2 The model category of D-stacks

In this Section we will present the construction of a model category of D-stacks. It will be our derived version of the category of stacks that is commonly used in moduli theory, and all our examples of derived moduli stacks will be objects of this category.

The main idea of the construction is the one used in [HAG-I], and consists of adopting systematically the functorial point of view. Schemes, or stacks, are sheaves over the category of commutative algebras. In the same way, D-stacks will be sheaves-like objects on the category of commutative differential graded algebras. This point of view may probably be justified if one convinces himself that commutative differential graded algebras have to be the affine derived moduli spaces, and that therefore they are the elementary pieces of the theory that one would like to glue to obtain global geometric objects. Another, more down to earth, justification would just be to notice that all of the wanted derived moduli spaces we are aware of, have a reasonable model as an object in our category of D-stacks.

Before starting with the details of the construction, we would like to mention that K. Behrend has independently used a similar approach to DAG that uses the 2-category of differential graded algebras (see [Be]) (his approach is actually the 2-truncated version of ours). It is not clear to us that the constructions and results we are going to present in this work have reasonable analogs in his framework, as they use in an essential way higher homotopical informations that are partially lost when using any truncated version.

Conventions. For the sake of simplicity, we will work over the field of complex numbers . The expression cdga will always refer to a non-positively graded commutative differential graded algebra over with differential of degree 1. Therefore, a cdga A looks like

2.1 D-Pre-stacks

We start by defining D - Aff := CDGA^op to be the opposite category of cdga's, and we consider the category SPr(D - Aff), of simplicial presheaves on D - Aff, or equivalently the category of functors from CDGA to SSet. The category SPr(D - Aff) is endowed with its objectwise projective model structure in which fibrations and equivalences are defined objectwise (see [Hi, 13.10.17]).

For any cdga A Î D - Aff, we have the presheaf of sets represented by A, denoted by

By general properties of left Bousfield localization (see [Hi]), the fibrant objects in D - Aff^Ù are the functors F : CDGA-®SSet satisfying the following two conditions

1. For any A Î CDGA, the simplicial set F(A) is fibrant.
2. For any quasi-isomorphism u : A-®B in CDGA, the induced morphism F(u) : F(A)-®F(B) is a weak equivalence of simplicial sets.

From this description, we conclude in particular, that the homotopy category Ho(D -Aff^Ù) is naturally equivalent to the full sub-category of Ho(SPr(D -Aff)) consisting of functors F : CDGA-®SSet sending quasi-isomorphisms to weak equivalences. We will use implicitely this description, and we will always consider Ho(D - Aff^Ù) as embedded in Ho(SPr(D - Aff)).

2.2 D-Stacks

Now that we have constructed the model category of D-pre-stacks we will introduce some kind of étale topology on the category D - Aff. This will allow us to talk about a corresponding notion of étale local equivalences in D - Aff^Ù, and to define the model category of D-stacks by including the local-to-global principle into the model structure.

We learned the following notion of formally étale morphism of cdga's from K. Behrend.

From Definition 2.4 we now define the notion of étale covering families. For this, we recall that a morphism of cdga's A-®B is said to be finitely presented if B is equivalent to a retract of a finite cell A-algebra (see for example [EKMM]). This is also equivalent to say that for any filtered systems {A-®C_i}_iÎI, the natural morphism

Definition 2.6 A finite family of morphisms of cdga's

is called an étale covering if it satisfies the following three conditions

1. For any i Î I, the morphism A-®B_i is finitely presented.
2. For any i Î I, the morphism A-®B_i is formally étale.
3. The induced family of morphisms of affine schemes is an étale covering.

The above definition almost defines a pre-topology on the category D - Aff. Indeed, stability and composition axioms for a pre-topology are statisfied, but the base change axiom is not. In general, the base change of an étale covering {A-®B_i}_iÎI along a morphism of A-®C will only be an étale covering if A-®C is a cofibration in CDGA. In other words, for the base change axiom to be satisfied one needs to replace fibered products by homotopy fibered products in D - Aff. Therefore, the étale covering families of Definition 2.6 do not satisfy the axioms for a pre-topology on D - Aff, but rather satisfy a homotopy analog of them. This is an example of a model pre-topology on the model category D - Aff, for which we refer the reader to [HAG-I, §4.3] where a precise definition is given.

In turns out that the data of a model pre-topology on a model category M is more or less equivalent to the data of a Grothendieck topology on its homotopy category Ho(M) (see [HAG-I, Prop. 4.3.5]). In our situation, the étale coverings of Definition 2.6 induce a Grothendieck topology, called the étale topology on the opposite of the homotopy category Ho(D - Aff) of cdga's. More concretely, a sieve S over a cdga A Î Ho(D -Aff) is declared to be a covering sieve if it contains an étale covering family {A-®B_i}_iÎI. The reader will check as an exercise that this defines a topology on Ho(D -Aff) (hint: one has to use that étale covering families are stable by homotopy pull-backs in D -Aff, or equivalentely by homotopy push-outs in CDGA). From now on, we will always consider Ho(D - Aff) as a Grothendieck site for this étale topology.

For a D-pre-stack F Î D - Aff^Ù (recall from Definition 2.3 that this implies that F sends quasi-isomorphisms to weak equivalences), we define its presheaf of connected components

As for the case of simplicial presheaves (see [Ja1]), one can also define higher homotopy sheaves, which are sheaves of groups and abelian groups on the sites Ho(D -Aff/A) for various cdga's A. Precisely, let F be a D-pre-stacks and s Î F(A)₀ a point over a cdga A Î D - Aff. We define the n-th homotopy group presheaf pointed at s by

The notion of homotopy sheaves defined above gives rise to the following notion of local equivalences.

One of the key results of "HAG" is the following theorem. It is a very special case of the existence theorem [HAG-I, §4.6], which extends the existence of the local model structure on simplicial presheaves (see [Ja1]) to the case of model sites.

The reason for calling D - Aff the model category of D-stacks is the following proposition. It follows from [HAG-I, 4.6.3], which is a generalization to model sites of the main theorem of [DHI].

Condition (3) is called the stack condition for the étale topology. Note that a typical étale hyper-covering of cdga's A-®B_* is given by the homotopy co-nerve of an étale covering morphism A-®B

The second part of the definition is justified because the homotopy category Ho(D -Aff) is naturally equivalent to the full sub-category of Ho(SPr(D -Aff)) consisting of objects satisfying conditions (2) and (3) of Proposition 2.9.

2.3 Operations on D-stacks

One of the main consequences of the existence of the model structure on D - Aff is the possibility to define several standard operations on D-stacks, analogous to the ones used in sheaf theory (limits, colimits, sheaves of morphisms ...).

First of all, the category D - Aff being a category of simplicial presheaves, it comes with a natural enrichement over the category of simplicial sets. This makes D -Aff into a simplicial model category (see [Ho, 4.2.18]). In particular, one can define in a standard way the derived simplicial Hom's (well defined in the homotopy category Ho(SSet)),

This simplicial structure also allows one to define exponentials by simplicial sets. For an object F Î D - Aff and K Î SSet, one has a well defined object in Ho(D - Aff)

The existence of the model structure D - Aff also implies the existence of homotopy limits and homotopy colimits, as defined in [Hi, §19]. The existence of these homotopy limits and colimits is the analog of the fact that category of sheaves have all kind of limits and colimits. We will use in particular homotopy pull-backs i.e. homotopy limits of diagrams , that will be denoted by

Finally, one can show that the homotopy category Ho(D -Aff) is cartesian closed (see [HAG-I, §4.7]). Therefore, for any two object F and G, there exists an object HOM(F,G) Î Ho(D - Aff), which is determined by the natural isomorphisms

If one looks at these various constructions, one realizes that D -Aff has all the homotopy analogs of the properties that characterize Grothendieck topoi. To be more precise, C. Rezk has defined a notion of homotopy topos (we rather prefer the expression model topos), which are model categories behaving homotopically very much like a usual topos. The standard examples of such homotopy topoi are model categories of simplicial presheaves on some Grothendieck site, but not all of them are of this kind; the model category D - Aff is in fact an example of a model topos which is not equivalent to model categries of simplicial presheaves on some site (see [HAG-I, §3.8] for more details on the subject).