Parallel transport and functors
Higher dimensional parallel transport generalizes parallel transport along curves to parallel transport along higher dimensional objects. Several mathematical objects have been used to define higher dimensional parallel transport, among them classes in Deligne cohomology, bundle gerbes with connection and curving, or 2-bundles with 2 connections. Bundle gerbes with connection can be obtained as a generalization of principal bundles with connection. However, in the case of both bundle gerbes and Deligne classes one encounters the obstruction that the structure group has to be abelian. Hence it is desirable to find a reformulation of fiber bundles with connection, that brings along a natural generalization for arbitrary structure group. A functor from the path groupoid of the base manifold \(M\) into a category that describes the fibers of the bundle arises from parallel transport of a connection in a smooth fiber bundle. The authors define two notions of local trivializations and smooth descent data to characterize the functor. These characterizations allow to substitute categories of functors with categories of smooth fiber bundles with connection. The authors point out that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport.
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