### 2-Connections on 2-Bundles

#### Posted by Urs Schreiber

We have a new, prettified and more concise version of our work on 2-connections in 2-bundles, prepared for the proceedings of the Streetfest conference last summer.

J. Baez & U. S.
**Higher Gauge Theory**

math.DG/0511710

Compared to our previous preprint the discussion has been made more transparent. The interpretation of 2-transitions of 2-connections as 2-functor $p$-morphisms is now included (as discussed here).

For a nice overview of the key ideas see the transparencies of the talk that John is giving next Sunday at the Union College Math Conference.

**Update (May, 28, 2006)**: We have now received the (anonymous) referee report on that paper. Here it is

Report on the paper “Higher Gauge Theory” by John Baez and Urs Schreiber.

This paper is an introduction to ongoing research by the authors of higher dimensional generalization of gauge theory (see [BS]). If to think of gauge theory as the study of parallel transport of 0- dimensional objects (points) the higher gauge theory will deal with parallel transport of 1-dimensional gadgets (strings, paths). The authors “categorify” such standard ingredients of the gauge theory as structure group G, principal G-bundle, connection on it. For some of them category analogs are well know, for others the authors work out what they should be to fit nicely in the gauge picture. The testing ground is the “holonomy functor” which is the assignment to a path in the base of a principal G-bundle with connection the holonomy along it. The higher version of this is the holonomy of paths between paths taking values in the structure 2-group. It should be noted that the whole picture is “smooth” meaning that everything is “internalized” in the category of smooth spaces.

Posted at December 2, 2005 2:37 PM UTCThe paper is very well written. The definitions and constructions are carefully motivated and explained. The (mostly omitted) proofs can be found in the preprint [BS]. Aimed at “categorically oriented” mathematical public the paper will fit perfectly into the volume and should be published in its present form.

## Re: 2-Connections on 2-Bundles

Hi Urs,

The premise that gauge theory is a theory of transporting points along paths in spacetime and that higher gauge theory is about transporting paths along surfaces in spacetime is pretty neat and makes perfect sense. But if you had asked me to develop a mechanism for transporting paths, the first thing that would come to my mind would be to draw disjoint paths and then draw a rectangular sheet connecting the two paths.

This might be a silly question, but why does it seem like you are holding the ends of the paths fixed so that you get these (American) football shaped surfaces, i.e. why are you restricting to a Path 2-Groupoid? Is that necessary? That is not what you do on loop space, right?

Just curious :)

Eric