A HOMOTOPY THEORY FOR STACKS SHARON HOLLANDER Abstract.We give a homotopy theoretic characterization of stacks on a si* *te C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. * *We compare different definitions of stacks and show that they lead to Quill* *en equivalent model categories. In addition, we show that these model struc* *tures are Quillen equivalent to the S2-nullification of Jardine's model struct* *ure on sheaves of simplicial sets on C. 1.Introduction Stacks arise as classifying objects for moduli problems in algebraic geometry. This means that, in some sense, maps from a scheme X into a stack correspond to isomorphism classes of families of certain objects over X. A standard example is the stack of all curves: a map from a scheme X into this stack corresponds to an isomorphism class of families of curves over X. Other examples include the stack representing vector bundles and the stack representing curves of genus g with n marked points. In algebraic geometry, stacks are regarded as a generalization * *of schemes, and many of the usual constructions for schemes are extended so as to make sense for stacks as well. For example, one can define cohomology groups for a stack. These groups yield important information about general properties of t* *he objects which the stack classifies. Recently, stacks have also come up in algebraic topology. Complex oriented cohomology theories give rise to Hopf algebroids which corepresent stacks on the category of affine schemes, and these stacks map to the moduli stack of formal groups. In recent work of Hopkins and Miller, it has been shown conversely that, in good situations, stacks over the moduli stack of formal groups give rise to * *spec- tra which are approximations (often localizations) of the sphere spectrum. These spectra play a key role in modern attempts at understanding and calculating the stable homotopy groups of spheres. There are many different definitions of stacks. The main purpose of this pape* *r is to show that all of these definitions can be interpreted in terms of homotopy t* *heory, and to show that from this point of view they are natural and easy to compare. One definition of stacks is based on the concept of category fibered in group* *oids [DM , Gi] and another based on the concept of lax presheaf of groupoids [Brn, B* *ry]. In each definition, part of the information encoded in a stack M is an assignme* *nt to each scheme X of a groupoid M(X). These assignments are required to satisfy `descent conditions', which are often somewhat cumbersome. We will show that, for the definitions of stack commonly in use, the descent conditions can be giv* *en a simple homotopy theoretic interpretation. ____________ Date: May 25, 2001. 1 2 SHARON HOLLANDER The descent conditions describe the circumstances under which we require that local data glue together to yield global data. Naively, one might require "isom* *or- phism classes of -" to satisfy the sheaf condition. However, for very fundament* *al reasons, this almost never happens in examples. Taking isomorphism classes is a localization process, and such processes rarely preserve limits such as those w* *hich arise in the statement of the sheaf condition. Instead, one can ask that an ass* *ign- ment of groupoids satisfy a sheaf condition with respect to the best approximat* *ion to the limit which is invariant under taking isomorphism classes. This is call* *ed the homotopy limit, and denoted holim. Stacks are assignments which satisfy this modified sheaf condition, so in this sense, stacks are the homotopy sheaves. We propose the following definition of stack as a reference point, as it is c* *oncep- tually the simplest: Definition 1.1. Let C be a Grothendieck topology. A presheaf of groupoids, F on C is a stack if for every cover {Ui! X} in C, we have an equivalence of categor* *ies ` Q Q Q ' F (X)____~_____//holim F (Ui)____+3 F (Uij)__`*4___F (Uijk) . . .. Here Ui0...indenotes the iterated fiber product Ui0xX . .x.XUin, and the homoto* *py limit is taken in the category of groupoids (see sections 3-4). We will show that all other definitions of stack commonly in use can be given similar homotopy theoretic interpretations. Not only the definition but many pr* *op- erties of stacks which are of interest are homotopy theoretic in nature, and th* *is homotopy theoretic perspective both simplifies the task of comparing the differ* *ent definitions as well as illuminates the sense in which stacks are the "right" cl* *assi- fying spaces for algebraic problems. In particular, the previous definition giv* *es an alternate category of stacks which is equivalent from the point of view of homo- topy theory but much easier to work with, and which is related in a simple way * *to familiar homotopy theoretic categories. In more detail, for each of the definitions of stack, we will construct a mod* *el category in which the stacks are the fibrant objects. In these model categories* *, con- structions that are commonly performed on stacks (such as 2-category pullbacks, stackification, sheaves over a stack and others) have easy homotopy-theoretic i* *n- terpretations [Holl]. Moreover, homotopy classes of maps from an object X 2 C to a stack M correspond to the isomorphism classes of M(X), and the homotopies themselves correspond to isomorphisms in M(X). We will see that all of these different model categories are Quillen equivalent. This is the formal way of sa* *ying they are all models for the same underlying homotopy theory. This equivalence makes precise the sense in which, when dealing with stacks, it is enough to con* *sider presheaves of groupoids satisfying descent conditions. More precisely, we will analyze three different categories in which stacks ca* *n be defined (see section 5 for the definitions) and prove the following results. Le* *t C be a Grothendieck topology. Theorem 1.2. There are adjoint pairs of functors between: categories fibered in groupoids over C, presheaves of groupoids on C, sheaves of groupoids on C, and * *lax presheaves of groupoids on C, _____~____--_________________________________________________* *______________________________________________________________@ lax - P (C, Grpd)nn_______Grpd=C____________________________________________* *______________________________________________________________@ ~ p sh A HOMOTOPY THEORY FOR STACKS 3 where the right adjoints point to presheaves. All of these functors take stack* *s as defined in the domain category to stacks as defined in the range category and t* *hus restrict to give adjoint pairs between the stacks in each of these categories. Theorem 1.3. There are simplicial model category structures on each of the above listed categories in which: (1) the stacks are the fibrant objects, (2) in P (C, Grpd) or Sh(C, Grpd), a weak equivalence is a map satisfying the local lifting conditions (see 8.2), (3) if the topology on C has enough points, the weak equivalences in P (C, G* *rpd) are the stalkwise equivalences of groupoids, (4) all of the adjoint pairs listed above are Quillen equivalences, (5) the fundamental groupoid of the simplicial Hom set between X 2 C and a stack M, the homotopy function complex h Hom (X, M) is equivalent to the groupoid M(X). In particular, [X, M] is the set of isomorphism classes of M(X). Presheaves of groupoids, which will be our preferred setting for talking about stacks, is closely related to presheaves of simplicial sets. The homotopy theor* *y of the latter has been developed by Jardine [Ja], and is the basis on which Morel * *and Voevodsky build the A1-homotopy theory of schemes, see [MV ]. Theorem 1.4. The local model structure on P (C, Grpd) is Quillen equivalent to Jardine's model category structure on P (C, sSet) localized with respect to the* * maps @ n X ! n X, for each X 2 C and n > 2. This theorem says that the homotopy theory of stacks is recovered from Jardin* *e's model category by eliminating all higher homotopies. 1.1. Notation and Assumptions. So as not run into set theoretic problems, we assume that the Grothendieck topology C is a small category. We also assume that the topology on C is subcanonical in order to construct the desired model struc* *ture on Sh(C, Grpd). For {Ui! X} a cover in C, and F a presheaf on C, o Ui0...indenotes the iterated fiber product Ui0xX . .x.XUin. o Uo denotes the simplicial diagram in P re(C) with a (Uo)n = Ui0xX . .x.XUin, I where the coproduct is taken over all multi-indices of length n, and the face and boundary maps are defined by the various projection and diagonal maps. This is referred`to as the nerve of the cover {Ui! X}. o To simplify notation` Ui will sometimes be denoted by U, the coproduct` Uijwill be denoted by U xX U, and Uijkby U xX U xX U. o FQ(Uo) = Hom (Uo, F ) denotes the cosimplicial diagram F (Uo)n = I F (Ui0xX . .x.XUin) with coface and codegeneracy maps dual to those for Uo. Q o We will sometimesQwrite F (U) for F (Ui), F (U xX U) for F (Uij),Q and F (U xX U xX U) for F (Uijk). o Similarly, a cover {Vi ! Y } may be denoted by V ! Y , and the nerve of this cover by {V ! Y }o. 4 SHARON HOLLANDER 1.2. Contents. The following is an outline of the contents of this paper: In section 2 we give necessary background information about groupoid* *s, monoidal categories, enriched categories, model categories and localization. In section 3 we construct a model structure on groupoids, and prove that it is Quillen equivalent to a localization of simplicial sets with respect to the * *map S2 ! *, called the S2 nullification of sSet. In section 4 we give some background on homotopy limits and colimits, and pro* *ve that the descent category is a model for the homotopy inverse limit of a cosimp* *licial diagram of groupoids. In section 5 we review the definition of categories fibered in groupoids over* * a fixed base category C. Then we construct an adjoint pair of functors between th* *is category and the category of presheaves of groupoids on C. We define stacks in each of these categories as the objects which satisfy a homotopy sheaf conditio* *n. Section 6 contains a discussion of the classical definition of stacks [DM ], * *and a proof that it is equivalent to our definition in terms of the homotopy sheaf co* *ndition. In section 7 we put model structures on the categories described in section 5 and on the category of sheaves of groupoids. The weak equivalences are defined to be object, respectively. fiberwise. We note that the pairs of adjoint func* *tors between the different categories that were defined previously are Quillen pairs* *. We also observe that these model structures can be localized with respect to the l* *ocal equivalences holimUo ! X, and in these local model structures the fibrant objec* *ts are the stacks. In section 8 we give a characterization of pointwise weak equivalences for presheaves of groupoids in terms of Dan Dugger's local lifting conditions. We use this to prove that the local model categories are all Quillen equivalent. * *We also obtain that the local model category structure on presheaves of groupoids * *is Quillen equivalent to the S2 nullification of Jardine's model category structur* *e on presheaves of simplicial sets, and conclude that when the Grothendieck topology* * on C has enough points, the weak equivalences in the local model category structure are precisely the pointwise equivalences of groupoids. Appendix A contains a discussion of limits and colimits in the category Grpd=C of categories fibered in groupoids. In appendix B we define the category of lax presheaves of groupoids and descr* *ibe the adjoint pair between lax presheaves and categories fibered in groupoids. Th* *is is an equivalence of categories and hence allows us to translate all the result* *s from categories fibered in groupoids to lax presheaves. 1.3. Acknowledgments. A great many thanks are owed to Dan Dugger, Gustavo Granja, and Mike Hopkins, for many helpful conversations and ideas, without whi* *ch this paper would not exist. The author has recently learned about the paper of Jardine [Ja2], which appea* *rs to treat some of the questions dealt with here. Although his approach is quite different, it is possible that there is some overlap in the results. A HOMOTOPY THEORY FOR STACKS 5 2.Preliminaries A groupoid is a small category in which all morphisms are invertible. Grpd denotes the full subcategory of Catwhose objects are groupoids. In this section* * we will define the notion of a category with a groupoid action. Many of the catego* *ries we will discuss in the future have groupoid actions, and many of their properti* *es follow from similar properties of groupoids. We show that such categories have a natural simplicial structure, determined by the action of the fundamental group* *oid of the simplicial sets. We also review the concept of a model category, and quo* *te the results about localization which we will need later. 2.1. Groupoids. Recall, the nerve embedding, N : Cat ! sSet. For C 2 Cat, N(C)n is the set of n-tuples of composable morphisms f0 f1 fn-2 fn-1 X0 oo___X1 oo___. .o.o__Xn-1 oo___Xn with the convention that a 0-tuple is just an object. For i 6= 0, n, the bound* *ary maps di send (f0, . .f.i, fi+1, . .,.fn-1) 7! (f1, . .f.i-1O fi, . .,.fn-1); d0* * leaves out f0, and dn leaves out fn. In particular N(G)1 d0-!N(G)0 is the domain function, and d1 is the range function. The degeneracy maps si insert an identity morphism in the ith position. We begin by noting that Grpd is complete and cocomplete since the (co)limit of a diagram of groupoids in Catis still a groupoid. Note 2.1. Recall that the limit of a diagram in Catis the category whose objects (morphisms) are the limit of the sets of objects (morphisms) in the diagram. To construct a colimit in Catone takes as objects the colimit of the sets of objec* *ts and for morphisms the formal compositions of elements in the colimit of the morphis* *ms, modulo the obvious relations. We will need the following characterization of the functors in Grpd that are equivalences of categories. Lemma 2.2. The functor G -F! H 2 Grpd is an equivalence of categories, if and only if the following two conditions hold: o F induces a bijection on isomorphism classes of objects. o For every a 2 G, the induced map AutG (a) ! AutH (F (a)) is an isomor- phism. Definition 2.3. [GZ ] Let sSet ßoid-!Grpdbe the functor which assigns to a sim- plicial set X the groupoid with objects X0 and morphisms freely generated under composition by the members of X1 and their formal inverses subject to the relat* *ions d2x O d0x = d1x for each x 2 X2. The proof of the following lemma is easy. Lemma 2.4. The functors ßoid: sSet $ Grpd : N are an adjoint pair, in which N is the right adjoint, and ßoidis the left adjoint. The composition of functo* *rs ßoidO N is naturally isomorphic to the identity functor of Grpd. Note 2.5. The definition of ßoidused here is the one given in [GZ ]. This group* *oid is naturally equivalent to the one defined via homotopy classes of paths [GJ , * *p. 39]. The [GZ ] definition is needed to form the adjoint pair (ßoid, N), and thus to * *define the simplicial structure on Grpd which is essential for many of our results. 6 SHARON HOLLANDER Note 2.6. It follows from the previous note, as ßoidO N ~=id, and NG is a Kan complex, that for any G 2 Grpd, the isomorphism classes of G are in bijective correspondence with ß0NG, and the automorphism group of an object a 2 G, is isomorphic to ß1(NG, a). The category Grpd has an internal Hom , written Grpd(G, H), where the objects of Grpd(G, H) are the functors G ! H, and the isomorphisms are the natural isomorphisms between these functors. Lemma 2.7. Let G be a groupoid, then Grpd(G, -) is the right adjoint to the functor G x (-). Recall [EK ] that a closed category (short for closed symmetric monoidal) is a category M with an internal Hom and an associative and commutative product with a unit S, such that for all X 2 M, the functor X (-) is the left adjoint* * of M(X, -). By lemma 2.7, Grpd is a closed category with the categorical product and the internal Hom defined above. Another example of a closed category is sSet. The tensor product is just given by the categorical product, and the internal Hom is given by the formula sSet(X, Y )n := Hom sSet( n x X, Y ) where o denotes the cosimplicial standard simplex [BK , p. 268]. Recall [Db ], that a category C is enriched over a monoidal category M, if there is a bifunctor from Cop x C ! M assigning to each X, Y 2 C an object MC(X, Y ) 2 M (which we also denote by M(X, Y )) for each object X an "iden- tity" S ! M(X, X), and for each triple of objects X, Y, Z 2 C a öc mposition" M(X, Y ) M(Y, Z) ! M(X, Z) which is associative and unital. Moreover it is required that Hom C(X, Y ) = hom M(S, M(X, Y )). C is said to be enriched with tensor and cotensor if for all G 2 M and X, Y 2 C there are objects X ~G and Y G 2 C such that M(X ~G, Y ) = M(X, Y G) = M(G, M(X, Y )). It then follows that this tensor and cotensor operations satisfy all the usual * *prop- erties. Note 2.8. In practice we will abuse notation and denote the tensor product of objects of C with objects M by . Any closed monoidal category M is enriched with tensor and cotensor over it- self. A category enriched with tensor and cotensor over simplicial sets is call* *ed a simplicial category. We will say that a category enriched with tensor and coten* *sor over groupoids has a groupoid action. Proposition 2.9. Let C be a category with a groupoid action. Then the assignment sSet(X, Y ) := N(Grpd(X, Y )) gives C the structure of a simplicial category. Moreover, the tensor and cotens* *or are given by the formulas Y S := Y ßoidS, Y S:= Y ßoidS, for any X, Y 2 C, S 2 sSet. This proposition follows immediately from the following lemma. A HOMOTOPY THEORY FOR STACKS 7 Lemma 2.10. Let X 2 sSet, G 2 Grpd, then the adjoint pair of functors ßoidand N satisfies N(Grpd(ßoidX, G)) = sSet(X, N(G)). In particular, given G, H 2 Grpd, N(Grpd(H, G)) = sSet(N(H), N(G)). Proof.The nerve of Grpd(ßoidX, G) has 0-simplices the functors ßoidX ! G. By lemma 2.4 these are the elements of Hom sSet(X, N(G)) = sSet(X, N(G))0. The n- simplices of N(Grpd(ßoidX, G)) are n-tuples of composable natural isomorphisms between such functors. They can be naturally identified with functors ßoidX x ßoid n = ßoid(X x n) ! G By another application of lemma 2.4 these are identified with the elements of Hom sSet(X x n, N(G)). The following examples of categories with a groupoid action will be used thro* *ugh- out the rest of the paper. Example 2.11 (Diagrams of Groupoids). Let X and Y be diagrams of groupoids indexed by a category D, and G a groupoid. Then we define Grpd(X, Y ), to be the groupoid with objects the natural transformations X ! Y , and with morphisms the natural isomorphisms X x ßoid 1 ! Y , where ßoid 1 denotes the constant diagram (which assigns to each object the groupoid with two objects and a unique isomorphism between them). Then we have (X G)(d) = X(d) x G, XG (d) = Hom (G, X(d)). When C is a Grothendieck topology, diagrams indexed by Cop are called presheaves of groupoids on C. The category of presheaves of groupoids on C is denoted P (C, Grpd). Example 2.12 (Sheaves of Groupoids on a Grothendieck Topology C). A sheaf of groupoids on C is a presheaf which satisfies the "sheaf condition": For eve* *ry covering {Ui! U}, Y Y F (U) ! F (Ui) ) F (UixU Uj) is an equalizer sequence. Equivalently, we could require F (U) to be the limit * *of the cosimplicial diagram determined by the nerve of the covering as, in any categor* *y, the limit of a cosimplicial diagram Xo is the equalizer of d0, d1 : X0 ) X1. The category Sh(C, Grpd) is the full subcategory of P (C, Grpd) whose objects are t* *he sheaves of groupoids on C. We list some of the important properties of sheaves and presheaves. (1) There is a "sheafification functor" P (C, Grpd) -sh!Sh(C, Grpd) which is* * the left adjoint to the inclusion functor of sheaves in presheaves. (2) The category Sh(C, Grpd) inherits a Grpd action via the inclusion into P (C, Grpd) as it is easy to check that for a sheaf F, the presheaves F * * G and FG are still sheaves. For further elaboration of these points see [MM ]. 8 SHARON HOLLANDER 2.2. Review of Model Categories. We recall the definition of a model category structure on a category C. Model categories are an abstract setting in which to* * do homotopy theory. A model category [Ho , Q, DS], is a category C, together with three distingui* *shed classes of morphisms in C, called cofibrations, fibrations, and weak equivalenc* *es, which are closed under composition and contain all identity morphisms, and sati* *sfy the following properties: o (MC1) Small limits and colimits exist in C. o (MC2) If f, g are morphisms with g O f defined, and two of the three mor- phisms f, g, g O f are weak equivalences then so is the third. o (MC3) If f is a retract of g and g is a fibration, cofibration, or weak * *equiv- alence, then so is f. o (MC4) Given a commutative square f A"_`___//X>>" i||l"" |p| fflffl|fflfflfflffl|g" B _____//Y. where either (a) i is a cofibration and p a trivial fibration (a fibrati* *on which is also a weak equivalence), or (b) i is a trivial cofibration (a cofibr* *ation which is also a weak equivalence), and p a fibration, then there exists a lifting l : B ! X making the above diagram commute. o (MC5) Any morphism f can be factored functorially in two ways: (a) f = pOi where i is a cofibration and p is a trivial fibration; and (b) f* * = pOi where i is a trivial cofibration and p is a fibration. An object X is called cofibrant if the map from the initial object ;, to X is a cofibration. An object X is called fibrant if the map from X to the final obje* *ct *, is a fibration. The category obtained from C by formally inverting the weak equivalences is called the homotopy category of C, and denoted Ho(C). A set of (trivial) cofibrations are said to generate if the trivial fibration* *s (fi- brations) are characterized by having the right lifting property (as in MC4) wi* *th respect to these morphisms. A simplicial model category is a model category C which has a simplicial stru* *cture compatible with the model structure in the sense that the following axiom holds: (SM7) Given a cofibration A -i! B and a fibration X -p! Y , the induced map sSet(B, X) ! sSet(A, X) xsSet(A,Ys)Set(B, Y ) is a fibration. In addition, if either i or p is a weak equivalence then the above map is a trivial fibration. A Quillen pair between model categories is an adjoint pair L : C $ D : R where the left adjoint L preserves cofibrations and trivial cofibrations, or equivale* *ntly the right adjoint R preserves fibrations and trivial fibrations. Under these condit* *ions, one can define the derived functors L_: Ho(C) ! Ho(D) and R_: Ho(D) ! Ho(C), and they form an adjoint pair. A Quillen pair is called a Quillen equivalence i* *f, for A 2 C cofibrant and B 2 D fibrant, a morphism LA ! B is a weak equivalence in D if and only if its adjoint A ! RB is a weak equivalence in C. A Quillen pair * *is A HOMOTOPY THEORY FOR STACKS 9 a Quillen equivalence if and only if it induces an equivalence of categories be* *tween Ho(C) and Ho(D), see [Ho , p. 19]. Under mild conditions there is a procedure called localization which formally adds weak equivalences to a model category (a good reference is [Dg ]). Let C b* *e a simplicial model category and S a set of morphisms between cofibrant objects in* * C. A fibrant object X 2 C is called S-local if for all f 2 S the induced map sSet(* *f, X) is a weak equivalence. A morphism f in C is called an S-equivalence if for all * *S-local X, we have that h Hom (f, X) is a weak equivalence, (where h Hom is the homotopy function complex, see [Hi]). A model category is left proper if pushouts of we* *ak equivalences along cofibrations are weak equivalences. Note 2.13. A model category C is combinatorial if it is cofibrantly generated a* *nd the underlying category is locally presentable [Sm ]. All the categories we wil* *l be working with here are locally presentable, (as they have underlying sets) and we will give explicit sets of generating cofibrations. Theorem 2.14. (J.Smith) [Sm ] Let C be a left proper, combinatorial, simplicial model category and S a set of morphisms between cofibrant objects in C. Then th* *ere exists a new model category structure on C in which o the weak equivalences are S-equivalences, o the cofibrations are the old cofibrations, o the fibrations are maps with the right lifting property with respect to * *the maps which are cofibrations and also S-equivalences. In addition the fibrant objects of C are precisely the S-local objects, and thi* *s new model structure is again left proper, combinatorial, and simplicial. This new model category is called the S-localization of C and denoted S-1C. Notice that all of the original weak equivalences in C are, by construction, S- equivalences. The following properties of localization will be used often. Note 2.15. (P. Hirschhorn) [Hi] Let S and C be as in the above theorem, D be a model category, and L : C $ D : R a Quillen pair such that L takes morphisms in S to weak equivalences in D. Then (a)The pair (L, R) is also a Quillen pair L : S-1C $ D : R. (b)In particular, if S0 is a set of morphisms between cofibrant objects in D, a* *nd L takes morphisms in S to S0-equivalences, there is a Quillen pair L : S-1C $ (S0)-1D : R. (c)[Hi, Theorem 3.4.20] If L : C $ D : R is a Quillen equivalence and S is a set of morphisms between cofibrant objects in C, then L : S-1C $ (LS)-1D : R is also a Quillen equivalence. (d)If S0, S are sets of morphisms in C then the two model structures S-1(S0)-1C* * = (S0)-1S-1C agree. 10 SHARON HOLLANDER 3. Model Category Structure on Groupoids In this section we will describe a model category structure on Grpdwhich appe* *ars in [An , Bo], a proof can be found in [St]. This model category structure will * *enable us to prove that the descent category, which appears prominently in the definit* *ion of stacks, is a model for the homotopy inverse limit of a cosimplicial groupoid. With this in mind the various definitions of stacks can be interpreted as diffe* *rent incarnations of presheaves of groupoids satisfying a `homotopy sheaf condition'. Under the nerve embedding, functors between categories become maps between simplicial sets, and natural transformations between functors give rise to homo- topies between the corresponding maps. If F -OE!G, and F - ,!H are natural transformations, we obtain homotopies between N(F ) and N(G), and from N(F ) to N(H). Though there is not necessarily a natural transformation from G to H corresponding to the composite homotopy. Thus, our intuitive notion of homo- topy in Cat, as a natural transformation between functors, does not correspond * *to the one defined via the nerve embedding in sSet. However, if our categories are groupoids, this problem does not arise since all natural transformations are na* *tural isomorphisms. This close relationship between our intuition for what homotopy should be in Grpdand the notion of homotopy defined via the nerve, motivates the model category structure on Grpdwe define here, where a map f in Grpdis a weak equivalence or fibration if and only if N(f) is one. We will sometimes abuse notation and denote the groupoid ßoid( i) by i. This is the groupoid with i + 1 objects with unique isomorphisms between them. Similarly, we will sometimes denote ßoid(@ i) by @ i. BG denotes the groupoid with one object whose automorphism group is the group G. Theorem 3.1. There is a left proper, simplicial, cofibrantly generated model ca* *te- gory structure on Grpd in which: o weak equivalences are functors which induce an equivalence of categories, o fibrations are the functors with the right lifting property with respect* * to the map 0 ! 1, o cofibrations are functors which are injections on objects. The generating trivial cofibration is the morphism 0 ! 1, and the generating cofibrations are the morphisms @ i! i, i = 0, 1, 2. Note 3.2. In this model category structure all objects are both fibrant and cof* *i- brant, so all weak equivalences are homotopy equivalences. Lemma 3.3. Let G -f!H be a map of groupoids. The following are equivalent: o f is a weak equivalence in Grpd o Nf is weak equivalence in sSet Similarly, the following are equivalent: o f is a (trivial) fibration in Grpd. o Nf is a (trivial) Kan fibration in sSet. o f has the right lifting property with respect to 0 ! 1 (with respect to @ n ! n for n = 0, 1, 2). Note that the morphisms @ i! i, i = 0, 1, 2, are o ; ! ?, o {?, ?} ! I A HOMOTOPY THEORY FOR STACKS 11 o 2 x (BZ ! ?). Proof.If f is a weak equivalence in Grpd, it is an equivalence of categories and so Nf is a homotopy equivalence in sSet. Since the nerve of a groupoid is a Kan complex, if Nf is a weak equivalence, it must be a homotopy equivalence, and so ßoidNf = f is an equivalence of categories. Kan fibrations of simplicial sets are characterized by having the right lifti* *ng property with respect to the maps Vn,k! n, n 1 and trivial Kan fibrations are characterized by having the right lifting property with respect to the maps @ n ! n. Given a morphism G ! H of groupoids, it is equivalent to construct a lifting in either of the diagrams Vn,k_____//N(G);; ßoidVn,k____//G;;w | ww | | w w | | w w | | w | fflffl|w fflffl| |fflfflw fflffl| n _____//N(H), ßoid n_____//H, so we can characterize the maps in Grpd whose nerves are fibrations as the maps which have the right lifting property with respect to ßoidVn,k ! ßoid n. Sim- ilarly, the maps whose nerves are trivial fibrations are characterized as the m* *or- phisms with the right lifting property with respect to the maps ßoid@ n ! ßoid* * n. Now notice that the inclusions ßoidVi,k! ßoid i are isomorphisms for i > 1, and that the inclusions ßoid@ i! ßoid i are isomorphisms for i > 2. Note 3.4. The previous lemma gives sets of generating cofibrations and trivial cofibrations for the model structure in Theorem 3.1. Proof of Theorem 3.1.For MC1-MC5 see [St]. For SM7, we need to show that given A -i! B a cofibration and X -p! Y a fibration, the induced map sSet(B, X) ! sSet(A, X) xsSet(A,Ys)Set(B, Y ) is a fibration of simplicial sets. In addition, we need to show that if either * *i or p is a weak equivalence, then the above map is a trivial fibration. The simplicial str* *ucture on Grpd is defined by taking the nerve of the internal Hom , and N commutes with limits, so we can rewrite the above map as N Grpd(B, X) ! Grpd(A, X) xGrpd(A,YG)rpd(B, Y;) which is a (trivial) fibration if and only if the map Grpd(B, X) ! Grpd(A, X) xGrpd(A,YG)rpd(B, Y ) is one. By lemma 3.3, this is the case if and only if this map has the right li* *fting property with respect to 0 ! 1 (@ i ! i, i = 0, 1, 2). In the first case, the desired lifting is equivalent to a lifting in the diagram A _____//A x 1____//X55kkkk;;w | |kkkkkkww | | kkk|k w | fflffl|kkfflffl|kkkfflffl|w B _____//B x 1____//Y. 12 SHARON HOLLANDER ` This follows since (Ax 1) A B ! B x 1 is a trivial cofibration.`Similarly, in * *the second case, the desired lifting exists because the map (Ax i) Ax@ i (Bx@ i) ! B x i is a cofibration. To show that the model category structure is left proper we must show that the pushout of a weak equivalence along a cofibration is again a weak equivalen* *ce. We have already observed above that this is true when the weak equivalence is a cofibration so, by MC5, it suffices to show that the pushout P of a trivial fib* *ration A -j!C along a cofibration A -i!B is a weak equivalence. This follows from the following more general proposition. Proposition 3.5. Let A, B, C be small categories, and A -i!B be a functor which is a monomorphism on objects, and A -j!C a surjective equivalence`of categories. Then the induced functor to the pushout in Cat, B ! P := C AB is also a surjective equivalence of categories. Proof.First note that the universal map B - p!P is surjective on objects. If b, b02 obB, then p(b) = p(b0) if and only if b = b0or there exist a, a02 obA wi* *th i(a) = b, i(a0) = b0 and j(a) = j(a0). In the latter case there is a unique map a -! a02 A which maps to the identity of j(a) and we will call the image of this map in B the canonical map b -! b0. For b not in the image of A the canonical map b -! b is defined to be the identity. It is clear that p induces an isomorphism on components so it remains to show that p induces an isomorphism Hom B(b, b0) -! Hom P(p(b), p(b0)). For fi, fi0 objects of P , let W (fi, fi0) denote the set of words formed by * *formal compositions of morphisms in B and C such that the first map in the word has domain fi, the last map has range fi0and consecutive maps have domains and rang* *es whose images in P agree. Recall that Hom P(fi, fi0) is the quotient of W (fi, f* *i0) by the equivalence relation generated by the composition in B, composition in C and i(f) ~ j(f) for f a morphism in A. Let b, b0be objects of B and write fi = p(b), fi0= p(b0). We will define func* *tions OEb,b0: W (fi, fi0) -! Hom B (b, b0) which are constant on the equivalence clas* *ses of W (fi, fi0) and so determine functions Hom P(fi, fi0) -! Hom B(b, b0). It w* *ill be immediate from the construction that these are inverse to p and this will compl* *ete the proof. The functions OEb,b0are defined by induction on the length of words as follow* *s. Let w be a word of length 1. If w is a morphism c -f!c02 C then let a, a0be the uni* *que objects in A such that i(a) = b, i(a0) = b0, j(a) = c, j(a0) = c0and let a -g!a* *0denote the unique morphism in A such that j(g) = f. Define OEb,b0(w) = i(g). If w is a morphism b1 -f!b2 2 B define OEb,b0(w) to be the composite b -! b1 -f!b2 -! b0 where the unlabeled arrows are canonical morphisms. Now suppose OEb,b0has been defined on words of length n and let w = w0f where w0 is a word of length n and f is a morphism in B or in C. Let b00be an arbitrary object of B mapping to the range of w0 and define OEb,b0(w) as the OEb,b00(w0)OEb00,b0(f) composite b -! b00 - ! b0. It follows from the construction that the value of OEb,b0is independent of the choice of b00and that OEb,b0is constant on the e* *quivalence classes of W (fi, fi0). A HOMOTOPY THEORY FOR STACKS 13 Corollary 3.6. With this model category structure on Grpd, the adjoint pair ßoi* *d: sSet$ Grpd: N is a Quillen pair. Remark 3.7. The previous corollary implies that ßoidpreserves trivial cofibrati* *ons, and hence is equivalent to the usual fundamental groupoid functor. To end this section we give an alternative description of the homotopy theory of groupoids. Consider the model category on sSet which is the localization of * *the usual model structure with respect to the map @ 3 ! 3. We will call this the S2 nullification of sSet, following [DF ]. Notice that th* *e maps @ n ! n, n > 2. are all weak equivalences in this localized model structure, so we could equiva* *lently localize sSet with respect to this set of maps. Lemma 3.8. In the S2 nullification of sSet, weak equivalences are the maps which induce an isomorphism on ß0 and ß1 at all base points. Theorem 3.9. The adjoint pair _ßoid,,________________________________________* *____________________ sSetkk___Grpd,_____________________________________* *_____________________ N is a Quillen equivalence between Grpd and the S2 nullification of sSet. 14 SHARON HOLLANDER 4. Homotopy Limits and Colimits It is well known how to define homotopy limits and colimits in simplicial mod* *el categories. One can write down explicit formulas going back to [BK ]. In this s* *ec- tion, we will give simplified formulas for homotopy (co)limits in case the simp* *licial structure comes from a groupoid action (2.9). Our main concern will be the homo- topy limit of a cosimplicial diagram, and dually the homotopy colimit of a simp* *licial diagram. Our simplified formula for the former will allow us in section 7 to in* *terpret the descent conditions for stacks in a homotopy-theoretic manner. Let C be a simplicial model category. The homotopy limit of an I-diagram X in C with each X(i) fibrant is the equalizer of the two natural maps Y Y X(i)N(I=i)) X(i)N(I=j), i j-ff!i where I=i denotes the category of objects over i. Similarly, the homotopy colim* *it of an objectwise cofibrant I-diagram X is the coequalizer of the two maps a a X(i) N(j=I) ) X(i) N(i=I), i-ff!j i where j=I denotes the category of objects under j. An exposition of these const* *ruc- tions for simplicial sets appears in [BK , GJ ], and for a general simplicial m* *odel category in [Hi]. For Y a fibrant object and X 2 CI objectwise cofibrant, the* *se functors satisfy the equation (4.1) sSet(hocolimX, Y ) = holimsSet(X, Y ). Note 4.2. When the simplicial structure on C is derived from a groupoid structu* *re, the above formula is obtained by applying N to the equality Grpd(hocolimX, Y ) = holimGrpd(X, Y ). Theorem 4.3. Let C be a simplicial model category whose simplicial structure derives from a groupoid action, and let Xo be a cosimplicial object in C, with * *each Xi fibrant. Then a model for the homotopy inverse limit of Xo is given by the equalizer of the natural maps Y2 i iY2,j 1 j (Xi) ) (Xi) . i=0 [j]![i] Proof.First, notice that writing sk2 o for the 2-skeleton of o, the inclusion ßoidsk2 o ! ßoid o is an isomorphism. It follows that T otXo, the space of maps from o to Xo, is isomorphic to T ot2Xo, the space of maps from the restriction o| [2]to Xo| [2]where [2] denotes the full subcategory of with objects [0],* * [1] and [2]. Since the map ßoidsk1 2 -! ßoid 2 is surjective, T ot2Xo is given by t* *he equalizer in the statement of the theorem. It now suffices to show that the homotopy limit of Xo is naturally homotopy equivalent to T otXo. Using the definition of the homotopy limit in a simplici* *al model category given above, this is an easy consequence of the following propos* *ition. A HOMOTOPY THEORY FOR STACKS 15 Proposition 4.4. There is a homotopy equivalence of cosimplicial groupoids F : ßoidN( =[o]) $ ßoid o : G. Proof.Morphisms in ßoidN( =[n]) are generated by the commutative triangles [k]____________//_[m] @@ "" @@@ """ @@__ ~~"" [n] and their formal inverses. Let ßoidN( =[n]) Fn-!ßoid n be the functor which sen* *ds o the object [m] ! [n] to the vertex [0] em-![m] ! [n], where ek : [0] ! [k] denotes the map which sends 0 to k. o a generating morphism as above to the 1-simplex in n which is the unique map filling in the following diagram 1 d0 [0]_d__//[1]Øoo_[0] ek|| !Ø |em| fflffl| Ø fflffl| [k]____________//Ø[m] AA Ø __ AAA Ø ___ A__Afflffl""__ [n]. One can check easily that F is well defined and natural in n, and so defines a morphism ßoidN( =[o]) -F!ßoid o 2 cGrpd. Let Gn be the functor which is defined o on objects by including [0] ! [n] in =[n]. o on generating morphisms [1] ! [n] as the composition 1 (d0)-1 [0]_d__//[1]___//[0] AA | "" AAA | """ A__Afflffl|~~"" [n]. Again it is easy to check that Gn is well defined and natural in [n], and so de* *fines a morphism ßoid o -G!ßoidN( =[o]) in cGrpd. The composition F OG is the identity. There is a natural transformation GOF -* *ff! id defined on objects [m] ! [n] 2 ßoidN( =[n]) by the triangle [0]_____em______//[m] AA __ AAA ___ AA__ ""__ [n]. 16 SHARON HOLLANDER The groupoid Tot2(Xo) will also be called the descent category of Xo. From now on, when we refer to the homotopy limit of a cosimplicial groupoid Xo we will m* *ean the simpler model Tot2(Xo). The following corollary gives an explicit descripti* *on of this groupoid. Corollary 4.5. The homotopy inverse limit of a cosimplicial groupoid Xo is the groupoid whose o objects are pairs (a, d1(a) -ff!d0(a)), with a 2 obX0, ff 2 morX1, such * *that s0(ff) = idx, and d0(ff) O d2(ff) = d1(ff), o morphisms (a, ff) ! (a0, ff0) are maps a -fi!a0, such that the following diagram commutes d1(fi) d1(a)____//_d1(a0) |ff| |ff0| fflffl|d0(|fflfflfi) d0(a)____//_d0(a0) Dually we have the following theorem giving a formula for homotopy colimits of simplicial diagrams. Theorem 4.6. Let C be a simplicial model category whose simplicial structure derives from a groupoid action and let Xo 2 sC, be such that each Xi is cofibra* *nt. Then the homotopy colimit of Xo is naturally homotopy equivalent to the coequal* *izer of the maps n 1,ma2 a2 Xm n ) Xn n. [n]-![m] n=0 A HOMOTOPY THEORY FOR STACKS 17 5.Categories Fibered in Groupoids There are different categories in which the descent condition can be formulat* *ed, and in which stacks can be defined. In this section we will discuss the categor* *y of categories fibered in groupoids over C, [DM , Gi]. This category is denoted Grp* *d=C. After discussing some important properties of Grpd=C, we will be able to defi* *ne an adjoint pair of functors ____..__________________________________________* *_______________________________________ Grpd=C mm___P_(C,_Grpd)________________________________* *________________________________________________ p satisfying the following properties: o For F in P (C, Grpd), the map F (X) ! pF (X) is an equivalence of groupoids, for all X 2 C. o For E 2 Grpd=C, the map p E ! E is an equivalence of categories over C. When C has a Grothendieck topology, we will define stacks in both categories so that the pair (p, ) restricts to an adjoint pair between the subcategories of * *stacks. In section 7, we will define model structures on these categories such that the adjunction above induces a Quillen equivalence. 5.1. Categories Fibered in Groupoids over C. One should think of a category fibered in groupoids over C as the analogue in Catof a fibration over C with fi* *bers which are groupoids. Recall that if X -f! Y is a fibration of topological space* *s, given a path I in Y , and x 2 X such that f(x) = I(1), we can lift I to a path * *I0 in X, with I0(1) = x. One can use these liftings to define a map from the fiber over I(1) to the fiber over I(0). This map is only determined up to homotopy but a homotopy between two liftings is again determined up to homotopy and so on. Similarly, a category fibered in groupoids over C, E -F! C, satisfies a * *path lifting condition, where the lift is unique only up to isomorphism. However, si* *nce in groupoids there are no nontrivial homotopies between homotopies, this isomorphi* *sm is unique. More precisely, a morphism X ! Y 2 C, determines a pullback functor from the fiber over Y to the fiber over X, which is unique up to a unique natur* *al isomorphism. Here is a standard example to motivate the definition. Example 5.1 (Vector Bundles on Top). Let V ec(Top) be the category whose ob- jects are vector bundles EY i Y , and whose morphisms are pullback squares EY _____//EX | | | | fflffl| fflffl| Y _____//_X. The projection functor V ec(Top) ! Top is an example of a category fibered in groupoids over Top. Here are some ways in which it resembles a fibration: 18 SHARON HOLLANDER o The fact that we can pull back vector bundles tells us that there is `pa* *th lifting' (EiZ) *__________//V7ec(Top)7 nn |1| n n || fflffl|Ynf-!Znn fflffl| I____________//_Top. A lifting in this diagram is a choice of a bundle E0 i Y and an isomor- phism E0 -~! f*E. Two different choices will necessarily be canonically isomorphic. o All the fibers of this functor are groupoids. Now we give the definition of a category fibered in groupoids, which formaliz* *es the `path lifting' condition described above. Definition 5.2. [DM ] The category Grpd=C is the full subcategory of Cat=C whose objects are functors E -F!C satisfying the following properties: (1) Given Y -f! X 2 C, and X0 2 E such that F (X0) = X, there exists 0 Y 0-f!X02 E such that F (f0) = f. (2) Given a diagram in E, over the commutative diagram in C, Y 0 __F__+3 Y ~~ | |f0| h~~~ |f g0 fflffl| F ~~~~g fflffl| Z0____//_X0 _____+3 Z ____//_X, with F (f0) = f, F (g0) = g, there exists a unique h0 such that g0O h0 =* * f0 and F (h0) = h. This definition may seem involved but it becomes very simple when we look at the functors FX0 induced by F on the over categories E=X0-FX0!C=X, where X0 2 E, and F (X0) = X. The conditions for E -F! C to be a category fibered in groupoids over C are equivalent to the following simple requirements* * of the functors FX0: (1) FX0 is surjective. (2) For every pair of objects Y 0, Z0 2 E=X0 with FX0(Y 0) = Y, FX0(Z0) = Z the induced map Hom E=X0(Y 0, Z0) ! Hom C=X(Y, Z) is a bijection. Together these conditions are equivalent to saying that the functors FX0 are surjective equivalences of categories. Let EX denote the fiber category over X in E. This has objects those of E lying over X and morphisms those of E lying over idX . It is easy to see that * *if E ! C 2 Grpd=C, the fiber categories EX are groupoids. Example 5.3. The simplest examples of categories fibered in groupoids over C are the projection functors C=X ! C for each X 2 C. If Y -f! X is an object of C=X, A HOMOTOPY THEORY FOR STACKS 19 then (C=X)=f ~= C=Y , and so conditions 1. and 2. above are trivially satisfied. Notice that (C=X)Y is the discrete groupoid whose set of objects is Hom C(Y, X). Another class of simple examples are C x G -pr!C, for G 2 Grpd. Here the fibe* *rs over each X 2 C are canonically isomorphic to G. Categories fibered in groupoids are enriched over Grpd in a natural way. Lemma 5.4. Grpd=C is enriched with tensor and cotensor over Grpd. The objects of Grpd(E, E0) are the functors E ! E0 over C, and the morphisms are the natural isomorphisms between such functors covering the identity natural automorphism of idC. Moreover, the tensor is given by the formula E G := E xC (C x G), and the cotensor EG is the category of functors from (G ! *) to (E ! C). Proposition 5.5. Let E -F!C 2 Grpd=C, X02 E, and let X = F (X0). Then (1) there is a section _E== G__ F|| _ fflffl| C=X _____//C such that G(idX ) = X0. (2) If G, G0 : C=X ! E are two such sections and G(idX ) -f! G0(idX ) a morphism in EX , then there is a unique natural isomorphism G -OE!G0 over idC, with OE(idX ) = f. Proof.First notice that giving a section C=X -G! E over C with G(idX ) = X0 is the same as giving a section E=X0I_____//EI Ø|~ | )| |F fflffl| fflffl| C=X _____//C. sending idX to idX0. 1) Define G on objects Y 2 C=X, to be an arbitrary choice of Y 02 E=X0 with FX0(Y 0) = Y , (this is possible since E=X0 ! C=X is a surjection). For a pair * *of objects Y, Z 2 C=X, define Hom C=X(Y, Z) -G!Hom E=X0(Y 0, Z0) to be the inverse of the bijection Hom E=X0(Y 0, Z0) FX0-!HomC=X(Y, Z). To show this construction gives a functor, consider a pair of composable morphi* *sms f, g 2 C=X. The morphisms G(f) O G(g) and G(f O g) have the same domain and range and the same image, f O g, in C=X, therefore they must be equal. 20 SHARON HOLLANDER 2) Suppose G0 is another such functor. Then for each object (Y ! X) 2 C=X there is a unique isomorphism G(Y ! X) ______//G(idX ) 9!|| |f| fflffl| fflffl| G0(Y ! X) ____//_G0(idX ). lying over the identity of Y . By uniqueness, this collection of isomorphism fo* *rms a natural isomorphism G -OE!G0, and OE is the unique natural isomorphism G ! G0 over idC which evaluated at idX is f. Corollary 5.6. For each X 2 C, the natural map Grpd(C=X, E) ! EX given by evaluation at idX is a surjective equivalence of groupoids. There is a* * left inverse which is unique up to unique natural isomorphism. This corollary says that given E ! C there is a functorial "rigidificationö * *f the fibers. Later we will use this method of rigidification to construct a functor * *from Grpd=C to P (C, Grpd). In a similar fashion we can prove: Proposition 5.7. Let E ! C be a category fibered in*groupoids, and Y -f! X morphism in C. There are üp llback" functors EX -f! EY which are unique up to a unique natural isomorphism covering idY . Proof.To construct the functor on objects X0 2 EX , we arbitrarily lift Y ! X using condition 1 of Definition 5.2. Once the functor has been defined on objec* *ts, condition 2 of Definition 5.2 yields a map Y 0! Y 00for each morphism X0! X002 EX . Finally, the uniqueness in condition 2 implies that this assignment is a f* *unctor and that any two assignments are naturally isomorphic over idY . Now we can give a definition of stack in Grpd=C. Definition 5.8. Let C be a category with a Grothendieck topology. A category fibered in groupoids E -F!C is a stack if for all covers {Ui! X} the map Grpd(C=X, E) ! holimGrpd(C=Uo, E) is an equivalence of groupoids. We will compare this definition with the usual definition [DM ] in the next s* *ection. 5.2. Adjoint Pair Between Grpd=C and P (C, Grpd). Let E ! C be a category fibered in groupoids. By Corollary 5.6, the assignment to each X 2 C of the sec* *tions Grpd(C=X, E) is a functor such that Grpd(C=X, E) -~!EX . Definition 5.9. Let : Grpd=C ! P (C, Grpd) be the functor which sends E ! C to the presheaf E(X) := GrpdGrpd=C(C=X, E). Let p : P (C, Grpd) ! Grpd=C be the functor defined by setting pF to be the category whose o objects are pairs (X, a) with a 2 F (X), o morphisms (X, a) ! (Y, b) are pairs (f, ff) where X -f! Y 2 C and a -ff! F (f)b is an isomorphism in F(X). A HOMOTOPY THEORY FOR STACKS 21 The composition of two morphisms (X, a) (f,ff)-!(Y, b) (g,fi)-!(Z, c) is the pa* *ir (g O f, F (f)(fi) O ff). It is easy to check that both p and preserve the groupoid action on their d* *omain categories. Under p presheaves of groupoids sit inside Grpd=C as the "trivializ* *able bundles" (see example 5.1). Theorem 5.10. The functors ____p_____--____________________________________* *______________________________________________________________@ P (C, Grpd)mm_________Grpd=C,_____________________________* *______________________________________________________________@ form an adjoint pair with p the left adjoint. The unit of the adjunction is an objectwise equivalence, and the counit is a fiberwise equivalence of groupoids. Proof.We will define natural transformations j : id ! p, and ffl : p ! id. It* * will be clear from their definition that they satisfy the equations required to form* * the the unit and counit of an adjunction. Define ffl : p E ! E on objects by sending (X, OE : C=X ! E) to OE(idX ) 2 E, and on morphisms by sending (f : X ! X0, , : OE ! f*(OE0)) to the composite OE0(f) O ,(idX ). It follows from Corollary 5.6, that ffl is a fiberwise equiva* *lence. Define j : F ! pF to be the map of presheaves which sends an object a 2 F (X) to the section OEa : C=X ! pF defined by OEa(Y - f!X) = (Y, F (f)a); OEa(Y - g! Z) = (g, id), and a morphism a -ff!b 2 F (X) to the natural transformation , : OEa ! OEb defined by ,(Y -f! X) = F (f)(ff). By construction F (X) is the f* *iber over X in pF . Another application of Corollary 5.6 shows that Grpd(C=X, pF ) -* *~! pFX = F (X), and so j is an objectwise equivalence. The existence of this adjoint pair now motivates the following definition of * *stack in P (C, Grpd). Definition 5.11. A presheaf F of groupoids on C is a stack if for all covers {U* *i! X} the map F (X) ! holimF (Uo) is an equivalence of groupoids. With this definition, a category fibered in groupoids E -F! C is a stack if a* *nd only if E is a stack in P (C, Grpd), so our adjoint pair restricts to one betw* *een the stacks in Grpd=C and the stacks in P (C, Grpd). 22 SHARON HOLLANDER 6.Stacks In this section we will discuss the usual definition of stacks in Grpd=C [DM * *] used in algebraic geometry, and show that it is equivalent to the definition we have* * given using homotopy limits (Definition 5.8). We start with an example that will hopefully provide intuition for the de- scent/homotopy sheaf condition. Example 6.1 (Principal G-bundles on X). Consider the functor ß0BG which assigns to a space the set of isomorphism classes of principal G bundles over i* *t. Locally all bundles are trivial, so gluing together isomorphism classes via the* * sheaf condition yields only the isomorphism class of the trivial bundles. The sheafif* *ication of ß0BG is just the constant assignment of the isomorphism class of the trivial bundle. In particular, ß0BG is not generally a sheaf. Yet there is a sense in which isomorphism classes of principal G-bundles are determined locally. A cover, principal G-bundles on each member of the cover, a* *nd coherent isomorphisms between their restrictions to the intersections determine* * a G-bundle on the total space. More precisely, given an open cover {Ui X} and o G-bundles Ei! Ui, o isomorphisms we call gluing data ffij: Ei|Ui\Uj! Ej|Ui\Uj, o satisfying ffjkO ffij= ffikwhen restricted to Ui\ Uj\ Uk, there is a principal G-bundle E ! X, and isomorphisms fi: E|Ui ! Ei, compatible with the gluing data: Ei|Ui\UjLofio_E|Ui\Uj LLLL f| ffijLL&&LLLjfflffl|| Ej|Ui\Uj. Let BG(X) denote the groupoid of principal G-bundles on X and isomorphisms between them, and Uo the nerve of the cover {Ui X}. We can translate the above property as saying: Q ff Given an object a 2 BG(Ui), and an isomorphism d1a -! d0a, which is coherent in the sense that d0(ff) O d2(ff) = d1(ff), then up to isomorphism a is in the * *image of BG(X). This is essentially what it means for BG(X) to be the homotopy inverse limit * *of the cosimplicial diagram of groupoids BG(Uo). Let E ! C be a category fibered in groupoids,*and assume that for each X -f! Y we have chosen pullback functors EY -f! EX . Given a morphism Ui! U 2 C, we will sometimes abuse notation and denote the pullback of an element a 2 EU to EUi by a|Ui. In defining some of the maps below, we will also make implicit use* * of the natural isomorphisms (a|Ui)|Uij~=a|Uij. Definition 6.2. [Gi] [DM ] A stack in Grpd=C is an object E ! C which satisfies the following properties for any cover {Ui! X} : (1) given a, b 2 EX , the following is equalizer sequence Y Y Hom EX(a, b) ! Hom EUi(a|Ui, b|Ui) ) Hom EUij(a|Uij, b|Uij), A HOMOTOPY THEORY FOR STACKS 23 (2) given ai2 EUi and isomorphisms ai|Uij-ffij!aj|Uij, satisfying the cocycle condition ffjk|UijkO ffij|Uijk= ffik|Uijk, then there exist a 2 EX , and isomorphisms a|Ui fii-!ai, such that the foll* *owing square commutes fii|Uij (6.3) a|Uij____//_ai|Uij = || |ffij| fflffl|fijfflffl||Uij a|Uij____//_aj|Uij. In this case, we say that E ! C satisfies descent. Note 6.4. Note that pulling back the square 6.3 along the diagonal map : Ui! Uiishows that the family of isomorphisms ffijmust satisfy the added condition *(ffii) = idUi and so we might as well have added this requirement to the cocy* *cle condition. This definition seems very complicated, but it can be considerably simplified* * if we recall the description of the homotopy inverse limit of a cosimplicial group* *oid given in Corollary 4.5. Proposition 6.5. A category fibered in groupoids E ! C is a stack in the sense * *of Definition 6.2, if and only if for all covers {Ui! X} (6.6) Grpd(C=X, E) ! holimGrpd(C=Uo, E) is an equivalence, i.e. if E ! C is a stack in the sense of Definition 5.8. Proof.We begin by showing that condition 1. in Definition 6.2 is equivalent to * *the requirement that for objects Fa, Fb 2 Grpd(C=X, E), the set of morphisms Fa ! Fb is in bijective correspondence with the set of morphisms between their images in holimGrpd(C=Uo, E). Consider objects Fa, Fb 2 Grpd(C=X, E), and let a = Fa(idX ) and b = Fb(idX ) in EX . Evaluation at id(-)induces bijections Q Q Hom (Fa, Fb)____//Hom (Fa|Ui, Fb|Ui)___+3_Hom (Fa|Uij, Fb|Uij) |~=| ~=|| |~=| fflffl| Q fflffl| Q fflffl| Hom EX(a, b)___//_Hom EUi(a|Ui, b|Ui)__+3Hom EUij(a|Uij, b|Uij). It follows that the top line is an equalizer if and only if the bottom one is. * * By corollary 4.5, the top line is an equalizer if and only if Hom (Fa, Fb) is in b* *ijective correspondence with the set of maps from the image of Fa to the image of Fb in holimGrpd(C=Uo, E). The requirement that the bottom line be an equalizer is condition 1. in Definition 6.2. To finish the proof we have to show that condition 2. is equivalent to the re* *quire- ment that every object in holimGrpd(C=Uo, E) be isomorphic to one in the image * *of Grpd(C=X, E). This follows from the description of morphisms in Corollary 4.5 o* *nce 24 SHARON HOLLANDER we show that specifying an object in holimGrpd(C=Uo, E) is equivalent to specif* *ying descent datum as in condition 2. of Definition 6.2. Q By corollary 4.5, an object of holimGrpd(C=Uo, E), consists of an object Fc 2 Grpd(C=Ui, E), and an isomorphism d1Fc -ff!d0Fc, satisfying d0(ff) O d2(ff) = d1(ff) and s0(ff) = idFc. For any U -f! V , and Fa 2 Grpd(C=V, E) with Fa(idV ) = a, the evaluation Fa|U (idU ) is a choice of pullback of a along f, * *and so Fa|U (idU ) is canonically isomorphic toQthe pullback f*a, which we chose in* * ad- vance. Evaluating at idUi determines c 2 EUi, and isomorphisms ffij= ff(idUij) satisfying the cocycle condition. Composing with the canonical isomorphisms c|Uij~= Fc|Uij(idUij), we obtain isomorphisms c|Ui ~ffij-!c|Uj, satisfying the * *cocy- cle condition. Q Conversely, given c 2 EUi and ffij, as in condition 2.Qsatisfying *(ffii) = idUi (see Note 6.4), we can lift them to an object Fc 2 Grpd(C=Ui, E), and an isomorphism d1Fc -ff!d0Fc. Since these lifts are essentially unique they must a* *lso satisfy the cocycle condition and s0(ff) = idFc and hence determine an object of holimGrpd(C=Uo, E). A HOMOTOPY THEORY FOR STACKS 25 7. Model Structures In this section we put model structures on P (C, Grpd), Sh(C, Grpd), and Grpd* *=C. In the first two subsections, we describe model structures on (pre)sheaves and categories fibered in groupoids. A morphism in (sh)P (C, Grpd) will be a fibrat* *ion or weak equivalence if it is one when evaluated at each object. In Grpd=C, the weak equivalences are the maps which induce an equivalence of groupoids on the fibers or, equivalently, maps which become weak equivalences in P (C, Grpd) aft* *er applying . The above model category structure on P (C, Grpd) is not very interesting bec* *ause it does not see the topology on C. In a Grothendieck topology there is a notion of locality. Just as sheaves are isomorphic if they are locally isomorphic, so* * too stacks should be equivalent if they are locally equivalent. Thus, there should* * be a model structure for which weak equivalences are those maps which locally are weak equivalences of groupoids. The most basic local equivalences are the maps hocolimUo ! X, as stacks can be defined to be those presheaves which see this as an equivalence. This suggests that we should declare these to be new weak equivalences. In the third subsection, we use Theorem 2.14 to localize the model structures* * on P (C, Grpd), Sh(C, Grpd), and Grpd=C, with respect to the set of maps hocolimUo ! X, where {Ui! X} a cover in C, We then observe that in these local model structures, the fibrant objects are t* *he stacks. In the next section we will prove that all these local model category structu* *res on P (C, Grpd), Sh(C, Grpd), and Grpd=C are Quillen equivalent. We will also pr* *ove that the weak equivalences in the local model structure on P (C, Grpd) are the * *maps which, locally, are weak equivalences. 7.1. Model Category Structure on (Pre)Sheaves of Groupoids. In this sub- section we construct a model category on both sheaves and presheaves of groupoi* *ds on a Grothendieck topology C, using a set of "generators". More precisely, we w* *ill give a collection of objects X and define a map f to be a weak equivalence or a fibration if and only if the map of groupoids Grpd(X, f) is one for all X. This definition of weak equivalences and fibrations together with the smallness of t* *he generators X implies that the sets of maps {X G ! X H}, where X is a generator and G ! H is a generating (trivial) cofibration of groupoids, form se* *ts of generating (trivial) cofibrations. In our case the "generators" X will be t* *he representable functors. Henceforth we will abuse notation and denote by X the sheaf Hom C(-, X) of discrete groupoids represented by the object X 2 C. Theorem 7.1. There are left proper, cofibrantly generated, model category struc- tures on P (C, Grpd), and Sh(C, Grpd), where o f is a weak equivalence or a fibration if Grpd(X, f) is one for all X 2 C, o cofibrations are the maps with the left lifting property with respect to tr* *ivial fibrations. The maps of the form X ! X 1, for X 2 C, form a set of generating trivial cofibrations. The maps of the form X @ i ! X i for X 2 C and i = 0, 1, 2 form a set of generating cofibrations. 26 SHARON HOLLANDER Corollary 7.2. The adjoint pair _i__--________________________________________* *________________________________ Sh(C, Grpd)nn___P_(C,_Grpd)______________________________* *________________________________________ sh is a Quillen pair. Proof.Presheaves: For MC1, note that limits and colimits are defined objectwise in P (C, Grpd). MC2-MC4a are obvious. For X 2 C, the functor GrpdP(C,Grpd)(X, -) is evaluation at X, which commutes with all limits and colimits in P (C, Grpd).* * It follows that X is small in P (C, Grpd), hence the domains of the generating (tr* *iv- ial) cofibrations are small. This implies MC5a. Now note that cofibrations ar* *e, in particular, objectwise cofibrations. Since colimits are computed objectwise* *, it follows that pushouts and directed colimits of trivial cofibrations are again t* *rivial cofibrations, which proves MC5b. Similarly, left properness follows from the l* *eft properness of Grpdand the fact that cofibrations are objectwise cofibrations. M* *C4b now follows by the same argument used in the proof of Theorem 3.1. SM7 follows immediately from SM7 for Grpd. Sheaves: MC1-MC4a, are obvious. The inclusion of sheaves in presheaves pre- serves filtered colimits so the domains of the generating (trivial) cofibration* *s are also small in sheaves, and MC5a follows. For MC5b, it suffices to show that the pushout in presheaves, of a sheaf along a generating trivial cofibration is sti* *ll a sheaf. Consider the diagram X _____________//F | | | | fflffl| fflffl|` X 1 ____//_(X 1) X F, ` where F is a sheaf and X 2 C. The presheaf of groupoids`X 1 X F has: o object presheaf, the presheaf of objects in F X and` ` o morphism`presheaf, the presheaf of objects in F 1 F 1xF X X xF F 1 X xF F 1xF X. ` The presheaves`of objects and morphisms of (X 1) X F are sheaves, so (X 1) X F is a sheaf. MC4b follows by the same argument given in the proof of Theorem 3.1. SM7 follows immediately from SM7 for Grpd. Since P (C, Grpd) is left proper, to show left properness for sheaves it suff* *ices to show that the pushout in P (C, Grpd) of a weak equivalence along a cofibration * *of sheaves is again a sheaf. Since we have already proven that the pushout of a sh* *eaf along a trivial cofibration is a trivial cofibration whose range is a sheaf, we* * can assume that our weak equivalence is a trivial fibration. We begin by noting that cofibrations of sheaves are, in particular, objectwise cofibrations, as sheafification preserves monomorphisms (and N and ßoidpreserve cofibrations). Trivial fibrations in Grpd are the surjective equivalences of categories, and* * so pushouts of trivial fibrations along objectwise cofibrations in P (C, Grpd) are* * again A HOMOTOPY THEORY FOR STACKS 27 trivial fibrations in P (C, Grpd). Consider the diagram in P (C, Grpd) A "___~__////_`F | | | | fflffl|~ fflffl|` B ___////_B A F. ` Let P denote the pushout B A F . The argument given above to show that cofi- brations are objectwise cofibrations shows also that the pushout in presheaves * *of a sheaf along a cofibration of sheaves is a sheaf on objects. Hence P is a shea* *f on objects. To see that the morphisms of P are a sheaf, recall that for each X 2 C, the m* *ap B(X) i P (X).is a surjective equivalence of categories. Given a presheaf G, let I(G) be the presheaf with I(G)(X) the category with objects, the objects of G(X) and a unique morphism between each pair of objects G(X). There is a canonical map G ! I(G) and if G is a sheaf on objects, then I(G) is a sheaf. Since B ! P is a trivial fibration, it is easy to check that B* * ~= I(B) xI(P)P . Using the following facts: o the set of morphisms of a fiber product is the fiber product of the morphis* *ms, o the map I(B) ! I(P ) is a surjection on objects and morphisms, it is not hard to check that P satisfies the sheaf condition. 7.2. Categories Fibered in Groupoids over C. In this subsection we construct a model category on Grpd=C relative using the set of "generators" C=X ! C. Theorem 7.3. There is a left proper, cofibrantly generated, simplicial model ca* *te- gory structure on Grpd=C in which o f is a weak equivalence or a fibration if GrpdGrpd=C(C=X, f) is one for all X 2 C, o cofibrations are the maps with the left lifting property with respect to tr* *ivial fibrations. The maps of the form C=X ! (C=X 1), for X 2 C, form a set of generating trivial cofibrations. The maps of the form (C=X @ i) ! (C=X i), for X 2 C and i = 0, 1, 2 form a set of generating cofibrations. Proof.For MC1, see Appendix A. MC2-MC4a are obvious. In order to apply the small object argument to prove MC5, we need to check that the objects C=X G ! C with G = (@) i, i = 0, 1, 2, are small with respect to the colimits which app* *ear in the small object argument. First notice that sequential colimits in Grpd=C a* *gree with sequential colimits in Cat =C. For convenience, in the construction of the factorization for MC5a we will take pushouts along both the generating cofibrat* *ions and the generating trivial cofibrations. Let Ei ! Ei+1 be constructed as usual, using the small object argument, and let consider a map F : C=X -! colimEi. F (idX ) lifts to some element X0 in some Ei, and we can extend this to a map Fi0: C=X -! Ei. Let F 0be the composition C=X ! Ei ! colimEi. Then F 0(idX ) = F (idX ), and so there is a unique natural 28 SHARON HOLLANDER isomorphism OE : F -! F 0making the following diagram commute F0i C=X __________//MEi | MMMMF0M | | MMM | fflffl|OEMM&&fflffl| C=X ____//____________33___________________________________* *______________________C=X//``1`colimEi. ____________________________________________________* *______________________________________________________________@ F The map C=X ! C=X 1 is one of the generating trivial cofibrations, so by construction we obtain a lift F0i C=X __________//Ei | | | | | fflffl| | 4Ei+14 | h h h88q | h h q | h h |h qq | h hh fflffl|OEq fflffl| C=X ____//____________33___________________________________* *______________________C=X//_c1olimEi. ____________________________________________________* *______________________________________________________________@ F Thus C=X is small with respect colimEi. Since natural transformations between sections are determined uniquely by their evaluation on idX , a similar argument shows that C=X (@) iis small with respect to colimEi. This completes the proof of MC5a. For MC5b, note that if E ! E0 has the left lifting property with respect to a* *ll fibrations,1then in1particular it has the left lifting property with respect to* * E ! C and (E0) ! (E0)@ , and therefore it is an equivalence of categories over C. * *An equivalence of categories over C is clearly a weak equivalence. It follows that* * the cofibration constructed using the small object argument for MC5b is also a weak equivalence. MC4b now follows by the same argument given in the proof of Theorem 3.1. SM7 follows immediately from the definition of (trivial) fibration in Grpd=C and the adjunction formulas given by the simplicial structure. To show left properness, it suffices to show that the pushout of a trivial fi* *bration along a cofibration is a weak equivalence. We begin by noting that trivial fibr* *ations are surjective equivalences of categories. Let F : E0 -! E00be a trivial fibrat* *ion and let X0, Y 02 E, X00= F (X0), Y 00= F (Y 0). Clearly F is surjective on obje* *cts and morphisms. We will show that the map Hom E0(X0, Y 0) ! Hom E00(X00, Y 00) is a bijection. If F (f0) = F (g0) then f0 and g0 have the same image in C and * *so there is a unique isomorphism h0filling in the following triangle in E0: X0ØC Ø CCf0CC Øh0 fflfflC!!Cg0 X0 _____//Y 0. By the uniqueness of the lifting h0, F (h0) = idX002 E00. Since F is a trivial * *fibration it follows that h0= idX0. A HOMOTOPY THEORY FOR STACKS 29 Now note that cofibrations in Grpd=C are inclusions on objects as this is the case for the generating cofibrations. Proposition 3.5 implies that the pushout* * in Cat=C of a surjective equivalence of categories along an inclusion on objects is still an equivalence of categories over C. This simultaneously implies that the pushout in Cat=C coincides in this case with the pushout in Grpd=C (see the pro* *of of Proposition A.1) and completes the proof. Corollary 7.4. The adjoint pair p : P (C, Grpd) $ Grpd=C : is a Quillen equiv* *a- lence. Proof.This follows immediately from the definition of the model structures and Theorem 5.10. 7.3. Local Model Category Structures. Recall that given X 2 C we also denote by X the (pre)sheaf represented by X. For convenience, we will sometimes also denote by X the category fibered in groupoids C=X ! C. In any of the categories P (C, Grpd), Sh(C, Grpd) or Grpd=C, we denote by S the set of maps S = {hocolimUo ! X : {Ui! X} is a cover inC} where Uo denotes (as usual) the nerve of the covering {Ui! X}. Proposition 7.5. Let M be one of the categories P (C, Grpd), Sh(C, Grpd) or Grpd=C. There is a model category structure on M which is the localization of the model structure of Theorems 7.1 or 7.3 with respect to the set of maps S. Proof.Since homotopy colimits of cofibrant objects are cofibrant, the domains a* *nd ranges of the morphisms in the localizing set are cofibrant. By Theorems 7.1 and 7.3, the model category structures on P (C, Grpd), Sh(C, Grpd) and Grpd=C satis* *fy the hypothesis of Theorem 2.14, so the proposition follows. Let M be one of the categories P (C, Grpd), Sh(C, Grpd) or Grpd=C. We will wr* *ite ML for the category M with the model structure given by the previous propositio* *n. Since in the old model structure on M every object is fibrant, and X 2 C is cofibrant, an object F 2 ML is fibrant if and only if Grpd(X, F ) ! Grpd(hocolimUo, F ) = holimGrpd(Uo, F ) is a weak equivalence for all covers. By definition of stack, this happens if * *and only if F is a stack. It follows that a fibrant replacement functor for ML is a stackification functor. Remark 7.6. Since stacks are the fibrant objects, and representables are cofibr* *ant, it follows that when M is a stack, h Hom (X, M) is equivalent to the groupoid M(X). In particular, [X, M] is the set of isomorphism classes of M(X). Remark 7.7. It is not hard to check that a small presentation (in the sense of * *[Dg , Definition 6.1]) of P (C, Grpd)L is given by the Yoneda embedding of C in P (C,* * Grpd) and the set of maps X @ n ! X n, for allX 2 C, n > 2 hocolimUo ! X for all covers{Ui! X} inC. This means that the local model category structure is the üq otientö f the uni* *versal model category generated by C by the relations given by the maps above. 30 SHARON HOLLANDER 8.Characterization of Local Equivalences In this section we prove that a morphism f is a local weak equivalence if and only if it satisfies one of the following equivalent properties: o f is an isomorphism on sheaves of homotopy groups, o f satisfies the local lifting conditions, o f is a stalkwise weak equivalence (when C has enough points). Furthermore we prove that our local model structure P (C, Grpd)L is Quillen equ* *iv- alent to the S2 nullification of Jardine's model structure on presheaves of sim* *plicial sets [Ja]. In subsection 8.1 we describe Jardine's model structure on presheaves of sim- plicial sets and show that it is the localization of the Heller model structure* * with respect to a set of maps Sß*. There is an analogue of the Heller model structur* *e for presheaves of groupoids which we denote by P (C, Grpd)H . We prove that its loc* *al- ization with respect to ßoidSß* has weak equivalences the isomorphisms on sheav* *es of homotopy groups, and is Quillen equivalent to the S2 nullification of Jardin* *e's model structure. The main theorem in this subsection is that the identity adjoi* *nt pair induces a Quillen equivalence (8.1) P (C, Grpd)L $ (ßoidSß*)-1P (C, Grpd)H . It follows that P (C, Grpd)L is Quillen equivalent to the S2 nullification of J* *ardine's model structure. We prove that (8.1) is a Quillen pair, and leave the proof tha* *t the weak equivalences are the same till 8.2. In subsection 8.2 we introduce Dan Dugger's local lifting conditions, and pro* *ve that they are satisfied by a map OE 2 P (C, Grpd) if and only if OE induces an * *iso- morphism on sheaves of homotopy groups, and if and only if OE is a local weak equivalence. This completes the proof that (8.1) is a Quillen equivalence. In subsection 8.3 we apply the characterization of local weak equivalences to show that the adjoint pairs sh : P (C, Grpd) $ Sh(C, Grpd) : i and p : P (C, Grpd) $ Grpd=C : are Quillen equivalences between the local model structures on each of these ca* *te- gories. 8.1. Jardine's Model Structure. In this subsection we compare the local model structure on presheaves of groupoids to Jardine's model structure on simplicial presheaves [Ja]. In order to define this model structure we will need the noti* *on of sheaves of homotopy groups. Note that for a simplicial set X, and basepoint a 2 X0, ßn(X, a) denotes the n-th homotopy group of the fibrant replacement of X with basepoint the image of a. Definition 8.2. [Ja] Let F be a presheaf of simplicial sets or groupoids. Then o ß0F is the presheaf of sets defined by (ß0F )(X) := ß0(F (X)). o For F 2 P (C, sSet) and a 2 F (X)0, ßn(F, a) is the presheaf of groups on C* *=X defined by ßn(F, a)(Y -f! X) = ßn(F (Y ), f*a). For F 2 P (C, Grpd) and a 2 obF (X), ßn(F, a) := ßn(NF, a). A HOMOTOPY THEORY FOR STACKS 31 We say that a map F -OE!G of presheaves of simplicial sets or groupoids is an isomorphism on sheaves of homotopy groups if the induced maps shß0(OE) and shßn(OE, a) are isomorphisms for all a 2 F (X), and all X 2 C. Note that if F is a presheaf of groupoids then ßi(F, a) = 0 for i > 1, and ß1* *(F, a) is the presheaf of groups AutF(a) on C=X, where AutF(a)(Y -f! X) := AutF(Y )(f*a). Note also that if F ! G is an objectwise weak equivalence, then the induced map of presheaves of homotopy groups is an isomorphism. Reference 8.3 (Jardine's Model Structure [Ja]). There is a left proper, cofibra* *ntly generated, simplicial model structure on P (C, sSet) where o cofibrations are the maps which are objectwise cofibrations, o weak equivalences are the maps which are isomorphisms on sheaves of homo- topy groups, o fibrations are the maps with the right lifting property with respect to the trivial cofibrations. The Jardine model category will be denoted by P (C, sSet)J. Proposition 8.4. (a)There is a model structure on P (C, Grpd), denoted (ßoidSß*)-1P (C, Grpd)H , in which the cofibrations are objectwise and the * *weak equivalences are the isomorphisms on sheaves of homotopy groups. (b) The adjoint pair (ßoid, N) induces a Quillen equivalence between (ßoidSß*)-1P (C, Grpd)H and the S2 nullification of P (C, sSet)J. To prove the proposition we will make use of the following model structure: Reference 8.5 (Heller Model Structure [He , Sm ]). There are left proper, cofi- brantly generated, simplicial model structures on P (C, sSet) and P (C, Grpd) w* *here o cofibrations are the maps which are objectwise cofibrations, o weak equivalences are the objectwise weak equivalences, and o fibrations are the maps with the right lifting property with respect to the trivial cofibrations. Proof.A proof for presheaves of simplicial sets is contained in [He ], while th* *e general case of a left proper combinatorial model category is contained in [Sm ]. The categories of presheaves of simplicial sets and groupoids with the Heller model structure will be denoted P (C, sSet)H and P (C, Grpd)H respectively. The following lemma will also be needed in the proof of Proposition 8.4. Lemma 8.6. (1) Let Sß* be a set of generating trivial cofibrations in P (C, sS* *et)J. Then the identity adjoint pair is an isomorphism (Sß*)-1P (C, sSet)H = P (C, sSet)J. (2) Consider the set of morphisms in P (C, sSet): @ n X ! n X, forn > 2, X 2 C and let (S2)-1P (C, sSet)H denote the localization of the Heller model stru* *cture with respect to these morphisms. The Quillen pair (ßoid, N) induces a Quill* *en equivalence: ßoid: (S2)-1P (C, sSet)H $ P (C, Grpd)H : N. 32 SHARON HOLLANDER Proof of Proposition 8.4.Applying Theorem 2.15(c) and (d) we see that after lo- calizing the above Quillen equivalences we still have Quillen equivalences (S2)-1(Sß*)-1P (C, sSet)H = (S2)-1P (C, sSet)J (S2)-1(Sß*)-1P (C, sSet)H $ (ßoidSß*)-1P (C, Grpd)H It follows that (ßoidSß*)-1P (C, Grpd)H is Quillen equivalent to (S2)-1P (C, sS* *et)J. Now we will show that the weak equivalences in (S2)-1P (C, sSet)J are the iso- morphisms on sheaves of homotopy groups in dimensions 0 and 1. As F ! NßoidF is a weak equivalence (because it is one in (S2)-1P (C, sSet)H ), morphisms whi* *ch are isomorphisms on sheaves of homotopy groups in dimensions 0 and 1 are weak equivalences. We claim that the fibrant replacement functor in (S2)-1(Sß*)-1P (C, sSet)H can be constructed as a transfinite composition of fibrant replacement functors of (S2)-1P (C, sSet)H and (Sß*)-1P (C, sSet)H [Dg2 ]. The desired number of compositions is a cardinal c such that all the generating trivial cofibrations * *in (S2)-1P (C, sSet)H and (Sß*)-1P (C, sSet)H are small with respect to c. As fibr* *ant replacement in (S2)-1P (C, sSet)H and in (Sß*)-1P (C, sSet)H are isomorphisms on sheaves of homotopy groups in dimensions 0, 1, the same is true for fibrant re- placement in (S2)-1(Sß*)-1P (C, sSet)H . Now let A -f! B be a weak equivalence and let P denote a fibrant replacement functor in (S2)-1(Sß*)-1P (C, sSet)H . As P f, A ! P A, and B ! P B are isomorphisms on sheaves of homotopy groups in dimensions 0, 1, so is f. We now show that weak equivalences in (ßoidSß*)-1P (C, Grpd)H are the iso- morphisms on sheaves of homotopy groups. As ßoidpreserves weak equivalences between cofibrant objects it preserves all weak equivalences. It follows that * *all isomorphisms on sheaves of homotopy groups are weak equivalences. Since ßoid induces a surjective equivalence of categories Ho((S2)-1(Sß*)-1P (C, sSet)H ) ! Ho((ßoidSß*)-1P (C, Grpd)H ), all the weak equivalences in (ßoidSß*)-1P (C, Grpd)H are the image under ßoidof weak equivalences in (S2)-1(Sß*)-1P (C, sSet)H and therefore are isomorphisms on sheaves of homotopy groups. Theorem 8.7. The identity adjoint pair induces a Quillen pair P (C, Grpd)L $ (ßoidSß*)-1P (C, Grpd)H . Proof.The cofibrations in the model structure on P (C, Grpd) of Theorem 7.1 are in particular objectwise cofibrations, and the weak equivalences agree with tho* *se in P (C, Grpd)H . So there is an induced Quillen pair P (C, Grpd) $ P (C, Grpd)H $ (ßoidSß*)-1P (C, Grpd)H . To complete the proof, by Theorem 2.15, it suffices to show that the maps hocolimUo ! X are isomorphisms on sheaves of homotopy groups. Note that in the model structure of Theorem 7.1 the homotopy colimit of the simplicial objec* *ts Uo agrees with the geometric realization |Uo|, as the homotopy colimit of objec* *twise cofibrant diagrams can be constructed objectwise. Let Y 2 C, and consider the map Grpd(Y, |Uo|) = |Grpd(Y, Uo)| ! Grpd(Y, X), A HOMOTOPY THEORY FOR STACKS 33 where the equality above holds because both the simplicial action and colimits * *are defined objectwise and Y is a discrete presheaf of groupoids. Using the fact th* *at the Yoneda embedding preserves limits we see that Grpd(Y, Uo) is the nerve of t* *he map Grpd(Y, U) ! Grpd(Y, X), that is, the simplicial groupoid: . ._._`*4___Grpd(Y, U) xGrpd(Y,X)Grpd(Y,_U)+3_Grpd(Y, U). As Grpd(Y, U) and Grpd(Y, X) are discrete groupoids, it follows that the simpli* *cial set Grpd(Y, Uo) has contractible components indexed by the image of Grpd(Y, U) in Grpd(Y, X). In other words Grpd(Y, |Uo|) is homotopy equivalent to the discr* *ete set of maps Y ! X which factor through U ! X. It follows that ß0|Uo| is the presheaf of sets defined by the image of U in X, and the presheaves ß1(|Uo|, a)* * are trivial for all base points. Therefore the induced maps on ß1 are isomorphisms. One checks easily that |Uo| ! X induces an isomorphism on shß0. Theorem 8.8. The identity adjoint pair induces a Quillen equivalence P (C, Grpd)L $ (ßoidSß*)-1P (C, Grpd)H . Furthermore, the weak equivalences in these two model structures agree. Proof.To see that the left adjoint preserves weak equivalences, i.e. that the l* *ocal weak equivalences are isomorphisms on sheaves of homotopy groups, factor a weak equivalence f 2 P (C, Grpd)L as a cofibration i followed by a trivial fibration* * p. The cofibration i is a weak equivalence and so, by Theorem 8.7, its image is a triv* *ial cofibration in (ßoidSß*)-1P (C, Grpd)H . As p is an objectwise weak equivalence* *, it is also a weak equivalence in (ßoidSß*)-1P (C, Grpd)H . To complete the proof, it suffices to show that the weak equivalences in (ßoidSß*)-1P (C, Grpd)H are also weak equivalences in P (C, Grpd)L. We use the characterization of weak equivalences in the next subsection to prove this in T* *heo- rem 8.13. Corollary 8.9. If the Grothendieck topology on C has enough points, a morphism f 2 P (C, Grpd) is a local weak equivalence if and only if it is a stalkwise we* *ak equivalence of groupoids. Proof.We have characterized the weak equivalences as those maps which induce isomorphisms on sheaves of homotopy groups, so the proof is exactly the same as the proof in [Ja] of the analogous result for P (C, sSet). Corollary 8.10. The local model structure on presheaves of groupoids P (C, Grpd* *)L is Quillen equivalent to the S2-nullification of Jardine's model struc* *ture on presheaves of simplicial sets (S2)-1P (C, sSet)J. 8.2. Characterization of Local Weak Equivalences. In this subsection we give a characterization of the weak equivalences in P (C, Grpd)L in terms of Dan Dugger's local lifting conditions. This characterization allows us to complete * *the proof of Theorem 8.8, and prove in subsections 8.3 that the local model structu* *res P (C, Grpd)L, Sh(C, Grpd)L and Grpd=CL of section 7.3 are Quillen equivalent. Definition 8.11. [Dg2 ] A map F -OE!G 2 P (C, Grpd) is said to satisfy the local lifting conditions if: 34 SHARON HOLLANDER (1) (Surjectivity on ß0). Given an isomorphism class in G(X), not necessarily represented in F (X), there is a cover U ! X such that it is represented in F (U). cb ` " [ h f e YX V** ; _____//_F (X) 0 oo____; _____//_F (X)__//_F (U) | | | | | | | | ) 9 | | | | fflffl| fflffl| fflffl| fflffl| fflffl| fflffl| 0 _____//G(X) 1 oVoX_Y [0____//G(X)___h33//_G(U). " ` b c ef (2) (Injectivity on ß0). If two isomorphism classes in F (X) become identified * *in G(X), there is a cover U ! X such that they become identified in F (U). @ 1 _____//F (X) @ 1 _____//F (X)___//_F5(U)5k | | | |k k k | | | ) 9 | k k| | fflffl| fflffl| fflffl|kkkfflffl| fflffl| 1 _____//G(X) 1 _____//G(X)____//_G(U). (3) (Surjectivity on ß1). If an element of the automorphism group of an object * *in G(X) is not in the image of the automorphism group of an object lying over * *it in F (X), then there is a cover U for which it is. 0 _____//F (X) 0 _____//F (X)____//F5(U)5k | | | |k k k | | | ) 9 | k k |k | fflffl| fflffl| fflffl|kkfflffl| fflffl| BZ _____//G(X) BZ ____//_G(X)___//_G(U). (Recall that BZ ' S1.) (4) (Injectivity on ß1). If two elements in the automorphism group of some obje* *ct in F (X) become identified in G(X), there is a cover U such that they become identified in F (U). BZ _____//F (X) BZ ____//_F (X)___//F5(U)5k | | | |k k k | | | ) 9 | k |k | fflffl| fflffl| fflffl|kkfflffl|k fflffl| 0 _____//G(X) 0 _____//G(X)____//_G(U). Theorem 8.12. [Dg2 ] A map F -OE!G 2 P (C, Grpd) is an equivalence on sheaves of homotopy groups if and only of it satisfies the local lifting conditions. Proof.Recall that for F a presheaf, its sheafification shF , can be constructed* * by setting shF (X) = colim(limF (U) ) F (V )) where the colimit is taken over all covers U ! X and V ! U xX U. It follows that if a 2 shF (X) then there exists a cover U ! X such that a lifts to an ele* *ment of F (U). Similarly if a, b 2 F (X) have the same images in shF (X) there exist* *s a cover U ! X so that they have the same image in F (U). Conversely these two properties are enough to characterize the sheafification. It follows that condi* *tions 1. and 2. are equivalent to shß0OE being an isomorphism, and conditions 3. and * *4. are equivalent to sh AutOE(a) being an isomorphism for all a 2 F (X), X 2 C. We use this theorem to prove the following result which completes the proof of Theorem 8.8. A HOMOTOPY THEORY FOR STACKS 35 Theorem 8.13. A map F ! G 2 P (C, Grpd) satisfies the local lifting conditions if and only if it is a local equivalence. Proof.We may assume F and G are fibrant, as fibrant replacement is a local weak equivalence, and we have already seen that the local weak equivalences are isomorphisms on sheaves of homotopy groups. In this case, we need to show that F ! G is an objectwise weak equivalence. Consider a map F ! G between stacks in P (C, Grpd) which satisfies the local lifting conditions. First we show F (X) ! G(X) is injective on automorphism groups. We are in the situation of 8.11(4), so we are guaranteed that there is* * a cover U ! X and a lift in the diagram of 8.11(4). The descent condition for the cover U ! X gives a commutative diagram BZ ____//_F (X) | | | | fflffl| fflffl| ____ 0 _____//F (U)___+3_F (U xX U)_`*4__F (U xX U xX U). The image of BZ in each F (Ui) is an identity morphism. Since F (X) ~-! holimF (Uo), the image of BZ in F (X) must be trivial also. To show that F (X) ! G(X) is surjective on automorphism groups, suppose we have a diagram as in 8.11(3). Consider again the descent condition for the cover U ! X, and the commutative diagram 0 ____//_F (X) | | | | fflffl| fflffl| ____ BZ _____//F (U)___+3_F (U xX U)_`*4__F (U xX U xX U). Let OE denote the image of BZ in F (U). Then d0(OE) and d1(OE) are automorphisms of the same object in F (U xX U), and they have the same image in G(U xX U). Since F ! G is an injection on automorphism groups, d0(OE) = d1(OE), which gives us a lift of OE to holimF (Uo). Since F (X) -~! holimF (Uo), there is a unique * *lift BZ ! F (X). Next we show that F (X) ! G(X) is an injection on connected components. Let a, b 2 F (X), be objects with isomorphic images in G(X). By 8.11(2), we have a commutative diagram @ 1 _____//F (X)___//_F5(U)5k | ff |k k k | | k k| | fflffl|kkkfflffl| fflffl| 1 _____//G(X)____//_G(U). i(ff) We also have two maps 1 d-! F (U xX U), whose composition to G(U xX U) is the same. Since F ! G is injective on automorphism groups, it follows that d1(ff) = d0(ff). This data gives a lifting of ff to holimF (Uo). Since F (X) * *-~! holimF (Uo), and the domain and range of ff lie in F (X), this in turn lifts un* *iquely to a morphism in F (X). 36 SHARON HOLLANDER Lastly, we show that F (X) ! G(X) is surjective on isomorphism classes. Con- sider the diagram from 8.11(1) _____a_____________________________________________* *______________________________________________________________@ _______________________**________________________________* *______________________________________________________________@ 0 oo____;_____//_F (X)___//F (U) | | | | | | | | fflffl| fflffl|b fflffl| fflffl| 1 _______________________33________________________________* *______________________________________________________________@ ___________________________________________________* *______________________________________________________________@ Let a 2 F (U) be the image of 0, b 2 G(X), be the image of 0 in G(X), and fi : im(a) -! im(b) be the image of 1 in G(U). Since F ! G is an surjection on automorphism groups, we can lift (d1fi)-1 O (d0fi) : im(d0(a)) ! im(d1(a)), * *to some ff : d0(a) -! d1(a) 2 F (U2). Since F ! G is an injection on automorphism groups, this lifting is unique. The image of d1(ff-1) O d0(ff) O d2(ff) is tri* *vial in G(U3), so it is also trivial in F (U3). Hence (a, ff) is an element of holimF * *(Uo), which determines a lifting in the diagram ;_____//_F (X)~__//holimF4(Uo)4j | | j j j | | j|j | fflffl|bjfflffl|jj~ fflffl| 0 ____//_G(X)____//holimG(Uo). Pick a0 2 F (X) whose image in holimF (Uo) is isomorphic to (a, ff). Then the image of a0in G(X) is isomorphic to b, so we can fill in the following diagram c a a0`] [ ,, 0 eoo___;______//F (X) | | | | | | fflffl| fflffl|b fflffl| 1 Yo[o]_`0_a_c22//_G(X). which completes the proof. Corollary 8.14. Let F ! G be an objectwise fibration, then the first of the loc* *al lifting conditions of 8.11 can be simplified to 10. (Surjectivity on ß0). ; _____//_F (X) ; _____//_F (X)__//_F5(U)5k | | | |k k k | | | ) 9 | kk| | fflffl| fflffl| fflffl|kkkfflffl| fflffl| 0 _____//G(X) 0 _____//G(X)____//_G(U). The local lifting conditions 10, 2, 3, 4 are preserved under pullbacks, so the * *pullback of an objectwise fibration which is a local weak equivalence is again an object* *wise fibration which is a local weak equivalence. 8.3. Comparison of the Local Model Category Structures. Proposition 8.15. The adjoint pairs _____.._________________________________________________* *_______________________________iqq____________________________@ Grpd=C mm___P_(C,_Grpd)________________________________________* *________________________________________00____________________@ p sh induce Quillen equivalences between the local model structures. A HOMOTOPY THEORY FOR STACKS 37 Proof.Let S denote the sets of morphisms hocolimUo ! X, for{Ui! X} a cover in P (C, Grpd), Sh(C, Grpd) and Grpd=C. Since homotopy colimits commute with the left adjoint in a Quillen pair, the set S 2 P (C, Grpd) is mapped by sh and* * p to the sets S in Sh(C, Grpd) and Grpd=C respectively. By Theorem 2.15, the adjoint pairs (sh, i), and (p, ) are still Quillen pairs between the local model categ* *ory structures, and (p, ) is still a Quillen equivalence. It remains to show that (sh, i) is a Quillen equivalence. By construction of * *the sheafification functor, the map F ! shF satisfies the local lifting conditions,* * and so is a weak equivalence in P (C, Grpd)L. Similarly, it is easy to check that i* *f a map f 2 P (C, Grpd) satisfies the local lifting conditions then so does sh(f). We will now prove that sh preserves weak equivalences. Let A ! B be a weak equivalence in P (C, Grpd)L, and P denote a fibrant replacement functor on P (C, Grpd)L. One can check directly that the sheafification of a stack F is a * *stack and so sheafification preserves fibrant replacement. We have the following comm* *ut- ing diagram A"______//`B" ` shA"________//`shB" ` |~| |~| sh) |~| |~| |fflffl fflffl| fflffl| fflffl| P A ____//_P B sh(P A)____//_sh(P B). In P (C, Grpd)L, the morphism sh(P A) ! sh(P B) is a weak equivalence between fibrant objects (as P A -~! P B is a weak equivalence in P (C, Grpd)L) and so is an objectwise weak equivalence. It follows that sh(P A) ! sh(P B) is a weak equivalence in Sh(C, Grpd)L, and therefore, so is shA ! shB. Now we show that the forgetful functor i also preserves weak equivalences. Let f be any weak equivalence in Sh(C, Grpd)L, and P f its fibrant replacement in P (C, Grpd)L. As sh(P f) is the fibrant replacement of f in sheaves it is also * *a weak equivalence, and so also an objectwise weak equivalence. It follows that sh(P f* *) is a weak equivalence in P (C, Grpd)L, and therefore f is a weak equivalence also. As both i and sh preserve weak equivalences, and the unit and counit are weak equivalences, the Quillen pair (sh, i) is a Quillen equivalence. Corollary 8.16. A morphism X -f! Y 2 Sh(C, Grpd)L is a weak equivalence if and only if i(f) is a weak equivalence in P (C, Grpd)L. It follows that the wea* *k equiv- alence in Sh(C, Grpd)L are the maps which satisfy the local lifting conditions.* * In particular, the weak equivalences in Sh(C, Grpd)L are the maps which are object* *wise full and faithful, and satisfy 8.11(1). Proof.We show that if a morphism X -f! Y 2 Sh(C, Grpd)L, is such that i(f) is a weak equivalence in P (C, Grpd)L then f was already a weak equivalence in Sh(C, Grpd)L. Let C denote a cofibrant replacement functor in P (C, Grpd)L, and let F be a fibrant sheaf. Then the map h Hom (f, F ) = sSet(Cf, F ) is a weak equivalence. As sheafification preserves fibrant replacement sSet(Cf, F ) = sSet(sh(Cf), F ), and so the map sSet(sh(Cf), F ) is also a weak equivalence. As sh(CX) and sh(CY ) are cofibrant as sheaves it follows that sh(Cf) is a weak equivalence in 38 SHARON HOLLANDER Sh(C, Grpd)L. We have the following commutative diagram in Sh(C, Grpd) sh(Cf) sh(CX) _____//_sh(CY ) ~|| |~| fflffl|f fflffl| shX ~=X _____//shY ~=Y. where the vertical arrows are weak equivalences because they are the sheafifica* *tion of weak equivalences in P (C, Grpd)L. By a 2 out of 3 argument, it follows that X -f! Y is also a weak equivalence in Sh(C, Grpd)L. To complete the proof, notice that for a morphism X -f! Y of sheaves, the loc* *al lifting conditions 2. - 4. are equivalent to f being objectwise full and faithf* *ul. A HOMOTOPY THEORY FOR STACKS 39 Appendix A. Limits and Colimits in Grpd=C Theorem A.1. Categories fibered in groupoids over C are closed under small limi* *ts and colimits. In order to prove this, we will need a few preliminaries. Definition A.2. F : E ! C 2 Cat=C is pre-fibered in groupoids if (1) Given f : Y ! X 2 C and X0 2 E such that F (X0) = X, there exists f0 2 E such that F (f0) = f. (2) Given a diagram in E, over the commutative diagram in C, Y 0 __F__+3 Y ~~ | |f0| h~~~ |f g0 fflffl| F ~~~~g fflffl| Z0____//_X0 _____+3 Z ____//_X, with F (f0) = f, F (g0) = g, there exists h0 such that g0O h0= f0 and F (h0* *) = h. Moreover, given two such maps h01, h02, there exists an automorphism OE 2 AutE(Y 0) such that F (OE) = idY and h01O OE = h02. Thus, the difference between fibered and pre-fibered is that categories which* * are pre-fibered in groupoids only satisfy the uniqueness in condition 2) of Definit* *ion 5.2 in a weak form. Proposition A.3. Let I be a small category, and F : I -! Grpd=C, a diagram. Then the colimit of F in Cat=C is pre-fibered in groupoids. Proof.The coproduct in Cat=C of a set of objects in Grpd=C is again in Grpd=C so it suffices to consider the case of a coequalizer diagram. Consider the diagram _F1_ E0_F2_?+3E___//_~E ?? | ??? | ?ØØfflffl| C where F1, F2 2 Grpd=C and ~Eis the coequalizer of the two arrows in Cat. Recall that the coequalizer in Cat has objects the coequalizer of the sets of objects,* * and morphisms the formal compositions of the coequalizer of the morphisms, modulo the relations given by composition in E. Thus the map ~E-! C clearly satisfies condition 1. of definition A.2. We now prove that it also satisfies condition 2. with an induction argument. Consider the diagrams Y~ Y " |~f| h""""|f| ~g fflffl| """"gfflffl| Z~_____//~X Z ____//_X where the bared objects and morphisms represent objects and morphisms in ~E projecting to the corresponding objects and morphisms in C. Using the construct* *ion of ~E, we can factor ~fand ~gas formal compositions of maps in the image of E i* *n ~E. Let ~f= (f~0, ~f1, . .,.~fn) and ~g= (~g0, ~g1, . .,.~gm), with domain(f~i) = r* *ange(f~i-1), domain(~gi) = range(~gi-1), and range(f~n) = range(~gm) = ~Xin ~E. 40 SHARON HOLLANDER Firstly, consider the case when n = m = 0. Let Y1 f1-!X1 Z2 g2-!X2 2 E be representatives of the maps ~fand ~grespectively. If there is X0 2 E0such th* *at F1(X0) = X1 and F2(X0) = X2, lift f, g to morphisms f0, g0 in E0 whose range is X0. Since E02 Grpd=C, there is a unique h02 E0, projecting to h 2 C, such that g0O h0= f0. Since E 2 Grpd=C, there are unique isomorphisms in E, projecting to identity morphisms in C, filling in the diagrams Y1-Ø Ø- !Ø-- ff--lffl F1(Y 0--) F2(Y 0) 44-- 44 F1(h0)|44-f1--4| F2(h0)|444| ffl44--4ffl| fflf4F2(f0)44fl| F1(Z0) 4--4 F2(Z0) 44 GGGG4--4 ØØGGGG44 0GG4--4G !Ø 0G44GG F1(g ) G-~~ßß4## fF2(gf)G4ææ##lffl X1 Z2 __g2__//_X2. Then the map ~h, defined as the formal composition Y1 -~!F1(Y 0) ~ F2(Y 0) ! F2(Z0) -~! Z2, is such that ~gO ~h= ~f2 ~E. In general, there will not be an ob* *ject X0 such that F1(X0) = X1 and F2(X0) = X2, but a finite sequence of objects in E0 such that their images under F1 and F2 form a chain connecting X1 and X2. The above argument is easily generalized to deal with this case. This completes* * the proof in the case when n = m = 0. If n = 0 then we can use the previous case and induction on m to lift as indi* *cated in the following diagram ~YXXXXXXXXXVVRRGØ Ø GRVRXXXXXXXXXf~XXXXXVVVG ~hØ G R RR V XXXXXXXXXVV fflffl G## R R V XXXXXXXXXXV ~Z= ~Z0____//~____((//_._V**.XXXXXX,,,,XX.//_~//_~ ~g0 Z1 ~g1 ~gm-1Zm ~gm X . so the result is true in the case when n = 0 and m is arbitrary. It is not hard to check that one can choose the lift ~hso that it is the imag* *e in ~E of a formal composition of isomorphisms in EY followed by a lift of h to E. To complete the proof, notice that there is a lift of f 2 C to a map ~Y00~f0-!~X2 ~E which is in the image of E. Then by the previous case, there is an isomorphism OE 2 ~E, projecting to idY 2 C, as well as a map ~h02 ~Esuch that the following diagrams commute in ~E Y~00 Y~0 Ø ?? ~0 0ØAAA~0 OEØ ??f?? Ø~h0AfAA fflfflØØØ? fflfflØ__A ~Y__~_//_~X ~Z____//_~X. f ~g A HOMOTOPY THEORY FOR STACKS 41 We can now take ~h= ~h0O OE-1. Notice that if h is the identity, we can choose * *~hto be an isomorphism. Proposition A.4. Let E ! C be pre-fibered in groupoids. Let ~ be the equivalence relation on E generated by setting ff ~ id for the automorphisms ff 2 E which satisfy: (1) ff maps to an identity morphism in C, (2) there exists f 2 E such that f O ff = f. Then E= ~! C is also pre-fibered in groupoids. Proof.The map E -! E= ~ is surjective on morphisms and bijective on objects so this is obvious. Proof of Theorem A.1.Colimits: Let I be a small category and F : I -! Grpd=C be a diagram. We denote by F 0the composite I -F! Grpd=C -! Cat=C. Let Ecolim denote the colimit of F 0in Cat. We will show that the colimit of F is the dire* *cted colimit of categories in Cat=C, (A.5) Ecolim! Ecolim= ~-! (Ecolim= ~)= ~-! . . . Denote the i-th category in this diagram Eicolimand the colimit E := colimi(Eic* *olim). Propositions A.3 and A.4 imply that Condition 1) and the existence part in Con- dition 2) of Definition 5.2 are still satisfied by E. To show the uniqueness part in Condition 2), suppose given a commutative diagram in E: f Y ____//_X>>~ h2|h1|g~~~~ |ff'|~~ Z such that h1 and h2 project to the same map in C. Pick lifts h01and h02of h1 and h2 in some Eicolim. Then they also project to the same map in C so by Propositi* *on A.3, there is an automorphism ff of Y in Eicolimmapping to an identity in C such that h02O ff = h01. It follows that h01= h022 Ei+1colimand so h1 and h2 agree i* *n E. We still need to show that E is the colimit in Grpd=C, but this follows becau* *se any map F ! E02 Cat=C, with E02 Grpd=C factors uniquely through F= ~. Limits: Let F : I ! Grpd=C be a diagram, and let limF 0denote its inverse limit in Cat=C. If limF 02 Grpd=C then it is the limit in Grpd=C as this is a f* *ull subcategory of Cat=C. The objects and morphisms of limF 0are the inverse limits of the sets of obje* *cts and morphisms, so for each object X0 2 limF 0, the category (limF 0)=X0, is the limit of categories F (i)=X0i, i 2 I. It is easy to see that the map (limF 0)=X* *0! C=X o is a bijection on Hom -sets, since this is the case for each of the constit* *uent functors F (i)=X0i! C=X, o but it is not necessarily a surjection on objects even though each of the f* *unctors F (i)=X0i! C=X is. It follows that if limF 0is not fibered in groupoids over C, this is due to the* * failure of Condition 1) in Definition 5.2. However, in this case, the full subcategory of * *limF 0 with objects all those X0 such that (limF 0)=X0 ! C=X is surjective on objects, clearly is fibered in groupoids and satisfies the universal property of the lim* *it. 42 SHARON HOLLANDER Appendix B. Lax Presheaves of Groupoids In this section we will define the category of lax presheaves of groupoids, d* *enoted lax - P (C, Grpd), and we will give an equivalence between this category and the category Grpd=C. When C has a Grothendieck topology, lax - P (C, Grpd) is also used as an ambient category in which to define stacks and we observe that the t* *wo different definitions of stack agree under this equivalence. Using this equivalence of categories, all the results proved in this paper for Grpd=C can be transfered to lax - P (C, Grpd). One should think of a lax presheaf on C as a category fibered in groupoids together with a choice of pullback functors. The morphisms between lax presheav* *es are sufficiently flexible so that different choices of pullback functors for th* *e same category fibered in groupoids correspond to canonically isomorphic lax presheav* *es. The relation between the categories Grpd=C and lax - P (C, Grpd) is analogous to the relation between two different ways of defining principal G-bundles. One* * can define a bundle on X as a space over X which is locally trivial, or one can def* *ine the bundle to be the space over X together with a set of local trivializations. Wh* *en the trivializations are part of the definition, one has to add morphisms to the cat* *egory which give equivalences between the different choices of trivializations. Definition B.1. [Bry, Brn] The objects of lax - P (C, Grpd) are the assignments: o for each object X 2 C, a groupoid F(X), o for each morphism Y -f! X 2 C, a functor F(X) F(f)-!F(Y ), o for each pair of composable morphisms Z -g! Y -f! X 2 C, a natural trans- formation F(g) O F(f) `g,f-!F(f O g), such that o for every triple of composable morphisms W -h! Z -g! Y - f!X 2 C, the following diagram commutes F(h)O`g,f F(h) O F(g) O F(f)____//_F(h) O F(f O g) `h,gOF(f)|| `h,fOg|| |fflffl `gOh,f fflffl| F(g O h) O F(f)______//F(f O g O h). A morphism OE : F -! F02 lax - P (C, Grpd) is an assignment: o for each object X 2 C, a map F(X) OE(X)-!F0(X), o for each morphism Y -f! X 2 C, a natural isomorphism OE(Y ) O F(f) OE(f)-! F0(f) O OE(X), OE(X) F(X) ____//_F0(X)6> vvvv F(f)||OE(f)vvvvF0(f)||vvv fflffl|vvvvfflffl|v F(Y )_OE(Y/)/_F0(Y ), such that A HOMOTOPY THEORY FOR STACKS 43 o for each pair of composable morphisms Z -g! Y -f!X 2 C, the following diagram commutes OE(Z) O F(fSO g) OE(Z)O`g,fkkkkkk SSSSOE(fOg)SS kkk SSSSS uukkkk SS)) OE(Z) O F(g) O F(f) F0(f O g)OOOOE(X) OE(g)OF(f)|| |`0g,fOOE(X)| fflffl| | F0(g) O OE(Y ) O F(f)___F0(g)OOE(f)____//_F0(g) O F0(f) O OE(X). There is a natural groupoid action on lax - P (C, Grpd), in which: o the groupoid of maps has objects maps, and morphisms the coherent natural isomorphisms, o the tensor and cotensor are defined objectwise. There is an obvious inclusion i : P (C, Grpd) -! lax - P (C, Grpd) which preser* *ves the groupoid action. Example B.2 (Vector Bundles on TopRevisited). Consider the assignment Top! Grpd which sends Y to the groupoid of vector bundles over Y , and a map f to the pullback function f?. This assignment is not a functor because given Z -g! Y -f! X 2 Top and E -! X a vector bundle, the pullbacks g*f*E and (f O g)*E are not equal. There is, however, a canonical isomorphism g*f*E ! (f O g)*E so the assignment above together with the canonical isomorphisms is an example of a lax presheaf on Top. Instead of working with this lax presheaf, we can consider its associated cat* *egory of pairs, or Grothendieck construction. This has objects the pairs (Y, E -! Y ), where E is a vector bundle over Y 2 Top, and morphisms (Y, E) ! (Z, E0), the pairs formed by a map f : Y - ! Z, and an isomorphism ff : E -! f*E0. It is easy to check that this category is isomorphic to the category V ec(Top) 2 Grpd* *=C of Example 5.1. Just as in the bundle case, there is a öf rgetful functor" lax - P (C, Grpd) * *-! Grpd=C which sends lax presheaves corresponding to different choices of pullback functors for E ! C, to objects in Grpd=C which are canonically isomorphic to E * *! C. Definition B.3. Given F 2 lax - P (C, Grpd), let pF 2 Cat=C be the category with o objects, the pairs (X, a) with X 2 C and a 2 F(X), o morphisms (X, a) ! (Y, b), the pairs (f, ff) where f : X -! Y is a morphism in C and ff : a -! F(f)b is an isomorphism in F(X). The composition of two morphisms (X, a) (f,ff)-!(Y, b) (g,fi)-!(Z, c) is the pa* *ir (g O f, `f,gO F(f)(fi) O ff). It is not hard to show that pF is a category fibered in groupoids over C, and* * that p defines a functor lax - P (C, Grpd) ! Grpd=C. Theorem B.4. The functor p : lax - P (C, Grpd) $ Grpd=C is an equivalence of categories. Proof.Let E 2 Grpd=C. A choice of a pullback functor f* : EX ! EY for each Y -f! X 2 C, determines a lax presheaf F with F(X) := EX , and F(f) = f*. Given two such choices of lax presheaves F, F0, there is a canonical isomorphism 44 SHARON HOLLANDER OE : F -! F0, where OE(X) = idEX for each X 2 C, and OE(f) is the canonical nat* *ural isomorphism from F(f) ! F0(f). For each E 2 Grpd=C make an arbitrary choice of pullback functors, and let L(E) denote the resulting lax presheaf. For each X -f! Y 2 C, a map E -F!E0determines squares f* EX _____//EY9A ---- F|| -!----F||- fflffl|--fflffl|-- E0X__f*_//E0Y. where the unique natural isomorphism follows from condition 2. of Definition 5.* *2. The uniqueness of the natural isomorphism in the square above guarantees that these squares patch together to give a morphism L(F ) : L(E) ! L(E0) 2 lax - P (C, Grpd) and that L is indeed a functor. It is now easy to check that there are canonical natural isomorphisms L O p ~= idlax-P and p O L ~=idGrpd=C. Note B.5. It is easy to check directly from the definition of stacks in lax pre* *sheaves [Brn, Pg.5] that F 2 lax-P (C, Grpd) is a stack if and only if pF is a stack in* * Grpd=C. Thus, the equivalence of categories between lax - P (C, Grpd) and Grpd=C restri* *cts to an equivalence between the subcategories of stacks. A HOMOTOPY THEORY FOR STACKS 45 References [An] D.W. Anderson, Fibrations and Geometric Realizations, Bull. Am. Math Soc. * *84, 765-786, (1978), 765-786. [Bo] A. K. Bousfield, Homotopy Spectral Sequences and Obstructions, Israel Jour* *nal of Math., Vol.66, Nos.1-3, (1989), 54-105. [BK] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions, and Localizat* *ions, Lecture Notes in Math. 304, Springer, (1972). [Brn]L. Breen, On the Classification of 2-Gerbes and 2-Stacks, Asterisque 225, * *(1994). [Bry]J.-L. Brylinski, Loop Spaces, Characteristic Classes,and Geometric Quantiz* *ation, Progress in Mathematics 107, Birkhaüser, Basel, (1993). [DM] P. Deligne and D. Mumford, The Irreducibility of the Space of Curves of Gi* *ven Genus, Publ. Math. IHES 36, (1969), 75-110. [Db] Eduardo J. Dubuc. Kan Extensions in Enriched Category Theory, Lecture Note* *s in Math. 145, Springer, (1970). [DF] Emmanuel Dror-Farjoun, Cellular spaces, null spaces and homotopy localizat* *ion, Lecture Notes in Mathematics 1635, Springer, (1995). [Dg] Dan Dugger, Universal homotopy theories, preprint (2000), to appear in Adv* *ances in Math- ematics. [Dg2]Dan Dugger, Simplicial Presheaves, Revisited, preprint (2000). [DK] W. G. Dwyer and D. M. Kan, Homotopy Commutative Diagrams and their Realiza* *tions, J. of Pure and Applied Algebra 57, (1989), 5-24. [DS] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Hand* *book of Algebraic Topology, Elsevier Science, (1995). [EK] S. Eilenberg and G. M. Kelly, Closed categories, Proc. Conf. on Categorica* *l Algebra (La Jolla 1965), Springer-Verlag. [Gi] J. Giraud, Cohomologie non-abelienne, Springer Verlag, Berlin Heidelberg N* *ew York, (1971). [GJ] P. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathe* *matics 174, Birkhaüser, Basel, (1999). [GZ] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Sprin* *ger Verlag, New York, (1967). [He] A. Heller, Homotopy Theories, Memoirs Amer. Math. Soc. Vol. 71, No. 383, (* *1988). [Hi] P. S. Hirschhorn, Localization of model categories, 2000 Preprint. [Ho] Mark Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer* *ican Math- ematical Society, (1999). [Holl]S. Hollander, work in progress. [Ja] J.F. Jardine, Simplicial Presheaves, J. of Pure and Applied Algebra 47 (19* *87), 35-87. [Ja2]J.F. Jardine, Stacks and the Homotopy Theory of Simplicial Presheaves, pre* *print. [Ml] Saunders MacLane, Categories for the Working Mathematician, Grad. Texts in* * Math., Vol.5, Springer Verlag, New York, (1971). [MM] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introdu* *ction to Topos Theory, Springer Verlag, Berlin Heidelberg New York, (1992). [May]J. Peter May, Simplicial objects in Algebraic Topology, Chicago Lectures i* *n Mathematics, (1969). [MV] F. Morel and V. Voevodsky, A1-homotopy theory of schemes, preprint. [Q] Daniel J. Quillen Homotopical Algebra, Lecture Notes in Math. 43, Springer* *, (1967). [Sm] Jeff Smith Combinatorial Model Categories, preprint. [St] Neil Strickland K(n)-local duality for finite groups and groupoids, Topolo* *gy 39, (2000). E-mail address: sharon@math.mit.edu+ Department of Mathematics, Massachusettes Institute of Technology