What is “the” Gerbe of a 2-Bundle?
Posted by Urs Schreiber
For some time I was puzzled by how exactly gerbes and 2-bundles fit together conceptually. It is known from studying their properties that they encode the same information (when appropriate qualifications are added). But the underlying conceptual reason for that has been unclear to me, for the following reason:
The obvious guess was that a gerbe is to a 2-bundle like a sheaf of sections is to a bundle. Principal 2-bundles have categories of 2-sections over open patches that happen to be groupoids. Hence it was tempting to speculate that these groupoids over each open set form a gerbe.
If done correctly this should even be true. But the trouble is that the stack in groupoids obtained this way can not be the one that we want to call ‘the gerbe of the 2-bundle’. The reason is that taking the collection of groupoids (to state it carefully) obtained this way, feeding it into the standard machinery and producing cocycles or whatnot from it, we do not get back to the 2-bundle data that we started with.
Now I have thought a little harder. It now seems to me that there is another natural groupoid structure on the set of 2-sections (local 2-trivializations, really) of a 2-bundle. And this does seem to be the right one.
Discussing this requires drawing some diagrams. I have done this in these notes:
Notes on the Relation between 2-Bundles and Gerbes .
Namely, given two trivializations and over a patch we can ask for all ‘retrivializations’
that take us from to .
For an ordinary bundle there is precisely one such retrivialization, which we could write as
( and take values in torsors and their ‘difference’ as above is a group element).
Hence, already for an ordinary bundle we get a groupoid structure on the set of sections for each . This groupoid however is special in that it has only precisely one morphism for any given ordered pair of objects – its vertex groups (automorphism groups of its objects) are trivial.
This collection of groupoids is plausibly already the shadow of the full gerbe that we are looking for, since we may interpret the bundle as a degenerate case of a 2-bundle.
Motivated by this observation, there is an obvious generalization to 2-bundles. There, between given trivializations and we have not just a 1-morphism , but a (2-torsor) 2-morphism
Composition now should be the horizontal product of these 2-morphisms. Declaring this to be the case we get a groupoid
for each .
This collection of groupoids should nicely fit into a non-empty and transitive stack of groupoids, hence a gerbe.
Moreover, it has the right properties to qualify as the gerbe of our 2-bundle. In particular note that its vertex groups are now non-trivial and in fact isomorphic to the group if our structure 2-group is . Hence this does have a chance of being the correct -gerbe.
To check this further, one can go ahead and see what the diagrams which are written down when working out the local description of a gerbe yield when its arrows are taken to be morphisms of the above retrivialization groupoid. It turns out that they become equivalent to the respective 2-group diagrams that one draws when working out the local description of the 2-bundle.