## 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 ($\to$)

Toby Bartels
Categorified gauge theory: 2-bundles
math.CT/0410328

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

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

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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/784

Read the post Castellani on FDA in SuGra: gauge 3-group of M-Theory
Weblog: The String Coffee Table
Excerpt: Castellani uses free differential algebra, and implictly Line n-algebra connections, for understanding the symmetries of 11D supergravity.
Tracked: July 24, 2006 10:07 PM
Read the post Bakovic on 2-Groupoid 2-Torsors
Weblog: The n-Category Café
Excerpt: Igor Bakovic defines and classifies categorified groupoid bundles.
Tracked: October 31, 2006 10:36 PM
Weblog: The n-Category Café
Excerpt: Bruce Bartlett reports on Rick Jardine's concept of 'cocycle categories' and their relation to anafunctors.
Tracked: January 24, 2007 11:00 AM
Read the post Oberwolfach CFT, Tuesday Morning
Weblog: The n-Category Café
Excerpt: On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.
Tracked: April 3, 2007 2:08 PM
Weblog: The n-Category Café
Excerpt: A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.
Tracked: April 19, 2007 8:49 PM
Read the post The inner automorphism 3-group of a strict 2-group
Weblog: The n-Category Café
Excerpt: On the definition and construction of the inner automorphism 3-group of any strict 2-group, and how it plays the role of the universal 2-bundle.
Tracked: July 4, 2007 12:55 PM
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 27, 2007 5:30 PM
Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 11:01 AM

Post a New Comment