The n-Category Café

2-Groups and Algebras

Posted by Urs Schreiber

In another thread #, I am talking with Jim Stasheff and David Roberts about the question how to reconstruct a 2-bundle with connection from its local transition data #.

There are $1\frac{1}{2}$ examples where I have some idea at least about certain aspects of the answer.

And there seems to be a pattern:

Example 1 is this: start with transition data on some space $X$ with respect to the 2-group $G_2$ coming from the crossed module $U(1)\to 1$ (characterizing an abelian gerbe #). It is well known that this is equivalent to a $(P U(H) \simeq K(\mathbb{Z},2))$-bundle on $X$. $P U(H)$ happens to be the automorphism group of the algebra of compact operators on $H$. Hence we can find the associated algebra bundle. Regarding each fiber not as a mere algebra, but as the category of modules of that algebra, we do obtain a 2-bundle of sorts. I think one can show that this is the 2-bundle whose local trivializations yields the 3-cocycle we started with #.

Example 2 is the string bundle with string connection by Stolz & Teichner #.

In both cases one can, I think, understand the algebra that the nerve acts on by automorphisms as the 2-vector space on which the 2-group is represented by its canonical 2-representation #.

So, clearly, there is some general mechanism at work which should generalize the above table from $(U(1)\to 1)$ and $(\hat \Omega G \to P G)$ to any strict 2-group. Which mechanism is that?