Categorical aspects of generating functions (I): Exponential formulas and Krull-Schmidt categories

In this paper, we study formal power series with exponents in a category. F or example, the generating function of a category E with finite hom sets is defined by E(t) = Sigmat(X)/\Aut(X)\, where the summation is taken over al l isomorphism classes of objects of E. We can use such power series to enum erate the number of E-structures along a faithful functor (Theorem 4.6). Ou r theory is closely related to the theory of species (Joyal, 1981). A speci es can be identified with a faithful functor from a groupoid to the categor y of finite sets (Theorem 3.6). We use mainly the concept of faithful funct ors with finite fibers instead of species, so that we can separate the role s 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 rath er classical enumerations. (C) 2001 Academic Press.