Higher categories, higher gauge theory

Higher categories, higher gauge theory

[John Baez]

Hi -

Some of you may enjoy these lecture notes, especially given how people
have been drumming up interest in my talks.

Best,
jb

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Higher Gauge Theory, Higher Categories
http://math.ucr.edu/home/baez/namboodiri/

Abstract:

The work of Eilenberg and Mac Lane marks the beginning of a trend in
which mathematics based on sets is generalized to mathematics based on
categories and then higher categories.  We illustrate this trend towards
"categorification" by a detailed introduction to "higher gauge theory".

Gauge theory describes the parallel transport of point particles using
the formalism of connections on bundles.  In both string theory and loop
quantum gravity, point particles are replaced by 1-dimensional extended
objects: paths or loops in space.  This suggests that we seek some kind
of "higher gauge theory" that describes the parallel transport as we move
a path through space, tracing out a surface.  Surprisingly, this requires
that we "categorify" concepts from differential geometry, replacing smooth
manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras
by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and
so on.  The basic tool used here is Ehresmann's notion of "internalization".

To explain how higher gauge theory fits into mathematics as a whole, we
begin with a lecture reviewing the basic principle of Galois theory and
its relation to Klein's Erlangen program, covering spaces and the fundamental
group, Eilenberg-Mac Lane spaces, and Grothendieck's ideas on fibrations.

The second lecture treats connections on trivial bundles and 2-connections
on trivial 2-bundles, explaining how they can be described either in terms
of their holonomies or in terms of Lie-algebra-valued differential forms.
For a clean treatment of these concepts, we recall Chen's theory of
"smooth spaces", which generalize smooth finite-dimensional manifolds.

The third lecture explains connections on general bundles and 2-connections
on general 2-bundles, explaining how they can be described either in terms
of holonomies or local data involving differential forms.  We also explain
how 2-bundles are classified using nonabelian Cech 2-cocycles, and how the
theory of 2-connections relates to Breen and Messing's theory of "connections
on nonabelian gerbes".