I recently went to California to visit Derek Wise at UC Davis – we were talking about expanding the talk he gave at Perimeter Institute into a more developed paper about ETQFT from Lie groups. Between that, the end of the Winter semester, and the beginning of the “Summer” session (in which I’m teaching linear algebra), it’s taken me a while to write up Emre Coskun’s two-part talk in our Stacks And Groupoids seminar.

Emre was explaining the theory of gerbes in terms of stacks. One way that I have often heard gerbes explained is in terms of a categorification of vector bundles – thus, the theory of “bundle gerbes”, as described by Murray in this paper here. The essential point of that point of view is that bundles can be put together by taking trivial bundles on little neighbourhoods of a space, and “gluing” them together on two-fold overlaps of those neighbourhoods – the gluing functions then have to satisfy a cocycle condition so that they agree on triple overlaps. A gerbe, on the other hand, defines line bundles (not functions) on double overlaps, and the gluing functions now live on triple overlaps. The idea is that this begins a heirarchy of concepts, each of which categorifes the previous (after “gerbe”, one just starts using terms like “2-gerbe”, “3-gerbe”, and so on). The levels of this hierarchy are supposed to be related to the various (nonabelian) cohomology groups $H^n(X,G)$ of a space $X$. I’ve mostly seen this point of view related to work by Jean-Luc Brylinski. It is a very differential-geometric sort of construction.

Emre, on the other hand, was describing another side to the theory of gerbes, which comes out of algebraic geometry, and is closely related to stacks. There’s a nice survey by Moerdijk which gives an account of gerbes from a similar point of view, though for later material, Emre said he drew on this book by Laumon and Moret-Bailly (which I can only find in the original French). As one might expect, a stack-theoretic view of gerbes thinks of them as generalizations of sheaves, rather than bundles. (The fact that there is a sheaf of sections of a bundle also generalizes to gerbes, so bundle-gerbes are a special case of this point of view).

Gerbes

So the setup is that we have some space $X$ – Emre was talking about the context of algebraic geometry, so the relevant idea of space here is scheme (which, if you’re interested, is assumed to have the etale topology – i.e. the one where covers use etale maps, the analog of local isomorphisms).  In the second talk, he generalized this to $S$-spaces: for some chosen scheme $S$.  That is, the category of “spaces” is based on the slice category $Sch/S$ of schemes equipped with maps into $S$, with the obvious morphisms.  This is a site, since there’s a notion of a cover over $S$ and so forth; an $S$-space is a sheaf (of sets) on this site.  So in particular, a scheme $X$ over $S$ determines an $S$-space, where $X : Sch/S \rightarrow Sets$ by $X(U) = Hom(U,X)$.  (That is, the usual way a space determines a representable sheaf).  There are also differential-geometric versions of gerbes.

So, whatever the right notion of space, a stack $\mathbb{F}$ over a space $X$ (in the sense of a sheaf of groupoids over $X$, which we’re assuming has the etale topology) is a gerbe if a couple of nice conditions apply:

1. There’s a cover $\{ U_i \rightarrow X \}$, such that none of the $\mathbb{F}(U_i)$ is empty.
2. Over any open $U$, all the objects $\mathbb{F}(U)$ are isomorphic (i.e. $\mathbb{F}(U)$ is connected as a category)

Notice that there doesn’t have to be a global object – that is, $\mathbb{F}(X)$ needn’t be empty – only some cover such that local objects exist – but where they do, they’re all “the same”.  These conditions can also be summarized in terms of the fibred category $\mathcal{F} \rightarrow X$.  There are two maps from $\pi, \Delta: \mathcal{F}\rightarrow \mathcal{F} \times_X \mathcal{F}$ – the projection and the diagonal.  The conditions respectively say these two maps are, locally, epi (i.e. surjective).

Emre’s first talk began by giving some examples of gerbes to motivate the rest. The first one is the “gerbe of splittings” of an Azumaya algebra. “An” Azumaya algebra $\mathcal{A}$ is actually a sheaf of algebras over some scheme $X$. The defining property is that locally it looks like the algebra of endomorphisms of a vector bundle. That is, on any neighborhood $U_i \subset X$, we have:

$\mathcal{A}(U_i) \cong End(\mathcal{E}_i)$

for some (algebraic) vector bundle $\mathcal{E}_i \rightarrow U_i$. A special case is when $X = Spec(\mathbb{R})$ is just a point, in which case an Azumaya algebra $\mathcal{A}$ is the same thing as a matrix algebra $M_n(\mathbb{R})$. So Azumaya algebras are not too complicated to describe.

The gerbe of splittings, $\mathbb{F}_{\mathcal{A}}$ for an Azumaya algebra is also not too complicated: a splitting is a way to represent an algebra as endomorphisms of a vector bundle – which in this case may only be possible locally. Over an given $U$, its objects are pairs $(E, \alpha)$, where $E$ is a vector bundle over $U$, and $\alpha : End(E) \rightarrow \mathbb{F}_{\mathcal{A}}(U)$ is an isomorphism. The morphisms are bundle isomorphisms that commute with the $\alpha$. So, roughly: if $\mathcal{A}$ is locally isomorphic to endomorphisms of vector bundles, the gerbe of splittings is the stack of all the vector bundles and isomorphisms which make this work. It’s easy to see this is a gerbe, since by definition, such bundles must exist locally, and necessarily they’re all isomorphic.

(This example – a gerbe consisting, locally, of a category of all vector bundles of a certain form – starts to suggest why one might want to think of gerbes as categorifying bundles.)

Another easily constructed gerbe in a similar spirit is found from a complex line bundle $\mathcal{L}$ over $X$ (and $n \in \mathbb{N}$). Then $\mathcal{X} \rightarrow X$ is a gerbe over $X$, where the groupoid $\mathcal{X}(U)$ over a neighborhood $U$ has, as objects, pairs $(\mathcal{M},\alpha)$ where $\alpha : \mathcal{M}^n \rightarrow \mathcal{L}$ is an isomorphism of line bundles. That is, the objects locally look like $n^{th}$ roots of $\mathcal{L}$. The gerbe is trivial (has a global object) if $\mathcal{L}$ has a root.

Cohomology

One says that a gerbe is banded by a sheaf of groups $\mathbb{G}$ on $X$ (or $\mathbb{G}$ is the band of the gerbe, or $\mathbb{F}$ is a $\mathbb{G}$-gerbe), if there are isomorphisms between the group $\mathbb{G}(U)$ and the automorphism group $Aut(u)$ for each object $u$ over $U$ (the property of a gerbe means these are all isomorphic). (These isomorphisms should also commute with the group homomorphisms induced by maps $\psi : V \rightarrow U$ of open sets.) So the band is, so to speak, the “local symmetry group over $U$” of the gerbe in a natural way.

In the case of the gerbe of splittings of $\mathcal{A}$ above, the band is $\mathbb{G}_m$, where over any given neighborhood, $\mathbb{G}_m(U) = Hom(U, G_m)$, where $G_m$ is the group of units in the base field: that is, the group $\mathbb{G}_m(U)$ consists of all the invertible sections in the structure sheaf of $X$. These get turned into bundle-automorphisms by taking a function $f$ to the automorphism that acts through multiplication by $f$. The gerbe $\mathcal{X}$ associated to a line bundle is banded by the group of $n^{th}$-roots of unity in sections in the structure sheaf.

From here, we can see how gerbes relate to cohomology. In particular, a $\mathbb{G}$-gerbe $\mathbb{F}$, we can associate a cohomology class $[F] \in H^2(X,\mathbb{G})$. This class can be thought of as “the obstruction to the existence of a global object”. So, in the case of an Azumaya algebra, it’s the obstruction to $\mathcal{A}$ being split (i.e. globally).

The way this works is, given a covering with an object $x_i$ in $\mathbb{F}(U_i)$, we take pull back this object along the morphisms corresponding to inclusions of sub-neighbourhoods, down to a triple-overlap $U_{ijk} = U_i \cup U_j \cup U_k$. Then there are isomorphisms comparing the different pullbacks: $u_{ij}^k : {x^i}_j^k \rightarrow x_i^{jk}$, and so on. (The lowered indices denote which of the $U$ we’re pulling back from).

Then we get a 2-cocycle in $\mathbb{G}(U_{ijk}$ (an isomorphism corresponding to what, for sheaves of sets, would be an identity). This is $c_{ijk} = u^i_{jk} ({u_i}^k_k)^{-1} u_{ij}^k$. The existence of this cocycle means that we’re getting an element in $H^2(X,\mathbb{G}$, which we denote $[\mathbb{F}]$. If a global object exists, then all our local objects are restrictions of a global one, the cocycle will always turn out to be the identity, so this class is trivial. A non-trivial class implies an obstruction to gluing the local objects into global ones.

Moduli Spaces

In the second talk, Emre gave some more examples of gerbes which it makes sense to think of as moduli spaces, including one which any gerbe resembles locally.

The first is the moduli space of all vector bundles $E$ over some (smooth, projective) curve $C$.  (Actually, one looks at those of some particular degree $d$ and rank $r$, and requires a condition called stability).

Actually, as discussed earlier in the seminar back in Aji’s talk, the right way to see this is that there is a “fine” moduli space – really a stack and not necessarily a space (in whichever context) – called $\mathcal{M}_C(r,d)$, and also a “coarse” moduli space called $M_C(r,d)$.  Roughly, the actual space $M_C(r,d)$ has points which are the isomorphism classes of vector bundles, while the stack remembers the whole groupoid of bundles and bundle-isomorphisms.  So there’s a map, which squashes a bundle to its isomorphism class: $\mathcal{M}_C(r,d) \rightarrow M_C(r,d)$ making the fine moduli space into a category fibred in groupoids – more than that, it’s a stack – and more than that, it’s a gerbe.  That is, there’s always a cover of $C$ such that there are some bundles locally, and (stable) bundles of a given rank and degree are always isomorphic.  In fact, this is a $\mathbb{G}_m$-gerbe, as above.

The next example is the gerbe of $G$-torsors, for a group $G$ (that is, $G$-sets which are isomorphic as $G$-sets to $G$ – the intuition is that they’re just like the group, but without a specified identity). The category $[\star / G ] = BG$ consists of $G$-torsors and their isomorphisms.  This is a gerbe over the point $\star$.  More interesting, when we’re in the context of $S$-spaces (and $S$ has a trivial action of $G$ on it), it becomes a $G$-gerbe over $S$.  Part of the point here is that any trivial gerbe (i.e. one with a section) is just such a classifying space for some group.  In particular, for the group of isomorphisms from a particular object to itself, crossed with $X$.

Since any gerbe has sections locally (that is, objects in $\mathbb{F}(U)$ for some $U$), every gerbe locally looks like one of these classifying-space gerbes.  This is the analog to the fact that any bundle locally looks like a product.