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 F̂, in such a way that Ẑ_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.
Keywords: Day convolution, Dirichlet series, species, zeta function
2020 MSC: 11M38, 11R42, 14G10, 18M80
Theory and Applications of Categories, Vol. 44, 2025, No. 39, pp 1316-1336.
Published 2025-12-31.
TAC Home