Algebraic geometry over model categories A general approach to derived algebraic geometry Betrand Toen Gabriele Vezzosi Laboratoire J. A. Dieudonn'e Dipartimento di Matematica UMR CNRS 6621 Universit`a di Bologna Universit'e de Nice Sophia-Antipolis Italy France October 9, 2001 Abstract For a (semi-)model category M, we define a notion of a öh motopy" Groth* *endieck topology on M, as well as its associated model category of stacks. We use* * this to define a notion of geometric stack over a symmetric monoidal base model ca* *tegory; geometric stacks are the fundamental objects to öd algebraic geometry ove* *r model categories". We give two examples of applications of this formalism. The f* *irst one is the interpretation of DG-schemes as geometric stacks over the model cat* *egory of complexes and the second one is a definition of 'etale K-theory of E1 -rin* *g spectra. This first version is very preliminary and might be considered as a det* *ailed research announcement. Some proofs, more details and more examples will be added i* *n a forthcoming version. Key words: Stacks, model categories, E1 -algebras, DG-schemes. Contents 1 Introduction 3 2 Stacks over model categories 13 2.1 The Yoneda embedding for semi-model categories . . . . . . . . . . . . * *. . . 14 2.2 Grothendieck topologies on semi-model categories . . . . . . . . . . . .* * . . . 23 2.3 Homotopy hypercovers . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . 24 2.4 The model category of stacks . . . . . . . . . . . . . . . . . . . . . * *. . . . . 25 2.5 Exactness properties of the model category of stacks . . . . . . . . . * *. . . . 28 2.6 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . 31 1 3 Stacks over E1 -algebras 33 3.1 Review of operads and E1 -algebras . . . . . . . . . . . . . . . . . .* * . . . . 34 3.2 Geometric stacks over a monoidal model category . . . . . . . . . . . . * *. . . 37 3.3 An example: Quotient stacks . . . . . . . . . . . . . . . . . . . . . .* * . . . . 40 4 Applications and perspectives 42 4.1 An approach to DG-schemes . . . . . . . . . . . . . . . . . . . . . . * *. . . . 42 4.2 E'tale K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . 47 2 1 Introduction By definition, a scheme is obtained by gluing together affine schemes for the Z* *ariski topol- ogy. Therefore, algebraic geometry is a theory which is based on the two funda* *mental notions of affine scheme and Grothendieck topology. It was observed already a l* *ong time ago that these two notions still make sense in more general contexts, and that * *schemes can be defined in very general settings. This has led to the theory of relative* * algebraic ge- ometry, which allows one to do algebraic geometry over well behaved symmetric m* *onoidal base categories (see [De1 , De2, Ha ]); usual algebraic geometry corresponds th* *en to the ä bsolute" case where the base category is the category of Z-modules. The goal of the present work is to start a program to develop algebraic geometr* *y relatively to symmetric monoidal 1-categories. Our motivations for starting such a program* * come from several questions in algebraic geometry and algebraic topology and will be* * clari- fied in the two entries Examples and applications and Relations with other work* *s of this introduction. It is well known that model categories give rise in a natural way to 1-categ* *ories. Indeed, B. Dwyer and D. Kan defined a simplicial localization process, which st* *arting from a model category M, constructs a simplicial category LM, the simplicial lo* *calization of M (see [D-K1 ]). As simplicial categories may be viewed as 1-categories for* * which i-morphisms are invertible up to (i + 1)-morphisms for all i > 1 (see [H-S , x2* *]), this suggests that model categories are a certain kind of 1-categories. In the same * *way, sym- metric monoidal model categories (as defined in [Ho , x4]) are a certain kind o* *f symmetric monoidal 1-categories (e.g. in the sense of [To1 ]). As a first step in our p* *rogram we would like to present in this paper a setting to do algebraic geometry relative* *ly to sym- metric monoidal model categories. For this, we will concentrate on defining a c* *ategory of geometric stacks over a base symmetric monoidal model category, whose construct* *ion will be the main purpose of this work. Review of usual algebraic geometry In order to explain our approach, we shall first present in detail a constru* *ction of the usual category of schemes, or more generally of algebraic stacks and of n-geome* *tric stacks (see [S1]), emphasizing the categorical ingredients needed in each step, in suc* *h a way that the generalization that will follow should look fairly natural. The starting point is the category Aff of affine schemes. By definition, we * *will take Aff to be the opposite of the category of commutative and unital rings. We cons* *ider its Yoneda embedding h : Aff -! Aff^ , where Aff^ is the category of presheaves of simplicial sets on Aff (i.e. Aff* *^ := SPr(Aff)) and h maps an affine scheme X to hX := Hom(-, X) (here a set is alwa* *ys considered as a constant simplicial set). From a categorical point of view, the* * embedding h : Aff -! Aff^ is obtained by formally adding homotopy colimits to Aff (see [D* *u2 ] for more details on this point of view). This process is relevant to our situat* *ion, as gluing objects in Aff will be done by taking certain formal homotopy colimits of objec* *ts in Aff 3 (i.e. taking homotopy colimits in Aff^ of object in Aff). The category Aff^ i* *s in a natural way a model category, where equivalences are defined objectwise, and th* *e functor h induces a Yoneda embedding on the level of the homotopy categories h : Aff -! Ho(Aff^ ). Throughout this introduction, the homotopy category Ho(C) of a category C with * *a dis- tinguished class of morphisms w will denote the category obtained from C by for* *mally inverting all morphisms in w; when C is a model category, we will implicitly as* *sume that w is the set of weak equivalences and when C does not come naturally equipped w* *ith a model category structure we will consider it as a trivial model category where * *w consists of all the isomorphisms. The next step is choosing a Grothendieck topology on Aff, that will be used * *to glue affine schemes. For the purpose of schemes, the Zariski topology is enough, but* * 'etale or even faithfully flat and quasi-compact (for short ffqc) topologies also proves * *very useful in order to define more general objects as algebraic spaces or algebraic stacks* *. We will choose here to work with the ffqctopology though the construction will be valid* * for any Grothendieck topology on Aff. The ffqctopology makes Aff into a Grothendieck si* *te and therefore one can consider its category of (1-)stacks, denoted by Ho(Aff~,f* *fqc). For us, the category Ho(Aff~,ffqc) is the full sub-category of Ho(Aff^ ) consis* *ting of simplicial presheaves satisfying the descent condition for ffqchyper-covers 1. * *The category Ho(Aff~,ffqc) is precisely the homotopy category of a certain model category st* *ructure on Aff^ , and the category Aff^ together with this model structure will be called * *the model category of stacks, and denoted by Aff~,ffqc. Finally, it is known that the top* *ology ffqc is sub-canonical, or in other words that the Yoneda embedding h : Aff -! Ho(Aff* *^ ) factors through Ho(Aff~,ffqc). Therefore we have an induced fully faithful func* *tor h : Aff -! Ho(Aff~,ffqc). A stack in the essential image of the functor h will be called by extension an * *affine scheme. Let us consider now a simplicial object X* : op- ! Aff~,ffqc(i.e. X* is a b* *i-simplicial presheaf) and suppose that X* satisfies the following three conditions> 1. The simplicial object X* is a Segal groupoid in Aff~,ffqc(see Def. 3.3.1); 2. The image of each Xn in Ho(Aff~,ffqc) is a disjoint union of affine scheme* *s; 3. The two morphisms (source and target) X1 ' X0 are faithfully flat and affi* *ne morphisms (this makes sense since the Xn's are disjoint union of affine sc* *hemes). To such a groupoid we associate its homotopy colimit |X*| 2 Ho(Aff~,ffqc), w* *hich can be defined to be the stack associated to the diagonal of the bi-simplicial pres* *heaf X*. It is not difficult to check that the full sub-category of Ho(Aff~,ffqc), consisti* *ng of objects isomorphic to some |X*|, with X* satisfying conditions (1), (2) and (3), is equ* *ivalent to the homotopy category of algebraic stacks (in the sense of Artin, see [La-Mo ])* * having an affine diagonal. In particular, it contains the category of separated scheme* *s as a full _______________________________1 For us the word stack will always mean a stack of 1-groupoids, adopting the * *point of view of [S2] and [To2], according to which stacks of 1-groupoids are modelled by simplicial pres* *heaves. 4 sub-category. Iterating this constructions as in [S1], one can also construct t* *he homotopy category of geometric n-stacks (which for n big enough contains the homotopy ca* *tegory of general algebraic stacks as a full sub-category ). This is precisely the construction we will imitate in defining our category of * *geometric stacks over a symmetric monoidal model category. Geometric stacks over symmetric monoidal model categories The previous construction of the homotopy category of algebraic stacks is pu* *rely cat- egorical. Indeed, it starts with the symmetric monoidal category (Z - mod, ),* * of Z- modules. The category Aff of affine schemes is then the opposite of the categor* *y of com- mutative and unital monoids in the symmetric monoidal category (Z-mod, ), whic* *h is a categorical notion. Furthermore, the notion of a topology on Aff is also catego* *rical. Our goal is to extend this categorical construction to the case where (Z - mod, ) * *is replaced by a general symmetric monoidal model category C (in the sense of [Ho , x4]). O* *f course, we want to keep track of the homotopical information contained in C and we will* * therefore require our constructions to be invariant when replacing C by a Quillen equival* *ent sym- metric monoidal model category. Let us start with a base symmetric monoidal model category (C, ) and try to* * imitate the construction of algebraic stacks we have presented. The first step is to fi* *nd a reasonable analog of the category of commutative and unital rings. It has been known since* * a long time by topologists that the correct analog of commutative rings in a homotopical co* *ntext is the notion of E1 -algebra (see for example [E-K-M-M , Hin, Sp]). This notion is a * *generalization of the notion of commutative monoid adapted to the case of symmetric monoidal m* *odel categories. In particular it makes sense to consider the category E1 - Alg(C),* * of E1 - algebras in C. Furthermore, it is proved in [Ber-Moe , Hin, Sp] that E1 - Alg(C* *) carries a natural model category structure2. Then, by analogy with the case of usual al* *gebraic geometry, we simply define the model category C - Aff, of affine stacks over C,* * to be the opposite of the model category of E1 -algebras in C. It is reasonable to denote* * by Spec A the object of C - Aff corresponding to a E1 -algebra A. The reader should note * *that if the model structure on C is trivial (i.e. equivalences are isomorphisms), then * *C - Aff is nothing else than the usual category of commutative and unital monoids in C, to* *gether with the trivial model structure. In particular, if C = Z - mod (endowed with t* *he trivial model structure), the category C - Aff is the usual category of affine schemes. Our next step is to define an analog of the Yoneda embedding for C - Aff. M* *ore generally, the problem is to find a good analog of the Yoneda embedding for a m* *odel category M. Of course, as an abstract category M possesses the usual Yoneda emb* *edding, but this construction is not suited for our purposes as it is not an invariant * *of the Quillen equivalence class of M (for example, it does not induces an embedding of the ho* *motopy category Ho(M)). To solve this problem, we define a model category M^ which tak* *es into account and depends on the model structure on M. The underlying category of M^ * *is as _______________________________2 To be very precise, at this point one needs the weaker notion of semi-model * *category, but we will neglect this technical subtlety in this introduction. 5 usual the category of simplicial presheaves SPr(M); however, the model category* * structure we consider on M^ is such that its fibrant objects are exactly objectwise fibra* *nt simplicial presheaves F : Mop -! SSet sending weak equivalences in M to weak equivalences of simplicial sets. Technically, M^ is defined as the left Bousfield localiza* *tion of the objectwise model structure with respect to the equivalences in M (see Def. 2.1.* *1). The construction M 7! M^ has then the property of sending Quillen equivalences to Q* *uillen equivalences (see Prop. 2.1.5). Furthermore, using mapping spaces in the model * *category M, we construct a functor h_: M -! M^ , which roughly speaking sends an objects* * x to the simplicial presheaf y 7! MapM (y, x), MapM (-, -) denoting the mapping spac* *e. This functor can be right derived into a fully faithful functor Rh_: Ho(M) -! Ho(M^ * *), which will be our öh motopical" Yoneda embedding for the model category M (see Thm. 2* *.1.13). To go further one has to introduce a good notion of Grothendieck topology on* * C - Aff and an associated notion of stack. As in the previous step, we approach, more g* *enerally, the problem of defining what is a homotopy meaningful Grothendieck topology ø o* *n a general model category M and what is the associated model category of stacks M~* *,fi, in such a way that for trivial model structures (i.e. when the weak equivalences a* *re exactly the isomorphisms) one finds back the usual notions. For this, we introduce a n* *otion of öh motopy" Grothendieck topology on a model category using the point of view of* * pre- topologies (see Def. 2.2.1). The idea is to give the usual data of ø-coverings * *at the level of the homotopy category Ho(M) and require the usual conditions of stability with * *respect to isomorphisms and composition in Ho(M) itself while the requirement of stabil* *ity under fibred products in Ho(M) is replaced with the requirement of stability under ho* *motopy fibred products. Therefore, the data of coverings for a topology ø on M are de* *fined in Ho(M) while the conditions these data have to satisfy are given at the "higher * *level" in M itself. This is completely natural from an homotopic point of view and one o* *btains almost, but not exactly, a usual Grothendieck topology on Ho(M). We may call th* *e pair (M, ø) a model site. A stack is then naturally defined as an object in Ho(M^ ) * *satisfying a reasonable descent condition with respect to ø-hypercoverings (see Def. 2.3.* *1). We actually define a model category of stacks M~,fias a certain left Bousfield loc* *alization of the model category M^ with respect to a set Sfiof maps in M^ determined by t* *he topology. Let us come back to our model category of affine stacks C - Aff. We suppose * *that we have chosen a topology ø on C - Aff which is sub-canonical, in the sense tha* *t the Yoneda embedding Rh_factors through Ho(C - Aff~,fi) ,! Ho(C - Aff^ ). Therefor* *e, Rh_induces a full embedding Rh_: Ho(C - Aff) -! Ho(C - Aff~,fi), and objects in the essential image of this functor will naturally be called affine stacks over* * C. The definition of geometric stacks over C is then straightforward. One consider si* *mplicial objects X* : op -! C - Aff~,fi, which are Segal groupoids such that X0 is a di* *s- joint union of affine stacks and with X1 ' X0 affine ø-coverings. The homotopy * *colimit |X*| 2 Ho(C -Aff~,fi) of such a simplicial object will be called a 1-geometric * *stack over C for the topology ø. Iterating this construction as in [S1], one also defines n-* *geometric stacks over C. The sub-category of Ho(C - Aff~,fi), consisting of n-geometric stacks f* *or some n, will be our setting to do algebraic geometry over the symmetric monoidal model * *category C. The following table offers a synthesis of our construction showing how it pa* *rallels the classical constructions in algebraic geometry. 6 Algebraic_Geometry_over___Z-mod__ ||Algebraic_Geometry_over_a_model_c* *ategory___ | | Base Category: (Z - mod, ) | Base Category: C = (C, ) | alg = Commutative algebras in(Z - mod, ) | Alg = E1 - algebrasinC | Aff = algop= Affine Schemes over(Z - mod, ) | C - Aff := (Alg)opp= Affine sta* *cks overC | Aff^ | C - Aff^ | Yoneda embedding : | öH motopy" Yoneda embedding* * : Aff ,! Aff^ | Ho(C - Aff) ,! Ho(C - Aff^ ) | ø : Grothendieck topologyAonff | ø : öH motopy" topology onC * *- Aff | Category of stacks: | Category of öh motopy" sta* *cks: Ho(Aff~,fi) | Ho(C - Aff~,fi) | Algebraic stacksHino(Aff~,fi) : |X*| |Geometric stacks inHo(C - Aff~* *fi) : |X*| Examples and applications The construction outlined above of the category of n-geometric stacks over a* * symmet- ric monoidal model category C has found his motivations in various questions co* *ming from algebraic geometry, algebraic topology an the recent rich interplay between the* *m . Among them, we describe below those which were the most influential for us. The first* * two are investigated in this paper. 1. Extended or derived moduli problems. Some of the moduli spaces arising in* * Alge- braic Geometry turns out to be non smooth (e.g. the moduli stack of vector* * bundles over a variety of dimension greater than one) and this maybe considered as* * a non- natural phenomenon. To overcome this difficulty, the current general phil* *osophy (see [Ko ],[Ka ], [Ci-Ka1]) teach us to consider the usual moduli spaces c* *onsidered so far as a truncation of an extended or derived moduli space. The non smoot* *hness would then arise from the fact that one is only considering this truncatio* *n instead of the whole object. The usual approach to these extended moduli spaces is* * through DG-schemes (e.g. [Ka , Ci-Ka1]). However, it was already noticed that the * *homo- topy category of DG-schemes might be not very well suited for the functori* *al point of view on derived algebraic geometry. Quoting [Ci-Ka2], 7 "Similarly to the case of the usual algebro-geometric moduli spaces, it wo* *uld be nice to characterize RHilb and RQuot in terms of the representability of some f* *unctors. This is not easy, however, as the functors should be considered on the der* *ived cat- egory of dg-schemes (with quasi-isomorphisms inverted) and for morphisms i* *n this localized category there is currently no explicit description. The issue s* *hould be prob- ably addressed in a wider foundational context for dg-schemes in our prese* *nt sense by means of gluing maps which are only quasi-isomorphisms on pairwise inte* *rsec- tions, satisfying cocycle conditions only up to homotopy on triple interse* *ction etc." Our personal way of understanding the "issue" referred to in this quotatio* *n is by stating that DG-schemes should be interpreted as geometric stacks over the* * symmet- ric monoidal 1-category of complexes. As a first evidence for this, we wil* *l produce a functor : Ho(DG - Sch) -! Ho(C(k) - Aff~,ffqc), where C(k) is the symmetric monoidal model category of complexes of k-modu* *les (for any commutative and unital ring k), Ho(DG - Sch) is the homotopy cate* *gory of DG-schemes over k 3 and ffqcis a certain extension of the faithfully fl* *at and quasi-compact topology from usual k-algebras to E1 -algebras in C(k). We * *prove furthermore that takes values in the category of geometric stacks and we* * conjecture it is fully faithful. In a forthcoming version, we will also give an interpretation in our setti* *ng of the notion of injective resolution of BG defined in [Ka ]. 2. Brave New Algebraic Geometry. Since the recent progress in stable algebrai* *c topol- ogy that led to a satisfying theory of spectra as a monoidal model categor* *y (see [E-K-M-M ], [Ho-Sh-Sm ], [Ly ]), it has become clear that one is actually* * able to do usual commutative algebra on commutative monoid objects in these categorie* *s, the so called brave new rings. It seems therefore natural to try to embed this* * brave new commutative algebra in a brave new algebraic geometry i.e. in a kind of al* *gebraic geometry over (structured) spectra. This could give new insights in Ellipt* *ic Coho- mology and Topological Modular Forms, theories for which the interplay bet* *ween geometry and topology already proved rich and powerful (see for example [G* *-H ], [Hop ], [AHS ], [Str]). As an example of application of our theory to this circle of ideas, we wil* *l use the category of stacks over the symmetric monoidal model category of symmetric* * spectra in order to define the notion of 'etale K-theory of an E1 -ring spectrum. * *We are not sure to deeply understand the issue of such a construction but it certainl* *y gives an answer to a question pointed out to us by P.A. Ostvær. To be more precise,* * we will define an 'etale topology on Sp - Aff, the model category of affine stack* *s over the model category of symmetric spectra. Then, sending each E1 -ring spectrum * *to its K-theory space (as defined for example in [E-K-M-M , xV I]) gives rise to* * a simplicial _______________________________3 Using E1 -algebra structures, the definition of DG-schemes given in [Ci-Ka1]* * can be generalized over an arbitrary ring k (see Def. 4.1.3). 8 presheaf K : Sp - Aff -! SSet Spec A -! K(A), and therefore to an object K 2 Sp - Aff~,'et. This object is in general * *not fi- brant (because it does not satisfy the descent condition for 'etale hyperc* *overings) and therefore we define for Spec A 2 Sp - Aff, K'et(A) := RK(Spec A), whe* *re RK is a fibrant replacement of K in the model category Sp - Aff~,'et. The* * space K'et(A) comes equipped with a natural localization morphism K(A) -! K'et(A* *). 3. Higher Tannakian duality. In the preliminary manuscript [To1 ], 1-Segal (o* *r simpli- cial) Tannakian categories were introduced in order to extend to higher ho* *motopy groups the algebraic theory of fundamental groups. The general idea was t* *o re- place in the usual Tannakian formalism the base symmetric monoidal categor* *y of vector spaces by the symmetric monoidal 1-category of complexes. Furtherm* *ore, as relative algebraic geometry has found interesting applications in the T* *annakian formalism (see [De1 ]), it should not be surprising that algebraic geometr* *y over the 1-category of complexes is relevant to higher Tannakian theory. As an exam* *ple of this principle, we will use our notion of geometric stacks over the symmet* *ric monoidal model category of complexes over some ring k in order to define the notion* * of affine 1-gerbes. We start with the symmetric monoidal model category C(k) of complexes over* * k, together with a Grothendieck topology ø on C(k) - Aff that will be assumed* * to be sub-canonical. In practice, the choice of the topology ø is a very imp* *ortant issue, but we will avoid going into these kind of considerations here. We* * consider Gp(C(k) - Aff~,fi), the category of group objects in the model category of* * stacks. For each G 2 Gp(C(k) - Aff~,fi), one can form its classifying simplicial p* *resheaf BG 2 Ho(C(k) - Aff~,fi). In the case where the underlying stack of G is a* *ffine and the morphism G -! * is a ø-covering, the classifying stack BG is a 1-g* *eometric stack. Stacks of the form BG for G satisfying the above conditions will b* *e called neutral affine gerbes over C(k), or neutral affine 1-gerbes over k (this d* *efinition depends on the topology ø). As the usual neutral Tannakian duality study n* *eutral affine gerbes (see [Sa]), neutral affine gerbes over C(k) are the basic ob* *ject of study of higher Tannakian duality. In the future, the higher Tannakian formalism* * will be developed consistently as a certain kind of algebraic geometry over C(k). Relations with other works The first work we would like to mention is K. Behrend's recent work on diffe* *rential graded schemes (see [Be ]). We learned about it in one of his talks during fall* * 2000 at the MPI in Bonn. Though our interpretation of DG-schemes, as geometric stacks over* * the model category of complexes, is similar to his own approach, the two works seem* * totally independent and it is not clear to us how the two approaches can be compared an* *d to which extent they are really equivalent. There are also some relations with several works on E1 -algebras, in which a* *lready some standard geometrical constructions were investigated, as for example the c* *otangent 9 complex, the tangent Lie algebra, the K-theory and Hochschild cohomology spectr* *um, Andr'e-Quillen cohomology etc. (see [E-K-M-M , Hin, G-H ]). We are quite convi* *nced that all these constructions can be generalized naturally to our setting of geometri* *c stacks over general model categories and will allow in future to talk about the cotangent c* *omplex, the Lie algebra, the cohomology or K-theory, Andr'e-Quillen cohomology etc. of * *a general geometric stack. We have already mentioned at the beginning of this introduction that our app* *roach is a first approximation of what we think algebraic geometry over symmetric mon* *oidal 1-categories should be. Using the theory of Segal categories introduced by Z. T* *amsamani and C. Simpson ([Ta , H-S]), it is possible to develop such a theory without an* *y consider- ations on model categories. However, in order to compare construction of the ca* *tegory of geometric stacks presented in this paper to a purely 1-categorical construction* *, one needs very strong version of strictification results (as for example in [H-S , x18]).* * These results are already partially proved, and are part of the foundational development of t* *he theory of higher categories. There is no doubt that the combined two approaches, toget* *her with a comparison theorem allowing to pass from the world of 1-categories to the world* * of model categories, will be a very powerful tool, allowing much more naturality and man* *ageability. As an example, let us mention that a first consequence of such a unified theory* * would be a 1-categorical interpretation of the theory of E1 -algebras as commutative mon* *oids in symmetric monoidal 1-categories. Such considerations appeared already in T. Lei* *snter's work on up-to-homotopy monoid structures ([Le]) and in the first author's prepr* *int [To1 ]. As we have already stressed, there are some applications of our theory to th* *e conjectural higher Tannakian formalism described in [To1 ]. In particular, the theory of af* *fine stacks and schematic homotopy types of [To2 ], as well as its application to non-abeli* *an Hodge theory in [Ka-Pa-To ], can be interpreted in terms of algebraic geometry over t* *he model category of complexes (at least in characteristic zero). Finally, as explained in his letter [M ], Y. Manin's idea of a "secondary qu* *antization of algebraic geometry" seems to be part of algebraic geometry over the symmetric m* *onoidal model category of motives (for example that defined in [Sp]) but our ignorance * *of this subject does not allow us to say more. However, using our notion of geometric * *stacks over a model category, we are able to define an interesting candidate for the m* *otive of an algebraic stack (in the sense of Artin) as a 1-geometric stack over the mode* *l category of motives. This construction, which was suggested to us by the lecture of [M * *], might be closely related to the subject of secondary quantized algebraic geometry and* * will be hopefully investigated in a future work. 10 Organization of the paper In the first Section of the paper we develop the theory of öh motopical" Gro* *thendieck topologies over model categories and the associated theory of stacks. We first * *define the Yoneda embedding of a model category, then we introduce the notion of topology * *and construct the model category of stacks. In the second Section, we define and investigate the notion of geometric sta* *ck over a symmetric monoidal model category. For this, we apply the theory of stacks deve* *loped in the first Section to the model category of E1 -algebras in a base symmetric mon* *oidal model category. We define inductively the notion of n-geometric stack and give a char* *acterization by means of Segal groupoids as explained in this introduction. Finally, in the third Section, we give two applications of the present theor* *y of algebraic geometry over model categories. We first explain how DG-schemes may be interpre* *ted as geometric stacks over the model category of complexes and we conclude by defini* *ng of the 'etale K-theory space of an E1 -ring spectra. Acknowledgements First, we would like to thank very warmly Markus Spitzweck for a very exciti* *ng dis- cussion we had with him in Toulouse a year ago, which turned out to be the star* *ting point of our work. We wish especially to thank Carlos Simpson for precious conversati* *ons and friendly encouragement: the debt we owe to his deep work on higher categories a* *nd higher stacks will be clear throughout this work. We are very thankful to Yuri Manin f* *or many motivating questions on the subject and in particular for his letter [M ]. Than* *ks to him, we were delighted to discover how much mathematics lies behind the question "Wh* *at is the motive of BZ=2 ?". For many comments and discussions, we also thank Kai Beh* *rend, Peter May, John Rognes and Paul-Arne Ostvær. It was Paul-Arne who pointed out t* *o us the possible relevance of defining 'etale K-theory of ring spectra. The second author wishes to thank the Max Planck Institut für Mathematik in * *Bonn and the Laboratoire J. A. Dieudonn'e of the University of Nice for providing a * *particularly stimulating atmosphere during his visits when part of this work was conceived, * *written and partly tested in a seminar. In particular, Andr'e Hirschowitz's enthusiasm * *was positive and contagious. 11 Notations and conventions: Throughout all this work, U and V will be two universes, with U 2 V, and we * *will assume that U contains the set of natural integers, N 2 U. We will use the exp* *ression U-set (resp. U-group, U-simplicial set, . . . ) to denote sets (resp. groups, r* *esp. simplicial sets . . . ) belonging to U. The corresponding categories will be denoted by U-* *Set, U-Gp, U-SSet . . . . The words set (resp. group, resp. simplicial set . . . ) will al* *ways refer to sets (resp. groups, resp. simplicial sets . . . ) belonging to the universe V. The c* *orresponding categories will simply be denoted by Set, Gp, SSet . . . . We will make the following exceptions when referring to categories. A U-cat* *egory (resp. a V-category) will refer to a category C such that for every pair of obj* *ects (X, Y ) in C, the set Hom(X, Y ) belongs to U (resp. V). By convention, all categorie* *s will be V-categories. We will say that a category is U-small (resp. V-small) if it be* *longs to U (resp. to V). Our references for model categories are [Ho , Hi]. For the weaker notion of * *semi-model category we refer to [Sp]. An opposite category of a semi-model category will * *again be called a semi-model category. We will not make a difference between the origina* *l notion and its dual. For any simplicial semi-model category M, we will denote by Hom__M its simpl* *icial Hom set. It will also be denoted simply by Hom__when the reference to M is clea* *r. The derived version of these simplicial Hom will be denoted by RHom__(see [Ho , Thm* *. 4.3.2]). The set of morphisms in the homotopy category Ho(M) will be denoted by [-, -]M * *, or simply by [-, -] when the reference to M is clear. For a general semi-model category M, its mapping complexes will be denoted b* *y MapM (or Map when the reference to M is clear), and will always be considered in the* * homotopy category of simplicial sets (see [Ho , 5.5.4], [Hi, x18], [Sp, I.2]). The homo* *topy fibred products in Ho(M) will be denoted`by x xhzy. In the same vein, the homotopy cof* *ibered products will be denoted by x hzy (see [Hi, x11]). By the expression V-cellular model categories (resp. V-combinatorial) model* * cate- gories) we mean a model category satisfying the conditions of definitions [Hi] * *(resp. [Sm ]), expect that all sets have to be understood as V-sets (in particular the ordinal* *s appearing in the definition belong to V). As usual, the standard simplicial category will be denoted by . For any sim* *plicial object F 2 C op in a category C, we will use the notation Fn := F ([n]). Simila* *rly, for any co-simplicial object F 2 C , we will use the notation Fn := F ([n]). 12 2 Stacks over model categories In this first section, we will present a theory of stacks over (semi-)model cat* *egories (we will be using [Sp, x2] as a reference for semi-model categories). For this, we* * will start by defining the Yoneda embedding of a model category, whose idea essentially go* *es back to some fundamental work of B. Dwyer and D. Kan (see [D-K2 ]). We do not claim * *any originality in this first paragraph, and the results stated are probably well k* *nown. Then, we introduce the notion of a Grothendieck topology on a model category, which a* *s far as we know is a new notion. The definition we give is very close to the usual * *one, and essentially one only needs to replace in the usual definition isomorphisms by e* *quivalences and fibred products by homotopy fibred products. Using this notion, we define h* *omotopy hypercovers, which are a straightforward generalization of hypercovers in Groth* *endieck's sites, and use them to define a model category of stacks over a model category * *endowed with a topology. In this first version of the paper, we have not detailed the standard propertie* *s of the model category of stacks, but we have included some statements concerning homot* *opy sheaves and computations of homotopy fibred products. These exactness propertie* *s are fundamental to do elementary manipulations in the model category of stacks. Fin* *ally, we end the section by discussing the functoriality properties of the given constru* *ctions. Setting. Throughout this section we will consider a semi-model category M, t* *ogether with a sub-semi-model category MU M. By this we mean that a morphism in MU is an equivalence (resp. a fibration, resp. a cofibration) if and only if it is * *an equivalence (resp. a fibration, resp. a cofibration) in M. Moreover, we will suppose that M* *U is stable under the functorial factorization in M (i.e. the functorial factorization func* *tor of M can be chosen such that the factorization of a morphism in MU stays in MU). We will also assume that MU is a U-category, which is furthermore a V-small * *cate- gory. The typical example of such a situation the reader should keep in mind i* *s when M is the model category of V-simplicial sets (respectively, V-simplicial groups* *, complexes of V-abelian groups, . . . ) and MU is the sub-category of U-simplicial sets (* *respectively, U-simplicial groups, complexes of U-abelian groups, . . . ). We will make a systematic use of the left Bousfield localization technique f* *or which we refer to [Hi, Ch. 3, 4]. In particular, the following elementary result will* * be used very frequently and we state it here merely for reference's convenience. Proposition 2.0.1 ([Hi, Prop. 3.6.1]) Let M be a model V-category which is left* * proper and V-cellular (see [Hi, 14.1]) or V-combinatorial (see [Sm ]). If S is a V-set* * of morphisms in M and LS(M) denotes the left Bousfield localization of M with respect to S, * *then an object W in LS(M) is fibrant iff it is fibrant in M and is S-local i.e. for any* * map f : A ! B in S, the induced map between homotopy mapping spaces f*(W ) : MapM (B, W ) ! MapM (A, W ) is an equivalence in SSet. Note that we use here a slightly different notion of local objects from that* * used in [Hi, Def. 3.2.4]: in Hirschhorn's terminology fibrant objects in LS(M) are exactly * *what he calls S-local objects in M. 13 2.1 The Yoneda embedding for semi-model categories In this first paragraph we will construct the analog of the Yoneda embedding fo* *r semi- model categories. We will start with the more general situation of a V-small ca* *tegory C together with a set of morphism S in C and define a model category (C, S)^. The* * model category (C, S)^ has to be thought as a homotopy analog of the category of pres* *heaves of sets D^ on a small category D, and is constructed as a certain left Bousfield l* *ocalization of the model category of simplicial presheaves on C. This model category will b* *e shown to be functorial in (C, S) and will only depend on the weak equivalence class o* *f (C, S) (see Prop. 2.1.5). Then, if (C, S) is the semi-model category MU M together w* *ith its equivalences, we will define a Quillen adjunction Re : M^U-! M M^U- M : h_. This adjunction will be shown to induce a fully faithful functor Rh_: Ho(MU) -! Ho(M^U), which will be our final Yoneda embedding. Let us start with the general situation of a V-small category C, together wi* *th a subset of morphisms S in C. Let SP r(C) be the category of V-simplicial presheaves on* * C, which by [Hi, Thm. 13.8.1] is a model category where fibrations and equivalenc* *es are defined objectwise. This model category is furthermore proper and simplicial a* *nd the corresponding simplicial Hom will simply be denoted by Hom__. Recall that the s* *implicial structure on SP r(C) is the data for any simplicial set K and any F 2 SP r(C), * *of simplicial presheaves K x F and F K defined by the following formulas (K x F )(x) := K x F (x) (F K)(x) := Hom__SSet(K, F (x)). These formulas allow one to define the simplicial set of morphisms between two * *simplicial presheaf F and G by the formula Hom__(F, G)n := Hom( n x F, G). As usual, one has the natural adjunction isomorphisms Hom__SPr(C)(K x F, G) ' Hom__SSet(K, Hom__(F, G)) ' Hom__SPr(C)(F, GK ), for any simplicial set K and simplicial presheaves F, G 2 SP r(C). The model category SP r(C) is also V-cellular and V-combinatorial, therefore* * the left Bousfield localization techniques of [Hi] or [Sm ] can be used to invert any V-* *set of maps. Let h- : C -! SP r(C) be the functor mapping an object x 2 C to the simplici* *al presheaf it represents. In other words, hx : Cop -! SSet send y to the constant* * simplicial set Hom(y, x) of morphisms from y to x in C. We will denote by hS the image of * *the set of morphism S by the functor h. The reader should note that hS is a V-set. Definition 2.1.1 o The simplicial model category (C, S)^ is the left Bousfiel* *d local- ization of SP r(C) along the set of maps hS. When the set of map S is clea* *r from the context, we will simply write C^ for (C, S)^. 14 o The derived simplicial Hom of (C, S)^ will be denoted by RSHom__(-, -) : Ho((C, S)^)opx Ho((C, S)^) -! Ho(SSet). Important remark. By definition, for F, G 2 (C, S)^, one has RSHom__(F, G) '* * RHom__(F, RG), where RG is a fibrant model for G in (C, S)^. This implies that in general, if * *G is not fibrant in (C, S)^, then the natural morphism RHom__(F, G) -! RSHom__(F, G) is not an isomorphism. This is why we need to mention the set S in the notation* * RSHom__. We will call a simplicial presheaf F 2 SP r(C) hS-local, if for any hx -! hy* * in hS the induced morphism RHom__(hy, F ) -! RHom__(hx, F ) is an isomorphism. The reader is warned that this is a slightly different noti* *on of local object with respect to [Hi, Def. 3.2.4]. Now, the Yoneda lemma implies that one has RHom__(hy, F ) ' F (y); therefore* *, the previous morphism is isomorphic, in the homotopy category of simplicial sets, t* *o the transition morphism F (y) -! F (x). This implies that F is an hS-local object i* *f and only if for any morphism x -! y in C which is in S, the induced morphism F (y) -! F * *(x) is an equivalence. Using this and proposition 2.0.1 one finds immediately the * *following result. Lemma 2.1.2 An object F 2 (C, S)^ is fibrant if and only if it satisfies the * *following two conditions: 1. For every object x 2 C, the simplicial set F (x) is fibrant (i.e. F is fi* *brant as an object in SP r(C)); 2. For any morphism x -! y in S, the induced morphism F (y) -! F (x) is an eq* *uiv- alence. Proof: It is a straightforward application of proposition 2.0.1. * * 2 The previous lemma implies that the homotopy category Ho((C, S)^) can be nat* *urally identified to the the full sub-category of Ho(SP r(C)) consisting of simplicial* * presheaves F : Cop -! SSet sending morphisms of S to equivalences in SSet. Furthermore, th* *e fi- brant replacement functor in (C, S)^ induces a functor r : Ho(SP r(C)) -! Ho((C* *, S)^), which is a retraction of the natural inclusion Ho((C, S)^) ,! Ho(SP r(C)). Let (C, S) and (D, T ) be two V-small categories with distinguished subsets * *of mor- phisms and f : C -! D a functor which sends S into T . The functor induces a di* *rect image functor on the categories of simplicial presheaves f* : SP r(D) -! SP r(C), 15 defined by f*(F )(x) := F (f(x)), for F 2 SP r(D) and x 2 C. This functor has * *a left adjoint f* : SP r(C) -! SP r(D), characterized by the property that f*(hx) ' hf(x), for any x 2 C. Lemma 2.1.3 The adjunction f* : (C, S)^ -! (D, T )^ (C, S)^ - (D, T )^ : f* is a Quillen adjunction. Proof: It is clear that the functor f* preserves levelwise equivalences and * *fibrations; therefore, the adjunction (f*, f*) is a Quillen adjunction between the model ca* *tegories SP r(C) and SP r(D). Using the general properties of left Bousfield localizatio* *n of model categories (see [Hi, Ch. 3, 4]), it is then enough to prove that the functor f** * sends fibrant objects in (D, T )^ to fibrant objects in (C, S)^. But this is clear by lemma 2* *.1.2 and the fact that f(S) T . 2 Definition 2.1.4A functor f : (C, S) -! (D, T ) between two V-small categories * *with subsets of morphisms is a weak equivalence if it satisfies the following three * *conditions: o f(S) T ; o There exists a functor g : D -! C with g(T ) S and natural transformatio* *ns fg ( A ) Id gf ( B ) Id, with A (resp. B) an endofunctor of D (resp. of C); o For any object y 2 D (resp. x 2 C), the induced morphisms fg(y) - A(y) -! y (resp. gf(x) - B(x) -! x) are in T (resp. in S). Proposition 2.1.5 Let f : (C, S) - ! (D, T ) be a weak equivalence between V-s* *mall categories with subsets of morphisms and g : D -! C be a functor like in defint* *ion 2.1.6. Then, the two Quillen adjunctions f* : (C, S)^ -! (D, T )^ (C, S)^ - (D, T )^ : f*, g* : (D, T )^ -! (C, S)^ (D, T )^ - (C, S)^ : g*, are Quillen equivalences. Proof: We will prove that the induced functor Lf* : Ho((C, S)^) -! Ho((D, T )^) 16 is an equivalence of categories, with quasi-inverse Lg*. This will be enough to* * show that (f*, f*) and (g*, g*) are Quillen equivalences. If F 2 Ho((C, S)^), let us prove that the natural morphisms Lg*Lf*(F ) - LB*(F ) -! F are isomorphisms in Ho((C, S)^). One can find a simplicial object L* of (C, S)* *^, such that for any [n] 2 , Ln is isomorphic to a coproduct of simplicial presheaves * *of the form hx with x 2 C, together with an isomorphism in Ho((C, S)^) F ' hocolim[n]2Ln. Then, since Lg*, Lf* and LB* commute with homotopy colimits (because g*, f* and* * B* are left Quillen functors), one is reduced to the case where F = hx, for some x* * 2 C. But, as objects of the form hz are always cofibrant, one has Lg*Lf*(hx) ' hgf(x) LB*(hx) ' hB(x), and it remains to show that the natural morphism hgf(x)- hB(x)-! hx is an isom* *or- phism in Ho((C, S)^). But this is true by definition of the model structure on* * (C, S)^ and by the fact that the morphisms gf(x) - B(x) -! x belong to S. In the same way, we prove that for any F 2 Ho((D, T )^), the morphisms Lf*Lg*(F ) - LA*(F ) -! F are isomorphisms in Ho((C, S)^). 2 Remark. In their paper [D-K1 ], Dwyer and Kan proved that the model category* * (C, S)^ is an invariant up to Quillen equivalences of the simplicial localization categ* *ory L(C, S). Proposition 2.1.5 is only a special case of this result. We now come back to the basic setting of this section i.e. to our semi-model* * categories MU M. The set of equivalences in MU will be denoted by WU. Definition 2.1.6Let McU(resp. MfU, resp. McfU) be the sub-category of MU consis* *ting of cofibrant (resps. fibrant, resp. cofibrant and fibrant) objects. We will note M^U:= (MU, WU)^ (McU)^ := (McU, WU \ McU)^ (MfU)^ := (MfU, WU \ MfU)^ (McfU)^ := (McfU, WU \ McfU)^. These are simplicial model categories and the corresponding derived simplicial * *Hom will be denoted by RwHom__(-, -) Rw,cHom__(-, -) Rw,fHom__(-, -) Rw,cfHom_(-, -). 17 Lemma 2.1.7 The natural inclusions ic : McU MU if : MfU MU icf: McfU MU, induce equivalences of categories R(ic)* : Ho(M^U) ' Ho((McU)^) R(if)* : Ho(M^U) ' Ho((MfU)^) R(icf)* : Ho(M^U) ' Ho((McfU)^). These equivalences are furthermore compatible with derived simplicial Hom, in t* *he sense that there exist natural isomorphisms Rw,cHom__(R(ic)*(-), R(ic)*(-)) ' RwHom__(-, -) Rw,fHom__(R(if)*(-), R(if)*(-)) ' RwHom__(-, -) Rw,cfHom_(R(icf)*(-), R(icf)*(-)) ' RwHom__(-, -). Proof: It is an application of proposition 2.1.5. Let us prove for example t* *hat R(ic)* is an equivalence. For this, let Q : MU -! McUbe a cofibrant replacement functo* *r (see [Ho , p. 5]). By definition, there exist natural transformations Qic ) Id icQ ) Id, such that for any x 2 MU the induced morphisms Qic(x) -! x, icQ(x) -! x are equ* *iva- lences in MU. Proposition 2.1.5 then implies that the derived functor R(ic)* is* * an equiva- lence, which preserves derived simplicial Hom (as any Quillen equivalence does). The same proof (applied to the opposite category MopU) implies that R(if)* i* *s an equiv- alence. Finally, to prove that R(icf)* is an equivalence one applies propositio* *n 2.1.5 first to a cofibrant replacement functor Q : MU -! McUand then to the restriction of * *a fibrant replacement functor R : McU-! McfU. 2 Lemma 2.1.7 is useful to establish functorial properties of the homotopy cat* *egory Ho(M^U). Indeed, if f : MU - ! NU is a functor whose restriction to McfUpreser* *ves equivalences, then f induces well defined functors Rf* : Ho(N^U) -! Ho((McfU)^) ' Ho(M^U), Lf* : Ho(M^U) ' Ho((McfU)^) -! Ho(N^U). The functor Rf* is clearly right adjoint to Lf*. For example, let f : MU -! NU be a right Quillen functor. Then, the restrict* *ion of f on McfUpreserves equivalences and therefore induces a well defined functor Rf* : Ho(N^U) -! Ho(M^U). The same argument applies to a left Quillen functor g : MU -! NU, which by rest* *riction to the subcategory of cofibrant and fibrant objects induces a well defined func* *tor Rg* : Ho(N^U) -! Ho(M^U). 18 Definition 2.1.8Let MU and NU be two V-small semi-model categories and f : MU -! NU be a functor whose restriction to McfUpreserves equivalences. The previously* * defined functor Rf* : Ho(N^U) -! Ho(M^U) will be called the inverse image functor. Its left adjoint Lf* : Ho(M^U) -! Ho(N^U) will be called the direct image functor. Remark. The reader should be warned that the direct and inverse image funct* *ors' construction is not functorial in f. In other words, if one does not add some h* *ypotheses on the functor f, then in general Rf* O Rg* is not isomorphic to R(g O f)*. How* *ever, one has the following easy proposition, which ensures in many cases the functoriali* *ty of the previous construction. Proposition 2.1.9 1. Let MU, NU and PU be V-small semi-model categories and f g MU ____//_NU____//PU be two functors preserving fibrant objects and equivalences between them. * *Then, there exist natural isomorphisms R(g O f)* ' Rf* O Rg* : Ho((PU)^) -! Ho((MU)^), L(g O f)* ' Lg* O Lf* : Ho((MU)^) -! Ho((PU)^). These isomorphisms are furthermore associative and unital in the arguments* * f and g. 2. Let MU, NU and PU be V-small semi-model categories and f g MU ____//_NU____//PU be two functors preserving cofibrant objects and equivalences between them* *. Then, there exist natural isomorphisms R(g O f)* ' Rf* O Rg* : Ho((PU)^) -! Ho((MU)^), L(g O f)* ' Lg* O Lf* : Ho((MU)^) -! Ho((PU)^). These isomorphisms are furthermore associative and unital in the arguments* * f and g. Proof: The proof is the same as that of the usual property of composition fo* *r derived Quillen functors (see [Ho , Thm. 1.3.7]), and is left to the reader. * * 2 Examples of functors as in the previous proposition are given by right or le* *ft Quillen functors. Therefore, given f g MU ____//_NU___//_PU 19 a pair of right Quillen functors, one has R(g O f)* ' Rf* O Rg* L(g O f)* ' Lg* O Lf*. In the same way, if f and g are left Quillen functors, one has R(g O f)* ' Rf* O Rg* L(g O f)* ' Lg* O Lf*. Proposition 2.1.10 If f : MU -! NU is a (right or left) Quillen equivalence bet* *ween V-small semi-model categories, then the induced functors Lf* : Ho(M^U) -! Ho(N^U) Ho(M^U) - Ho(N^U) : Rf*, are equivalences, quasi-inverse of each others. Proof: Let us prove the proposition in the case where f is a right Quillen f* *unctor. The case where f is left Quillen is proved similarly. Let g : NU -! MU be the left adjoint to f; let us show that Lg* is quasi-inv* *erse to Lf*. For this, let us consider the following two functors Q c g R f f : MfU-! NU RgQ : NU ____//_NU___//_MU____//MU, where Q is a cofibrant replacement functor and R is a fibrant replacement funct* *or. The natural transformations Id ) R, Q ) Id and gf ) Id, induce natural transformati* *ons RgQf ( gQf ) gf ) Id. By hypothesis, for any x 2 MfU, the induced morphisms RgQf(x) - gQf(x) -! x are equivalences. The same proof as in proposition 2.1.5 shows that Lg*Lf* is i* *somorphic to the identity. The dual argument then shows that Lf*Lg* is isomorphic to the * *identity. 2 To finish this paragraph, we will define a Quillen adjunction Re : M^U-! M M^U- M : h_ and show that the functor h_induces a fully faithful embedding on the level of * *homotopy categories (see Thm. 2.1.13). >From now on, let ( : MU -! MU , i) be a fixed cofibrant resolution functor* * (see [Hi, 17.1.3]). This means that for any object x 2 MU, (x) is a co-simplicial objec* *t in MU, which is cofibrant for the Reedy model structure on MU , together with a natura* *l weak equivalence i(x) : (x) -! c(x), c(x) being the constant co-simplicial object i* *n MU at x. In the case that the semi-model category M is simplicial, one can use the s* *tandard cofibrant resolution functor (x) := * x. At the level of model categories, the construction of the functor h_will dep* *end on the choice of , but after passing to the homotopy categories it will be shown that* * possibly 20 different choices give the same Yoneda embedding. We define the functor h_-: M -! SP r(MU), by sending each x 2 M to the simpl* *icial presheaf h_x: MopU -! SSet y 7! Hom( (y), x), where (y) is the cofibrant resolution of y induced by the functor . To be mor* *e precise, the presheaf of n-simplices of h_xis given by the formula (h_x)n(-) := Hom( (-)n, x). Lemma 2.1.11 The functor h_: M -! SP r(MU) is a right Quillen functor. Proof: The fact that h_is right Quillen is a direct verification and is prov* *ed in detail in [Du2 , 9.5]. * * 2 Corollary 2.1.12 The adjunction Re : M^U-! M M^U- M : h_ is a Quillen adjunction. Proof: By the general properties of Bousfield localization of model categori* *es (see [Hi, Ch. 3, 4]) and by lemma 2.1.11 it is enough to show that the functor h_preserve* *s fibrant objects. But, by definition of h_this follows immediately from the standard pro* *perties of mapping spaces (see [Hi, x18]) and lemma 2.1.2. * * 2 Remark. The reader should notice that if ( 0, i0) is another cofibrant resol* *ution func- tor, then the two derived functor Rh- and Rh0-obtained using respectively and* * 0are naturally isomorphic. Therefore, our construction does not depend on the choice* * of once one is passed to the homotopy category. The main result of this first paragraph is the following one, that will play* * the role of the Yoneda embedding in our theory. Theorem 2.1.13 For any object x 2 Ho(M) which is isomorphic to an object in Ho* *(MU), the adjunction morphism LReRh_x-! x is an isomorphism in Ho(M). Equivalently, the restriction of Rh_: Ho(M) -! Ho(M^U) to the full subcatego* *ry of objects isomorphic to an object of Ho(MU), is fully faithful. Proof: Let x be a fibrant and cofibrant object in MU and x - ! x* a simplic* *ial resolution of x in MU (see [Hi, 17.1.2]). We consider the following two simplic* *ial presheaves hx* : (McU)op -! SSet y 7! Hom(y, x*), 21 h_x*: (McU)op - ! SSet y 7! Hom( (y), x*). The augmentation (-) -! c(-) and co-augmentation x -! x* induce a commutative diagram in (McfU)^ hx __a__//_h_x b|| d|| fflffl|c fflffl| hx*____//_h_x*. By the properties of mapping spaces (see [Hi, x18]), both morphisms c and d are* * equiva- lences in SP r(McU). Furthermore, the morphism hx -! hx*is isomorphic in Ho(SP * *r(McU)) to the induced morphism hx -! hocolim[n]2hxn. As each morphism hx -! hxn is an equivalence in (McU)^, this implies that d is an equivalence in (McU)^. We ded* *uce from this that the natural morphism hx -! h_xis an equivalence in (McU)^. Let us show how this implies that for any x 2 MU, the natural morphism hx -!* * Rh_x is an isomorphism in M^U. Indeed, if F is a fibrant object in M^Uand Fc is its * *restriction to McU, then one has RHom__M^U(Rh_x, F ) ' RHom__(McU)^(h_RQ(x), Fc) ' RHom__(McU)^(hRQ(x), Fc) ' F* * (RQ(x)) ' RHom__M^U(hRQ(x), F ) ' RHom__(hx, F ), where RQ(x) is a fibrant and cofibrant model for x in MU. This shows, by the Yo* *neda lemma for Ho(M^U), that hx -! h_xis an equivalence in M^U. Now, let x 2 MfUand let us consider the natural morphism in Ho(M), Re(hx) -! LRe(h_x) -! x. As hx -! h_xis an equivalence and hx is cofibrant in MU, the first morphism Re(* *hx) -! LRe(h_x) is an isomorphism in Ho(M). Therefore, to finish the proof of the theo* *rem, it remains to show that Re(hx) -! x is an isomorphism in Ho(M). But, by adjunction* *, for any fibrant object y 2 M, one has [Re(hx), y]M ' [hx, h_y]M^U ' ß0(Hom__( (x), y)) ' [x, y]M , showing that Re(hx) -! x is indeed an isomorphism in Ho(M). 2 To finish this paragraph, let us notice that for any object x 2 MU, the natu* *ral mor- phism i : (-) -! c(-) induces in the obvious manner a morphism in M^ , hx -! h* *_x. Corollary 2.1.14 For any object x 2 MU, the natural morphism hx -! h_x is an equivalence in the model category M^U. Proof: This follows immediately from the proof of the previous theorem. * * 2 22 2.2 Grothendieck topologies on semi-model categories In this paragraph, we present the notion of a Grothendieck topology on a semi-m* *odel cat- egory. The definition is quite natural as it is formally obtained by replacing * *isomorphisms by equivalences and fibred products by homotopy fibred products in the usual de* *finition. Recall that for any diagram x__a__//zobo_y in a semi-model category M, one * *can define a homotopy fibred product x xhzy 2 Ho(M) (see [Hi, x11]). Explicitly, it* * is defined by x xhzy := x0xy0z0, 0 b0 where x0_a__//_z0oo_y0is an equivalent diagram such that the two morphisms a0 * *and b0 are fibrations and the objects x0, y0 and z0 are fibrant. The object x xhzy* * only de- pends, up to a natural isomorphism in Ho(M), on the equivalence class of the di* *agram x __a_//_zbooy_. Furthermore it only depends, up to a non-natural isomorphism,* * on the isomorphism class of the image of the diagram x __a_//_zbooy_ in Ho(M). In * *other words, for any diagram, x __a_//_zbooy_ in Ho(M), the isomorphism class of the * *object x xhzy 2 Ho(M) is well defined. In the same way, the isomorphism classes of th* *e two projections x xhzy -! x and x xhzy -! y are well defined. Definition 2.2.1A topology ø on a V-small semi-model U-category M is the data f* *or any object x 2 M, of a V-set Covfi(x) of U-small family of objects in Ho(M)=x, * *called covering families of x, satisfying the following three conditions: o (Stability) For all x 2 M and any isomorphism y ! x in Ho(M), the family {* *y ! x} is in Covfi(x). o (Composition) If {ui ! x}i2I2 Covfi(x), and for any i 2 I, {vij! ui}j2Ji, * *the family {vij! x}i2I,j2Jiis in Covfi(x). o (Homotopy base change) For any {ui ! x}i2I 2 Covfi(x), and any morphism in Ho(M), y ! x, the family {uixhxy ! y}i2Iis in Covfi(y). A V-small semi-model U-category M together with a topology ø will be called a (* *V-small) semi-model (U-)site. Remark. For any semi-model category M, one can form its homotopy 2textit-cat* *egory D 2(M) (see [Ga-Zi] and [Sp, x2]). This 2-category is a fine enough invariant o* *f M to be able to recover the homotopy fibred products. It is then easy to check that the* * data of a topology on M only depends on the 2-category D 2(M), up to a 2-equivalence. It * *seems to us that this is the reason why the homotopy 2-category of differential grade* *d algebras is used in [Be ]. We warn the reader that however the homotopy category of stac* *ks we will define in Def. 2.4.1 depends on more than just D 2(M), as higher homotopies in * *M enter in the definition. Before going further in the study of topologies on semi-model categories, we* * would like to present three examples. 23 o Trivial model structure. Let M be a V-small U-category with the trivial m* *odel structure (i.e. equivalences are isomorphisms and all morphisms are fibrat* *ions and cofibrations). Then, Ho(M) = M and the homotopy fibred products are just f* *ibred products. Therefore, a topology on the model category M in the sense of de* *finition 2.2.1 is the same thing as a usual Grothendieck topology on the category M. o Topological spaces. Let us take as M the model category of U-topological * *spaces, T op, and let us define a topology ø in the following way. A family of mor* *phism in Ho(T`op), {Xi ! X}i2I, I 2 U, is defined to be in Covfi(X) if the induced * *map i2Iß0(Xi) -! ß0(X) is surjective. The reader will check immediately tha* *t this defines a topology on T op in the sense of definition 2.2.1. o Negatively graded CDGA (see [Be ]). Let k be a field of characteristic ze* *ro and M = CDGAopkbe the opposite model category of commutative and unital differ* *ential graded k-algebras in negative degrees which belong to U (see for example [* *Hin] for the description of its model structure). Let ø0 be one of the usual topolo* *gies defined on k-schemes (e.g. Zariski, Nisnevich, 'etale, ffpf or ffqc). Let us defin* *e a topology ø on CDGAopkin the sense of Def. 2.2.1, as follows. A family of morphism* *s in Ho(CDGAk), {B ! Ai}i2I, I 2 U, is defined to be in Covfi(B) if it satisfie* *s the two following conditions: 1.The induced family of morphisms of affine k-schemes {Spec H0(Ai) ! SpecH0(B)}i2I is a ø-covering. 2.For any i 2 I, one has H*(Ai) ' H*(B) H0(B)H0(Ai). The reader can check as an exercise that this actually defines a topology * *on the model category CDGAopk. We will come back to this very important example i* *n the last section of the paper. 2.3 Homotopy hypercovers Using Reedy model structures ([Ho , 5.2]) on the category of simplicial objects* * in a model category, we generalize the definition of hypercovers to the case of (semi-)mod* *el sites. Let MU be a V-small semi-model U-category and let us consider sMU the catego* *ry of simplicial objects in MU. By definition, the category MU has all U-limits and a* *ll U-colimits so that the category sMU is naturally enriched in U - SSet. Recall that for K 2* * U - SSet and x* 2 sMU, one has by definition K x x* : op -! ` MU [n] 7! Kn xn. For x* and y* objects in sMU, we define Hom__(x*, y*) : op -! U - SSet [n] 7! HomsMU( n x*, y*). 24 Finally, the exponential object yK*, for K 2 U - SSet and y* 2 sMU, is characte* *rized by the adjunction isomorphism Hom(K x*, y*) ' Hom(x*, yK*), for all x* 2 sMU. The category sMU is endowed with its Reedy structure described in [Ho , Thm.* * 5.2.5] and [Sp, Prop. 2.7], which makes it into a V-small semi-model U-category. Let u* *s recall that equivalences in sMU are defined to be levelwise equivalences. Recall also* *, that a morphism f : x* -! y* is defined to be a fibration if for all n the morphism in* *duced by the inclusion @ n ,! n, n n n x* -! x@* xy@*ny* , is a fibration in MU. For a U-simplicial set K, the functor (-)K : sMU -! MU x* 7! (xK*)0, which sends a simplicial object x* to the 0-th level of the exponential object * *xK*, is a right Quillen functor. Its right derived functor will be denoted by (-)RK : Ho(sMU) -! Ho(MU). We will consider objects of MU as constant simplicial objects via the constant * *simplicial functor MU -! sMU. In particular, for x 2 MU and K 2 U - SSet, we will consider* * the object xRK 2 Ho(MU). Definition 2.3.1Let x 2 MU be an object in a semi-model site (MU, ø). A homoto* *py ø-hypercover of x in MU, is a simplicial object u* 2 Ho(sMU), together with a m* *orphism u* -! x in Ho(sMU), such that for any n 0, the natural morphism n R@ n h R n uR* -! u* xxR@ nx is a ø-covering in MU. 2.4 The model category of stacks In this paragraph we will use the notion of homotopy hypercover defined previou* *sly in or- der to construct the model category of stacks over a model site. Our constructi* *on is based on a recent result of D. Dugger identifying the model category of simplicial pr* *esheaves of [Ja] as the left Bousfield localization of the model category of simplicial pre* *sheaves for the trivial topology by formally inverting hypercovers (see [Du1 ]). By definition,* * our model category of stacks over a model site (M, ø) will be the left Bousfield localiza* *tion of the model category M^ by formally inverting ø-hypercovers. In this first version of* * the paper, we will also state without proof a generalization of Dugger's theorem by introd* *ucing the notion of homotopy sheaves in our setting. This result is fundamental to contro* *l elementary manipulations in the model category of stacks (as for example, homotopy fibred * *products). 25 We come back to the basic setting of the present section, i.e. to an inclusi* *on of semi- model categories MU M, together with the associated Yoneda embedding Rh_: Ho(MU) -! Ho(M^U) defined in the first paragraph. We will suppose that MU is endowed with a topol* *ogy ø in the sense of definition 2.2.1. We define two V-sets of morphisms in M^Uin the following way. For this, reca* *ll the functor h : MU -! M^U= SP r(MU), which maps an object x 2 MU to the constant simplicial presheaf it represents. For any U-set I and any family of cofibrant objects {xi}i2I2 (McU)I, we cons* *ider the following natural morphism in SP r(MU) a hxi- ! h` i2Ixi. i2I When I varies in the set of U-sets and the xi's vary in the set of object in Mc* *U, we find a V-set of morphisms in M^U. a Ssum := { hxi- ! h` i2Ixi| I 2 U, {xi}i2I2 (McU)I}. i2I Now, for any fibrant object x 2 MfU, let HHC(x) be the V-set of simplicial o* *bjects u* 2 s(M=x), whose image in Ho(sM)=x is a homotopy ø-hypercover of x in MU (see* * Def. 2.3.1). For any u* 2 HHC(x), [n] 7! hun is a simplicial presheaf on MU defined * *by the following formula: hu* : MopU -! SSet y 7! ([n] 7! hun(y)). The augmentation u* -! x gives then a morphism of simplicial presheaves hu* -! * *hx. When x varies in MfUand u* varies in HHC(x), we find a V-set of morphisms in M^U Shhc:= {hu* -! hx | x 2 MfU, u* 2 HHC(x)}. Definition 2.4.1The simplicial model category of stacks on MU for the topology * *ø is the left Bousfield localization of the simplicial model category M^Ualong the V-set* * of morphisms Sfi:= Ssum [ Shhc. It will be denoted by M~,fiU, or simply by M~U when the topo* *logy ø is clear. The derived simplicial Hom of M~,fiUwill be denoted by Rw,fiHom_(-, -) : Ho(M~,fiU)opx Ho(M~,fiU) -! Ho(SSet). The following characterization of fibrant objects in M~,fiUis an immediate a* *pplication of the general criterion in Prop. 2.0.1. Lemma 2.4.2 An object F 2 M~,fiUis fibrant if and only if it satisfies the fo* *llowing four conditions: 26 1. For any x 2 MU, the simplicial set F (x) is fibrant; 2. For any equivalence y ! x in M, the induced morphism F (x) - ! F (y) is an equivalence of simplicial sets; 3. For any U-set I and any family of cofibrant objects {xi}i2I in MU, the nat* *ural morphism of simplicial sets a Y F ( xi) -! F (xi) i2I i2I is an equivalence; 4. For any fibrant object x 2 MfUand any simplicial object u* 2 s(MU=x), whos* *e image in Ho(sMU)=x is a homotopy ø-hypercover, the natural morphism in Ho(SSet) F (x) -! holim[n]2F (un) is an isomorphism. Proof: It is a direct application of proposition 2.0.1. * * 2 >From the previous lemma we immediately deduce that the homotopy category Ho* *(M~,fiU) can be identified with the full subcategory of Ho(SP r(MU)) of simplicial presh* *eaves sat- isfying conditions (2), (3) and (4) of lemma 2.4.2. Furthermore, the natural i* *nclusion Ho(M~,fiU) -! Ho(SP r(MU)) has a left adjoint which is a retraction. This retra* *ction will be denoted by a : Ho(SP r(MU)) -! Ho(M~,fiU); note that a2 is naturally isomorp* *hic to a. Definition 2.4.3 o A stack on MU for the topology ø is an object F 2 Ho(SP r(* *MU)) such that the natural morphism F -! a(F ) is an isomorphism. o For any F 2 Ho(SP r(MU)), the stack associated to F is the stack a(F ). o The topology ø is sub-canonical if for any x 2 Ho(MU), the object Rh_x2 Ho* *(M^U) is stack. Remarks: o If MU is endowed with the trivial model structure, then the model category* * M~,fiU is Quillen equivalent to the model category of simplicial presheaves defin* *ed by J.F. Jardine in [Ja]. This is proved in [Du1 ]. We will also state a more gen* *eral result which, for not necessarily trivial model structures, identifies the equiva* *lences in M~,fiU as local equivalences (see Thm. 2.5.5). Note however, that the model struc* *ture we use is not the one defined in [Ja] but rather its projective analog descri* *bed in [H-S , x5] and [Bl]. o When the topology ø is trivial, then the model category M~,fiUis equivalen* *t to M^U. In particular, a stack for the trivial topology is a simplicial presheaf F : * *MopU-! SSet which preserves equivalences. 27 o When the topology ø is sub-canonical, one obtains a fully faithful functor Rh_: Ho(MU) -! Ho(M~,fiU), which embeds the homotopy theory of MU into the homotopy theory of stacks * *over MU. The following criterion for the topology ø to be sub-canonical can be deduce* *d imme- diately from lemma 2.4.2. Corollary 2.4.4 A topology ø on a semi-model category MU is sub-canonical if an* *d only if for every homotopy ø-hypercover u* -! x in MU, the natural morphism hocolim[n]2 opun -! x is an isomorphism in Ho(MU). Proof: It is a direct application of lemma 2.4.2 and of the universal proper* *ty of homo- topy colimits. 2 2.5 Exactness properties of the model category of stacks In this paragraph we will present a more classical definition of weak equivalen* *ces in M~,fiU, closer to the one used in [Ja]. The comparison theorem 2.5.5 is an (easy) exten* *sion of a result of D. Dugger. Therefore, we will not give all details and the interested* * reader may consult [Ja, Du1 ] for further materials. In the whole paragraph a topology ø is fixed on MU. Let us start by saying a few words on the notion of sheaves of sets in our s* *etting of semi-model sites. As usual, any V-set will be considered as a constant V-si* *mplicial set (note that such simplicial sets are always fibrant in SSet) and therefore a* *ny functor MopU-! Set will be regarded as an object in SP r(MU). Definition 2.5.1 1.A sheaf of sets on the model site (MU, ø) is a functor F :* * MopU-! Set which is stack when considered as an object in Ho(SP r(MU)). 2. A morphism of sheaves (of sets) F -! F 0is a natural transformation. The category of sheaves (of sets) on the model site (MU, ø) will be denoted by * *Sh(MU, ø). By definition, a functor F : MopU-! Set is a sheaf when it satisfies the fol* *lowing three conditions: 1. For every equivalence x ! y in MU, the induced morphism F (y) -! F (x) is * *an isomorphism; 28 2. For every U-small family of objects in MU, {xi}i2I, I 2 U, the induced mor* *phism ah Y F ( xi) -! F (xi) i2I i2I is an isomorphism; 3. For any fibrant object x 2 MfUand any simplicial object u* 2 s(MU=x), whos* *e image in Ho(sMU)=x is a homotopy ø-hypercover, the natural morphism in Ho(SSet) F (x) -! lim[n]2F (un) ' Ker (F (u0) ' F (u1)) is an isomorphism. Remark. By the universal property of the homotopy category, the category of * *sheaves on (MU, ø) is naturally equivalent to a full sub-category of the category of pr* *esheaves of sets SetHo(MU)op. However, as the topology ø on the semi-model category MU doe* *s not induce in general a topology on Ho(MU), the category Sh(MU, ø) is a priory not * *a cate- gory of sheaves in the usual sense. Lemma 2.5.2 The natural functor Sh(MU, ø) -! Ho(SP r(MU)) factors through the* * full sub-category of stacks Ho(M~,fiU) and it is fully faithful. Proof: Let F 2 Sh(MU, ø) be a sheaf and let us consider it as an object in t* *he model category M~,fiU. The sheaf conditions together with lemma 2.4.2 show that F is * *a fibrant object in M~,fiUand in particular that its image in Ho(SP r(MU)) is a stack. Furthermore, as a sheaf is always a fibrant object in M~fiU, one checks imme* *diately that for two sheaves F and G, the set of morphisms [F, G] in Ho(M~,fiU) is isom* *orphic to the set of natural transformations between F and G. * * 2 Let us consider (MU, triv), the semi-model site with trivial topology. Then,* * the cat- egory Sh(MU, triv) is equivalent to the category of functors F : MopU-! Set sen* *ding equivalences to isomorphisms. In particular, there exists a natural fully faith* *ful functor Sh(MU, ø) -! Sh(MU, triv). Lemma 2.5.3 The natural functor Sh(MU, ø) -! Sh(MU, triv) has a left adjoint a0 : Sh(MU, triv) -! Sh(MU, ø). Moreover, the functor a0 is * *left exact (i.e. it commutes with finite limits). 29 Sketch of Proof: The idea is to imitate the usual associated sheaf functor c* *onstruction, replacing fibred products by homotopy fibred products. The proof of the left ex* *actness of a0 is very similar to the usual one. * * 2 Remark. We have denoted a0 the associated sheaf functor in order to make a d* *ifference with the associated stack functor a. However, we will show (see Thm. 2.5.5) tha* *t they coincide when applied to a sheaf, considered as an object in Ho(M^U). Note also* * that a0 is only defined for presheaves MopU-! Set sending equivalences in MU to isomorp* *hisms (i.e. for sheaves for the trivial topology on MU). In the next definition, ßn(K), for K 2 SSet and n 0 denote the set of homo* *topy classes of morphisms n -! K which sends @ n to a point. More precisely, i j ßn(K) := ß0 RHom__( n, K) xhRHom_(@ n,K)K0, where RHom__( n, K) -! RHom__(@ n, K) is induced by the restriction to @ n @ n and K0 -! RHom__(@ n, K) is adjoint to the natural projection K0x@ n -! K0 -! K. The natural projection RHom__( n, K) xhRHom_(@ n,K)K0 -! K0 induces morphisms ßn(K) -! K0, which make the ßn(K) for n > 0 (resp. for n > 1) group objects (resp. abelian group objects) over the set of 0-simplices K0. Definition 2.5.4 1.Let F 2 SP r(MU) be a stack for the trivial topology on MU* * (i.e. F sends equivalences in MU to equivalences in SSet). The homotopy groups pre* *sheaves of F are defined by ßprn(F ) :MopU- ! Set x 7! ßn(F (x)). They come equipped with a natural projection ßprn(F ) -! F0. The associate* *d sheaves of ßprn(F ) are denoted by ßfin(F ) := a0(ßprn(F )). 2. Let F, F 02 SP r(MU) be two stacks for the trivial topology on MU. A morp* *hism F -! F 0in SP r(MU) is called a ßfi*-equivalence if for all n 0 the foll* *owing square is cartesian in Sh(MU, ø) ßfin(F_)__//ßfin(F 0) | | | | fflffl| fflffl| a0(F0) ____//_a0(F00). The main theorem of this paragraph is the following generalization of the ma* *in result proved in [Du1 ]. Its proof will not be given in this version of the paper. 30 Theorem 2.5.5 Let F and F 0be stacks for the trivial topology on MU (i.e. they* * preserve equivalences) and f : F -! F 0be a morphism in SP r(MU). Then, f is an equivale* *nce in M~,fiUif and only if it is a ßfi*-equivalence. Besides its own interest, the previous theorem implies the following corolla* *ry which will be crucial for the study of geometric stacks in the next paragraph. Corollary 2.5.6 Let F _____//F1 | | | | fflffl|fflffl| F2_____//F0 be a homotopy cartesian diagram in SP r(MU). If F1, F2 and F0 are stacks for th* *e trivial topology on MU, then the natural morphism F -! F1 xhF0F2 is an isomorphism in Ho(M~,fiU). In other words, the associated stack functor a, when restricted to the full * *sub-category of stacks for the trivial topology, commutes with homotopy fibred products. Proof: It is an application of theorem 2.5.5, lemma 2.5.3, the long exact se* *quence in homotopy for a homotopy fibred product of simplicial sets and an extended versi* *on of the five lemma. 2 Remark. The functor a : Ho(SP r(MU)) -! Ho(M~,fiU) will not commute with ho- motopy fibred products in general. Indeed, suppose that ø is the trivial topol* *ogy and let x be an object of MU. The object hx 2 SP r(MU) represented by x, is such t* *hat a(hx) ' h_x. Therefore, if a would commute with homotopy fibred products, the n* *atural functor MU -! Ho(MU) would send fibred products to homotopy fibred products, wh* *ich is not the case in general. 2.6 Functoriality We finish this first section by the standard functoriality properties of the ca* *tegory of stacks i.e. with direct and inverse images functors. We have seen in x1.1 that any functor f : MU -! NU between V-small semi-model categories, whose restriction to McfUpreserves equivalences, gives rise to a pa* *ir of adjoint functors Lf* : Ho(M^U) -! Ho(N^U) Ho(M^U) - Ho(N^U) : Rf*. Definition 2.6.1Let f : MU -! NU be a functor between two V-small semi-model ca* *te- gories with topologies øM and øN , respectively. Let us suppose that the restr* *iction of f to McfUpreserves equivalences. Then, f is said to be continuous if the inverse ima* *ge functor Rf* : Ho(N^U) -! Ho(M^U) preserves the categories of stacks. 31 It is immediate to check that if f is a continuous functor, then the functor Rf* : Ho(N~,fiNU) -! Ho(M~,fiMU) has a left adjoint L(f*)~ : Ho(M~,fiMU) -! Ho(N~,fiNU). Explicitly, it is defined by the formula L(f*)~ (F ) := a(Lf*(F )), for F 2 Ho(M~,fiMU) Ho(M^U), a being the associated stack functor. Proposition 2.6.2 Let (MU, øM ), (NU, øN ) and (PU, øP ) be V-small semi-model * *sites. 1. Let f g MU ____//_NU____//PU be two continuous functors preserving fibrant objects and equivalences bet* *ween them. Then, there exist natural isomorphisms R(g O f)* ' Rf* O Rg* : Ho((PU)~,fiP) -! Ho((MU)~,fiM), L((g O f)*)~ ' L(g*)~ O L(f*)~ : Ho((MU)~,fiM) -! Ho((PU)~,fiP). These isomorphisms are furthermore associative and unital in the arguments* * f and g. 2. Let f g MU ____//_NU____//PU be two continuous functors preserving cofibrant objects and equivalences b* *etween them. Then, there exist natural isomorphisms R(g O f)* ' Rf* O Rg* : Ho((PU)~,fiP) -! Ho((MU)~,fiM), L((g O f)*)~ ' L(g*)~ O L(f*)~ : Ho((MU)~,fiM) -! Ho((PU)~,fiP). These isomorphisms are furthermore associative and unital in the arguments* * f and g. Proof: It is immediate from proposition 2.1.9 and the basic properties of th* *e functor a. 2 The following criterion gives some examples of continuous functors. Lemma 2.6.3 Let f : MU -! NU be a right Quillen functor between two V-small s* *emi- model categories with topologies øM and øN , respectively. Suppose that f sat* *isfies the following two conditions: 32 1. For any x 2 Ho(MU) and any covering family {ui! x}i2I2 CovfiM(x), the indu* *ced family {Rf(ui) ! Rf(x)}i2I is in CovfiN(Rf(x)). 2. The functor Rf : Ho(MU) -! Ho(NU) commutes with coproducts. Then, the functor f is continuous. Proof: Let F 2 Ho(N~,fiNU) be a stack and let us prove that Rf* is a stack. * *For this, we use lemma 2.4.2. The reader should notice that conditions (1) and (2) are alway* *s satisfied by Rf*(F ), if they are by F . Recall that by definition of the functor Rf*, for F 2 Ho(N~U) and x 2 MU, on* *e has a natural isomorphism Rf*(F )(x) ' F (Rf(x)) in Ho(MU). Now, by condition (2) on f, one has ah Y Rf*(F )( xi) ' R Rf*(F )(xi), i2I i2I for any family of objects {xi}i2Iin Ho(NU). This show that Rf*(F ) satisfies (3* *) of lemma 2.4.2. Furthermore, as f is right Quillen it commutes with the functors (-)RK (* *introduced just before definition 2.3.1), for any simplicial set K. Using this and conditi* *on (1) on f, one checks immediately that Rf*(F ) satisfy condition (4) of lemma 2.4.2. * * 2 3 Stacks over E1 -algebras In this second section we present the construction of the category of geometric* * stacks over a base symmetric monoidal model category, as sketched in the Introduction. For * *this, we will start by recalling the homotopy theory of E1 -algebras and modules over th* *em in general symmetric monoidal model categories. The references for this part are t* *he foun- dational papers [E-K-M-M ], [Kr-Ma ], [Hin] and especially [Sp] where the gene* *ral case is studied (in particular the case where the monoid axiom does not hold). We will * *then ap- ply our theory of stacks to (semi-)model categories of E1 -algebras to give a d* *efinition of geometric stacks. In the last paragraph we give the groupoid approach to the co* *nstruction of geometric stacks. Setting. Throughout this section we will consider a left proper V-cofibrantl* *y generated symmetric monoidal model category C. The unit of C will be denoted by 1 and wil* *l always be assumed to be a cofibrant object. We also assume C satisfies assumption [Sp,* * 9.6], i.e. that the domains of the generating cofibrations of C are cofibrant. We will also consider CU C a sub-monoidal model category. By this we mean * *that CU is stable under the monoidal structure and is a sub-model category as alread* *y ex- plained at the beginning of section 1. We will assume that CU is a U-cofibrantl* *y generated model category which is V-small, and that the domains and codomains of the gene* *rating cofibrations and trivial cofibrations in C belong to CU. 33 Finally, we assume that C is an algebra over the model category SSet (of V-s* *implicial sets) or over C(Z) (the category of complexes of V-abelian groups). The sub-mod* *el cate- gory CU is then assumed to be stable under external products by U-simplicial se* *ts or by complexes of U-abelian groups. 3.1 Review of operads and E1 -algebras The main reference for this paragraph is [Sp, x1 - 10], as well as the foundati* *onal papers [E-K-M-M , Hin, Kr-Ma ]. The reader may also consult [Ber-Moe ]. Recall first that an operad O in C is the data, for each integer n 2 N, of a* *n object O(n) 2 C, together with an action of the symmetric group n, a unit 1 -! O(1) a* *nd structural morphisms X O(k) O(n1) . . .O(nk) -! O( ni), i for all integers k 1 and n1, . .,.nk. These structural morphisms are require* *d to sat- isfy suitable associativity, commutativity and unity rules that the reader may * *find in [Kr-Ma , I.1.1]. A morphism between two operads O and O0in C is the data of mor* *phisms O(n) -! O0(n) commuting with the unit and the structural morphisms. With these * *def- initions, operads in C form a well defined category that will be denoted by Op(* *C). Following [Hin], [Sp] and [Ber-Moe ], a morphism f : O -! O0 of operads in C* * is a fibration (resp. an equivalence) if for all n 2 N, the induced morphism fn : O(* *n) -! O0(n) is a fibration (resp. an equivalence) in C. Theorem 3.1.1 ([Sp, Thm. 3.2]) The category Op(C) of operads in C, together wi* *th the class of fibrations and equivalences defined above, is a cofibrantly generated * *semi-model category. It is important to remark that Op(CU) is a sub-model category of C, which is* * U- cofibrantly generated and V-small. The fundamental operad we are interested in, is the operad COM, classifying * *commu- tative and unital monoids in C. Explicitly, it is defined by COM(n) = 1 for any* * n 0, with the trivial action of n. Definition 3.1.2([Sp, Def. 8.1]) A unital E1 -operad in C is an operad O 2 Op(* *C) satisfying the following conditions: o There exists an equivalence u : O -! COM; o The induced morphism u : O(0) -! COM(0) = 1 is an isomorphism; 34 o For any n 0, the object O(n) is cofibrant in the semi-model category C n* * of n- equivariant objects in C. It is important to remark that unital E1 -operads in C always exist. This fo* *llows from [Sp, Lem. 8.2] and our assumptions on C which include that 1 is cofibrant and t* *hat C is left proper. For any operad O 2 Op(C), one can define the category of O-algebras in C. By definition, an O-algebra is the data of an object A 2 C and structural morphism* *s O(n) A n - ! A, for all n 0. These structural morphisms are required to satisfy c* *ertain associativity, commutativity and unit rules that the reader may find in [Kr-Ma * *, I.2.1]. A morphism of O-algebras is the data of a morphism A -! A0 in C, commuting with t* *he structural morphisms. These definitions allow to define the category of algebr* *as over a fixed operad O in C, that will be denoted by Alg(O). Let f : O -! O0 be a morphism in Op(C). Then, there exists a natural restri* *ction functor f* : Alg(O0) -! Alg(O). This functor has a left adjoint f* : Alg(O) -! Alg(O0). As for the case of operads, a morphism f : A -! A0of O-algebras in C, is a f* *ibration (resp. an equivalence) if it is a fibration (resp. an equivalence) in when cons* *idered as a morphism in C. Theorem 3.1.3 ([Sp, Thm. 4.7] and [Sp, Cor. 6.7]) 1. Let O 2 Op(C) be a unital E1 -operad in C. Then the category Alg(O) of O-a* *lgebras in C, together with the class of fibrations and equivalences defined above* *, is a cofi- brantly generated semi-model category. 2. Let f : O -! O0 be an equivalence between two operads in C. If for every n* * 0, O(n) and O0(n) are cofibrant in M n , then the induced Quillen adjunction f* : Alg(O) -! Alg(O0) Alg(O) - Alg(O0) : f* is a Quillen equivalence. Corollary 3.1.4 The semi-model category Alg(O) of algebras over a unital E1 -al* *gebra in C, is independent, up to a Quillen equivalence, of the choice of O 2 Op(C). Proof: This follows from part (2) of Theorem 3.1.3 and the fact that two uni* *tals E1 - operads in C are isomorphic in Ho(Op(C)) (because they are both isomorphic to C* *OM). 2 The previous corollary justifies the following definition 35 Definition 3.1.5The semi-model category of E1 -algebras in C is defined to be A* *lg(O), where O is a unital E1 -operad in C. It will be denoted by E1 - Alg(C). The opposite semi-model category (E1 - Alg(C))op will be called the semi-mod* *el cate- gory of affine stacks over C and will be denoted by C - Aff. An E1 -algebra A considered as an object in C - Aff will be symbolically den* *oted by Spec A. The same notations and terminology will be used for the category CU - Aff :=* * (E1 - Alg(CU))op. Remarks: o By conventions, the model category C is an algebra over the monoidal model* * cat- egory SSet of simplicial sets (i.e. is a simplicial monoidal model categor* *y) or over the category C(Z) of complexes of abelian groups (i.e. is a complicial mo* *noidal model category). There are well known and famous E1 -operads in SSet and C* *(Z), for example the singular realizations of the little n-cubes operad and of * *the linear isometries operad, as well as their homology complexes (see [Kr-Ma , I.5])* *. These operads can be transported to C via the unit of the algebra structure SSet* * -! C or C(Z) -! C and give rise to unital E1 -operads in C. This implies that in p* *ractice, there exist natural choices for the unital E1 -operad in C. o It is important to note that E1 - Alg(CU) is a sub-model category of E1 - * *Alg(C), which is U-cofibrantly generated and V-small, as soon as the E1 -operad ha* *s been chosen in CU. The category Aff(CU), opposite to the category E1 - Alg(CU)* * is really the category we will be mostly interested in. o If the model structure on C is trivial, then so is the model structure on * *E1 -Alg(C). Actually, a unital E1 -operad is then automatically isomorphic to the oper* *ad COM. Therefore in this case, E1 -Alg(C) is just the trivial model category of c* *ommutative and unital monoids in C. Let O be an operad in C and A be an O-algebra in C. An A-module in C is the * *data of an object M 2 C and structural morphisms O(n) An-1 M -! M. These structural morphisms are required to satisfy certain associativity, commutativity and unit* * rules that the reader may find in [Kr-Ma , I.4.1]. A morphism of A-modules is the data of * *a morphism M -! M0 in C, commuting with the structural morphisms. These definitions allow* * to define the category of modules over a fixed operad algebra A over a fixed opera* *d O in C, which will be simply denoted by Mod(A). Let O 2 Op(C) be an operad in C and f : A -! A0be a morphism in Alg(O). Then, there exists a natural restriction functor f* : Mod(A0) -! Mod(A). This functor has a left adjoint f* : Mod(A) -! Mod(A0). 36 As for the case of operads and algebras, a morphism f : M - ! M0 of A-modules in C is a fibration (resp. an equivalence) if it is a fibration (resp. an equiv* *alence) when considered as a morphism in C. Theorem 3.1.6 ([Sp, Thm. 6.1] and [Sp, Cor. 6.7]). Let O 2 Op(C) be a unital E1 -operad in C and A 2 Alg(O) a cofibrant E1 -algebr* *a in C. Then 1. The category Mod(A) of A-modules in C, together with the classes of fibrat* *ions and equivalences defined above, is a cofibrantly generated model category; 2. If f : A -! A0 is an equivalence between two cofibrant E1 -algebras in C, * *the ad- junction f* : Mod(A) -! Mod(A0) Mod(A) - Mod(A0) : f* is a Quillen equivalence. Again, if O 2 Op(CU) is a unital E1 -algebra and A 2 Alg(O) is an E1 -algebr* *a in CU, then the category of A-modules in CU is a sub-model category of Mod(A). It is f* *urther- more U-cofibrantly generated and V-small. The compatibility between the pushforward and pullback functors above, on th* *e model categories of modules, is expressed through the following base change formula. Proposition 3.1.7 ([Sp, Prop. 9.12]). Let f A _____//B g || |g0| fflffl|fflffl| A0__f0//_B0 be a homotopy co-cartesian diagram of cofibrant E1 -algebras in C. Then, for an* *y M 2 Ho(Mod(B)), the natural base change morphism L(g)*Rf*(M) -! Rf0*L(g0)*(M) is an isomorphism in Ho(Mod(A0)). 3.2 Geometric stacks over a monoidal model category For this paragraph, recall our basic setting of this section: an inclusion CU * * C of monoidal model categories, satisfying the conditions explained at the beginning of this * *section. We will assume that the semi-model category CU -Aff of affine stacks over CU is en* *dowed with a topology ø. This semi-model site will be denoted by (CU -Aff, ø) and the corr* *esponding model category of stacks will simply be denoted by CU -Aff~,fi. For the sake of* * simplicity we assume the topology ø is sub-canonical (see Def. 2.4.3). 37 Recall from Section 1 the existence of a Quillen adjunction Re: CU - Aff~,fi-! C - Aff CU - Aff~,fi- C - Aff : Spec, where we denote by Spec the functor we have called h_in Section 1, because this* * seems more natural when dealing with E1 -algebras. This Quillen adjunction, as we sa* *w in Section 1, induces a fully faithful functor (the Yoneda embedding) RSpec_: Ho(CU - Aff) = Ho(E1 - Alg(CU))op- ! Ho(CU - Aff~,fi). In order to give the definition of n-geometric stacks, we will need to add t* *he following hypothesis on our topology ø: Hypothesis 3.2.1 Let {Spec Bi -! Spec A}i2I be a U-small family of morphisms in Ho(CU - Aff) and, for each i 2 I, let {Spec Cj -! Spec Bi}j2Jibe a ø-covering f* *amily. If the induced family in Ho(CU - Aff), {Spec Cj -! Spec A}i2I,j2Jiis a ø-coveri* *ng, then so is {Spec Bi- ! Spec A}i2I. The definition of an n-geometric stack over CU is given by induction on n. W* *e define simultaneously the notion of n-geometric stack and the notion of n-covering fam* *ily by induction on n. Note that the both definitions depend on the topology ø, and on* *e should probably use the expression nfi-geometric stacks. However, as we will not consi* *der different topologies at the same time, we will omit the reference to ø. Definition 3.2.2 o The category of 0-geometric stacks over CU is the essentia* *l image of the functor RSpec. It will be denoted by 0-GeSt(CU) and is equivalent t* *o CU-Aff via the Yoneda embedding. Note also that it does not depend on ø. The category 0 - GeSt will also be called the category of affine stacks ov* *er CU. o A morphism f : F -! F 0in Ho(CU - Aff~,fi) is 0-representable if for any * *0- geometric stack H and any morphism H -! F 0, the homotopy pull-back F xhF0* *H is a 0-geometric stack (this is again independent of ø). o A U-small family of morphisms {fi: Fi- ! F 0}i2I, I 2 U, in Ho(CU - Aff~,f* *i), is a 0-covering if it satisfies the two following conditions: - For any i 2 I, the morphism fi is 0-representable; - For any 0-geometric stack H, any morphism H - ! F 0and any i 2 I, the homotopy pull-back family {FixhF0H - ! H}i2I (which is a U-small family of morphisms of 0-geometric stacks by the first condition), correspond* *s to a ø-covering family in Ho(CU - Aff). Let us suppose that the full sub-category (n - 1) - GeSt(CU) Ho(CU - Aff~,* *fi) of (n-1)-geometric stacks has been defined, as well as the notion of a (n-1)-cover* *ing family in Ho(CU - Aff~,fi). o A morphism f : F -! F 0in Ho(CU - Aff~,fi) is (n - 1)-representable if, fo* *r every 0-geometric stack H and any morphism H - ! F 0, the homotopy fibred product F xhF0H 2 Ho(CU - Aff~,fi) is an (n - 1)-geometric stack. 38 o A stack F 2 Ho(CU - Aff~,fi) is n-geometric if it satisfies the following two* * condi- tions: - The diagonal morphism F -! F x F is (n - 1)-representable. - There exists a U-small (n - 1)-covering family {fi: Fi- ! F }i2I, I 2 U, * *such that each Fi is a 0-geometric stack. Such a family will be called a (n - * *1)-atlas for F . The full sub-category of Ho(CU - Aff~,fi) consisting of n-geometric stacks wi* *ll be denoted by n - GeSt(CU). o Let F be a 0-geometric stack. A U-small family of morphisms {fi: Fi- ! F }i2I* *in n - GeSt(CU) is a special n-covering if, for any i 2 I, there exists a (n - 1* *)-atlas {Hi,j-! Fi}j2Ji, such that the induced family {Hi,j-! F }i2I,j2Jiis a 0-cover* *ing in Ho(CU - Aff~,fi). o A U-small family of morphisms {fi : Fi -! F }i2I in Ho(CU - Aff~,fi) is a n- covering if it satisfies the following two conditions> - For any i 2 I, the morphism fi is n-representable; - For any 0-geometric stack H and any morphism H -! F 0, the homotopy pull- back family {FixhF0H - ! H}i2I (which is a family of morphisms from n- geometric stacks to a 0-geometric stack), is a special n-covering family * *(as defined before). A stack will be simply called geometric if it is n-geometric for some integer n. Remarks: o For F an n-geometric stack, the integer n refers to the complexity of the geo* *metry of F and not to its homotopical complexity as the usual expression n-stack re* *fers to. In general, the notion of n-geometric stack has nothing to do with the no* *tion of n-stack i.e. of n-truncated simplicial presheaf. These two notions relate eac* *h others only when the model structure on C is trivial and the reason is that in this * *case affine stacks are 0-truncated (i.e. are presheaves of constant simplicial set* *s). o The reader should be warned that when CU is the monoidal trivial model catego* *ry of U-abelian groups, then our notion of n-geometric stacks for n = 0, 1 is not e* *quivalent to the notion of algebraic spaces and algebraic stacks as commonly used (e.g.* *, in [La-Mo ]), say with ø the ffqc-topology. For example, a non-affine scheme is* * not a 0-geometric stack in our sense but it is a 1-geometric stack if it is separat* *ed. To get non-separated schemes one needs to consider 2-geometric stacks. In the same w* *ay, an Artin stack with a non-affine diagonal is not a 1-geometric stack in our s* *ense. It is a 2-geometric stack if it is quasi-separated but only a 3-geometric stack * *in general. 39 Using a big induction argument on n like it is done in [S1], one proves the * *following basic proposition. We will not rewrite the argument in this version of the pap* *er. Note however that the proof uses in an essential way the hypothesis 3.2.1 on our top* *ology. This hypothesis is precisely used to prove independence of the choice of atlases. Proposition 3.2.3 With the notations as above: 1. There are natural inclusions n - GeSt(CU) (n + 1) - GeSt(CU); 2. The set of n-representable morphisms is stable by composition and base cha* *nge. Any isomorphism is n-representable for any n; 3. The set of n-covering families is stable by compositions and base changes.* * Any isomorphism is a n-covering family for any n; 4. If f : F -! F 0is a n-representable morphism and F 0is a n-geometric stack* *, then so is F ; 5. The sub-category n - GeSt(CU) Ho(CU - Aff~,fi) is stable under homotopy * *fibred products. Proof: See [S1]. 2 3.3 An example: Quotient stacks The model category of stacks CU - Aff~,fiis a U-cofibrantly generated model cat* *egory and therefore, for any U-small category I, the category of I-diagrams (CU - Aff* *~,fi)I is a model category with the so-called projective model structure (see [Hi, Thm. 13.* *8.1]). Let us recall that fibrations and equivalences in (CU - Aff~,fi)I are defined level* *wise. We will be interested in the case I = opi.e. in the category of simplicial objects in * *CU - Aff~,fi. As usual we will denote sCU - Aff~,fi:= (CU - Aff~,fi) op and, for X* 2 sCU - A* *ff~,fi, Xn := X([n]). Recall that for any integer n > 0 and any X* 2 sCU - Aff~,fi, there exists a* * Segal morphism Sn : Xn -! X1 xhX0X1 xhX0. .x.hX0X1 ___________-z__________" n times induced by the morphisms in , ffi: [1] -! [n] do : [0] -! [1] d1 : [0] -! [1], which are defined by ffi(0) = i ffi(1) = i + 1, 0 i < n. The following definition is a generalization of Segal's op-spaces to the ca* *se where the space of objects is not contractible. Definition 3.3.1An object X* 2 sCU - Aff~,fiis called a Segal groupoid object i* *f it satisfies the following two conditions: 40 1. For every integer n > 0, the Segal morphism Sn : Xn -! X1 xhX0X1 xhX0. .x.hX0X1, ___________-z__________" n times is an equivalence in CU - Aff~,fi; 2. The natural morphism d2 x d1 : X2 -! X1 xhd1,X0,d1X1 is an equivalence in CU - Aff~,fi. Remark. Condition (2) implies that the induced simplicial object in Ho(CU - * *Aff~,fi) is a groupoid object. The colimit functor colim : sCU - Aff~,fi-! CU - Aff~,fiis clearly a left Qu* *illen functor and can then be left derived to a functor at the level of homotopy cate* *gories | - | := hocolim op : Ho(sCU - Aff~,fi) -! Ho(CU - Aff~,fi). Definition 3.3.2If X* 2 Ho(sCU - Aff~,fi) is a Segal groupoid, then |X*| 2 Ho(C* *U - Aff~,fi) is called the quotient stack of X*. The fundamental theorem of this section is the following generalization of t* *he criterion of [S1, Prop. 4.1]. We will only provide a sketch of its proof in this version * *of the paper. Theorem 3.3.3 A stack F 2 Ho(CU - Aff~,fi) is n-geometric if and only if there* * exists a Segal groupoid X* 2 Ho(sCU - Aff~,fi) such that F ' |X*| and satisfying the f* *ollowing two conditions: o The stack X0 is a U-small disjoint union of (n - 1)-geometric stacks; o Each of the two natural morphisms d0, d1 : X1 -! X0 is a (n - 1)-covering * *family (with one element). Sketch of proof: Suppose that X* is a Segal groupoid satisfying the two cond* *itions of the theorem. Then, using [Se, Prop. 1.6] and corollary 2.5.6, one checks that t* *he natural morphism X0 -! |X*| is such that X0 xh|X*|X0 ' X1. From this one deduces easily* * that if {Hi -! X0}i2Iis an (n - 1)-atlas for X0, then {Hi -! X0 -! |X*|}i2Iis again * *an (n-1)-atlas for |X*|. One can also check that the following diagram is homotopy* * cartesian |X*|_____//|X*|OxO|X*|OO | | | | | | X1 ______//X0 x X0. This implies that the diagonal of |X*| is (n - 1)-representable and finally tha* *t |X*| is n-geometric. 41 For the other implication,`let F be an n-geometric stack and {Hi- ! F }i2Ian* * (n-1)- atlas. Let X0 := i2IHi -! F be the induced morphism. The homotopy nerve of X0 -! F is a simplicial object X* 2 Ho(sCU - Aff~,fi) such that |X*| ' F . Fur* *ther- more, the fact that F is n-geometric implies that X* satisfies the two conditio* *ns of the theorem. 2 4 Applications and perspectives In this last section, we present two applications of our theory. The first one * *is an approach to DG-schemes in which we interpret them as geometric stacks over the model cat* *egory of complexes. The second application is a definition of 'etale K-theory of E1 -* *ring spectra. Several other applications will appear in a forthcoming version. 4.1 An approach to DG-schemes For this paragraph let k be a commutative ring with unit, C(k) the symmetric mo* *noidal model category of complexes of k-modules in V and C(k)U the full sub-category o* *f C of objects belonging to U. We adopt the convention that complexes are Z-graded co-* *chain complexes (i.e differentials increase degrees). As explained in the previous s* *ection, we will work with a fixed unital E1 -operad O in C(k)U. For the sake of simplicity* *, we will assume that for each n, one has O(n)i= 0 for i > 0 (i.e. the operad O is concen* *trated in non-positive degrees). Applying definition 3.1.5, we can consider the semi-model categories of affi* *ne stacks in C(k) and in C(k)U C(k) - Aff C(k)U - Aff. In this special case, it is known that C(k) - Aff and C(k)U - Aff are actually * *model categories (see [Hin]). Let us fix one of the standard Grothendieck topologies ø0 on the category of* * k-schemes (e.g. Zariski, Nisnevich, 'etale, faithfully flat, . . . ). Starting from ø0,* * we construct a topology ø on the model category C(k)U - Aff (Def. 2.2.1) in the following way.* * Recall that for Spec A 2 C(k) - Aff, one can consider its cohomology algebra H*(A) = * *Hi(A) which is in a natural way a graded commutative k-algebra. The construction A 7!* * H*(A) is of course functorial and therefore defines a functor from Ho(C(k) - Aff)op t* *o graded commutative k-algebras. The following definition was inspired by the work of K. Behrend [Be ], where* * an 'etale topology on differential graded algebras is used. Definition 4.1.1A U-small family of morphisms in C(k)U - Aff {fi: Spec Ai- ! Spec B}i2I is a ø-covering if it satisfies the following two conditions: 42 o The induced family of morphisms of (usual) affine schemes {fi: Spec H0(Ai) -! Spec H0(B)}i2I is a ø0-covering; o For any i 2 I, the induced morphism H*(B) H0(B)H0(Ai) -! H*(Ai) is an isomorphism. The topology ø on C(k)U - Aff, associated to the Grothendieck topology ø0 on* * k- schemes, will be called the strong ø0-topology. Covering families in (C(k)U - A* *ff, ø) will be called strongly ø0-covering families. The above definition allows one to introduce the strong Zariski (resp. Nisne* *vich, 'etale, faithfully flat and quasi-compact, . . . ) topology on C(k)U-Aff. The correspon* *ding model site will be denoted by (C(k)U - Aff, Zar) (resp. (C(k)U - Aff, Nis), resp. (* *C(k)U - Aff, 'et), resp. (C(k)U - Aff, ffqc), . . . ). The associated model categories * *of stacks will be naturally denoted by C(k)U - Aff~,Zar C(k)U - Aff~,Nis C(k)U - Aff~,'et C(k)U - Aff~,ffqc and so on Proposition 4.1.2 For any Grothendieck topology ø0 on k-Sch which is coarser th* *an the faithfully flat and quasi-compact topology, the induced strong ø0-topology ø on* * C(k)U -Aff is sub-canonical. Proof: Using lemma 2.4.2, it is enough to show that for any ø-hypercover Spe* *c B* -! Spec A in C(k)U - Aff, the natural morphism A -! holim[n]2Bn is an equivalence of E1 -algebras. As the forgetful functor from the category * *of E1 - algebras to the category of complexes commutes with homotopy limits, it is enou* *gh to show that A -! holim[n]2Bn is a quasi-isomorphism of complexes of k-modules. Further* *more, in the model category C(k) the homotopy limits along can be computed using to* *tal complexes and therefore it is enough to show that the natural morphism of compl* *exes A -! T ot(B*) is a quasi-isomorphism. To prove this, we use the spectral sequence computing t* *he coho- mology of a total complex as described e.g. in [We , 5.6], Ep,q2= Hp(Hq(B*)) ) Hp+q(T ot(B*)), where Hq(B*) is the normalized complex associated to the co-simplicial k-module* * ([n] 7! Hq(Bn)). Now, by definition of a ø-hypercover and by the hypothesis on ø0, one * *can use 43 the T or spectral sequence (see [Kr-Ma , thm. V.7.3]) to prove that the co-simp* *licial algebra ([n] 7! H*(Bn)) corresponds to a faithfully flat hypercover of affine schemes Spec H*(B*) -! Spec H*(A). By the usual faithfully flat descent (see [Mi , xI]), the above spectral sequen* *ce degenerates and satisfies Ep,q2= 0 forp 6= 0, E0,q2= Hq(A). This in turns implies that A -! T ot(B*) is a quasi-isomorphism. * * 2 We recall from [Ci-Ka1] the notion of DG-scheme. We will actually adopt a sl* *ightly different definition which is adapted to the case of an arbitrary base ring k. * *In the case k is a field of characteristic zero, our notion and that of [Ci-Ka1] are homotopical* *ly equivalent (see below). Let X be a k-scheme (all schemes will be separated and quasi-compact) and CQ* *Coh(OX ) its category of complexes of quasi-coherent OX -modules. This category is an al* *gebra over the symmetric monoidal category C(k), therefore it makes sense to talk about E1* * -algebras in CQCoh(OX ) (see [Sp]). Definition 4.1.3A (separated and quasi-compact) DG-scheme is a pair (X, AX ) wh* *ere X is a (separated and quasi-compact) k-scheme and AX is a E1 -algebra in CQCoh(* *OX ) satisfying the following two conditions: o AX is concentrated in non-positive degrees (i.e. AiX= o for i > 0); o The unit morphism OX -! A0Xis an isomorphism. A morphism between DG-schemes f : (X, AX ) -! (Y, AY ) is the data of a morp* *hism of schemes f : X - ! Y together with a morphism of E1 -algebras in CQCoh(OX ), f*(AY ) -! AX . For a DG-scheme (X, AX ), the cohomology sheaf H0(AX ) is a quasi-coherent O* *X - algebra whose associated X-affine scheme will be denoted by H 0(X, AX ) := Spec H0(AX ) -! X. Actually, as A0X ' OX and A1X = 0, the scheme H 0(X, AX ) is a closed sub-sche* *me of X. The cohomology sheaves H*(AX ) are naturally quasi-coherent H0(AX )-modules * *and therefore correspond to quasi-coherent sheaves on the sub-scheme H0(X, AX ). Th* *ey will still be denoted by H*(AX ). Definition 4.1.4A morphism of DG-schemes f : (X, AX ) - ! (Y, AY ) is a quasi- isomorphism if it satisfies the following two conditions: o The induced morphism of schemes H0(f) : H0(X, AX ) -! H0(Y, AY ) is an iso* *mor- phism; 44 o The natural morphism of quasi-coherent sheaves on H0(X, AX ) ' H0(Y, AY ) H*(AY ) -! H*(AX ) is an isomorphism. The homotopy category of DG-schemes is the category obtained from the catego* *ry of DG-schemes belongings to U by formally inverting the quasi-isomorphisms. It wi* *ll be denoted by Ho(DG - Sch). Remarks: o The category of DG-schemes in U is a V-small category. Therefore, Ho(DG - * *Sch) is also a V-small category but it is not clear a priory that it is a U-sma* *ll category. o When k is a field of characteristic zero, the definition of DG-scheme give* *n in [Ci-Ka1] is not strictly equivalent to 4.1.3. However, it is well known that in th* *is case the homotopy theory of commutative differential graded algebras is equivalent * *to the ho- motopy theory of E1 -algebras. This fact implies easily that the homotopy * *category of DG-schemes as defined in [Ci-Ka1] (and called by the authors, the right* * derived category of k-schemes) is equivalent to our Ho(DG - Sch). o Let A be a E1 -algebra in U such that Ai = 0 for i > 0. As the operad O is concentrated in non-positive degrees, the k-module A0 carries an induced E* *1 -algebra structure. As it is a complex concentrated in degree zero, this is then n* *othing else than a commutative and unital algebra structure. Moreover, it is cle* *ar that the natural morphism of complexes A0 -! A is a morphism of E1 -algebras. * *In particular, A is naturally a complex of A0-modules. This implies that for * *any E1 - algebra A such that Ai = 0 for i > 0, one can define a DG-scheme X := Spec* *_A, whose underlying scheme is Spec A0 and with AX := eA2 QCoh(X). It is clear* * that any DG-scheme (X, AX ) such that X is an affine scheme is of the form Spec* *_A for some E1 -algebra in non-positive degrees A (in fact, one has A ' (X, AX )* *). Proposition 4.1.5 There exists a functor : Ho(DG - Sch) -! Ho(C(k)U - Aff~,ffqc) such that, for any E1 -algebra in non-positive degrees A, one has (Spec_A) ' RSpec A. Moreover, for every DG-scheme (X, AX ), the stack (X, AX ) is 1-geometric. Sketch of Proof: Let (X, AX ) be a DG-scheme and let {Ui}i2I be a finite Za* *riski covering of X by affine schemes. Taking the nerve of this covering yields a si* *mplicial diagram of affine schemes a [n] 7! Ui0,...,in, i0,...,in2In+1 45 where Ui0,...,in:= Ui0\ . .\.Uin. By restricting AX on each Ui0,...,in, one act* *ually obtains a simplicial diagram of DG-schemes a a [n] 7! ( Ui0,...,in, AUi0,...,in). i0,...,in2In+1 ` Moreover, as each i0,...,in2In+1Ui0,...,inis an affine scheme, this diagram i* *s actually the image by Spec_of a co-simplicial diagram of E1 -algebras or, equivalently, of a* * simplicial diagram in C(k)U - Aff F (U, X) : op ____//_C(k)U - Aff [n]Ø______//_Spec An Considering its image by RSpec, this diagram induces a well defined object i* *n Ho(s(C(k)U- Aff~,ffqc)), the homotopy category of simplicial objects in C(k)U - Aff~,ffqc F (U, X) : op ____//_C(k)U - Aff~,ffqc [n]Ø_______//RSpec An. We then define (U, X) := hocolim[n]2 opRSpec An as an object in Ho(C(k)U - Aff~,ffqc). With some work, it is not difficult to v* *erify that the stack (U, X) does not depend on the choice of the affine covering {Ui}i2Ia* *nd that (X, AX ) 7! (U, X) defines a functor : Ho(DG - Sch) -! Ho(C(k)U - Aff~,ffqc). By construction, it is clear that (Spec_A) ' RSpec A. Finally, to prove that (X, AX ) is a 1-geometric stack, one applies the cri* *terion 3.3.3. The conditions of 3.3.3 are satisfied because by construction (X, AX ) is the * *geomet- ric realization of the Segal groupoid [n] 7! RSpec An, for which the natural mo* *rphisms RSpec A1 -! RSpec A0 are clearly strong Zariski coverings and a fortiori coveri* *ngs in C(k)U - Aff~,ffqc. 2 We make the following Conjecture 4.1.6 The functor of proposition 4.1.5 is fully faithful. This conjecture says that the homotopy theory of DG-schemes can be embedded * *into the homotopy theory of geometric stacks over the model category of complexes. I* *n other words, the theory of DG-schemes should be a part of algebraic geometry over the* * model category of complexes. We propose the model category of stacks C(k)U - Aff~,ffq* *cas a natural setting for the theory of DG-schemes and more generally, for the theory* * of DG- stacks. One of the reasons why we believe this is a natural candidate is that i* *n this way DG-schemes would appear naturally as a part of a fully-fledged homotopy theory,* * in the abstract modern sense of Quillen model categories. Instead, trying to obtain in* * a complete 46 elementary way a homotopy structure out of usual DG-schemes (e.g., defining the* * weaker structure of a category with fibrations and equivalences, by declaring smooth m* *aps to be fibrations and quasi-isomorphisms to be equivalences, as it seems to be suggest* *ed in [Ka ]) seems to run into difficulties and moreover it is not a priori clear what kind * *of flexibility such a construction could have. 4.2 E'tale K-theory The problem of definition 'etale K-theory was raised by P.A. Ostvær and we give* * below a possible answer. We were very delighted by the question since it looked as a pa* *rticularly good test of applicability of our theory. Let Sp be the model category of symmetric spectra in V and SpU its sub-model category of objects in U (see [Ho-Sh-Sm ]). The wedge product of symmetric spe* *ctra makes Sp and SpU into symmetric monoidal model categories. Applying definition* * 3.1.5, we may consider the semi-model categories SpU - Aff of affine stacks over SpU. * *Again, it is know that SpU - Aff is actually a model category. For each object Spec A 2 SpU - Aff, one can consider the category of A-modul* *es in SpU, Mod(A)U as defined in the previous section. As Sp satisfies the monoid a* *xiom, Mod(A)U is actually a model category (with fibrations and equivalences defined * *on the underlying objects) which is moreover Quillen equivalent to Mod(QA0)U, where QA* *0is a cofibrant replacement of A. Therefore, in theorem 3.1.6 one does not need to as* *k A to be a cofibrant object in order to get a good theory of modules. Recall from [Sp, Prop. 9.10] that the homotopy category Ho(Mod(A)U) is a cl* *osed symmetric monoidal category. One can therefore define the notion of strongly d* *ualiz- able objects in Ho(Mod(A)U) (following [E-K-M-M , xIII.7]). The full sub-cate* *gory of Mod(A)cUconsisting of strongly dualizable objects will be denoted by Mod(A)sdU,* * and will be equipped with the induced notion of cofibrations and equivalences comin* *g from Mod(A)U. It is not difficult to check that with this structure, Mod(A)sdUis the* *n a Wald- hausen category (see [E-K-M-M , xV I]). Furthermore, if A -! B is a morphism o* *f E1 - algebras in Sp , then the base change functor f* : Mod(A)sdU-! Mod(B)sdU, being the restriction of a left Quillen functor, preserves equivalences and cof* *ibrations. This makes the lax functor Mod(-)sdU: SpU -! Cat Spec A 7! Mod(A)sdU (f : A ! B) 7! f* into a lax presheaf of Waldhausen V-small categories. Applying standard strict* *ification techniques we deduce a presheaf of V-simplicial sets of K-theory K(-) : SpU - ! SSet Spec A 7! K(Mod(A)sdU). 47 Definition 4.2.1The previous presheaf will be considered as an object in SpU - * *Aff^ and will be called the presheaf of K-theory over the symmetric monoidal model c* *ategory SpU. For any Spec A 2 SpU - Aff, we will write K(A) := K(Spec A). Remark. The same construction as above works if one replaces SpU by a genera* *l sym- metric monoidal model category allowing therefore to define the spectrum K(A) f* *or any E1 -algebra A in a general symmetric monoidal model category. It could be inter* *esting to look at this construction for the motivic categories considered in [Sp, 14.8]. Definition 4.2.2Let ø be a topology on the model category SpU -Aff and SpU -Aff* *~,fi the associated model category of stacks. Let K -! Kfibe a fibrant replacement o* *f K in SpU - Aff~,fi. The Kfi-theory space of an E1 -algebra A in SpU is defined by Kfi(A) := Kfi(Spec A). The natural morphism K -! Kfiinduces a natural augmentation (localization morph* *ism) K(A) -! Kfi(A). Remark. Note that we have Kfi(A) ' RHom__w,fi(hSpec A, K) ' RHom__w,fi(RSpec A, K). An application: 'etale K-theory of E1 -ring spectra. One defines an 'etale topology on SpU - Aff by stating that a family {fi : Spec* * Bi -! Spec A}i2Iis an 'etale covering if it satisfies the following three conditions: 1. For all i 2 I, the morphism A -! Bi is a formally 'etale morphism of E1 -r* *ing spectra (in the sense that the corresponding co-tangent complex LBi=A of [* *Hin, 7] vanishes); 2. For all i 2 I, the A-algebra Biis finitely presented (in any reasonable se* *nse, see e.g. [Ma-Re ] or [Ro , p. 7] in the ä bsolute" case, for connective, p-complete* * spectra)4; 3. The family of base change functors {Lf*i: Ho(Mod(A)U) -! Ho(Mod(Bi)U)}i2I is conservative i.e. a morphism in Ho(Mod(A)U) is an isomorphism if and on* *ly if, for any i 2 I, its image in Ho(Mod(Bi)U) is an isomorphism. 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