Parallel Transport and Functors

Abstract

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.

73 pages, 1 figure. Version 2 contains several improvements, in particular concerning the proofs of Lem. 2.15 and of Prop. 4.7, and the discussion in Appendix A2. Some notation has also been changed. Version 3 comes with an extended discussion of groupoid bundles with connection. Version 4 is the published version. In Version 5 we have replaced parts of the proof of Prop. 4.7 by better arguments

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