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6 Final comments on deriving moduli functors

In this Subsection, keeping in mind the examples presented before, we would like to discuss, from a more general point of view, the problem of derivation of moduli functors, with the aim of at least making explicit some general features shared by the examples.

Suppose M : (Aff)op = (C - alg)-®Set is a functor arising from some geometric moduli problem e.g., the problem of classifying isomorphism classes of families of (pointed) curves of a given genus. Very often, the moduli functor M is not representable and only admits a coarse moduli space. As its name says, when passing to a coarse moduli space some information is lost. The theory of stacks in groupoids was originally invented to correct this annoyance, by looking at natural extensions of M, i.e. to functors M1, from algebras to groupoids, such that the following diagram commutes

Af fop -M----Set | M1 p0 | Grpd,
Here the vertical arrow assigns to a given groupoid its set of isomorphisms classes of objects. Of course, the point of the theory of stacks in groupoids is precisely to develop a geometry on this kind of functors.

More generally, other natural higher moduli problems are not representable even when considered as stacks in groupoids, e.g. the 2-stack perfect complexes of length 1, the 2-stack of linear categories ...; the theory of higher stacks precisely says that one should consider M extended as follows

op -M---- Aff Set M1 p0 | M Grpd P 1 SSet,
where the functor P1 maps a simplicial sets to its fundamental groupoid. The notion of geometric n-stacks of [S1] can then be used in order to set up a geometry over these kind of objects, in pretty much the same way one is doing geometry over stacks in groupoids.

The idea of derived algebraic geometry is to seek for derived extensions of M, M1 and M i.e. to extend not (only) the target category of this functors but more crucially the source category in a "derived" direction. More precisely, we define a derived extension of a functor M : Affop-®SSet, as above, to be a functor RM : (D - Aff)op-®SSets making the following diagram commute

op M Aff| --------Set | M1 p0 | | j | Grpd | M | | P1 D - Af fop-----SSet RM
where j denotes the natural inclusion (a C-algebra viewed as a cdga concentrated in degree zero). The above diagram shows that, for any derived extension RM, we have
p0RM(j(SpecR)) -~ M (SpecR)
and moreover
P1RM(j(SpecR)) -~ M1(SpecR)
for any commutative C-algebra R. In other words, the 0-truncation of RM gives back M when restricted to the image of j, while its 1-truncation gives back M1.

What about the existence or uniqueness of a derived extension RM ? First of all, extensions always exists, as one can use the trivial one given by the functor i of §3.2. But of course, this extension is far from being unique and usually does not give the expected answer. However, there is no canonical choice for an extension which could be nicer than others. This tells us that the choice of the extended moduli functor RM highly depends on the geometrical meaning of the original moduli functor M, M1 of M. We would like to give here a clear example to show this.

Let S2 be the 2-dimensional sphere, and let us consider M1 := Loc n(S2), the moduli stack of rank n local systems on S2. We clearly have M1 -~ BGLn. If one thinks of M1 simply as BGln, and forget about the fact that it is the moduli stack of local systems on S2, then a reasonable extension of M1 is simply iBGln -~ RBGln as described in §3.4. However, if one remembers that M1 is Locn(S2), then the correct (or at least expected) extension is RLocn(S2) presented in Theorem 5.1. Definitely, these two extensions are very different. This example shows that the expected extension RM depends very much on the way we think of the original moduli problem M. In a way we are more deriving our interpretation of the moduli functor rather than the moduli functor itself. Another example of the existence of multiple choices can be found in [Ci-Ka2], in which the derived Hilbert dg-scheme is not the same as the derived QuotO dg-scheme.

Nevertheless, the derived extension of a moduli functor that typically occurs in algebraic geometry, is expected to satisfy certain properties and this gives some serious hints in order to guess the correct answer.

First of all, in general, one knows a priori what is the expected derived tangent stack TRM (or, at least, the disembodied derived tangent complexes at the points, the (WRM,x1)*'s in the notations of §4.3); namely, this is true in the case where M classifies vector bundles over a scheme, local systems over a topological space, families curves or higher dimensional algebraic varieties, stable maps from a fixed scheme and so on. For some examples of the expected derived tangent spaces we refer again to [Ci-Ka1, Ci-Ka2]. To put it slightly differently, it is always the case that one looks for a derived extension by simply requiring it to have the expected derived tangent stack. This is essentially due to the fact that the correct derived deformation theory of the moduli problem has already been guessed, and the corresponding, already established, formal theoy is based on this guess (see [Hin2, Ko-So, So], to quote a few).

Even if this does not say exactly how to construct a derived extension, it certainly puts some constraints on the possible choices. To go a bit further, one may notice that all the usual moduli functor occurring in algebraic geometry classify families of geometric objects over varying base schemes. To produce a derived extension RM, the main principle is then the following

Main principle: Let Mbe a moduli stack classifying certain kind of families of geometric objects over varying commutative algebras A. In order to guess what the extended moduli stack RMshould be, guess first what is a family of geometric objects of the same type parametrized by a commutative dga A.

In the case, for example, where the classical notion of a family is defined through the existence of a map with some properties (like for example in the case of the stack of curves), the derived analog is more or less clear: one should say that a derived family over a cdga A is just a map of simplicial presheaves F-®RSpecA, having the same properties in the derived sense (e.g., as we extended the notion of étale morphism of schemes to cdga's, see §2.2, the same can be done with the notions of smooth, flat ...morphisms of schemes). Then, a natural candidate for a derived notion of family of geometric objects, is given by any derived analog of a family such that when restricted along Spec(H0(A))-®SpecA it becomes equivalent to an object coming from M(Spec(H0(A))). This condition, required in order to really get a derived extension, essentially says that the derived version of a family of geometric objects should reduce to a non-derived family of geometric objects in the non-derived or scheme-like direction, i.e. along Spec(H0(A))-®SpecA. A typical example of this case is the one of G-torsors given in §3.4. Another example would be that of the moduli stack of surfaces. One could say for example that a smooth projective family of surfaces over a cdga A, is a strongly smooth morphism of strongly geometric D-stacks F-®RSpecA, such that for any geometric point x : SpecC-®RSpecA, the pull-back F ×RSpecAhSpecC is equivalent to a smooth projective surface.

Though this gives perhaps only a vague recipe of a possible construction of derived extensions of some of the moduli functors occurring in algebraic geometry, we thought it was worthwhile presenting it, if not certainly to solve the problem at least to pose it in a general perspective.

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