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.

3 thoughts on “Towards Stacks 1

  1. Bonjour,

    Voir ma folie sur mon Blog:


    Clovis Simard

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