Dirichlet Functors are Contravariant Polynomial Functors

Polynomial functors are sums of covariant representable functors from the category of sets to itself. They have a robust theory with many applications – from operads and opetopes to combinatorial species. In this paper, we define a contravariant analogue of polynomial functors: Dirichlet functors. We develop the basic theory of Dirichlet functors, and relate them to their covariant analogues.