Stacks 3: Stacks on Sites

Today we’ll end the discussion on stacks for a bit. All we want to do is say what a stack on a general site is. But all of the pieces of this are already in place. We converted our topological space {X} into a site {\text{Top}(X)} as our first step and then only used properties of sites to define everything. It might have been easier to visualize things as actual coverings by open sets and things lying over open sets, but formally we always used the site language.
Let {\mathcal{C}} be a site. Then it is a category with a Grothendieck topology. Since it is a category, we know what it means to be fibered in groupoids over it. Let {\mathcal{S}} be a category fibered in groupoids over {\mathcal{C}}. Given an open set, {U}, (i.e. object in {\mathcal{C}}) and two objects {\eta} and {\eta '} over {U}, then we get a natural contravariant functor to Set, {\text{Isom}_{\eta, \eta'}}. If this functor (re: presheaf) is a sheaf, then {\mathcal{S}} is a prestack on {\mathcal{C}}.

A word should be said about “sheaf”. Recall that on a site, a sheaf is just a contravariant functor that also satisfies a particular exactness diagram {\displaystyle F(U)\rightarrow \prod_{i} F(U_i) \stackrel{\rightarrow}{\rightarrow} \prod_{i,j} F(U_i\times_U U_j)}. When it won’t cause confusion, I’ll probably just write an actual restriction or {\eta_{ij}} to mean the pullback since this is what most people will have in their heads anyway.

Lastly, a prestack is a stack if every descent datum is effect. Since we have a notion of covering built into our site, namely the Grothendieck topology specifies coverings, we can define a descent datum to be a collection of objects over each open set (object) in the covering along with isomorphisms that satisfy the cocycle condition. The descent datum is effective if there is an object over the open set (object) being covered that satisfy the same conditions as first defined.

For most of the time, if we have some scheme, {X}, floating around when we say stack we’ll mean stack on the Zariski site {X_{Zar}} or ├ętale site {X_{et}}.

Now that we have what a stack is, we’ll just throw a bunch of examples out there. If one of them interests you, then you can actually check the details of whether or not it is a stack. The important point here is that they occur all over the place, and not just in algebraic geometry. Recall that one of the points of constructing the notion of stack was to get a “generalized space” in some sense, but since many of these examples are clearly not geometric, we’ll probably want to specify later some more conditions to get it to look more like a geometric space.

A sort of canonical example is to take the site of topological spaces, Top, and consider the category of arrows Cont. So Cont just consists of continuous maps. The functor that sends an arrow to its codomain fibers it in groupoids and one can check that Cont is a stack on Top.

Next, there is a way in which we can consider any sheaf a stack. Given a (separated) presheaf on some site {F:\mathcal{C}^{op}\rightarrow \text{Set}}, we get a category fibered in groupoids, which we’ll just denote {X\rightarrow \mathcal{C}}. Here {X} can in some sense be thought of as the espace ├ętale of the presheaf as a category. It turns out that the presheaf is a sheaf if and only if the category fibered in groupoids associated to it is a stack. This just amounts to unraveling what each of those definitions are.

An immediate corollary to the above is that any scheme is a stack via its functor of points and hence stacks really are generalizations of spaces.

The category of quasi-coherent sheaves on a scheme {X_{Zar}} is a stack.

Most examples of moduli spaces are stacks (for instance {M_g}, the moduli space of curves of genus {g}).

A very important example for us is that the so-called Schlessinger deformation functor is a stack. Suppose we have some fixed scheme {Z} over {A}. Then {\text{Def}_Z(A')} is the set of (cartesian) diagrams that give deformations of {Z} over {Spec(A')}.

To prove my point that stacks come up all over the place, we’ve already talked about how they appear in differential geometry as bundles. A place where they may show up in anaylsis is to consider the category of (Radon?) measures on {\text{Top}(X)} in the same way as the vector bundle example. It consists of pairs {(U, \mu)} where {\mu} turns {U} as a subspace into a measure space. The morphisms are “isos” after restriction, so {(U, \mu)\stackrel{f}{\rightarrow} (V, \rho)} is a morphism if we have an automorphism {f:V\stackrel{\sim}{\rightarrow} V}, such that {f_{\sharp} \rho |_U = \mu}. This category has a natural forgetful fucntor to {\text{Top}(X)} the same way that {\text{Vect}^r(X)} did. I was talking to someone who does analysis to see if this really was a stack, and we decided it probably was, but we kept not understanding eachother’s language and so we aren’t sure. It would be interesting to see if it really is.

Lastly, since the point of this was to eventually get to groupoids I won’t talk anymore about stacks and all the various ways to think about them and all the extra conditions you can impose to get more rigid spaces. But a few words should be said about some of the major things I’ve left out and maybe later I’ll come back and talk more about them.

The collection of stacks actually forms a category (or better yet, a 2-category if you know what that is). So we maybe should have specified what the morphisms between them are. There is a beautiful way to think about stacks that involves forming the category of descent data. So the descent data we talked about actually forms a category which some people actually use to define what a stack is.

All the examples listed here are proven to be stacks in detail except the deformation example in Vistoli’s article in Fundamental Algebraic Geometry (aka FGA Explained) by Fantechi et al if you’re curious about seeing details. The deformation stack is proved in the article Beyond Schlessinger: Deformation Stacks by Brian Osserman available at his website. When it comes up later when talking about gerbes, I might explain it more thoroughly and prove it as well.

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