Two-dimensional parallel transport : combinatorics and functoriality

We extend the usual notion of parallel transport along a path to triangulated surfaces. A homotopy of paths is lifted into a fibered category with connection and this defines a functor between the fibers above the boundary paths. These "sweeping functors" transport fiber bundles with connection along a surface whereas usual connections transport a group element along a path. We show that to get rid of the parametrization, we must use Abelian degrees of freedom. In the general, non-Abelian case, we conjecture that the smooth limit of this construction provides us with representations of the group of diffeomorphisms of the swept surface. Applications to gauge theories are proposed.