  Parallel transport of a connection in a smooth fibre bundle yields a functor
from the path groupoid of the base manifold into a category that describes the
fibres of the bundle. We characterize functors obtained like this by two
notions we introduce: local trivializations and smooth descent data. This
provides a way to substitute categories of functors for categories of smooth
fibre bundles with connection. We indicate that this concept can be generalized
to connections in categorified bundles, and how this generalization improves
the understanding of higher dimensional parallel transport.
