A Diffeological Construction of Singer's Universal Connection

TL;DR

This paper constructs Singer's universal connection using diffeology and extends it to a functorial framework relating holonomy and bundle-connection pairs.

Contribution

It provides a rigorous diffeological construction of the universal connection and establishes an equivalence between holonomy representations and bundle-connection pairs.

Findings

Constructed Singer's universal connection via diffeology.

Generalized the universal connection to the diffeological setting.

Established an equivalence of categories between holonomy and bundle-connection pairs.

Abstract

We provide a rigorous construction of I.M. Singer's universal connection, a natural connection on a bundle of paths associated to any manifold, using the theory of diffeology. Furthermore, we generalize the universal connection to the diffeological setting, which enables the reconstruction of diffeological principal bundles with connections from their holonomy representations. We show that any two diffeological bundle-connection pairs with conjugate holonomy representations must be equivalent in a certain sense. These constructions are functorial in that, ultimately, our results can be summarized as an equivalence of categories between the so-called holonomy category and the category of diffeological bundle-connection pairs.