  In conventional gauge theory, a charged point particle is described by a
representation of the gauge group. If we propagate the particle along some
path, the parallel transport of the gauge connection acts on this
representation. The Lagrangian density of the gauge field depends on the
curvature of the connection which can be calculated from the holonomy around
(infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a
generalization, known as p-form electrodynamics, in which (p-1)-dimensional
charged objects can be propagated along p-surfaces and in which the Lagrangian
depends on a generalized curvature associated with (infinitesimal) closed
p-surfaces. In this article, we use Lie 2-groups and ideas from higher category
theory in order to formulate a discrete gauge theory which generalizes these
models at the level p=2 to possibly non-Abelian symmetry groups. An important
feature of our model is that it involves both parallel transports along paths
and generalized transports along surfaces with a non-trivial interplay of these
two types of variables. Our main result is the geometric picture, namely the
assignment of non-Abelian quantities to geometrical objects in a coordinate
free way. We construct the precise assignment of variables to the curves and
surfaces, the generalized local symmetries and gauge invariant actions and we
clarify which structures can be non-Abelian and which others are always
Abelian. A discrete version of connections on non-Abelian gerbes is a special
case of our construction. Even though the motivation sketched so far suggests
applications mainly in string theory, the model presented here is also related
to spin foam models of quantum gravity and may in addition provide some insight
into the role of centre monopoles and vortices in lattice QCD.
