## Gerbes 3: Another Example and Some Caution

This might be my last post on gerbes (explicitly for gerbe’s sake), so as in my last ‘stacks for stack’s sake’ post I’ll try to clarify some things with more examples and then give some cautions. Last time I mentioned the classifying stack ${BA}$. Let’s first actually construct it better than the quick idea I gave.

Let ${B}$ be a topological space, and ${A}$ a sheaf of abelian groups on ${B}$ (note that I’ll use ${A}$ instead of ${\mathcal{A}}$ to avoid typing the script, but it is a ${\mathit{sheaf}}$ and not just a group, otherwise we’ll just recover the classifying space).

Define a functor ${BA: \text{Top}(B)\rightarrow \text{Grpds}}$, where ${\text{Grpds}}$ is the category of groupoids, by ${BA(U)=}$ groupoid of ${A_U}$-torsors over ${U}$. This is a sheaf, say ${T}$, on ${U}$ with an action ${A_U\times T\rightarrow T}$ such that if ${T(V)\neq \emptyset}$, then ${A_U(V)}$ acts simply transitively on ${T(V)}$.

Again, this is just fancy language for something that is probably familiar to you. Since we have a sheaf of groups, just think an open set at a time. ${A_U(V)}$ is a group, call it ${G}$. Then ${G\times T\rightarrow T}$ is really just an honest group action, and “acting simply transitively” means that if we pick out some ${t\in T}$, then we have a way to identify ${G}$ with ${T}$, namely ${G\stackrel{\sim}{\rightarrow} T}$ (as sets) via the action ${g\mapsto g\cdot t}$.

You could also think of this as a “relative” principal bundle. The group that it is a principal bundle of gets to change locally, but if it is a constant sheaf and hence not changing, then we really do just get the classifying space.

I told you ${BA}$ as a functor to Grpds and not as a functor to ${\text{Top}(B)}$ which is how stacks were defined, but recall if we have a sheaf on ${B}$, then we can convert it to that form by taking our category to have objects the pairs ${\{(s, U)\}}$ where ${s\in BA(U)}$, and the maps in the category are inclusions and restricting to the right thing. If we were doing all the details we’d have to check all of this and then check it is actually a gerbe and that it is actually an ${A}$-gerbe, etc, but we’d be stuck here forever and these are all straightforward enough that it would make a great exercise if you don’t see it right away.

Recall last time that an ${A}$-gerbe, ${G}$, is isomorphic to ${BA}$ if and only if it has a global object. Recall that ${\text{Vect}^1}$, the stack of rank one vector bundles, was a ${\mathbb{G}_m}$-gerbe, and it has the trivial bundle as a global object, so ${B\mathbb{G}_m\simeq \text{Vect}^1}$.

Let’s actually prove it now. If ${G\simeq BA}$, then ${A(B) \in BA(B)}$ and hence ${G(B)\neq \emptyset}$. For the reverse direction, suppose there is some ${s\in G(B)}$, then we get a map ${G\rightarrow BA}$ which we’ll denote ${t\mapsto \text{Isom}(t,s)}$. One can check that this induces the isomorphism. In fact, one can check that whenever you have a map ${G_1\rightarrow G_2}$ in the category of ${A}$-gerbes, it will be an isomorphism.

This is why I wanted to bring this example up. Here are some of the cautions that jump to my mind. Something might feel fishy to you right now. That’s because I haven’t really told you the proper way to think about these things. When I say “isomorphism” what does that mean? Well, it really means as ${2}$-categories.

Also, suppose you are an algebraic geometer and you say you have a gerbe on the étale site of ${X}$. This isn’t precise enough, since funny differences can happen whether or not you’re on the big or small site. I guess because of all that I’ve left out in an attempt to bring the concept out, my main caution is to consult the literature and not any of these blog posts if you want to know if something is true.

## Gerbes 2: The Motivation

I’m going to make another definition, but I may as well get to the punchline first or else anyone reading this that doesn’t already know the punchline is going to skip reading it or tune out. If you have an abelian sheaf ${\mathcal{A}}$ on ${X}$, then there is a notion of ${\mathcal{S}}$ being not only a stack/gerbe over ${X}$, but an ${\mathcal{A}}$-gerbe. I’ll define this later.

Here’s the amazing part, Giraud did a lot of work for us and tells us that the global elements of the stack, ${\mathcal{S}}$, i.e. the objects lying over all of ${X}$ are in bijection with elements of ${H^1(X, \mathcal{A})}$. Take the line bundle example, then ${L(X)}$ is actually a ${\mathcal{O}_X^\times}$-gerbe and hence (iso classes of) line bundles on ${X}$ are in correspondence with ${H^1(X, \mathcal{O}_X^\times)}$. Wait! We already knew that since ${H^1(X, \mathcal{O}_X^\times)\simeq \text{Pic}(X)}$.

We also found in our two examples of deformations before that this is true. We found that infinitesimal extensions by coherent sheaves are classified by ${H^1(X, \mathcal{F}\otimes \mathcal{T})}$ here and here. It turns out this wasn’t a coincidence. Things are classified by ${H^1}$ all over algebraic geometry and this is the underlying thread connecting them.

But it turns out Giraud didn’t stop there and we get even more. We actually get an obstruction theory as well. Giraud tells us that the obstruction to constructing a global object lies in ${H^2(X, \mathcal{A})}$. We may have not gone through it for our deformations with a tedious cocycle argument like the ${H^1}$ exercises, but many books do go through this (see Hartshorne’s recent book on Deformation Theory). Good thing we didn’t go through it, we just get it from knowing that it is a gerbe.

I’ve tried to search for this, but can’t find it anywhere. This is why this theory is so cool and widespread. Think about that measure theory example from two times ago. If we knew that stack was an ${\mathcal{A}}$-gerbe for some ${\mathcal{A}}$, then we could use cohomology to determine whether certain measure theoretic constructions existed. I don’t think anyone has ever done this before. Knowing something is a gerbe is very powerful since it converts existence questions to cohomology computations.

Let’s get right to it now. Fix a sheaf of abelian (possibly not necessary) groups ${\mathcal{A}}$ on ${X}$. Then an ${\mathcal{A}}$-gerbe is a gerbe ${\mathcal{S}}$ on ${X}$ such that for any open ${U}$ on ${X}$ we have a functorial isomorphism ${\mathcal{A}(U)\stackrel{\sim}{\rightarrow} \text{Aut}(s)}$ for all ${s\in \mathcal{S}(U)}$.

Note that since ${\mathcal{S}}$ is a stack, ${\text{Aut}(s)}$ is a sheaf, so by isomorphism we mean an isomorphism as sheaves, and by functorial we mean given another object ${t\in \mathcal{S}(U)}$, the isomorphism commutes

$\displaystyle \begin{matrix} \mathcal{A}(U) & \rightarrow & \text{Aut}(s) \\ Id \downarrow & & \downarrow \\ \mathcal{A}(U) & \rightarrow & \text{Aut}(t) \end{matrix}$

In particular, we get that for any two objects ${C, D\in \mathcal{S}(U)}$ we have that the sheaf ${Isom(C,D)}$ is an ${\mathcal{A}}$-torsor. This gives that if there is some object over ${U}$, namely that ${\mathcal{S}(U)\neq \emptyset}$, then the set of isomorphism classes of obects in ${\mathcal{S}(U)}$ is in natural bijection with ${H^1(U, \mathcal{A}_U)}$, as was pointed out in the motivation above.

One can form the ${\mathit{classifying \ stack}}$ over ${B}$ (the stacky version of a classifying space), ${B\mathcal{A}}$ by taking ${B\mathcal{A}(U)=\mathcal{T}ors(\mathcal{A}(U))}$. So above an open set we get the category of ${\mathcal{A}(U)}$-torsors on ${U}$. A basic theorem about ${\mathcal{A}}$-gerbes is that an ${\mathcal{A}}$-gerbe, ${\mathcal{S}}$, is isomorphic to ${B\mathcal{A}}$ if and only if ${F(B)\neq \emptyset}$. This says that ${F}$ is isomorphic to the classifying stack if and only if it has a global object.

## Gerbes 1: The Definition

Here’s a nice short definitional post. If you think that defining stacks and now we have even more definitions is just completely absurd, abstract, solipsism bear with me for just one more post. In the next post we’ll see what the point of all of this is. It is not just pointless abstraction. Figuring out something is a gerbe actually gives you an amazingly powerful tool to work with.
A gerbe is just a special type of stack. Let’s go back to thinking topologically, since if I have non-AG readers, this probably feels most comfortable. Consider a stack ${\mathcal{S}}$ over ${\text{Top}(X)}$. For instance, line bundles on ${X}$.

So we get some stuff associated to every open set of ${X}$. Recall, we think of these as lying over these open sets. No part of the definition of stack guarantees that we must have things lying over open sets (i.e. the collection of things over a particular open set could be empty). The first condition for a stack to be a gerbe is that for any open set ${U}$, there is a covering ${\{V_i\subset U\}}$ such that ${\mathcal{S}(V_i)\neq \emptyset}$. In other words, we can always shrink our open set to get an object over it.

Let’s think to our line bundle example. Check. We at least always have the trivial bundle.

The other condition for a stack to be a gerbe is that everything is locally isomorphic in the following sense, whenever we have two objects ${\eta, \eta '}$ over some open set ${U}$, then there is some covering ${\{V_i \subset U\}}$ such that we get ${\eta|_{V_i} \stackrel{\sim}{\rightarrow} \eta '|_{V_i}}$.

Let’s think to our line bundle example. Check. By definition we have local trivializations, which are all isomorphic. So ${L(X)}$ is a gerbe. I didn’t do a good job at examples of non-stacks, but it might actually be useful to give examples of stacks that are not gerbes. The stack ${M_g}$ that was briefly mentioned last post is not a gerbe (in fact, I haven’t really told you what I mean by ${M_g}$, and it turns out if you define the moduli space with respect to the Zariski topology ${M_1}$ isn’t even a stack).

Again, most importantly for us the deformation stack (of a smooth scheme, ${Z}$) from last time is also a gerbe (mostly for the same reason as the bundle example, you have the trivial one and locally everything becomes the trivial one).

## 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.

## Fun Statistics From WordPress

WordPress emailed this to me. I definitely didn’t post as much as I had hoped to. Only 26 posts for the entire year. I don’t know why I even try. Since February of 2008, my post Lost in the Funhouse has been the top post every single week. I may as well shift the blog to a completely literary blog. Anyway, enjoy, hopefully later today I’ll do another Stacks post.

The stats helper monkeys at WordPress.com mulled over how this blog did in 2010, and here’s a high level summary of its overall blog health:

## Crunchy numbers

The average container ship can carry about 4,500 containers. This blog was viewed about 22,000 times in 2010. If each view were a shipping container, your blog would have filled about 5 fully loaded ships.

In 2010, there were 26 new posts, growing the total archive of this blog to 244 posts. There were 9 pictures uploaded, taking up a total of 244kb. That’s about a picture per month.

The busiest day of the year was March 3rd with 189 views. The most popular post that day was Lost in the Funhouse.

## Where did they come from?

The top referring sites in 2010 were terrytao.wordpress.com, wiki.henryfarrell.net, amathew.wordpress.com, en.wordpress.com, and onlinedegree.net.

Some visitors came searching, mostly for lost in the funhouse, lost in the funhouse analysis, john barth lost in the funhouse analysis, normal basis theorem, and galois descent.

## Attractions in 2010

These are the posts and pages that got the most views in 2010.

1

Lost in the Funhouse February 2009

2

Me? April 2008

3

Measure Decomposition Theorems July 2008