5, 2.2]. Frthermore, we can assme that the objects of A i are all cofibrant. (An explicit sch set can be fond, for example, sing Dgger s notion of a presentation of a combinatorial model category [6].) Gien a 1 A 1 and a 2 A 2, consider the class of all objects sch that there are pairs of weak eqialences F 1 (a 1 ) F 2 (a 2 ). Since A 1 and A 2 are sets, we can choose one representatie of for each pair a 1 and a 2 with F 1 (a 1 ) weakly eqialent to F 2 (a 2 ). Taking the nion of this set together with the generating set A 3 for M 3, we obtain a set which we denote B 3. For i = 1, 2, let B i = A i. In M, consider the following set of objects: {(x 1, x 2, ;, ) x i B i,, weak eqialences in M 3 }. By taking filtered colimits, we can obtain from this set all objects (x 1, x 2, ;, ) of M for which the maps and are weak eqialences; while arbitrary colimits do not necessarily presere these weak eqialences, filtered colimits do [5, 7.3]. Ths, we can take a right Bosfield localization of M with respect to this set of objects; if M is right proper, then this localization has a model strctre [13, 5.1.1], [1]. Unfortnately, it seems to be difficlt to describe conditions on the model categories M 1, M 2, and M 3 garanteeing that M is right proper. We can weaken this condition somewhat, sing a remark of Hirschhorn [13, 5.1.2]. Alternatiely, Barwick discsses the strctre which is retained after taking a right Bosfield localization of a model category which is not necessarily right proper [1]. Nonetheless, when the conditions of this theorem are not satisfied, we can still se the original leelwise model strctre on M and simply restrict to the appropriate sbcategory when we want to reqire and to be weak eqialences. In order to determine whether this constrction really gies a homotopy fiber prodct of homotopy theories, we need to translate it into the complete Segal space model strctre ia the fnctor L C. When we reqire the maps and to be weak eqialences, we can still take the associated complete Segal space een withot a model strctre, and we do get a homotopy pllback in the model category CSS. The proof of this statement is gien in the next section. Howeer, the more general constrction also has a precise description as well, which we gie in the following section. We conclde this section with a few examples.

8 8 J.E. BERGNER Example 3.2. We begin with some comments on the se of homotopy fiber prodcts of model categories as sed by Toën to proe associatiity of his deried Hall algebras [27]. In this sitation, we hae a stable model category; this extra assmption that the homotopy category is trianglated implies that or model category has a zero object so that the initial and terminal objects coincide. We denote this object 0. Let T be a dg category, or category enriched oer chain complexes oer a finite field k. Then a dg modle oer T is a dg fnctor T C(k), where C(k) denotes the category of chain complexes of modles oer k. There is a model strctre M(T ) on the category of sch modles oer a fixed T, where the weak eqialences and fibrations are gien leelwise [28, 3]. Gien an object of M(T ) [1], namely a map f : x y, let F : M(T ) [1] M(T ) be the target map, so that F (f : x y) = y. Let C : M(T ) [1] M(T ) be the cone map, so that C(f : x y) = y x 0. Using these fnctors, we get a diagram M(T ) [1] C M(T ) [1] F M(T ). To nderstand the homotopy fiber prodct M of this diagram, Toën ses the model strctre on the homotopy fiber prodct gien by leelwise maps; eentally in the proof he adds the additional assmption that the maps and in the definition be weak eqialences [27, 4]. The homotopy fiber prodct M gien by this diagram is eqialent to the model category M(T ) [2] whose objects are pairs of composable morphisms in M(T ). Example 3.3. Here we consider the following special case of a homotopy pllback, the homotopy fiber of a map. Therefore, this definition of homotopy fiber prodct of model categories leads to the following definition. Definition 3.4. Let F : M N be a left Qillen fnctor of model categories. Then the homotopy fiber of F is the homotopy fiber prodct of the diagram where the map N is necessarily the map from the triial model category to the initial object φ of N. Using or definition, the objects of this homotopy fiber are triples (, m, n;, ), where denotes the single object of the triial model category, m is an object of M, n is an object of N, : φ n is the niqe sch map, and : F (m) n. Imposing or condition that and be weak eqialences, we get that n mst be weakly eqialent to the initial object of N, and m is any object of M whose image nder F is weakly eqialent to the initial object of N. While this definition follows natrally from the sal notions, it is nsatisfactory for many prposes. The reqirement that the fnctors in the pllback diagram be left Qillen is a ery rigid one. One might perhaps prefer to look at the homotopy fiber oer some other object, bt here one cannot. M N F

9 HOMOTOPY FIBER PRODUCTS 9 Example 3.5. A frther specialization of this definition illstrates its particlarly odd natre. If we take the analoge of a loop space and define the loop model category as the homotopy pllback of the diagram for any model category M, we simply get the sbcategory of M whose objects are weakly eqialent to the initial object. M 4. Homotopy pllbacks of complete Segal spaces Consider the fnctor L C which takes a model category (or simplicial category) to a complete Segal space. Gien a homotopy fiber sqare of model categories as defined in the preios section (namely, where we reqire the maps and to be weak eqialences), we can apply this fnctor to obtain a homotopy commtatie sqare L C M L C M 2 L C M 1 L C M 3. Alternatiely, we cold apply the fnctor L C only to the original diagram and take the homotopy pllback, which we denote P, and obtain the following diagram: P L C M 2 L C M 1 L C M 3. Since P is a homotopy pllback, there exists a natral map L C M P. Theorem 4.1. The map L C M P = L C M 1 h L C M 3 L C M 2 is a weak eqialence of complete Segal spaces. To proe this theorem, we wold like to be able to se Theorem 2.5 which characterizes the complete Segal spaces that reslt from applying the fnctor L C to a model category. Howeer, this theorem only gies the homotopy type of each space in the simplicial diagram, not an explicit description of the precise spaces we obtain. Ths, we begin by npacking this characterization in order to obtain actal maps between these nice ersions of the complete Segal spaces L C M i. More details can be fond in [2, 7], where Theorem 2.5 is proed. We begin with the description of the space at leel zero. Gien a model category M, we can take its corresponding simplicial category LM gien by Dwyer- Kan simplicial localization. Denote by C the sb-simplicial category of LM whose morphisms are all inertible p to homotopy. Then there is a weak eqialence F (C) C, where F (C) denotes the free simplicial category on C [10, 2]. Then taking a gropoid completion of F (C) gies a simplicial gropoid F (C) 1 F (C). The characterization of the corresponding complete Segal space ses the fact that this

10 10 J.E. BERGNER simplicial gropoid is eqialent to one which is the disjoint nion of simplicial grops, say G. There is a fnctor F (C) 1 F (C) G collapsing each component down to one with a single object. Ths, we obtain a zig-zag of weak eqialences of simplicial categories G F (C) 1 F (C) F (C) C. Since all these constrctions are fnctorial, by sing Theorem 2.5, we are essentially passing from working with C to working with G. We can apply these same constrctions to the morphism category M [1] to nderstand the space at leel one, and, more generally, to M [n] to obtain the description of the space at leel n. Proof of Theorem 4.1. Since all the objects in qestion are complete Segal spaces, i.e., local objects in the model strctre CSS, it sffices to show that the map L C M P is a leelwise weak eqialence of simplicial sets. Let s begin by comparing the space at leel zero for each. The space P 0 looks like (L C M 1 ) 0 h (L C M 3) 0 (L C M 2 ) 0 = BAt h (x 1 ) h BAt h BAt h (x 2 ). (x x 1 3 ) On the other hand, (L C M) 0 looks like (x 1,x 2,;,) BAt h ((x 1, x 2, ;, )). Howeer, since the classifying space fnctor B commtes with taking the disjoint nion, this space is eqialent to B At h ((x 1, x 2, ;, )). (x 1,x 2, ;,) Ths, (L C M) 0 looks like the nere of the category whose objects are diagrams of the form F 1 (a 1 ) where each a i At h (x i ). In other words, a 3 F 2 (x 2 ) F 2 (a 2 ) F 2 (x 2 ) At h ((x 1, x 2, ;, )) consists of triples (a 1, a 2, a 3 ) sch that the aboe diagram commtes. For the moment, let s sppose that we hae no homotopy inariance problems and that P 0 can be gien by a pllback, rather than a homotopy pllback; frther explanation on this point will be gien shortly. x 2

11 HOMOTOPY FIBER PRODUCTS 11 Since B is a right adjoint fnctor (see [12, III.1] for details), it commtes with pllbacks, and we hae that P 0 BAt h (x 1 ) BAt h BAt h (x 2 ) ( ) x 1 x 2 B At h (x 1 ) At h At h (x 2 ) ( ) B x 1 x 1, x 2, x 2 At h (x 1 ) At h ( ) At h (x 2 ). Ths, P 0 also looks like the nere of the category whose objects are diagrams of the form gien aboe, since the leftmost and rightmost ertical arrows are indexed by maps in At(x 1 ) and At(x 2 ), not by their images in M 3. So, if F 1, for example, identifies two maps of At(x 1 ), we still cont two different diagrams. Howeer, if we are taking a strict pllback, the horizontal maps mst be eqalities. We claim that taking the homotopy pllback, rather than the strict pllback, gies precisely all the diagrams as gien aboe, withot this restriction, as follows. Since or diagram consists of fibrant objects in CSS, we can apply [13, ] and obtain a homotopy pllback by replacing one of the maps in the diagram with a fibration. In doing so, an object x 1, for example, is replaced by a pair gien by x 1 together with a map. Doing the same for the other map (since there is no harm in replacing both of them by fibrations) we obtain all diagrams of the form gien aboe. So, we hae shown that we hae the desired weak eqialence on leel zero. Now, it remains to show that we also get a weak eqialence of spaces at leel one. The argment here is essentially the same bt with larger diagrams. Again, we take ordinary pllbacks to redce notation, bt this isse can be resoled jst as in the leel zero case. The space P 1 = (L C M 1 LC M 3 L C M 2 ) 1 can be written as follows: f 1 : x 1 y 1 BAt h (f 1) f 3 : y 3 BAt h (f 3) B f i : x i y i f 2 : x 2 x 2 BAt h (f 2) ( ) At h (f 1) At h (f 3) Ath (f 2). Note that when we take f i : x i y i, the notation is meant to signify that we are arying x i and y i as objects, as well as maps between them, and then taking distinct weak eqialence classes. On the other hand, if we let f = (f 1, f 2, f 3 ): (x 1, x 2, ;, ) (y 1, y 2, y 3 ; w, z),

12 12 J.E. BERGNER the space (L C M) 1 can be written as BAt h (f) B At h (f). f f As aboe, let a i denote a homotopy atomorphism of x i, and let b i denote a homotopy atomorphism of y i. Then, both of the aboe spaces are gien by the nere of the category whose objects are diagrams of the form F 1 (a 1 ) F 1 (f 1 ) F 1(f 1) f 3 a 3 f 3 F 2 (x 2 ) F 2 (a 2 ) F 2 (x 2 ) F 2(f 2) w F 1 (y 1 ) z y 3 F 2 (y 2 ) b 3 F 2 (b 2 ) F 1(b 1) w F 1 (y 1 ) z y 3 F 2 (y 2 ) F 2 (f 2 ) One cold show that the higher-degree spaces of each of these complete Segal spaces are also weakly eqialent, bt since these spaces are determined by these two, the aboe argments are sfficient. 5. The more general constrction on complete Segal spaces In this section, we drop the condition that the maps and in the definition of the homotopy fiber prodct are weak eqialences in M 3 and gie a characterization of the reslting complete Segal space. Again, let M 2 M 1 F 1 M 3 be a diagram of model categories and left Qillen fnctors. Let N be the category whose objects are gien by 5-tples (x 1, x 2, ;, ), where x i is an object of M i for each i, and the maps and fit into a diagram F 2 F 2 (x 2 ). The 0-space of the complete Segal space L C N has the homotopy type BAt h ((x 1, x 2, ;, )). (x 1,x 2,;,) An element of the grop At h ((x 1, x 2, ;, )) looks like a diagram F 2 (x 2 ) F 2 (x 2 ).

13 HOMOTOPY FIBER PRODUCTS 13 Using this diagram as a gide, we can formlate a concise description of (L C N ) 0. Proposition 5.1. Let N 0 denote the nere of the category gien by ( ). The space (L C N ) 0 has the homotopy type of the pllback of the diagram Map(N t 0, L C M 3 ) Map( [0] t, L C M 1 ) Map( [0] t, L C M 2 ) where the horizontal map is gien by Map( [0] t, L C F 1 ) Map( [0] t, L C F 2 ) Map( [0] t, L C M 3 ) 2 and the ertical arrow is indced the pair of sorce maps (s 1, s 2 ): N 0 [0] [0]. Proof. For i = 1, 2, 3, let a i denote a homotopy atomorphism of x i in M i. The collection of diagrams of the form can be written as the pllback F 1(a 1) a 3 F 2 (x 2 ) F 2(a 2) F 2 (x 2 ). At h () At h ( ) Ath (). Taking classifying spaces and coprodcts oer all isomorphism classes of objects, we obtain the pllback (5.2) BAt h () BAt h ( ) : F 1(x 1) BAt h (). : F 2(x 2) Howeer, notice that the space BAt h () : F 1(x 1) is eqialent to the pllback BAt h (x 1 ) BAt h ( ) x 1 since an element of At h () looks like a diagram y 3 f 3 BAt h (f 3 ), f 3 : y 3 where y 3 =. Analogosly, the space y 3 f 3 BAt h () : F 2 (x 2 )

14 14 J.E. BERGNER is eqialent to the pllback BAt h (x 2 ) BAt h ( ) x 2 f 3 : y 3 BAt h (f 3 ). Ptting these two eqialences together, we get that the pllback (5.2) can be written as BAt h (x 1 ) BAt h BAt h (f 3 ) ( ) x 1 f 3 BAt h ( ) BAt h (x 2 ) BAt h BAt h (f 3 ) ( ). x 2 f 3 Howeer, this pllback can be written in a mch more manageable way sing or characterization of the complete Segal spaces corresponding to a model category. Ths, we get a pllback (L C M 1 ) 0 (LC M 3 ) 0 (L C M [1] 3 ) 0 (LC M 3 ) 0 (L C M 2 ) 0 (LC M 2 ) 0 (L C M [1] 3 ) 0. Rearranging terms in the pllback gies an eqialent formlation of this space as ((L C M 1 ) 0 (L C M 2 ) 0 ) (LC M 3) 2 0 ((L CM [1] 3 ) 0 (LC M 3 ) 0 (L C M [1] 3 ) 0). Howeer, this space is precisely the pllback of the diagram gien in the statement of the proposition, since Map( [0] t, L C M 1 ) = (L C M 1 ) 0 and analogosly for M 2, and the pllback on the right agrees with the space Map(N0, t L C M 3 ). Now we gie a characterization of the space (L C N ) 1. Proposition 5.3. If N 1 denotes the nere of the category gien by then the space (L C N ) 1 is weakly eqialent to the homotopy pllback of the diagram 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 where the maps are analogos to the ones in the preios proposition. Proof. Again, let f = (f 1, f 2, f 3 ): (x 1, x 2, ;, ) (y 1, y 2, y 3 ; w, z).

15 HOMOTOPY FIBER PRODUCTS 15 Notice that, by definition, the homotopy type of the space (L C N ) 1 is gien by BAt h (f). An element of the grop At h (f) is gien by a diagram f F 1(a 1) F 1 (f 1 ) f 3 a 3 F 2 (x 2 ) F 2(a 2) F 2 (x 2 ) F 2 (f 2 ) F 1 (f 1 ) w F 1 (y 1 ) z y 3 F 2 (y 2 ) b 3 F 2(b 2) F 1 (b 1 ) w F 1 (y 1 ) z y 3 F 2 (y 2 ). f 3 F 2(f 2) If we let α 1 : w and α 2 : z be maps in M [1], sch a diagram can also be regarded as an element of the homotopy pllback At h (f 1 ) At h (f 3 ) At h (α 1 ) At h (f 3 ) At h (α 2 ) At h (f 2 ) At h (f 3 ). Taking classifying spaces and coprodcts oer all possible classes of objects and morphisms, we obtain a pllback BAt h (f 1 ) BAt h BAt h (α 1 ) (f 3 ) f 1 α 1 f 3 BAt h (f 3 ) f 3 BAt h (f 2 ) BAt h BAt h (α 2 ) (f 3 ). f 2 α 2 f 3 This pllback can be rewritten in terms of the corresponding complete Segal spaces as ( ) ( ) (L C M 3 ) 1 (LC M 3) 1 (L C M [1] 3 ) 1 (LC M 3) 1 (L C M 2 ) 1 (LC M 3) 1 (L C M [1] 3 ) 1. At this point, notice that this space is also the pllback of the diagram (L C M [1] 3 ) 1 (LC M 3 ) 1 (L C M [1] 3 ) 1 (L C M 1 ) 1 (LC M 3 ) 1 (L C M 2 ) 1 (L C M 3 ) 2 1. Howeer, since the pper space is eqialent to Map(N1, t L C M 3 ), we hae completed the proof.

16 16 J.E. BERGNER Lastly, notice that the simplicial set N 1 from Proposition 5.3 is jst Map( [1], N 0 ) = N [1] 0, where N 0 is as in Proposition 5.1. We can se these reslts and the properties of complete Segal spaces to gie the following theorem. Theorem 5.4. Let N be the simplicial space gien by N n = N [n] 0. Then the complete Segal space L C N is weakly eqialent to the homotopy pllback of the diagram Map(N, L C M 3 ) L C M 1 L C M 2 L C M 3 L C M 3. References [1] Clark Barwick, On the dreaded right Bosfield localization, preprint aailable at math.at/ [2] J.E. Bergner, Complete Segal spaces arising from simplicial categories, Trans. Amer. Math. Soc. 361 (2009), [3] J.E. Bergner, A model category strctre on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), [4] J.E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), [5] Daniel Dgger, Combinatorial model categories hae presentations. Ad. Math. 164 (2001), no. 1, [6] Daniel Dgger, Uniersal homotopy theories, Ad. Math. 164 (2001), no. 1, [7] W.G. Dwyer and D.M. Kan, Calclating simplicial localizations, J. Pre Appl. Algebra 18 (1980), [8] W.G. Dwyer and D.M. Kan, Eqialences between homotopy theories of diagrams, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), , Ann. of Math. Std., 113, Princeton Uni. Press, Princeton, NJ, [9] W.G. Dwyer and D.M. Kan, Fnction complexes in homotopical algebra, Topology 19 (1980), [10] W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, J. Pre Appl. Algebra 17 (1980), no. 3, [11] W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Handbook of Algebraic Topology, Elseier, [12] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math, ol. 174, Birkhaser, [13] Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical Sreys and Monographs 99, AMS, [14] Mark Hoey, Model Categories, Mathematical Sreys and Monographs, 63. American Mathematical Society [15] Thomas Hüttemann, John R. Klein, Wolrad Vogell, Friedhelm Waldhasen, and Brce Williams, The fndamental theorem for the algebraic K-theory of spaces, I. J. Pre Appl. Algebra 160 (2001), no. 1, [16] Thomas Hüttemann and Olier Röndigs, Twisted diagrams and homotopy sheaes, preprint aailable at math.at/ [17] A. Joyal, Simplicial categories s qasi-categories, in preparation. [18] A. Joyal, The theory of qasi-categories I, in preparation. [19] André Joyal and Myles Tierney, Qasi-categories s Segal spaces, Contemp. Math. 431 (2007) [20] Sanders Mac Lane, Categories for the Working Mathematician, Second Edition, Gradate Texts in Mathematics 5, Springer-Verlag, [21] J.P. May, Simplicial Objects in Algebraic Topology, Uniersity of Chicago Press, [22] Daniel Qillen, Homotopical Algebra, Lectre Notes in Math 43, Springer-Verlag, 1967.

17 HOMOTOPY FIBER PRODUCTS 17 [23] C.L. Reedy, Homotopy theory of model categories, npblished manscript, aailable at [24] Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353(3) (2001), [25] Graeme Segal, Classifying spaces and spectral seqences, Pbl. Math. Inst. des Hates Etdes Scient (Paris) 34 (1968), [26] Alexandr E. Stanclesc, A model category strctre on the category of simplicial mlticategories, preprint aailable at math.ct/ [27] Bertrand Toën, Deried Hall algebras, Dke Math. J. 135, no. 3 (2006), [28] Bertrand Toën, The homotopy theory of dg-categories and deried Morita theory, Inent. Math. 167 (2007), no. 3, Department of Mathematics, Uniersity of California, Rierside, CA address: bergnerj@member.ams.org