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 étale 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 étale 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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Towards Stacks 1

Let’s start working towards what a stack is. I don’t usually like to skip a lot of material, but I know of at least two other blogs that have done some of the preliminary work I need. So today will be very sketchy. I’ll just blurt out a whole bunch of stuff without explaining it, but I’ll give references to other blogs.

First, recall that a site is a category equipped with a Grothendieck topology. You can read about these at Rigorous Trivialites or at Climbing Mount Bourbaki. This is just a way to extend the notion of a topology to a general category.

Some of the standard examples in AG are the Zariski site {X_{Za}}, which is just the category of open immersions to {X} with obvious morphisms (the ones that respect the immersion), and the coverings are open immersions {\{U_\alpha\stackrel{\phi_\alpha}{\rightarrow} V\}} such that {\cup \phi_\alpha(U_\alpha)=V}. Notice this is just a more formal way of saying that the coverings are Zariski open sets that actually cover the set. Likewise we can define the étale site or fppf site by requiring our maps to be étale or “faithfully flat and locally of finite presentation”.

Sometimes we may want to distinguish between “big” and “small” sites (we’ll see why later). The difference will be that in the big site we allow all scheme maps to be the objects in the category. The small site will be that in the category we only allow maps of the type specified by the site (which is the one I technically wrote above).

If a category, {\mathcal{C}}, has two Grothendieck topologies {\mathcal{T}} and {\mathcal{T}'}, then there is a notion of the two topologies being equivalent. An easy way to define this is that each of the topologies are refinements of eachother. Another way to define it is if there is a continuous map between the two sites {F: (\mathcal{C}, \mathcal{T})\rightarrow (\mathcal{C}, \mathcal{T}')} that satisfies three conditions:

1) {F^{-1}} is fully faithful.

2) Every open set in {U} in {\mathcal{T}} has a covering of the form {\{f^{-1}(V_\alpha)\rightarrow U\}} where {V_\alpha} are open in {\mathcal{T}'}.

3) A collection {\{V_\alpha\rightarrow V} in {\mathcal{T}'} is a covering if {\{f^{-1}(V_\alpha)\rightarrow f^{-1}(V)\}} is a covering in {\mathcal{T}}.

Note that an equivalence of topologies in not the same thing as the two sites being “isomorphic”. Equivalence is a weaker notion.

Now that we have a generalized notion of a topological space (on a category), we can try to generalize sheaves on sites. Again, this has been done in two other places (here and here), so we’ll hit the highlights.

Recall that a sheaf on a standard topological space, {X}, is just a contravariant functor from the category of open subsets of {X} plus some stuff that makes it “local”. Since all of these things were just stated categorically, it extends in a natural way to any site. Thus we get a category of sheaves on a site denoted {Sh(\mathcal{T})}.

It turns out that if you have a category with two equivalent topologies, then the pushforward induces an equivalence of categories {F_*: Sh(\mathcal{T})\rightarrow Sh(\mathcal{T}')}, and hence the natural map of cohomology is an iso {H^i(\mathcal{T}', F_*\mathcal{F})\rightarrow H^i(\mathcal{T}, \mathcal{F})}.

So for instance you could define the site {X_C} to be the category with objects holomorphic maps from analytic sets {U\rightarrow X(\mathbb{C})} to the {\mathbb{C}}-valued points that are local homeomorphisms and coverings to be if the union of the image actually covers {X}. Then we have a continuous map {F: X_C\rightarrow X_{et}} since given an \'{e}tale map {U\rightarrow X} if we look at the underlying analytic sets {U(\mathbb{C})\rightarrow X(\mathbb{C})} this is a local homeo. One can check that this is actually an equivalence of topologies. Thus we get that computing complex analytic cohomology or étale cohomology will give the same answer.

Here is why the big site is important. We can only compare Grothendieck topologies on a category if, well, the underlying category is the same. Taking the category as all scheme maps into {X}, and then designating certain ones as “special” by the topology allows us to compare the topologies. Notice the underlying categories in the small sites are not the same. The category of open immersions {U\rightarrow X} is not the same as the category of étale maps {U\rightarrow X}. I’ve never seen this reasoning for the big site pointed out, and it confused me for awhile, so that’s why I’m making a big deal out of it.

(New edit:) It seems that the above point I just made isn’t universal in the literature for the following reason. There are obviously continuous maps between two sites where the underlying categories aren’t the same. For instance, any of the big sites have a continuous maps to the (same) small sites just by sending the map to itself. It is possible for a continuous map between two sites with different underlying categories to be an equivalence. The comment above was mostly based on Vistoli’s stack notes, in which he only defines equivalence of Grothendieck topologies on the same category.

That seems enough for today. Just to reiterate, the ultimate goal is to figure out what a gerbe is, but in order to do that we need to know what a stack is.