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Connections on non-Abelian gerbes and their holonomy

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Urs Schreiber and Konrad Waldorf

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 2-group, which plays the role of a band, or structure 2-group. Upon
choosing certain examples of Lie 2-groups, our axiomatic framework
reproduces in a systematical way several known concepts of gerbes with
connection: non-abelian differential cocycles, Breen-Messing gerbes,
abelian and non-abelian bundle gerbes. These relationships convey a
well-defined 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 non-abelian gerbes. Several new features of surface holonomy are
exposed under its extension to non-abelian gerbes; for example, it carries
an action of the mapping class group of the surface.

Keywords:
Parallel transport, surface holonomy, path 2-groupoid, gerbes, 2-bundles,
2-groups, non-abelian differential cohomology, non-abelian bundle gerbes

2010 MSC:
Primary 53C08, Secondary 55R65, 18D05

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 17, pp 476-540.

Published 2013-07-17.

http://www.tac.mta.ca/tac/volumes/28/17/28-17.dvi

http://www.tac.mta.ca/tac/volumes/28/17/28-17.ps

http://www.tac.mta.ca/tac/volumes/28/17/28-17.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/17/28-17.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/17/28-17.ps

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