  The idea that gauge theory has 'surplus' structure poses a puzzle: in one
much discussed sense, this structure is redundant; but on the other hand, it is
also widely held to play an essential role in the theory. In this paper, we
employ category-theoretic tools to illuminate an aspect of this puzzle. We
precisify what is meant by 'surplus' structure by means of functorial
comparisons with equivalence classes of gauge fields, and then show that such
structure is essential for any theory that represents a rich collection of
physically relevant fields which are 'local' in nature.
