  Perhaps the most important contribution of gauge theory to general
mathematics is to point out the importance of association functors. Emphasizing
category theory we characterize association functors by two of their natural
properties and use this characterization to establish an equivalence between
the category of principal bundles and a suitably defined category of functors.
From the point of view of differential geometry we detail the specialization of
non-linear or Ehresmann to principal and linear connections and discuss the
widely known and very useful universality of principal curvature in order to
characterize the vector bundles in the image of a given association functor.
This is the second version.
