Parallel Transport and Functors

TL;DR

This paper characterizes the functors arising from parallel transport in smooth fibre bundles with connection, introduces local trivializations and descent data, and discusses generalizations to categorified bundles for higher-dimensional transport.

Contribution

It introduces a novel characterization of parallel transport functors using local trivializations and descent data, and extends the framework to categorified bundles for higher-dimensional cases.

Findings

Characterization of parallel transport functors via local trivializations.

Introduction of smooth descent data for classifying functors.

Extension of the framework to categorified bundles for higher-dimensional transport.

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.