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".Click on this to see the transparencies of the talks:
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.
To explain how higher gauge theory fits into mathematics as a whole, we start 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".
Our talks are based on these papers: