Details
morning and afternoon coffee breaks will take place in the Eckhart Tea Room

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".