## Different interpretations of moduli stacks

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I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to classify (smooth curves of genus g over ℂ, for example). Fine moduli spaces are defined to be the objects that represent the functor that takes an object and gives you the [set, for schemes; groupoid, as I understand it, for stacks] of ways that that object parametrizes families of the object we want to classify. Now, for schemes - this makes sense in the following way:

Let that functor be F, and let it be represented by M. Then F(Spec ℂ) are the families of (desired) objects parametrized by Spec ℂ (a point), so it corresponds to all desired objects (the ones we want to classify). But F(Spec ℂ) is also Hom(Spec ℂ, M), and so corresponds to the closed points of M. Thus M really does, intuitively, have as points the objects it wishes to classify.

Does this idea go through to moduli stacks? Of course, it probably does, and this is all probably trivial - but I feel like I need someone to assure me that I'm not crazy. So let me put the question like this: Can you formulate how to think of a fine moduli stack (as an object that represents an F as above; also: how would you define this F in the category of stacks?) in a way that makes it clear that it parametrizes the desired objects?

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Thanks for the question. It put clearly what my own problem with the algebraic geometric version of stacks. I know the non-Abelian cohomology motivation much better. – Tim Porter Feb 13 2010 at 21:39

## 2 Answers

I'll assure you that you're not crazy. Not only does the idea go through for stacks, but it's impossible (or at least very hard) to make sense of stacks without that idea.

If you're trying to parameterize wigits, you can build a functor F(T)={flat families of wigits over T}. If there is a space M that deserves to be called the moduli space of wigits, it should represent F. It's not just that the points of M must correspond to isomorphism classes of wigits, so we must have F(Spec ℂ)=Hom(Spec ℂ,M). The points are also connected up in the right way. For example, a family of widits over a curve should correspond to a choice of wigit for every point in the curve in a continuous way, so it should correspond to a morphism from the curve to M.

It happens that if wigits have automorphisms, there's no hope of finding a geometric object M so that maps to M are the same thing as flat (read "continuous") families of wigits. The reason is that any geometric object should have the property that maps to it can be determined locally. That is, if U and V cover T, specifying a map from T to M is the same as specifying maps from U and V to M which agree on U∩V. The jargon for this is "representable functors are sheaves." If a wigit X has an automorphism, then you can imagine a family of wigits over a circle so that all the fibers are X, but as you move around the circle, it gets "twisted" by the automorphism (if you want to think purely algebro-geometrically, use a circular chain of ℙ1s instead of a circle). Locally, you have a trivial family of wigits, so the map should correspond to a constant map to the moduli space M, but that would correspond to the trivial family globally, which this isn't. Oh dear!

Instead of giving up hope entirely, the trick is to replace the functor F by a "groupoid-valued functor" (fibered category), so the automorphisms of objects are recorded. Now of course there won't be a space representing F, since any space represents a set-valued functor, but it turns out that this sometimes revives the hope that F is represented by some mythical "geometric object" M in the sense that objects in F(T) (which should correspond to maps to M) can be determined locally. If this is true, we say that "F is a stack" or that "M is a stack." Part of what makes your question tricky is that as things get stacky, the line between M and F becomes more blurred. M isn't really anything other than F. We just call it M and treat it as a geometric object because it satisfies this gluing condition. We usually want M to be more geometric than that; we want it to have a cover (in some precise sense) by an honest space. If it does, then we say "M (or F) is an algebraic stack" and it turns out you can do real geometry on it.

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I'm sure Anton knows this, but to the questioner, in general, a fibered category is not fibered in groupoids.  A general fibered category is merely "fibered in categories".  Another way to say this (although it requires the axiom of choice) is that a fibered category is a pseudofunctor $\mathcal{F}:\mathcal{C}\to \mathcal{C}at$, and a category fibered in groupoids is a pseudofunctor $\mathcal{F}:\mathcal{C}\to \mathcal{G}pd$. – Harry Gindi Feb 13 2010 at 20:50
@fpqc: thanks for pointing that out. I was trying hard to drop the relevant terms you'd want to know while presenting the intuitive picture. I hope there aren't too many other technical lies I told in the process. – Anton Geraschenko♦ Feb 13 2010 at 20:58
Well, by pseudofunctor I meant contravariant pseudofunctor, so it should in fact be from $\mathcal{C}^{op}$.  Sorry for the confusion if the OP was confused.  The whole point of a fibered category is that it's a direct generalization of a presheaf, which is a contravariant functor into sets. – Harry Gindi Feb 13 2010 at 21:00

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Yeah, you've got the right idea! Say you've got some moduli problem -- for instance you have some class of objects X over C you're interested in -- and you want to say what the moduli space of X's is, as a variety (or stack or whatever) over C. The idea is that you can describe a variety by how other varieties map to it -- Yoneda's Lemma. So the moduli space M will be determined if you know what to call Hom(T,M) for every variety T. Certainly if T=Spec(C) is the point, this should be the original set (groupoid, whatever) of objects X over C; but in general Hom(T,M) should be the set of families of X's continuously parametrized by T, which usually is nicely encoded by some flat family over T whose fibers over C-points are X's.

So, to sum up, the notion of a moduli space as a functor exactly encodes the idea that the points of a moduli space should be the objects you're trying to parametrize. One just needs to use "points" in the more general context of maps from an arbitrary variety T, and be able to say what it means to give an object of the type you're interested in over an arbitrary such T. This is not formally determined by the case T=Spec(C) a point, but there is usually a natural extension to make.

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Sorry, I just reread your question, and it looks like your concern was exactly with the change from varieties to stacks, i.e., from sets to groupoids, which I handled with a couple of dismissive "whatever"s. I'm sorry for my misinterpretation and resulting uncouthness! What my "whatever"s were trying to convey, however clumsily, is that the formalism and ideas are exactly the same whether one requires sets or groupoids. – Dustin Clausen Feb 13 2010 at 20:07

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The reason one still only has to define one's functor on varieties, and not general stacks, is the same as why one only ever needs to define it on affine varieties, and not all varieties: by definition stacks can be "glued" from varieties, just like varieties can be "glued" from affines. Formally, any stack is a colimits of varieties, or affine varieties, and this implies that the functor of points restricted even to affines recovers the space. – Dustin Clausen Feb 13 2010 at 20:10

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While stacks can be glued from affines, there is no way to define a general algebraic stack without first defining an algebraic space. This is simply because the requirement that a sheaf of groupoids in the fppf topology be an algebraic stack is that its diagonal is representable by an algebraic space. There are other constructions you can use, but I think that most of them are equivalent to defining an algebraic space first. – Harry Gindi Feb 13 2010 at 20:38
What I meant was this: intuitively, we want stacks to be glued from affine varieties -- and this is exactly what justifies the formal definition of stack as a kind of sheaf on affine varieties, since the category of sheaves on a site is exactly what you get when you formally adjoin colimits to the site (colimits = gluing) subject to the restriction that the base X of a cover {U_i} --> X should be glued from the Cech nerve of the cover. – Dustin Clausen Feb 13 2010 at 20:48
Note that this has nothing to do with potential "algebraicity" of stacks, and is independent of what definition one takes there. – Dustin Clausen Feb 13 2010 at 20:51
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