John Baez

Minicourse at the Calgary summer school on Topics in Homotopy Theory

August 22-26, 2005

Higher Gauge Theory,
Homotopy Theory
and n-Categories

These are rough notes for four lectures on higher gauge theory, aimed at explaining how this theory is related to some classic themes from homotopy theory, such as Eilenberg-Mac Lane spaces. After a brief introduction to connections on principal bundles, with a heavy emphasis on the concept of `torsor', we describe how to build the classifying space BG of a topological group G starting from the topological category of its torsors. In the case of an abelian topological group A, we explain how this construction can be iterated, with points of BnA corresponding to `finite collections of A-charged particles on the n-sphere'. Finally, we explain how BnA can be constructed from the n-category of n-torsors of A. In the process, these notes give a quick introduction to the most basic notions of enriched category and strict n-categories. References provide avenues for further study. Unfortunately the above notes are missing some pictures which I have drawn by hand. Someday I'll rectify this - but right now, if you're desperate, you can download a ridiculously large (30 megabyte) file that includes scanned-in versions of the pictures:

Here's some extra reading material to get ready for the course:

I also suggest reading parts of this book: especially the parts where they describe the classifying space of a topological group, and the appendix where they construct characteristic classes using deRham cohomology by picking a connection on a vector bundle and constructing closed forms from its curvature. This assumes you know about deRham cohomology, connections, and their curvature! If you don't know this stuff, you should definitely learn it, and one simple way is to look in here:

© 2005 John Baez