On the origin of (combinatorial) species: a category-theoretical (de)construction of the theory of Joyal species

Abstract

Abstract. We give a brief overview of alternative theories of combinatorial
species other than that given by Joyal [1]. Then, we construct a general theory
of species which extends the concept of species for any two categories with
definite categories as objects, and invertible morphisms (functors) in the first
category, and this construct is equivalent to that of all the existing species
defined on categories as the aforementioned, so that ordinary species, all types
of weighted species, q-species [5], Mobius species [4], and L-species are just
special cases. We adapt to this new scenario the classic constructs of assemblies
of species, derivation and pointing. An analogue for exponentiation is
constructed. Adapting the concept of arithmetic product yields a new type of
function called "rational Dirichlet series".