Dirichlet Species and Arithmetic Zeta Functions
John C. Baez
TL;DR
The paper builds a bridge between combinatorial species and arithmetic zeta functions by introducing zeta species Z_X from multiplicative, tame functors X: CommRing -> Set and showing that their Dirichlet series recover arithmetic zeta functions. It defines Dirichlet products and exponentials for tame Dirichlet species, establishes an Euler-product calculus, and proves an isomorphism Z_X ≅ ∏^D_p exp_D(F_{X,p}) that categorifies the local-to-global structure of zeta functions. The Riemann zeta function arises as the Dirichlet series of the Riemann species Z, obtained from the terminal functor, and the framework yields a clean, categorical perspective on how arithmetic data over finite fields assemble into global zeta functions. The results open avenues for categorifying other arithmetic objects and relating zeta data to familiar categorical constructs like the big Witt ring and related comonads.
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.