Dirichlet Species and Arithmetic Zeta Functions

Abstract

Though Joyal's species are known to categorify generating functions in enumerative combinatorics, they also categorify zeta functions in algebraic geometry. The reason is that any scheme $X$ of finite type over the integers gives a "zeta species" $Z_X$, and any species $F$ gives a Dirichlet series $\widehat{F}$, in such a way that $\widehat{Z}_X$ is the arithmetic zeta function of $X$, a well-known Dirichlet series that encodes the number of points of $X$ over each finite field. Specifically, a $Z_X$-structure on a finite set is a way of making that set into a semisimple commutative ring, say $k$, and then choosing a $k$-point of the scheme $X$. This is an elaboration of joint work with James Dolan.