What’s a Stack? June 19, 2008Posted by A.J. Tolland in Algebraic Geometry, mathematical physics.
Algebraic stacks are essential to my research. This is more acceptable now than it was twenty years ago, but it still presents a bit of a language barrier. Most mathematicians, I think, don’t know what a stack is in the way that they know what a manifold or a scheme is. So I want to use this post to explain what stacks are, with an eye towards their appearance in mathematical physics. I won’t quite define them (see Vistoli’s notes for that), but I’ll get you a lot closer than Harris & Morrison do (see p. 139), hopefully close enough to be comfortable that you know what’s going on when someone says “stack”.
Let’s start by saying that a space is what you get when you start with a set and then add some geometry. Maybe make the set into a manifold, maybe make it into a scheme; you can choose your favorite category. The elements of the set become the points of your space.
A stack is what you get when you start with a groupoid instead of a set, and then add geometry. A groupoid, remember, is a category whose morphisms are all isomorphisms. This means that the points of a stack aren’t just elements of some set; they also come equipped with a bunch of relations, telling you which points are isomorphic to each other.
So why would anyone try to make a groupoid into a geometry?
Well, one reason is that it would be nice if your category of spaces was closed under quotients. This obviously isn’t true; for example, there’s no nice manifold or scheme structure on the orbit set of modulo . But it is true if we embed our category of spaces into a larger category of stacks. Every space is a stack in a tautological way, because any set can be made into a groupoid by simply adding the identity morphisms. And there’s a good way of taking quotients in the category of stacks: If we have space acted on by a group , then we can define the quotient to be the stack whose underlying groupoid is the action groupoid of on , i.e., the category which has an object for every element and a morphism from to for every such that . The quotient stack will be smooth if and the action of are. And its dimension will be what you would naively expect:
Obviously, I’m cheating here. I told you what the points of are, but I haven’t told you what the geometry is, how the points fit together. But hopefully it’s clear that the action groupoid is not a bad substitute for the orbit set. We’ve enriched the problem a little. We can recover the orbit set from the action groupoid, if we want it, by taking equivalence classes, but we’ve also remembered how the group acts. The orbit set of on doesn’t know how big the stabilizers of the points of are, but the action groupoid does. This is roughly why quotient stacks can be smooth when orbit set quotients aren’t; the degrees of freedom only add up correctly if you remember to count the stabilizers.
Hopefully that’s enough motivation to make it worth suffering through some definitions. We need to find a setting in which we can make sense of the idea that the points of a stack form a groupoid.
The usual way of doing this, at least in the world of algebraic geometry, is to use the functor of points. If you have a scheme , you can consider the functor from schemes to sets defined by . They call this the functor of points, because if is a point, then is just the set of points of . But the functor of points knows about more than points; it knows how all the points fit together, which of them lie on a given subscheme and so forth.
In fact, if you know the functor in some abstract terms, you can actually recover . For example, the functor which assigns to the set of units in the global sections of the structure sheaf is actually the functor of points of the multiplicative group. Morally speaking, you can recover from the functor because knows all there is to know about the open affine covers of . Which means that any geometry we can do with can also be done with the functor .
Which means that we can shift our perspective a bit. We can forget about spaces as much as possible, and try to use functors as our geometric objects.
This actually works. A little more precisely: The functor of points has a pretty remarkable property. It’s actually a sheaf on the category of schemes. This means two things: First, the category of schemes has something like a topology: a (Zariski) open set in the category of schemes is an open embedding . (You can check that this definition has all the properties you’d want in a topology.) Second, it means that we can recover the functor of points on any given scheme by knowing the functor of points on open subschemes ; the sheaf condition means that we can glue the local data together.
So we can try to study the following objects: sheaves for the Zariski topology on the category of schemes which have affine open covers, i.e., there is an affine scheme (the atlas) and a surjection , telling us how is glued together from the affine functors of points.
We don’t get anything new this way. These sheaves are exactly the schemes. But we can get new geometric objects if we choose a finer topology. For example, we can give the category of schemes the “etale” topology, declaring the open sets to be etale morphisms, not all of which are Zariski embeddings.
It’s a big theorem that the functor of points of a scheme is a sheaf in the etale topology. But not every etale sheaf with a cover by (the functors of points of) affine schemes arises from a scheme; there are new geometric objects here. (In the literature, these etale sheaves go by the slightly awful name “algebraic spaces”. I try to avoid this name, since I like to reserve the term “space” for when I’m being deliberately vague.)
In fact, you can also show that the functor of points is a sheaf for even finer topologies, such as when the open sets are smooth morphisms. But I’m not going to dwell on this fact, because I wanted to tell you about stacks.
We get stacks by replacing the set-valued functor of points with a new functor valued in groupoids. Doing this correctly — specifically, spelling out what it means to have a sheaf of groupoids — takes a little bit of care, and goes by the name “descent theory”. There are two issues: First, you need to know that you can patch together isomorphisms living over open sets and if they agree on the intersection . Second, you need to know that if you have two objects, one on and one on , then you can glue them together if their restrictions to are isomorphic.
Algebraic geometers usually make this notion — the groupoid-valued sheaf — precise by introducing “categories fibered in groupoids, satisfying the descent conditions”. But let’s just stick with the intuitive idea; this is a blog post! (Look at Vistoli’s notes if you’re curiosu.)
A groupoid-valued sheaf is a stack if it has a cover by affine schemes, i.e., if
1) there is a morphism to from (the stack associated to the functor of points of) an open scheme , and
2) the fibers of are schemes, meaning that, for any morphism from a scheme to , the fiber product is a scheme.
Generally, people speak of Deligne-Mumford stacks when the category of schemes has the etale topology, and of Artin stacks when the category of schemes has the smooth topology. (There are also some technical conditions I won’t get into.)
So what is all this good for?
I know of two good answers. First, like I said, you can construct quotient stacks. There’s some real fun to be had here. Consider the quotient map . The point is definitely an affine scheme. Can we convince ourselves, using the idea that is the action groupoid of on the point, that the morphism should have only schemes as fibers? Well, consider the fiber of over another map ; this fiber will consist of a point (only one), a point (only one), and a morphism in the action groupoid. Since such morphisms correspond to elements of , we see that the fiber of over the embedded point is precisely a copy of , although without a distinguished identity. From this, we can make one amusing conclusion: the dimension of the is equal to minus the dimension of .
What’s more, the fibers of fit together nicely; in fact, the morphism is actually a principal -bundle. Crazier yet, the morphism is actually the universal principal -bundle. (Liar’s proof: The total space is contractible!) This is actually taken to be a definition: the stack is the functor which assigns to a scheme the groupoid of principal -bundles on . In particular, the stack is the classifying stack of line bundles.
Note, however, that is not the familiar classifying space . The former has dimension ; the latter has dimension . The latter classifies line bundles; the former classifies line bundles together with their automorphisms.
The other thing stacks are good for is moduli problems. In algebraic geometry, we often want to parametrize objects of a given kind, modulo isomorphism. One of the best ways of doing this is to show that the set of isomorphism classes of objects forms a space, because then you can introduce parametrizations just by putting coordinate charts on this space. But it often happens that these sets don’t have nice geometric structures; they might not have any structure at all, or they might be very singular. Or they might be spaces, but not classifying spaces. In fact, these sort of annoyances crop up precisely when the objects in question have non-trivial automorphisms.
But this is precisely the situation where stacks are useful. (In fact, it’s what Deligne & Mumford invented them for. [Edit: Jason Starr says that stacks predate D & M. See his comment below.]) So what we do instead now, in moduli theory, is look at the groupoid valued functor which assigns to a scheme the groupoid of all families of relevant objects parametrized by . If we’re lucky, we can show that this functor is actually a stack, and then we can do geometry, thinking of the atlas as a collection of coordinate charts on the stack.
For example, there’s a moduli stack ; this is the stack which assigns to a scheme the groupoid of all families over of smooth genus complex curves. (A family of curves on is a flat morphism whose fibers are curves.) This is one of the nicest stacks there is. It’s smooth, connected, and has constant dimension . It’s of finite type, which means that it can be covered by finitely many coordinate charts, each of which is a smooth affine scheme of the appropriate dimension carrying a family of complex curves.
Even nicer, carries a universal curve . This is the stack assigning to a scheme the groupoid of pairs , where is a curve over , and is a morphism introducing a point onto each fiber of . The morphism forgets this marked point; thus the fiber of over any curve is a copy of . In fact, by definition, any family of curves is tautologically the pullback of along some morphism . In other words, lives up to its name; it’s the classifying stack for smooth complex curves of genus !
Whew! OK, did anyone actually read all that?