A Diffeological Construction of Singer's Universal Connection

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.