  We prove that isomorphism classes of principal bundles over a diffeological
space are in bijection to certain maps on its free loop space, both in a setup
with and without connections on the bundles. The maps on the loop space are
smooth and satisfy a "fusion" property with respect to triples of paths. Our
bijections are established by explicit group isomorphisms: transgression and
regression. Restricted to smooth, finite-dimensional manifolds, our results
extend previous work of J. W. Barrett.
