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).