  Just as gauge theory describes the parallel transport of point particles
using connections on bundles, higher gauge theory describes the parallel
transport of 1-dimensional objects (e.g. strings) using 2-connections on
2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where
the fiber is not a manifold but a category with a suitable smooth structure.
Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses
their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a
theory of 2-connections on principal 2-bundles and explain how this is related
to Breen and Messing's theory of connections on nonabelian gerbes. The
distinctive feature of our theory is that a 2-connection allows parallel
transport along paths and surfaces in a parametrization-independent way. In
terms of Breen and Messing's framework, this requires that the "fake curvature"
must vanish. In this paper we summarize the main results of our theory without
proofs.
