Connections on nonabelian Gerbes and their Holonomy
Abstract
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2group, which plays the role of a band, or structure 2group. Upon choosing certain examples of Lie 2groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: nonabelian differential cocycles, BreenMessing gerbes, abelian and nonabelian bundle gerbes. These relationships convey a welldefined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to nonabelian gerbes. Several new features of surface holonomy are exposed under its extension to nonabelian gerbes; for example, it carries an action of the mapping class group of the surface.
 Publication:

arXiv eprints
 Pub Date:
 August 2008
 DOI:
 10.48550/arXiv.0808.1923
 arXiv:
 arXiv:0808.1923
 Bibcode:
 2008arXiv0808.1923S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Category Theory;
 Mathematics  Geometric Topology;
 Primary 53C08;
 Secondary 55R65;
 18D05
 EPrint:
 57 pages. v1 is preliminary. v2 is completely rewritten, former Sections 1 and 2 have been moved into a separate paper (arxiv:1303.4663), and the discussion of nonabelian surface holonomy has been improved and extended. v3 is the final and published version with a few minor corrections