  The term higher gauge theory refers to the generalization of gauge theory to
a theory of connections at two levels, essentially given by 1- and 2-forms. So
far, there have been two approaches to this subject. The differential picture
uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of
a conventional gauge theory to the next level. The integral picture makes use
of curves and surfaces labeled with elements of non-Abelian groups and
generalizes the formulation of gauge theory in terms of parallel transports. We
recall how to circumvent the classic no-go theorems in order to define
non-Abelian surface ordered products in the integral picture. We then derive
the differential picture from the integral formulation under the assumption
that the curve and surface labels depend smoothly on the position of the curves
and surfaces. We show that some aspects of the no-go theorems are still present
in the differential (but not in the integral) picture. This implies a
substantial structural difference between non-perturbative and perturbative
approaches to higher gauge theory. We finally demonstrate that higher gauge
theory provides a geometrical explanation for the extended topological symmetry
of BF-theory in both pictures.
