Categorical Aspects of Generating Functions (I): Exponential Formulas and Krull–Schmidt Categories

Abstract

In this paper, we study formal power series with exponents in a category. For example, the generating function of a category E with finite hom sets is defined by E(t)=∑tX/|Aut(X)|, where the summation is taken over all isomorphism classes of objects of E. We can use such power series to enumerate the number of E-structures along a faithful functor (Theorem 4.6). Our theory is closely related to the theory of species (Joyal, 1981). A species can be identified with a faithful functor from a groupoid to the category of finite sets (Theorem 3.6). We use mainly the concept of faithful functors with finite fibers instead of species, so that we can separate the roles categories and functors play. For example, the exponential formula E(t)=exp(Con(E)(t)) means the unique coproduct decomposition property (Theorem 5.8). In the final section, we give some applications of our theory to rather classical enumerations.