Parallel transport and functors

We describe parallel transport of a connection in a vector bundle as a functor defined on the path groupoid of the base manifold. We introduce a notion of local triviality and smoothness for such functors by which we characterize parallel transport functors among arbitrary functors. The categorification of this concept yields a new interpretation of (non-abelian) bundle gerbes with connection and curving as descent data of parallel transport 2-functors.

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