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The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people think about some of the general stacks geometrically (instead of merely thinking about them as fibered categories).

If the paragraph still doesn't explain my purpose, let me be more specific: people talk about "locally around a curve in M_g" something happens. But they don't say "locally near a coherent sheaf in Coh", right?

I read Vistoli's notes on general stacks before I saw people talk about algebraic stacks geometrically, and I always feel like for me there is a gap between these two ways of thinking. I feel like e.g. when a birational geometer uses stacks as a tool he probably don't put general fibered categories in his mind, (although abstract nonsense shouldn't be a big problem for him) right?

[A related question is, is category over Schemes fibered in something other than groupoid useful in geometry? Why people only care about categories fibered in groupoids?]

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Not an application in algebraic geometry, but last Spring I attended a colloquium by Dan Freed, who was reporting on conversations with Mike Hopkins if I'm remembering correctly, about (infinity) stacks on the site of manifolds, and certainly his examples were not necessarily "algebraic". A central player is the "classifying" stack for G-bundles with connection, where G is a Lie group. One fun example from the talk was that Freed computed the de Rham cohomology of this stack. –  Theo Johnson-Freyd Oct 4 '11 at 7:31
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@Theo: In the context of algebraic geometry, "algebraic stack" means a stack satisfying certain conditions, probably the most relevant one being that there is a scheme (or an "algebraic space", something close to a scheme) that surjects onto the stack. This scheme is to be thought of as an "atlas" for the stack. While the example you give is not "algebraic" in the sense that it isn't defined in terms of algebraic geometry, it is still "algebraic" in the sense that it is covered by a "scheme" -- in this case a manifold, or at least some kind of infinite dimensional manifold. –  Kevin H. Lin Oct 4 '11 at 7:46
    
@ Kevin: No that is not true, at least for the large site of manifolds. The stack classifying G-bundles with connection does not admit a cover by a representable manifold, infinite dimensional or otherwise. Locally connections are given by lie algebra valued 1-forms and the problem persists there. The problem is essentially that the sheaf of 1-forms $\Omega^1 (-)$ is not representable. How could it be? $\Omega^1(pt) = 0$, so this sheaf has only one point! –  Chris Schommer-Pries Oct 4 '11 at 17:38
    
@Chris: Thanks for the correction. I was thinking of moduli stacks of holomorphic G-bundles on a Riemann surface, rather than classifying stacks. But you are right. –  Kevin H. Lin Oct 5 '11 at 19:12
    
You may like my questions mathoverflow.net/questions/56962 and mathoverflow.net/questions/57057 –  David Roberts Oct 6 '11 at 8:49
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2 Answers

In order to do geometry, you need to have some kind of global structure which has good local models (the "neighborhoods") and good gluing conditions. In algebraic geometry, the good local models are rings. If you want do geometry with a fibered category, you need gluing conditions (that is, you need your fibered category to be a stack), and you need local models, that is, you need your category to be locally, in some pre-topology, an affine scheme (this is not quite right, but I hope it gives a rough idea). The pre-topology must be such that if $X \to Y$ is a covering, the fact that $Y$ has certain "interesting" local properties implies that $X$ also has them. Étale coverings work very well, of course; smooth coverings also work, not quite as well.

So, you can't do geometry with the stack of coherent sheaves because this does not have good neighborhoods. See also my answer to Qcoh(-) algebraic stack? to see what can wrong.

As to why algebraic stacks are always assumed to be stacks in groupoids, there are several things I could say, but the honest answer is that I don't know the deep reason for this. I know that in practice it suffices, so there is no reason to give up the inversion map, which is quite useful. Just think of how much more you can say about group actions, than about actions of monoids.

Of course, this does not mean that in the future people will not feel the need to extend the theory of algebraic (or topological, or differentiable) stacks to the more general case.

[Edit]: So, why is a geometric stack a stack in groupoids? Well, the first reason is that the inversion map is very useful in proving results. Of course, if we needed to do without it, we would.

The second, more serious, reason, is that, in concrete examples, stacks with non-cartesian maps tend not to admit non-trivial map to spaces. For example, consider the stack $\mathcal M_{1,1}$ of elliptic curves. If we admitted all squares as morphisms, instead of only the cartesian ones, any map from $\mathcal M_{1,1}$ to a space would have to collapse an isogeny classes of curves to a point, and then one can see that it would map everything to a point. So, no moduli space.

As another example, take the stack of vector bundles on a projective variety $X$. There is a map between any two vector bundles, so no open substack could possibly admit a non-trivial map to a space.

Of course, if $F$ is a stack over a site $C$, there is substack $F^*$ with the same objects, whose arrows are the cartesian arrows in $F$; and if $X$ is an object of $C$, or a sheaf on $C$, any cartesian functor $X \to F$ would factor through $F^*$; so you could argue that a chart for $F$ would in fact come from a chart for $F^*$. In all the examples I know, $F^*$ is the right object to consider.

But, once again, none of these reasons is really compelling; for example, if monoid actions became important in geometry, I would bet that soon people would start working with geometric stacks that are not stacks in groupoids.

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This is a great answer. Thanks a lot. I'm still curious about what could be said on the issue of stacks in groupoids. What are the not-so-deep reason for it? ;) –  36min Oct 4 '11 at 15:02
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The answer is a qualified yes, and it depends on how broadly you define "algebraic geometry". Here is a class of examples:

A sheaf with flat connection on a smooth variety is equivalent to a sheaf on the de Rham stack of that variety, and de Rham stacks of positive dimensional varieties are not algebraic. You are basically taking a quotient with respect to the action of an infinitesimal symmetry, and the corresponding covering morphism satisfies only a formal version of smoothness. More generally, there are variations on the notion of sheaf with connection, e.g., action of an algebroid, that can be viewed functorially using sheaves on non-algebraic stacks which are stackifications of reasonably nice formal groupoids.

There is a sketch of a theory of such stacks in sections 2.9.10-2.9.12 of Chiral Algebras by Beilinson and Drinfeld, but in fact, the entire book is secretly about them, restricted to the language of $D$-modules. This is a situation where you can do some kind of "geometry", but it is a more degenerate environment than most algebraic geometers tend to consider. For example, you may end up using "compound tensor structures" because ordinary tensor products on quasicoherent sheaves, with their nice properties, are unavailable.

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