The n-Category Café

Aldrovandi on non-abelian gerbes and 2-bundles

Ettore Aldrovandi’s talk was titled Butterflies, morphisms between gr-stacks, and non-abelian cohomology There were lots of ideas in this talk, but what came through loudest and clearest for me is how the stacky picture of 2-bundles contributes two important additions which should really be added to the picture of 2-bundles John presented in his talk (page 17 of John’s slides). I’ll recall the theorem for you here:

If $\mathcal{G}$ is a strict 2-group and $X$ a manifold, then there is a 1-1 correspondence between

(a) equivalence classes of principal $\mathcal{G}$-2-bundles over $X$

(b) elements of the Cech cohomology $H(X, \mathcal{G})$

(c) homotopy classes of maps $f : X \rightarrow B |N \mathcal{G}|$

(d) elements of the Cech cohomology $H(X, |N \mathcal{G}|)$

(e) isomorphism classes of principal $|N\mathcal{G}|$-bundles over $X$.

What’s missing here are the following:

(f) isomorphism classes of gerbes which are torsors over the gr-stack represented by $\mathcal{G}$
(in the sense of Breen)

(g) isomorphism classes of gerbes bounded by the crossed module associated to $\mathcal{G}$ (in the sense of Debremaeker).

What is a gerbe bounded by a crossed module? It was defined by Debremaeker in the 70’s. A gerbe $E$ over $X$ banded by a crossed module $H \rightarrow G$ consists of a gerbe $E$ over $X$ and a functor $J : E \rightarrow G-\mathrm{Tor}$ compatible with the crossed-module structure maps. I think that’s a pretty neat definition… how does it relate to previous definitions of nonabelian gerbes? Actually Google has just shown me a paper by Jurco on Crossed module bundle gerbes; classification, string group and differential geometry which seems to answer these questions… mmm.

Anyhow, Ettore’s talk was very interesting to me because he spoke a language “closer to Brylinski” than I’ve experienced in other formulations of 2-bundles and gerbes. I like old-school geometry in the form of bundles, sections, vector fields, connections, and so on. I am a big fan of the higher gauge theory program, but it seems to me that none of the ways to talk about 2-bundles listed in (a)-(e) give a hands-on geometric way to handle them - and that’s what we need, if we want to go through with John and Urs’s programme of computing characteristic classes for String bundles using 2-connections, and so on.

Here’s another question about higher gauge theory I have: we know that the strict version string 2-group corresponds to the crossed module $\hat{P}_k(G) \rightarrow P_k(G)$ of paths in $G$. How are we going to do differential geometry on this big thing? For 2-connection purposes, should we rather be working with the smaller, weak model?

Coming back to Ettore’s talk. He recalled how these two descriptions (noabelian gerbes versus gerbes banded by a crossed module) were related, and he showed that the whole thing is one huge equivalence of 2-categories… in fact it’s so canonical this equivalence works locally so it’s really an equivalence of 2-stacks. He ended the talk by showing how one can use the beautiful language of butterflies to represent morphisms of weak 2-groups and hence make this whole thing functorial in the 2-group $\mathcal{G}$.

Anyhow sorry about the higgledy-piggledy nature of this post but I’m having some trouble with the server (probably the fault is on my side).