5 Further examples

In this Section we present three examples of geometric D-stacks: the derived stack of local systems on a space, the derived stack of vector bundles and the derived stack of associative algebra and A_¥-categorical structures. The derived moduli space of local systems on a space has already been introduced and defined in [Ka2] as a dg-scheme. In the same way, the derived moduli space of (commutative) algebra structures has been constructed in [Ci-Ka2] also as a dg-scheme. Finally, the formal derived moduli spaces of local systems on a space and of A_¥-categorical structures have been considered in [Hin2, Ko2, Ko-So].

The new mathematical content of this part is the following. First of all we give a construction of the derived moduli stack of vector bundles, that seems to be new, and we also define global versions of the formal moduli spaces of A_¥-categorical structures that were apparently not known. We also provide explicit modular descriptions, by defining various derived moduli functors, which were not known (and probably not easily available), for the constructions of [Ka2, Ci-Ka1, Ci-Ka2].

5.1 Local systems on a topological space

Throughout this subsection, X will be a CW-complex. For any cdga A, we denote by A-Mod_X the category of presheaves of dg-A-modules over X. We say that a map M®N in A-Mod_X is a quasi-isomorphism if it induces a quasi-isomorphism of dg-A-modules on each stalk. This gives a notion of equivalences in the category A - Mod_X, and of equivalent objects (i.e. objects which are isomorphic in the localization of the category with respect to equivalences).

A presheaf M of dg-A-modules on X will be said locally on X ×A_ét equivalent to Aⁿ if, for any x Î X, there exists an open neighborhood U of x in X and an étale cover A ® B, such that the presheaves of dg-B-modules M_|U Ä_AB and Bⁿ are equivalent. We will also say that a presheaf M of dg-A-modules is flat, if for any open U of X, the dg-A-module M(U) is cofibrant. By composing with a cofibrant replacement functor in A - Mod, one can associate to any dg-A-module an equivalent flat dg-A-module (since equivalences are stable by filtered colimits). The category wLoc_n(X;A) of rank n local systems of dg-A-modules has objects those presheaves of flat dg-A-modules on X which are locally on X × A_ét equivalent to Aⁿ, and morphisms quasi-isomorphisms between them. For morphisms of cdga's A-®B we obtain pull-back functors

We denote by Loc_n(X) the simplicial presheaf on D -Aff sending a cdga A to (the nerve of wLoc_n(X;A)). We call it the D-pre-stack of rank n derived local systems on X.

Obviously, the objects in wLoc_n(X;A) are a derived version of the usual local systems of R-modules on X, where R is a commutative ring. More precisely, if we consider such an R as a cdga concentrated in degree zero, then RLoc_n(X;R) is the closure under quasi-isomorphisms of the groupoid of rank n local systems of R-modules on X; in other words, if we invert quasi-isomorphisms in the category wLoc_n(X;R) then we obtain a category which is equivalent to the groupoid of rank n local systems of R-modules on X.

Note that the classical points of Loc_n(X) (i.e. morphisms from iSpeck, for some commutative ring k) coincide with the classical points of its truncation h⁰Loc_n(X) which coincides with the usual (i.e. not derived) stack of rank n local systems on X. So we have no new classical points, as desired.

Let us give only some remarks to show what the proof of Theorem 5.1 really boils down to. First of all notice that the first assertion is a consequence of the second one, once one knows that Loc_n(pt) iBGL_n and is a stack; so we are reduced to prove the absolute case (X = pt) of 1. and 2. The first two properties in 3. follows from 2., the finiteness of X and the analogous properties of BGl_n. Finally the dimension count in 3. is made by an explicit computaion of the tangent D-stack at some local system E. Explicitly, one finds that (in the notations of §4.3) (W_Locn(X),E¹)^* is the complex C^*(X,End (E))[1], which is a complex of -vector spaces concentrated in degrees [-1,¥[ whose Euler characteristic is exactly -n²c(X).

It is important to notice that the D-stack Loc_n(X) might be non-trivial even if X is simply connected. Indeed, the tangent at the unit local system is always the complex C^*(X,)[1]. This shows that Loc_n(X) contains interesting information concerning the higher homotopy type of X. As noticed in the Introduction of [K-P-S], this is one of the reasons why the D-stack Loc_n(X) might be an interesting object in order to develop a version of non-abelian Hodge theory. We will therefore ask the same question as in [K-P-S].

This question is of course somewhat imprecise, and it is not clear that the object Loc_n(X) itself could really support an interesting Hodge structure. However, we understand the previous question in a much broader sense, as for example it includes the question of defining derived versions of the moduli spaces of flat and Higgs bundles, and to study their relations from a non-abelian Hodge theoretic point of view, as done for example in [S2].

5.2 Vector bundles on a projective variety

We now turn to the example of the derived stack of vector bundles, which is very close to the previous one. Let X be a fixed smooth projective variety.

If A is a cdga, we consider the space X (with the Zariski topology) together with its presheaf of cdga O_X Ä A. It makes sense to consider also presheaves of dg-O_X Ä A-modules on X and morphisms between them. We define a notion of equivalences between such presheaves, by saying the f : M-®N is an equivalence if it induces a quasi-isomorphism at each stalks. Using this notion of equivalences we can talk about equivalent dg-O_X Ä A-modules (i.e. objects which become isomorphic in the localization of the category with respect to quasi-isomorphisms).

We say that a presheaf of dg-O_X Ä A-module M on X is a vector bundle of rank n, if locally on X_zar × A_ét it is equivalent to (O_X Ä A)ⁿ (see the previous Subsection for details on this definition). We consider the category wV ect_n(X,A), of dg-O_X Ä A-modules which are vector bundles of rank n and flat (i.e. for each open U in X, the O_X(U) Ä A-module M(U) is cofibrant), and equivalences between them. By the standard strictification procedure we obtain a presheaf of categories

We state the following result as a conjecture, as we have not checked all details. However, we are very optimistic about it, as we think that a proof will probably consist of reinterpreting the constructions of [Ci-Ka1] in our context.

The same remark as in the case of the derived stack of local systems holds. Indeed, the usual Artin stack of vector bundles on X is given by HOM(X,BGl_n), and our D-stack of vector bundles on X is HOM(iX,iBGl_n).

5.3 Algebras and A_¥-categorical structures

In this last Subsection we present the derived moduli stack of associative algebra structures and A_¥-categorical structures. These are global versions of the formal moduli spaces studied in [Ko2, Ko-So].

Associative algebra structures. We are going to construct a D-stack Ass, classifying associative dg-algebra structures.

Let A be any cdga, and let us consider the category of (unbounded) associative differential graded A-algebras A-Ass (i.e. A-Ass is the category of monoids in the symmetric monoidal category A-Mod, of (unbounded) dg-A-modules)4. This category is a model category for which the weak equivalences are the quasi-isomorphisms and fibrations are epimorphisms. We restrict ourselves to the category of cofibrant objects A - Ass^c, and consider the sub-category wA-Ass^c consisting of equivalences only. If A-®A' is any morphism of cdga's, then we have pull-back functors

We define a sub-simplicial presheaf Ass_n of Ass, consisting of associative dg-A-algebras B for which there exists an étale covering A-®A' such that the dg-A'-module B Ä_AA' is equivalent to (A')ⁿ.

From (3) we see that the geometric D-stack Ass_n is not fp-smooth. Indeed, Quillen gives in [Q, Ex. 11.8] an example of a point in Ass_n at which the dimension in the sense of Definition 4.10 is not defined.

The previous theorem can also be extended in the following way. Let V be a fixed cohomologically bounded and finite dimensional complex of -vector spaces. We define Ass_V to be the sub-simplicial presheaf of Ass consisting of associative dg-A-algebras B for which there exists an étale covering A-®A' such that the dg-A'-module B Ä_AA' is equivalent to A'Ä V .

One can show that Ass_V is again a D-stack, but it is not in general strongly geometric in the sense of Definition 4.1 (nor in the sense of Definition 4.11). However, we would like just to mention that Ass_V is still geometric in some sense when considered as a stack over unbounded cdga's (the reader will find details in the forthcoming paper [HAG-II]). The tangent D-stack of Ass_V at a point is given by the same formula as before.

The construction of Ass_V can also be extended to classify algebra structures over an operad on the complex V . One can check that the D-stacks one obtains in this way are again geometric. These are the geometric counterparts of the (discrete) moduli spaces described by C. Rezk in [Re].

A_¥-Categorical structures5. Let A by any cdga. Recall that a dg-A-category C consists of the following data

1. A set of objects Ob(C).
2. For each pair of object (x,y) in Ob(C), a (unbounded) dg-A-module C_x,y.
3. For each triplet of object (x,y,z) in Ob(C), a composition morphism C_x,yÄ_AC_y,z-®C_x,z which satisfies obvious associativity and unital conditions.

There is an obvious notion of morphism between dg-A-categories. There is also a notion of equivalences of dg-A-categories: they are morphisms f : C-®C' satisfying the following two conditions

1. For any pair of objects (x,y) of C, the induced morphism f_x,y : C_x,y-®C'_x,y is a quasi-isomorphism of dg-A-modules.
2. Let H⁰(C) (resp. H⁰(C')) be the categories having repectively the same set of objects as C (resp. as C'), and H⁰(C_x,y) (resp. H⁰(C'_x,y)) as set of morphisms from x to y. Then, the induced morphism is an equivalence of categories (in the usual sense).

Using these definitions, one has for any cdga A, a category A - Cat of dg-A-categories, with a sub-category of equivalences wA - Cat. Furthermore, if A-®A' is a morphism of cdga, one has a pull-back functor A - Cat-®A'- Cat, obtained by tensoring the dg-A-modules C_x,y with A'. With a bit of care (e.g. by restricting to cofibrant dg-A-categories), one gets a simplicial presheaf

We now fix a graph O of non-positively graded complexes of -vector spaces. This means that O is the datum of a set O₀, and for any (x,y) ÎO, of a complex O_x,y. We will assume that all the complexes O_x,y are bounded with finite dimensional cohomology. We consider the sub-simplicial presheaf Cat_O of Cat, consisting of all those dg-A-categories C such that locally on A_ét the underlying graph of C is equivalent to OÄ A; the underlying graph of C is defined to be the graph G(C) whose set of objects is a set of representatives of isomorphism classes of objects in H⁰(C), and whose complexes of morphisms are the ones of C. The simplicial presheaf Cat_O classifies dg-categorical structures on the graph O.

The following theorem identifies the tangent of Cat_O.

Let us suppose that O is now a graph of finite dimensional vector spaces (i.e. the complexes O_x,y are concentrated in degree 0 for any x,y). Then one can show that the D-stack is strongly 2-geometric. Here we use a notion of strongly n-geometric D-stacks obtained by iterating Definition 4.1. The reader will find details about higher geometric stacks in [HAG-II] and might also wish to consult [S1]. Note that the D-stack cannot be 1-geometric, as its truncation h⁰ is the (2-)stack of linear categories. As a 1-geometric (not derived) stack is always 1-truncated (as opposite to the derived case), this shows that must be at least 2-geometric.

As in the case of Ass, if the graph O is not a graph of vector spaces, then the D-stack is not strongly 2-geometric anymore, but is still geometric in some sense, when considered as a stack over unbounded cdga's.

Let V be a bounded complex with finite dimensional cohomology, also considered as a graph of complexes with a unique object. Then, there exists a natural morphism

The fact that the tangent D-stack of at a dg-category with only one object is the whole (shifted) Hochschild complex C^*(A,A)[2], where A is the dg-algebra of endomorphisms of the unique object, is also our way to understand the following sentence from [Ko-So, p. 266].

In some sense, the full Hochschild complex controls deformations of the A_¥-category with one object, such that its endomorphism space is equal to A.

We see that the previous results and descriptions produce global versions of the formal moduli spaces of A_¥-categories studied for example in [Ko-So, So]. This also shows that there are interesting higher geometric stacks, and probably even more interesting examples will be given by the D-stack of (n - 1)-dg-categories (whatever these are) as suggested by a higher analog of the exact triangle above (see [Ko2, 2.7 Claim 2]).