A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold

A classic result in the foundations of Yang-Mills theory, due to Barrett [Int. J. Theor. Phys. 30, 117111215 (1991)], establishes that given a generalized holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this recovery theorem yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of unique recovery in Barretts theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or loop, formulation.