Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps

TL;DR

This paper establishes a bijection between principal bundles over diffeological spaces and fusion maps on their loop spaces, extending previous results to a broader setting with explicit constructions.

Contribution

It introduces explicit transgression and regression isomorphisms linking bundles and fusion maps in diffeological spaces, generalizing prior work on manifolds.

Findings

Bijection between bundle classes and fusion maps established

Explicit group isomorphisms for transgression and regression provided

Results extend previous manifold-based theories to diffeological spaces

Abstract

We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections are established by explicit group isomorphisms: transgression and regression. Restricted to smooth, finite-dimensional manifolds, our results extend previous work of J. W. Barrett.