Zeta functions of categories (Q1357044)

This paper is a trial of the interpretation of zeta functions from the categorical point of view. Let \(C\) be a category with zero object. We say that an object \(X\) is simple if \(\Hom(X,Y)\) consists of monomorphisms and zero-morphisms for every object \(Y\). The norm \(N(X)\) of an object \(X\) is defined as the cardinality of endomorphisms of \(X\). We say that \(X\) is finite if \(N(X)\) is finite. We denote by \(P(C)\) the isomorphism classes of all finite simple objects of \(C\). The norm is defined for each isomorphism class. We define the zeta function \(\zeta(s,C)\) of \(C\) by \(\zeta(s,C): =\prod_{P \in P(C)} (1-N (P)^{-s})^{-1}\). The author gives some examples and how to unify Hasse zeta and Selberg zeta functions from his point of view.