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A classic result in the foundations of Yang-Mills theory, due to J. W.Barrett ["Holonomy and Path Structures in General Relativity and Yang-MillsTheory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a"generalized" holonomy map from the space of piece-wise smooth, closed curvesbased at some point of a manifold to a Lie group, there exists a principalbundle with that group as structure group and a principal connection on thatbundle such that the holonomy map corresponds to the holonomies of thatconnection. Barrett also provided one sense in which this "recovery theorem"yields a unique bundle, up to isomorphism. Here we show that something strongeris true: with an appropriate definition of isomorphism between generalizedholonomy maps, there is an equivalence of categories between the category whoseobjects are generalized holonomy maps on a smooth, connected manifold and whosearrows are holonomy isomorphisms, and the category whose objects are principalconnections on principal bundles over a smooth, connected manifold. This resultclarifies, and somewhat improves upon, the sense of "unique recovery" inBarrett's theorems; it also makes precise a sense in which there is no loss ofstructure involved in moving from a principal bundle formulation of Yang-Millstheory to a holonomy, or "loop", formulation.

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