Higher order principal bundles J.F. Jardine*and Z. Luo February 7, 2004 Introduction Suppose that G is a sheaf of groups on a topological space X, and that i : Y ! X is a principal G-bundle for X. The bundle is locally trivial, so there is an open covering {Uff} of X such that the restriction of the bundle to each member of the covering admits a section _Y>> __ ___ |i| __ fflffl| Uff_____//X On intersections Uff\Ufiof the covering family, the restrictions may be differe* *nt but they are related by multiplication by a unique element gfffiof the sections G(Uff\ Ufi) of the group G. The collection of all elements gfffidefine a cocycle for the covering with coefficients in the sheaf of groups G, and cohomologous cocycles correspond to isomorphic bundles. In this way, the set of isomorphism classes of G bundles i which trivialize over the covering {Uff} is isomorphic t* *o a set of naive homotopy classes of maps ß(Uo, BG) of simplicial sheaves from the C~ech resolution Uo corresponding to the covering to the classifying simplicial sheaf BG. This line of argument is classical and well known and, subject to placing oneself in the context of simplicial sheaves, is most of the proof of t* *he theorem that says that there is a natural bijection H1(X, G) ~=[*, BG] relating the non-abelian H1 invariant associated to G (aka. isomorphism classes of principal G-bundles) with morphisms in the homotopy category of simplicial sheaves on X from the terminal object * to the classifying simplicial sheaf BG. The link between H1(X, G) and the homotopy theory object [*, BG], or rather the desciption of it arising from the generalized Verdier hypercovering theorem, amounts to the observation that the fundamental groupoid of a hypercover is a C~ech resolution. ____________________________* This research was supported by NSERC. 1 The argument is also universal, in that it gives a bijection H1(E, G) ~=[*, BG] (1) relating isomorphism classes of G-torsors with morphisms in the simplicial sheaf homotopy category, which holds in any Grothendieck topos E and for all groups G in the topos. This result has been known for some time now [6]. The purpose of this note is to give a description of the corresponding homo- topy theoretic invariant ___ [*, W G] ~=[*, dBG] when G is either a presheaf of simplicial groups or G is a presheaf of groupoids enriched in simplicial sets. Of course, the last case is the most general _ a simplicial group is a simplicial groupoid with one object _ but a separate development is given for presheaves of simplicial groups in the first two secti* *ons of this paper. ___ The_objects W G and dBG are models for a classifying space construction for G: W G is the universal cocycles construction, and as such is a generalization * *of a classicial construction of Eilenberg-Mac Lane [5], while dBG is the diagonal * *of the bisimplicial object arising from standard nerve functor applied to a simpli* *cial groupoid G. These constructions are shown to be weakly equivalent in the final section of this paper. They have been used to create model structures for various flavours of sheaves and presheaves of simplicial groupoids [3], [9], [1* *1], all of which give models for the homotopy category of simplicial sheaves and presheaves. The following observation is a central idea of this paper. If G is an ordina* *ry sheaf of groups on a Grothendieck site C, then a G-torsor is a sheaf X admitting a free (or principal) G-action such that the coinvariant sheaf X=G is a copy of the terminal sheaf * up to isomorphism. Backing up a bit, one knows that, in the presence of a principal G-action on a sheaf Y , the corresponding Borel construction EG xG Y is a simplicial sheaf which is weakly equivalent to the discrete object Y=G. In fact, the converse is true: if EG xG Y ! Y=G is a local weak equivalence then the G-action on Y is principal. Thus a G-torsor is a sheaf X with a G-action such that the Borel construction EG xG X is locally weakly equivalent to the terminal sheaf * in the sense of simplicial sh* *eaf homotopy theory. The analogue of a group action for a presheaf of groupoids G enriched in sim- plicial sets is a simplicial functor X defined on G and taking values in simpli* *cial presheaves. Each such simplicial functor X has a homotopy colimit holim---!GX, and one says, by direct analogy with the Borel construction (aka. homotopy colimit) description of ordinary torsors, that X is a G-torsor if this homotopy colimit is locally weakly equivalent to the point *. There is a category of G- torsors G - Tors which has a class ß0(G - Tors) of path components, and the main result of this paper (Theorem 17) asserts that there is a natural bijection [*, dBG] ~=ß0(G - Tors). 2 The special case of Theorem 17 corresponding to the case where G is a presheaf of simplicial groups is given its own proof in the second section of t* *his paper, and appears as Corollary 10. This is done to display a quick application of our main new technical device, which is an expanded notion of cocyle which appears in Lemma 8. Cocycles have previously been interpreted (in the most general formulation) as maps defined on hypercovers. Hypercovers are most precisely defined as locally fibra* *nt presheaves which are weakly equivalent to the terminal object *. As such, they are examples of simplicial presheaves U which are weakly equivalent to a point, and the magic thing here is that when one looks at the path components of the category Triv=Y of all morphisms U ! Y (no homotopy classes), one gets a class ß0(Triv=Y ) which is isomorphic to [*, Y ]. Furthermore this result holds in great generality: Y can be any member of an arbitrary right proper model category having a cofibrant terminal object. We also give a new demonstration of the bijection (1) relating isomorphism classes of G-torsors with homotopy classes of maps [*, BG] for sheaves of groups G, in Remark 11. With the new approach to cocycle theory in hand, Theorem 17 is a rather easy consequence of a result of Joyal and Tierney [9] which asserts that their homotopy category of sheaves of simplicial groupoids is equivalent to the ho- motopy category of simplicial sheaves via the classifying space functor dB. The point is that when G is a simplicial groupoid with discrete objects and X is a G-torsor, the homotopy colimit holim---!GX can be taken apart and put back to- gether again with well known results of Quillen. There is no appeal to amenable objects [9] in the proof of Theorem 17. Also, our torsors do not coincide with the pseudo-torsors of [8] (see Remark 19). One can define G-torsors, and one has an analog of Theorem 17 for all presheaves of simplicial groupoids G. In the case where G has discrete objects, a result of Moerdijk [13] can be used to show that a G-torsor X is locally a copy of the loop space object dBG of the classifying space. In particular, in this case, any morphism of G-torsors is a local weak equivalence. This is the analogue of the well known observation that any morphism of ordinary torsors for a sheaf of groups is an isomorphism. There is another antecedent for our theory in the description of torsors for* * a sheaves of groupoids which appears in [7], and Theorem 17 is a generalization of [7, Th.14]. The reader should be aware that the proof of the older result conta* *ins an error which is fixed in the proof of Theorem 17 _ see the explanation in Remark 18 at the end of the third section. The final section of this paper contains a first application: Theorem 23 identifies the class of path components of the category of G-gerbes for a sheaf* * of groups G with set of isomorphism classes of right torsors over the automorphism 2-groupoid object Aut (G). In the world of ordinary groups, the automorphism 2-groupoid of a group H has one 0-cell, 1-cells given by the automorphisms of H and 2-cells given by their homotopies. A gerbe is a locally connected stack, and a G-gerbe is a gerbe which is locally equivalent to either G or its associated stack of G-torsors. Our classification theorem can be inferred from a 3 result of Breen [2, Prop.7.3] and its proof employs similar constructions, but * *we avoid a discussion of bitorsors and do not encounter homotopy coherence issues. Theorem 23 is a direct homotopy theoretic classification of G-gerbes up to local equivalence. The results of Sections 1-4 of this paper appeared in preliminary form in the thesis of Zhiming Luo (the second author), [12]. Contents 1 Torsor categories for presheaves of simplicial groups 4 2 Cocycles 7 3 Torsors for presheaves of simplicial groupoids 11 4 Universal cocycles for simplicial categories 16 5 Gerbes 18 1 Torsor categories for presheaves of simplicial groups Suppose that G is a presheaf of simplicial groups on a (small) site C, and write G-s Pre(C) for the category of all simplicial presheaves X admitting a G-action G x X ! X. We shall also call these objects simplicial G-presheaves. Lemma 1. There is a cofibrantly generated closed model structure on the cat- egory G - s Pre(C) of simplicial G-presheaves, where a map f : X ! Y is a fibration (respectively weak equivalence) if the underlying map of simplicial presheaves is a global fibration (respectively local weak equivalence). Proof. The product G x X is the free simplicial G-presheaf on a simplicial presheaf X. It follows that the free simplicial G-presheaf functor preserves cofibrations and weak equivalences. Colimits in simplicial G-presheaves are formed as in simplicial presheaves. Thus, if i : A ! B is a trivial cofibration* * of simplicial presheaves and the diagram G x A _____//X 1xi|| i*|| fflffl| fflffl| G x B ____//_Y is a pushout in the category of simplicial G-presheaves, then the map i* is a weak equivalence. It follows that the generating set A ! B for the class of trivial cofibrations of simplicial presheaves determines a generating family 4 G x A ! G x B for the trivial cofibrations of simplicial G-presheaves. The set of morphisms G x K ! G x LU ( n) of simplicial G-presheaf morphisms induced by the simplicial presheaf inclusions K LU ( n) is a generating set * *__ for the class of all cofibrations of simplicial G-presheaves. |* *__| As usual, LU ( n) denotes the simplicial presheaf which is freely generated by an n-simplex in U-sections. Say that G acts freely on X or that X is G-free if the simplicial group G(U) acts freely on the simplicial set X(U) for all U 2 C. Suppose that G acts freely on X. Let X=G be the quotient simplicial presheaf and let ß : X ! X=G be the canonical map. Then the lifting OE exists in the diagram t:X:t tt tOEttt ß|| tt fflffl| LU ( n)__x__//X=G for each simplex x 2 X=G(U) for all U 2 C. Multiplying the map OE by the action of G determines a simplicial presheaf map G x LU ( n) ! X, and it is easy to see that the freeness of the action implies this map factors through an isomorphism ~ OE* : G x LU ( n) =-!LU ( n) xX=G X. The cofibrant simplicial G-presheaves X in the model structure of Lemma 1 are (sectionwise) principal G-bundles X ! X=G, on account of the following: Lemma 2. Suppose that X is a cofibrant simplicial G-presheaf. Then G acts freely on X in all sections. Proof. The maps GxK ! GxLU ( n) generate the cofibrations of the category of simplicial G-presheaves, and any pushout G x K _______//_Z | | | | fflffl| fflffl| G x LU ( n)_____//W has the effect of adding some freely generated G(U)-space to Z(U) for each U 2 C. The cofibration ; ! X has a factorization ;_____//?V ?? | ??? |ß ?ØØ?fflffl| X where ß is a trivial fibration and the map ; ! V is a transfinite colimit of pushouts of the above form. It follows that G acts freely on V . But then, by a standard argument, X is a retract of V (since ß is a trivial fibration)_so that* * G acts freely on X. |__| 5 Remark 3. The model structure of Lemma 1 specializes to the standard model structure for G-spaces in the case where the site C is a point, and where G is a simplicial group. In the case of G-spaces, Lemma 2 has a converse [5, V.2.10]. It follows that a simplicial G-presheaf X having free G-action is a diagram of G(U)-spaces X(U), each of which is cofibrant. It is not clear that X itself is cofibrant. Write G - Torsfor the category of cofibrant simplicial G-presheaves X such that the canonical map X=G ! * is a hypercover (ie. a local trivial fibration). A morphism f : X ! Y of G - Tors is just a G-equivariant map of simplicial presheaves. Write G - Tors for the corresponding category. Choose a factorization ; ___i_//BEG BB | BBB ß| BB!!fflffl| * where i is a cofibration and ß is a trivial fibration in the category of simpli* *cial G-presheaves. Write BG = EG=G. Observe that BG is a presheaf of Kan complexes on account of Lemma 2, [5, V.2.7], and [5, V.3.7]. The homotopy type of BG is independent of the choice of the object EG. If E0G is a second such choice with quotient B0G = E0G=G then there is a G- equivariant homotopy equivalence EG ! E0G since both objects are fibrant and cofibrant. This homotopy equivalence induces a homotopy equivalence BG ! B0G of the quotients. Remark 4. The Eilenberg-Mac Lane object W G has a free G-action, and the map W G ! * is a sectionwise trivial fibration. There are maps of simplicial G-presheaves W G- pW~ G j-!EG such that p is a trivial fibration and W~G is cofibrant, and such that j is a trivial cofibration and EG is fibrant. The map p is in particular a sectionwise weak equivalence of sectionwise cofibrant_G-spaces, and therefore induces a sec- tionwise weak equivalence W~G=G ! W G. The map j is a trivial cofibration of cofibrant simplicial G-presheaves, and therefore induces a weak equivalence W~ G=G ! EG=G = BG. A similar argument works for the diagonal map d(EG) ! d(BG) induced by the standard bisimplicial sheaf map EG ! BG, because the induced map of diagonal simplicial objects is a sectionwise principal G-fibration and the obje* *ct d(EG) is weakly equivalent to a point. It follows that d(BG) ' BG for the two different senses of BG. Remark 5. Every trivial cofibration i : A ! B of simplicial G-presheaves induces a trivial cofibration i* : A=G ! B=G. In effect, i has the left lifting property with respect to all global fibrations p : X ! Y of simplicial presheav* *es with trivial G-action. 6 Suppose that X is a cofibrant simplicial G-presheaf such that the induced map X=G ! * is a local weak equivalence. Find a trivial cofibration j : X ! ~X in the category of simplicial G-presheaves such that X~ is fibrant. Then the induced map j* : X=G ! ~X=G is a trivial cofibration of simplicial presheaves, and ~X=G is a presheaf of Kan complexes so the map ~X=G ! * is a hypercover. Write G - Tors0 for the category of cofibrant simplicial G-presheaves X such that X=G ! * is a local weak equivalence. Then the inclusion G - Tors G - Tors0 induces an isomorphism ß0(G - Tors) ~=ß0(G - Tors0). Remark 6. Write G-Tors 1for the category of simplicial G-presheaves Y such that the canonical map d(EG xG Y ) ! * is a local weak equivalence, where d(X) denotes the diagonal of a bisimplicial object X. Then there is an inclusion G - Tors0 G - Tors1 since the canonical map d(EG xG Z) ! Z=G is a sectionwise weak equivalence if Z is cofibrant. On the other hand, if X is a simplicial G-sheaf such that d(EG xG X) ! * is a weak equivalence, there is a trivial fibration Z ! X of simplicial G-presheaves such that Z is cofibrant. The induced map d(EG xG Z) ! d(EG xG X) is a local weak equivalence, so that Z is an object of G - Tors0. It follows that there is an isomorphism ß0(G - Tors0) ~=ß0(G - Tors1). 2 Cocycles Suppose that M is a model category with a terminal object *, and let X be an object of M. Write Triv=X for the category whose objects are all morphisms W ! X of M such that the map W ! * is a weak equivalence. Observe that there is a function _X : ß0(Triv=X) ! [*, X] which is defined by associating to an object W ! X the composite * -' W ! X in the homotopy category. Lemma 7. Suppose that M is a right proper model category with terminal object *. Suppose that the map g : X ! Y is a weak equivalence. Then the induced function g* : ß0(Triv=X) ! ß0(Triv=Y ) is a bijection. 7 Proof. The function g* is induced by a functor which is defined by associating to the object W ! X the composite W ! X g-!Y. Suppose that v : U ! Y is an object of Triv=Y . Choose a factorization j U ____//_@@V @@ p| v@@ØØ@fflffl|| Y of v, where j is a trivial cofibration and p is a fibration. Form the pullback g* X xY V _____//V | p| | | fflffl| fflffl| X ___g___//_Y Then the map g* is a weak equivalence by the right properness assumption, so that the projection X xY V ! X is an object of Triv=X. The path component of this object is independent of the choices made, and is independent of the choice of representative for the path component of U ! Y . In effect, if U ______ff____//_@@U0 @@ """" v @@ØØ@~~v0""" Y is a morphism of Triv=Y and v0= p0. j0 is a factorization of v0with j0 a trivial cofibration and p0a fibration, then there is a commutative diagram j0ff U ____//_V>0>" j||!"""" |p0| fflffl|"fflffl|" V __p__//Y and so there is a commutative diagram X xY VH _____!*_____//_X xY V 0 HHH uuuu HHH uuu HH## zzuu X It follows that there is a well-defined function g0: ß0(Triv=Y ) ! ß0(Triv=X). The composite functions g0. g* and g* . g0 are both identities. |_* *__| 8 Lemma 8. Suppose that Y is an object of a right proper model category M in which the terminal object * is cofibrant. Then the function _Y : ß0(Triv=Y ) ! [*, Y ] is a bijection. Proof. By Lemma 7, it is enough to suppose that Y is fibrant. Then the function ß(*, Y ) ! [*, Y ] is a bijection since * is cofibrant. Here, ß(*, Y ) denotes homotopy classes of maps with respect to a fixed cylinder object I of *. If two maps f, g : * ! Y are homotopic, then there is a diagram * @ | @@@f d0| @@ fflfflØØ@| IO____//_YO??~ d1|| ~~g~~ | ~~ * Then the morphisms d0 and d1 are weak equivalences, so that f and g are in the same path component of Triv=Y . It follows that there is a well defined function OE : ß(*, Y ) ! ß0(Triv=Y ) and that the diagram ~= ß(*, Y )_______//[*, Y ] MMM OO MMM |_Y OEMMMM&&|| ß0(Triv=Y ) commutes. Finally, if U ! Y is an object of Triv=Y , there is a factorization j U ____//_AV AA | AAA p| AA__fflffl| * where j is a trivial cofibration and p is a trivial fibration. The fibration p * *has a section s : * ! V since * is cofibrant, and the map U ! Y extends to a map V ! Y since j is a trivial cofibration and Y is fibrant. It follows that the function OE is surjective, and is therefore a bijection. __ The map _Y is therefore a bijection if Y is fibrant. |__| 9 Lemma 9. Suppose that G is a presheaf of simplicial groups. Then there is a bijection [*, BG] ~=ß0(G - Tors0). Recall that G - Tors0 is the category of cofibrant simplicial G-presheaves X such that the map X=G ! * is a local weak equivalence. Proof. We establish the existence of a bijection ß0(Triv=BG) ~=ß0(G - Tors0). Then the desired result follows from Lemma 8. First of all, there is a function ß0(Triv=BG) ! ß0(G - Tors0) which is defined by associating a cofibrant model Z(X) of the simplicial G- presheaf X xBG EG to the object X ! BG of Triv=BG. Here, one means that a choice of trivial fibration Z(X) ! X xBG EG is made in the category of simplicial G-presheaves such that Z(X) is cofibrant. This can be done functorially since the model structure on the category of simplicial G-presheaves is cofibrantly generated. Observe that the induced map Z(X)=G ! X is a sectionwise weak equivalence since X xBG EG is G-free. Suppose that Z is a cofibrant simplicial G-presheaf such that Z=G ! * is a local weak equivalence. Then there is a G-equivariant map Z ! EG and an induced map Z=G ! BG. The class of the object Z=G ! BG in ß0(Triv=BG) is independent of the choices that have been made: any two G-equivariant maps Z ! EG are naively homotopic and so the induced maps Z=G ! BG are naively homotopic and hence represent the same element of ß0(Triv=BG). It follows that there is a well defined function ß0(G - Tors0) ! ß0(Triv=BG) and this function is the inverse of the function in ß0 which is induced by_the functor of the previous paragraph. |__| Corollary 10. There is a bijection [*, BG] ~=ß0(G - Tors1). Recall that the objects of the category G - Tors1are simplicial G-presheaves Z such that d(EG xG Z) ! * is a local weak equivalence. Lemma 9 is equivalent to Corollary 10, by Remark 6. Remark 11. If G is a sheaf of groups, then a G-torsor X is naturally a member of G - Tors1 after identification of X with a constant simplicial G-sheaf, and in this way the category G - torsof ordinary G-torsors imbeds in G - Tors1. We claim that the induced function ß0(G - tors) ! ß0(G - Tors1) (2) 10 is a bijection. Suppose that X is a simplicial G-presheaf such that the map d(EGxG X) ! * is a local weak equivalence, or that X is a member of G - Tors1. Then the canonical map d(EG xG X) ! BG is a local fibration with fibre X according to Lemma 12 below. The total space object d(EG xG X) is locally weakly equivalent to a point by assumption, so that X is non-equivariantly locally equivalent to BG ' G, where the sheaf of groups G is identified with a constant simplicial sheaf. It follows in particular that the G-equivariant map X ! ß0X is a local weak equivalence. The sheaf of groups G acts on the associated sheaf ß~0X, and the composite X ! ß0X ! ~ß0X is a G-equivariant local weak equivalence. The induced map d(EG xG X) ! EG xG ~ß0X is also a local weak equivalence, so that EG xG ~ß0X is locally equivalent to a point. This last statement means precisely that the sheaf ~ß0X is a G-torsor: the freeness of the G-action is the vanishing of the sheaf ~ß1(EG xG ~ß0X), and ß~0(EG xG ~ß0X) ~=(~ß0X)=G ~=* as a sheaf. All constructions are natural, so the function ß0(G - tors) ! ß0(G - Tors1) is a bijection with inverse specified by X 7! ~ß0X. 3 Torsors for presheaves of simplicial groupoids Write sGpd 0to denote the category of presheaves of groupoids enriched in simplicial sets, and write sGpd for the full category of presheaves of simplic* *ial groupoids. A groupoid enriched in simplicial sets is a simplicial groupoid with discrete objects, and the two ways of describing such an object will be used interchangeably. All sheaves or presheaves in this section are defined on a fix* *ed small Grothendieck site C. ___ The purpose of this section is to analyze the set of morphisms_[*, W G] for a presheaf of simplicial groupoids with discrete objects. Here, W G is the univer* *sal cocycle construction of [5] and [11] _ see also Section 4. It_is_also shown in Section 4 that there is a natural weak equivalence j : dBG ! W G, where dBG denotes the diagonal of the usual bisimplicial nerve BG. The homotopy type of dBG is also insensitive to whether or not G is a sheaf, and we shall therefore focus attention on computing [*, dBG] when G is a sheaf of groupoids with discrete objects. Joyal and Tierney have a model structure for sheaves of simplicial groupoids [9] for which a map G ! H is a weak equivalence if and only if the induced map dBG ! dBH is a local weak equivalence of simplicial sheaves. The Joyal- Tierney model structure is proper [9, Th.9]. They also show [9, Th.12] that the functor dB determines a functor dB : sGpd =G ! sShv =dBG 11 which induces an equivalence of homotopy categories. It follows from Lemma 8 that the functor dB induces an isomorphism dB : ß0(Triv=G) ~=ß0(Triv=dBG) for all sheaves of simplicial groupoids G. If one says that a map f : H ! H0 of presheaves of simplicial groupoids is a weak equivalence if the induced map dBH ! dBH0 is a local weak equivalence of simplicial presheaves, then it's clear that the functor dB and the associated sheaf functor H 7! H~ together induce a commutative diagram ß0(Triv=H) __dB//_ß0(Triv=dBH) ~=|| |~=| fflffl| ~= |fflffl ß0(Triv=H~)__dB//_ß0(Triv=dBH~) The function dB in the diagram is therefore a bijection for all presheaves of simplicial groupoids H. The following result is a restatement of a theorem of Moerdijk, specifically Theorem 2.1 of [13]. It can also be proved with the techniques used to prove the group completion theorem in [5]. As Moerdijk observes in [13], the group completion theorem is a consequence of this result. Lemma 12. Suppose that C is a category enriched in simplicial sets and that X : C ! S is a simplicial functor taking values in simplicial sets. Suppose that all arrows a ! b of C0 induce weak equivalences X(a) ! X(b). Then the map X(a) ! Fa taking values in the homotopy fibre over a of the simplicial set map d(holim---!CX) ! d(BC) is a weak equivalence. The object holim---!CX is the bisimplicial set with simplicial set G X(a0) x G(a0, a1) x . .x.G(an-1, an) (a0,a1,...,an) in horizontal degree n. In vertical degree m, it is the simplicial set holim---* *!GmXm . Corollary 13. Suppose that G is a groupoid enriched in simplicial sets, and that X : G ! S is a simplicial functor taking values in simplicial sets. Then the map X(a) ! Fa taking values in the homotopy fibre over a of the simplicial set map d(holim---!GX) ! d(BG) is a weak equivalence. A simplicialFfunctor X : G ! S can alternatively be described as simplicial set X = a2Ob(G) Xa fibred over the object set Ob (G) in the sense that there is a simplicial set map f : X ! Ob (G) which collapses summands to points. Suppose that the simplicial set X xsMor (G) is defined by the pullback diagram X xs Mor(G) _____//Mor(G) | |s | | fflffl| fflffl| X _____f____//_Ob(G) 12 where s is the source map. Then the other piece of data required for the sim- plicial functor X is a simplicial set map m : X xs Mor(G) ! X which fits into a commutative diagram X xs Mor(G) ___m___//X | | | |f fflffl| fflffl| Mor (G)____t___//Ob(G) where t is the target map. The map m must respect identities of G is an obvious way. There is a canonical diagram X _______//d(holim---!GX) (3) | | | | fflffl| fflffl| Ob (G)________//d(BG) and then Corollary 13 has the following equivalent formulation Corollary 14. Suppose that G is a groupoid enriched in simplicial sets, and that X : G ! S is a simplicial functor taking values in simplicial sets. Then the diagram (3) is homotopy cartesian. Lemma 12 can be expressed in terms of a similar homotopy cartesian dia- gram. Example 15. Suppose that H is a simplicial groupoid with discrete objects and let f : U ! H be a morphism of simplicial groupoids (U does not necessarily have discrete objects). Take a 2 Ob (H) and write f # a for the simplicial category given in degree n by the comma category fn # a arising from the functor fn : Un ! Hn. Then the functors Hn ! catgiven by a 7! fn # a define a simplicial functor dB(f # ) : H ! S. The forgetful functors fn # a ! Un also assemble to define a weak equivalence holim---!HB(f # ) ff-!BU The simplicial sets dB(f # a) therefore become identified with the homotopy fibres of the diagonal simplicial set map associated to the canonical bisimplic* *ial set map holim---!HdB(f # ) fi-!BH In öh rizontal degree" n, this map can be identified with the projection dB(f # a0) x H(a0, a1) x . .x.H(an-1, an) ! H(a0, a1) x . .x.H(an-1, an) 13 Remark 16. Suppose that C is a small category, and consider the simplicial set maps G fi G BC -ffd( B(C # x0) -! d( *) = BC x0!...!xn x0!...!xn arising from the simplicial set construction underlying Example 15. In other words the map ff is induced by the forgetful functors C # a ! C, while fi is the canonical map induced by the simplicial set maps B(C # x0) ! *. Both maps are weak equivalences. The object G X = d( B(C # x0)) x0!...!xn is the simplicial set consisting of strings (y, x) of arrows y0 ! . .!.yn ! x0 ! . .!.xn of length 2n+1 in C, and the map ff takes this string to the string y0 ! . .!.yn while fi maps this element to the string x0 ! . .!.xn. The n-simplices of the simplicial object X can therefore be identified with functors n * n ! C defined on the poset join n * n, and with simplicial structure maps induced by precomposition with maps ` * ` : m * m ! n * n. The maps ff and fi are induced by the inclusions n ! n * n of the left and right substrings of length n respectively. There is a poset map hn : n x 1 ! n * n which is defined by ( (i, ffl) 7! i if ffl = 0, and n + i if ffl = 1. As a picture, hn is the diagram y0 _____//y1___//_._._._//yn | | | | | | fflffl| fflffl| fflffl| x0 _____//x1___//_._._._//xn The maps hn are natural in ordinal numbers n. It follows that the composites n x 1 hn--!B(n * n) (y,x)---!BC together define a simplicial set map X x 1 ! BC from ff to fi. This construc- tion is natural in all small categories. Suppose that G is a presheaf of simplicial groupoids with discrete objects. A torsor for G is a simplicial functor X : G ! s Pre(C) taking values in sim- plicial presheaves such that the associated simplicial presheaf d(holim---!GX) * *is weakly equivalent to a point. A morphism f : X ! Y of G-torsors is a natural 14 transformation of simplicial functors; it may also be described as a simplicial presheaf morphism f X _________________//FY FF xxx FFF xxx F""F __xx Ob(G) which respects the G-structure. It is an immediate consequence of Corollary 14 that any such map f must be a local weak equivalence. This map also induces weak equivalences of all local choices of fibres. Write G - Tors for the category of G-torsors, and let ß0(G - Tors ) denote its set of path components. There is a well-defined function OE : ß0(G - Tors) ! ß0(Triv=dBG) which is induced by associating to a G-torsor X the element represented by the map d(holim---!GX) ! dBG. Note that the map OE is morally induced by the elts map of Joyal and Tierney [9, p.288], although it is defined on enriched diagrams of presheaves rather th* *an sheaves. There is a function _ : ß0(Triv=G) ~=ß0(Triv=dBG) ! ß0(G - Tors) which is defined as follows. Let f : U ! G be an object of Triv=G and perform the construction of Example 15 sectionwise to construct the diagram dBU oo'__ d(holim---!GdB(f # )) | | |f* | | fflffl|ff fflffl| dBG oo'__ d(holim---!GdB(G # )) '|fi| fflffl| dBG Then the simplicial G-functor a 7! dB(f # ) is a G-torsor. This construction is functorial and defines the function _. The composites fi .f* and ff.f* are homotopic by the construction of Remark 16. It follows that the canonical map fi . f* and the original map dBU ! dBG represent the same element of ß0(Triv=dBG), and so the composite OE . _ is the identity function. If X is a G-torsor, then the canonical map d(holim---!GX) ! dBG is induced by a morphism f : EG X ! G of presheaves of simplicial groupoids (where EG X is the translation category for the functor Xn : Gn ! Set in each degree. There is a G-natural functor f # a ! Xn(a) which induces a map (also a weak equivalence) dB(f # a) ! Xn(a) 15 for all n and a, and hence determines a map of G-torsors dB(f # ) ! X It follows that the composite _ . OE is the identity function. We have therefore proved the following Theorem 17. Suppose that G is a presheaf of simplicial groupoids with discrete objects. Then the natural function OE : ß0(G - Tors) ! ß0(Triv=dBG) ~=[*, dBG] is a bijection. Proof. The displayed isomorphism is a consequence of Lemma 8. The proof __ that OE is a bijection is given above. |__| Remark 18. Theorem 17 generalizes Theorem 14 of [7]. The proof of Theorem 17 also implicitly fixes an error in the proof of that result, which does not properly take into account the phenomenon discussed in Remark 16. Remark 19. A simplicial sheaf of groupoids G for which the coequalizer c(G) of the source and target maps s, t : Mor (G) ! Ob (G) Ob (G) ! ~ß0(Ob (G)) is simplicially discrete is said to be locally transitive in [8]. All sheaves * *of simplicial groupoids with discrete objects are locally transitive in this sense* *.Joyal and Tierney define a G-pseudo torsor for a locally transitive object G to be a simplicial sheaf X on which G acts freely (in each simplicial degree), and such that the colimit X=G is locally weakly equivalent to a point. All G-pseudo torsors are G-torsors in the sense of this paper for G with discrete objects, b* *ut the class of G-torsors is larger. Theorem 24 of [8] gives a homotopy classifica* *tion of pseudo-torsors, and implies that the categories of G-pseudo torsors and G- torsors have isomorphic presheaves of path components in the case where G is a simplicial sheaf of groupoids with discrete objects. 4 Universal cocycles for simplicial categories ___ The simplicial set W G for a groupoid enriched in simplicial sets (aka. simplic* *ial groupoid with discrete objects), is defined as a space of universal cocycles in [5, V.7]. We show here how to extend the definition of this construction_to all simplicial categories C, and we construct a_comparison_map j : dBC ! W C. We show in Lemma 20 that the map j : dBG ! W G is a weak equivalence for groupoids G enriched in simplicial sets. Suppose that C is a simplicial object in the category of small categories. Write EC for the following variant of the Grothendieck construction: the set of objects of EC consists of all pairs (x, n) with x 2 Cn, and a morphism (f, `) : (x, m) ! (y, n) is a pair consisting of an ordinal number map ` : m ! n and a morphism f : x ! `*y of Cm . There is an obvious forgetful functor ß : EC ! which takes values in the ordinal number category . 16 The segment category Seg(n) of subintervals [j, n] of n = [0, n] can be iden- tified with the opposite nop via the functor [j, n] 7! j. There is a functor cn : nop! which is defined by j 7! n - j. An n-cocyle taking values in the simplicial category C is a functor X : nop! EC which is a lifting of cn in the sense that the diagram of functors EC== zz Xzzzz ß|| zz fflffl| nop _cn__//_ commutes. This is a generalization of the definition of an n-cocycle taking values in a groupoid enriched in simplicial sets, in view of the identification* * of the categories Seg(n) and nop. The n-cocycle X : nop! EC is otherwise described as a string of arrows (x0, n) (x1, n - 1) . . .(xn, 0) each of which has the form (ffi, d0), with ffi : xn-i ! d0(xn-i-1). This means that the string consists of objects xi2 Cn-i and morphisms xi! d0(xi-1). Every ordinal number map ` : m ! n induces a commutative diagram ~= m - i_______//[i,_m]___//m | `i|| `i|| `| fflffl| fflffl| fflffl|| n - `(i)_~=_//[`(i),_n]_//_n and there is a corresponding diagram (`*0x`(0), m)oo___(`*1x`(1), m -o1)o_. .o.o____ (`*mx`(m), 0) (1,`0)|| (1,`1)|| (1,`m|)| fflffl| fflffl| fflffl| (x`(0), n - `(0))oo_(x`(1), n - `(1))oo._.o.o__(x`(m), n - `(m)) The string on top is denoted by `*X._ ___ In this way, a simplicial set W C is defined, with W Cn given by the set of n-cocycles in C. The functoriality follows from the relations `ø(i)øi= (`ø)i associated to composable ordinal number maps k ø-!m `-!n. There is a function ___ j : dBCn = (BCn)n ! W Cn 17 which sends a string x0 x1 . . .xn in Cn (note the Bousfield-Kan indexing [1, p.328]) to the cocycle consisting of the objects dj0xn-j 2 Cn-j and the induced morphisms dj0ffn-j : dj0xn-j ! dj0xn-j-1 = d0dn-j-10xn-j-1, or rather to the string (x0, n) (d0x1, n - 1) . . .(dn0xn, 0) in the Grothendieck construction EC . Suppose that ` : m ! n is an ordinal number map. One checks that the composite ___ `* ___ dBCn j-!W Cn -! W Cm sends the string of arrows x0 x1 . . .xn in Cn to the string (`*0d`(0)0x`(0), m) (`*1d`(1)0x`(1), m - 1) . . .(`*md`(m)0x`(m), 0) while the composite * ___ dBCn `-!dBCm -j!W Cm sends that same string in Cn to the string (`*x`(0), m) (d0`*x`(1), m - 1) . . .(dm0`*x`(m), 0). Then `*id`(i)0x`(i)= di0`*x`(i), and it follows that the maps j respect the sim* *pli- cial structure. Lemma 20. Suppose_that G is a groupoid enriched in simplicial sets. Then the map j : dBG ! W G is a weak equivalence. ___ Proof. The functors dB and W preserve homotopy equivalences_and disjoint unions. If H is a simplicial group, the map j : dBH ! W H classifies the H- bundle dEH ! dBH, and so j is a weak equivalence for simplicial groups. Every simplicial groupoid G is homotopy equivalent to a disjoint union of simplicial_ groups. |__| 5 Gerbes In homotopy theoretic terms, but according to the standard definition [2], [4], [10], a gerbe is a locally connected stack on a (small) Grothendieck site C. If G is a sheaf of groups on C, then a G-gerbe (following [2]) is a stack D such that there is a covering family U ! * of the terminal sheaf such that there are equivalences D|U ! St(G|U ) 18 for each U in the covering family, where St(G|U ) is the stack completion (stack of G|U -torsors) of the restricted sheaf of groups G|U . In this case, the stac* *k D is automatically locally connected. We can alternatively say that a stack D is a G-gerbe if there is a covering family V ! * such that there are local equivalences G|V ! D|V for each V in the cover. In effect, locally, there is an object x of (G|U ) - t* *ors which lifts to D|U up to isomorphism, any equivalence of groupoids induces isomorphisms of automorphism groups, and the sheaf of automorphisms of any G-torsor is isomorphic to G. A presheaf of groupoids E is said to be a G-gerbe if there is a covering W ! * of the terminal object by objects of C such that there are local weak equivalences G|W ! E|W for each W in the covering. Write G - gerbe for the corresponding category of G-gerbes and morphisms E ! E0 of presheaves of groupoids which are local weak equivalences. We shall only be interested in local weak equivalence classes of G-gerbes, so it will be irrelevant whether our gerbes are sheaves or preshea* *ves of groupoids. Note that the definition of G-gerbe works equally well when G is a presheaf of groups, and that the following is easily proved: Lemma 21. Suppose that G is a presheaf of groups, and let ~Gdenote its as- sociated sheaf. Then the natural functor ~G- gerbe ! G - gerbe defined by restriction of structure induces a bijection ß0(G~- gerbe) ~=ß0(G - gerbe). If H is a group, write Aut(H) for the 2-groupoid with one object, a 1-cell for each automorphism of H and a 2-cell for each homotopy (conjugation by an element of H) between automorphisms. One can check that the object Aut(H) is a group object in groupoids, so the simplicial groupoid (automorphisms and all their strings of homotopies) corresponding to Aut(H) is a simplicial group, and there is a natural inclusion Aut(H) hom (BH, BH). We shall identify the 2-groupoid Aut(H) with this simplicial group. In fact, Aut(H) can be characterized as the subcomplex of hom (BH, BH) which consists of those n-simplices (functors) H x n ! H whose vertices are automorphisms. Observe that the evaluation hom (BH, BH) x BH ! BH restricts to an action Aut(H) x BH ! BH 19 of the simplicial group Aut(H) on the nerve BH. Suppose that G and G0 are presheaves of groupoids. The simplicial set equi (G, G0) is the subobject of hom (BG, BG0) consisting of all functors G x n ! G0such that all restrictions to vertices G ~=G x 0 1xi--!G x n ! G0 are local equivalences of presheaves of groupoids. Any f : G ! G0 which is homotopic to a local weak equivalence must be a local weak equivalence, so it suffices that there is some restriction to a vertex which is a local weak equivalence. It follows also that the simplicial set equi(G, G0) is the nerve of a groupoid whose objects are the local weak equivalences G ! G0 and whose morphisms are the homotopies between them. The simplicial presheaf Equi(G, G0) is defined by Equi (G, G0)(U) = equi(G|U , G0|U ) for each object U of the underlying site C. Then Equi(G, G0) is the nerve of a presheaf of simplicial groupoids, in an obvious way. Write also Aut (G) = Equi(G, G), and aut(G) = equi(G, G). Then Aut (G)(U) = aut(G|U ) for each object U of the site C. Lemma 22. Suppose that G is a sheaf of groups with associated stack morphism j : G ! St(G). Then the map j induces local weak equivalences * Equi (G, G) j*-!Equi(G, St(G)) -j Equi (St(G), St(G)). Proof. Note first of all that any local weak equivalence G ! G0 of fibrant presheaves of groupoids is a homotopy equivalence, since the associated map BG ! BG0is a homotopy equivalence. It follows that if G ! G0is a different choice of fibrant model for G, then there is a map G0! St(G) which induces a homotopy equivalence equi(G, G0) ! equi(G, St(G)) It follows that the induced map Equi(G, G0) ! Equi(G, St(G)) is a sectionwise weak equivalence. 20 We can therefore assume that St(G) is a sheaf of groupoids. Write * for the image of the unique object of G under j (the trivial torsor). Then since j is a local weak equivalence and St(G) is a sheaf of groupoids the induced map j* : G ! hom (*, *) is an isomorphism of sheaves of groups. Suppose that f : G ! St(G) is a local equivalence of sheaves of groupoids, and let x = f(*) be the image of * in global sections. Then St(G) is locally connected, so there is a covering family of objects U ! * of C, and a morphism x|U ! * for each member of the covering family. Then f|U is homotopic to a composite of the form 0 G|U -f!G|U -j!St(G)|U for each U in the covering family. This is true in all sections, so it follows * *that the induced sheaf map ~ß0Equi(G, G) ! ~ß0Equi(G, St(G)) is an epimorphism. This map is also a monomorphism, on account of the sheaf isomorphism G ~=hom (*, *) which is induced by j. That same sheaf isomor- phism induces a sheaf of fundamental groups isomorphism ~ß1(Equi (G, G), ff) ~=~ß1(Equi (G, St(G), jff) for all (local) choices of base points ff. The map Equi(G, G) j*-!Equi(G, St(G)) is induced by a morphism of presheaves of groupoids, and is therefore a map of presheaves of Kan complexes. It follows that j* is a local weak equivalence. The diagram Equi (St(G), St(G))___//_Hom(B(St(G)), B(St(G))) j*|| j*|| fflffl| fflffl| Equi (G, St(G))________//Hom(BG, B(St(G))) is a pullback since j is a local equivalence. The map j* of function complex presheaves is a trivial fibration since j is a trivial cofibration and the simp* *licial presheaf B(St(G)) is fibrant. It follows that the map j* : Equi(St(G), St(G)) ! Equi(G, St(G)) is a trivial fibration, and is therefore a local weak equivalence. * *|___| 21 Suppose that H is a simplicial group and that simplicial sets X and Y are chosen such that X has a right H-action and Y has a left H-action. Then H acts on the left on X x Y via (g, (x, y)) 7! (xg-1, gy), and the resulting bisimplicial set EHxH (XxY ) has horizontal path components isomorphic to X xH Y (balanced product), where X xH Y = (X x Y )= and the indicated equivalence relation is generated by the relation (xg, y) ' (x, g* *y). Note that HxH Y ~=Y in the special case where the group H is interpreted as having a right H-action by the group multiplication. In this case, the canonical map d(EH xH (H x Y )) ! H xH Y ~=Y is a weak equivalence. In effect, the path component in EH xH (H x Y ) corre- sponding to a fixed vertex (e, x) has objects consisting of all pairs (g, g-1x), and the map g : (g, g-1x) ! (e, x) is uniquely determined. The function H ! H x Y which is defined by g 7! (g, g-1x) induces an isomorphism of categories of EH xH H (right action) with the path component of (e, x). It follows that if H acts freely on X, then the map d(EH xH (X x Y )) ! X xH Y is a weak equivalence. The corresponding (opposite) simplicial group Ho is obtained by reversing all arrows in H all simplicial degrees. A right (aka. contravariant) action of the simplicial group H on a simplicial set X corresponds to a left action Ho x X ! X. Suppose that G is a sheaf of groups and that F is a right Aut (G)-torsor, meaning (see Remark 5) that F is a cofibrant Aut (G)o-object, and the map F=Aut (G) ! * is a local weak equivalence.In particular, F has a free right Aut (G)-action. Remark 4 and Lemma 9 together imply that there are bijections ___ o o [*, dB(Aut (G)o)] ~=[*, W (Aut (G) )] ~=ß0(Aut (G) - Tors). The remainder of this section consists of the proof of Theorem 23, which asserts that these objects are in bijective correspondence with the set ß0(G - gerbe) of path components (ie. local equivalence classes) of G-gerbes. The simplicial sheaf of groups Aut (G) acts on the simplicial sheaf BG via the composition Aut(G) x BG ! Hom (BG, BG) x BG ev-!BG where ev is the evaluation map. The canonical map d(EAut (G) xAut(G)(F x BG)) ! F xAut(G)BG (4) is a local weak equivalence by the previous paragraphs. 22 Since the map F=Aut (G) ! * is a local weak equivalence, there is a covering family of maps U ! * with U 2 C such that there are sections F?? oe~~~|~ ~~ | ~ fflffl| U _____//* These sections induce Aut (G)-equivariant equivalences (maps of right torsors) oe* : Aut (G|U ) ! F |U for all maps U ! * in the covering family. The induced maps of balanced products BG|U ~=Aut (G|U ) xAut(G|U)BG|U ! F |U xAut(G|U)BG|U are local weak equivalences for all U ! * in the covering family by the previous paragraphs, so that F xAut(G)BG is locally equivalent to BG. It follows that the stack completion St(ß(F xAut(G)BG)) of the corresponding fundamental groupoid is a G-gerbe. The fact that the maps (4) are weak equivalences for all right Aut (G)-torsors also implies that any map F ! F 0of Aut (G)-torsors induces a local weak equivalence St(ß(F xAut(G)BG)) ! St(ß(F 0xAut(G)BG)) of G-gerbes. Suppose that E is a G-gerbe, interpreted as a stack which is locally equival* *ent to G. Then there is a covering family U ! * by objects U 2 C such that there are equivalences ffU : G|U ! E|U for all U ! * in the covering family. Since E|U is a stack there are equivalences G|U -Tors ! E|U such that the diagrams G|U ___ffU__//_E|U99 sss j || ssss0 fflffl|ffUsss G|U - Tors commute. In the composite 0 Equi(G|U , G|U ) j*-!Equi(G|U , G|U - Tors) ffU*--!Equi(G|U , E|U ) the map ff0U*is a homotopy equivalence since ff0Uis a weak equivalence of stack* *s, and the map j* is a local weak equivalence by Lemma 22. These maps are equivariant for the action by Equi (G|U , G|U ) on the right. It follows that t* *he map EAut (G) xAut(G)Equi (G, E) ! * is a local weak equivalence, so that Equi (G, E) represents a right Aut (G)- torsor. The corresponding torsor is an Aut (G)-cofibrant model ß : Equi(G, E)c ! Equi(G, E). 23 Note that the cofibrant object Equi (G, E)c has a free Aut (G)-action, so that any restriction Equi(G, E)c|U has a free Aut (G|U )-action. Suppose that there is a covering family U ! * of objects U 2 C such that there are local weak equivalences ffU : G|U ! E|U for all U ! * in the covering family. Then there is a diagram Equi(G, G)c|U xAut(G|U)BG|U___'__//Equi(G, E)c|U xAut(G|U)BG|U ' || || fflffl| fflffl| Equi (G|U , G|U ) xAut(G|U)BG|U___//Equi(G|U , E|U ) xAut(G|U)BG|U ~=|| || fflffl| fflffl| BG|U ______________'_____________//_BE|U The top horizontal map is a local weak equivalence since the map Equi (G, G)c ! Equi(G, E)c is a local weak equivalence of simplicial presheaves having free Aut (G)-action* *s. Similarly, the map Equi(G, G)c ! Equi(G, G) is a weak equivalence of simpli- cial presheaves having free Aut (G) actions, so the corresponding vertical map is a weak equivalence. It follows that the induced composite Equi (G, E)cxAut(G)BG ! Equi(G, E) xAut(G)BG ! BE determined by the evaluation map is a local weak equivalence. In particular, there is an induced natural local equivalence St(ß(Equi (G, E)cxAut(G)BG)) ! E of stacks. Suppose that F is a right Aut (G)-torsor. Then there is an Aut (G)-equivar- iant map F ! Equi(G, St(ß(F xAut(G)BG))) (5) which is adjoint to the canonical map F xAut(G)BG ! B St(ß(F xAut(G)BG)). Locally, the map (5) has the form Aut (G) ! Equi(G, St(ß(Aut (G) xAut(G)BG))) (6) Thus, if we show that all instances of (6) are local weak equivalences, then all instances of (5) are local weak equivalences. The evaluation isomorphism Aut (G) xG BG ! BG is adjoint to the iso- morphism (identification) Aut (G) ! 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