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 article, 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 that are ‘local’ in nature. 1Introduction2Theories as Categories 2.1Relations between models2.2Relations between theories3Gauge Theory as a Category 3.1Gauge theory on contractible manifolds3.2Other candidates for representing U(1) gauge theory3.3Surplus and inter-theoretical comparisons4Gauge Theory as a Functor 4.1Richness and locality4.2Richness and locality imply surplus*5ConclusionAppendix

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