Path space connections and categorical geometry

TL;DR

This paper develops a categorical framework for path space connections using Chen integrals, introducing decorated principal bundles and parallel transport, with specific examples illustrating the approach.

Contribution

It introduces a novel categorical approach to path space connections, including decorated bundles and parallel transport, expanding the theoretical understanding of such geometric structures.

Findings

Categorical framework for path space connections established

Decorated principal bundles defined and analyzed

Examples demonstrating the application of the theory provided

Abstract

We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop parallel transport on such bundles, and explore specific examples in the context of pathspaces.