open access

https://doi.org/10.14943/83562

https://hdl.handle.net/2115/69166

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

英語

1998-06-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

416

1-44

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) = :Etx /IAut(X)I, where the summation is taken over all isomorphism classes of obejects of E. We can use such power series to enumerate the number of £-structures along a faithful functors(Theorem 4.6). Our theory is closely related to the thory 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 that of species, by which we can separate the roles which categories play and which 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.

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