Parallel Transport and Functors

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.

💡 Research Summary

The paper revisits the classical relationship between a smooth fibre bundle (E\to M) equipped with a connection (\nabla) and the notion of parallel transport, but it does so entirely within a categorical framework. The authors begin by recalling that the set of points of the base manifold (M) together with smooth paths (considered up to thin homotopy) form a path groupoid (\Pi_1(M)). Objects are points of (M); morphisms are equivalence classes of smooth paths with fixed endpoints, and composition is given by concatenation of paths.

Given a connection (\nabla) on (E), each path (\gamma:x\to y) determines a linear (or more generally, fibre‑preserving) isomorphism (P_\gamma:E_x\to E_y) obtained by solving the parallel‑transport equation along (\gamma). Collecting all these maps yields a functor \