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April 4, 2006

More by Bartels on 2-Bundles

Posted by Urs Schreiber

A while ago, Toby Bartels had a paper on the arXiv in which a notion of a categorified bundle, a 2-bundle, was defined ()

Toby Bartels
Categorified gauge theory: 2-bundles

Meanwhile this material has evolved. A draft of a refined version is now available:

Toby Bartels
Higher gauge theory: 2-Bundles (draft)

There are various refinements of the original definition. As far as I am aware, the most central one is that concerning the concept of 2-bundles associated to 2-transitions.

This is a matter of picking one of several a priori possible ways of encoding the structure of an ordinary bundle in terms of commuting diagrams, and then internalizing these diagrams in some 2-category of of “2-spaces”, where a 2-space is, essentially, a smooth category.

In this respect the crucial diagram now is (32) (beware that since this is a draft, I cannot guarantee that this number remains meaningul in the future). This encodes how a given transition between trivial bundles on local patches of a good covering may be associated (in a slightly nonstandard sense) to a bundle on the entire base space.

The categorification of this to 2-bundles is given in diagram (119), which of course (that is the whole point of this internalization approach to categorification) looks precisely as the original one, with the only exception that the commutativity condition is replaced with the existence of a coherent 2-isomorphism filling the diagram.

There is a more or less obvious way to define a 2-category of the 2-bundles thus defined. The punchline of the whole enterprise is then supposed to be theorem 3 (currently in section 3.3 “Gerbes”), which says that the 2-category of 2-bundles over some base space is equivalent to a suitable 2-category of nonabelian gerbes over that space.

The basic idea here (to my mind) is that a 2-bundle should be to a gerbe what an ordinary bundle is to its “sheaf of local retrivializations”. I have tried to sketch how this should work here.

Unfortunately, the proof of this important theorem is not yet given in the present stage of Toby’s draft.

Apparently an important new ingredient necessary to make this work is that in the definition of morphisms of 2-bundles one uses, instead of naive smooth functors, so-called smooth anafunctors. These are functors between smooth categories which locally are naturally isomorphic to ordinary smooth functors.

There would probably be more to say, but I need to get this entry here finished. If I find the time I might try to elaborate a little on how the notion of 2-transition developed by Toby Bartels relates to the concept of transition which I am using in the theory of 2-transport, as described in section 1.2 of these notes.

The kind of (2-)transition defined there immediately gives a relation for instance to bundle gerbes with connection and curving, but also to other structures involving higher order transport. In fact, one can understand the curious definition of a bundle gerbe with connection as defining precisely a 2-trivialization with 2-transition for a line 2-bundle. This is described in these notes.

There would be more to say. But I have to run now.

Posted at April 4, 2006 7:48 PM UTC

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