Path space connections and categorical geometry

We develop a new differential geometric structure using category theoretic tools that provides a powerful framework for studying bundles over path spaces. We study a type of connection forms, given by Chen integrals, over path spaces by placing such forms within a category-theoretic framework of principal bundles and connections. A new notion of ‘decorated’ principal bundles is introduced, along with parallel transport for such bundles, and specific examples in the context of path spaces are developed.