Categorified gauge theory
Categorification is the process of taking concepts defined using sets,
functions and equations and generalizing them to concepts defined using
categories, functors, and natural transformations. If we apply this
process to gauge theory we obtain a theory of "2-connections," which allow
parallel transport not only along paths in the base manifold, but also
paths of paths. These arise naturally from the differential geometry
of gerbes, which has recently begun to play a role in string theory.
However, there are also many other examples. After an overview of
these ideas we describe a categorified version of the Yang-Mills equations.
2002 CTS/PNGS meeting