Zeta functions. No.8

Yoshifumi Tsuchimoto

DEFINITION 8.1   A category $\mathcal{C}$ is a collection of the following data

1. A collection $\operatorname{Ob}(\mathcal{C})$ of objects of $\mathcal{C}$ .
2. For each pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$ , a set
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)$$
   of morphisms.
3. For each triple of objects $X,Y,Z \in \operatorname{Ob}(\mathcal{C})$ , a map(``composition (rule)'')
   $$\operatorname{Hom}_{\mathcal{C}}(X,Y)\times \operatorname{Hom}_{\mathcal{C}}(Y,Z)\to \operatorname{Hom}_{\mathcal{C}}(X,Z)$$

satisfying the following axioms

1. $\operatorname{Hom}(X,Y)\cap \operatorname{Hom}(Z,W) =\emptyset$ unless $(X,Y)=(Z,W)$ .
2. (Existence of an identity) For any $X\in \operatorname{Ob}(\mathcal{C})$ , there exists an element $\operatorname{id}_X\in \operatorname{Hom}(X,X)$ such that
   $$\operatorname{id}_X \circ f=f,\quad g \circ \operatorname{id}_X=g$$
   holds for any $f \in \operatorname{Hom}(S,X), g \in \operatorname{Hom}(X,T)$ ( $\forall S,T \in \operatorname{Ob}(\mathcal{C})$ ).
3. (Associativity) For any objects $X,Y,Z,W \in \operatorname{Ob}(\mathcal{C})$ , and for any morphisms $f\in \operatorname{Hom}(X,Y), g\in \operatorname{Hom}(Y,Z), h \in \operatorname{Hom}(Z,W)$ , we have
   $$(f\circ g)\circ h =f\circ (g\circ h).$$

DEFINITION 8.2  

1. A morphism $f: X\to Y$ in a category is said to be monic if for any object $Z$ of $\mathcal{C}$ and for any morphism $g_1,g_2: Z\to X$ , we have
   $$f\circ g_1 =f \circ g_2 \implies g_1=g_2$$

2. A morphism $f: X\to Y$ in a category is said to be epic if for any object $Z$ of $\mathcal{C}$ and for any morphism $g_1,g_2: Y\to Z$ , we have
   $$g_1\circ f = g_2 \circ f \implies g_1=g_2$$

DEFINITION 8.3   An object $X$ is called initial (resp. terminal) if $\operatorname{Hom}(X, Y) (resp. \operatorname{Hom}(Y, X))$ consists of only one element for every object $Y$ . We say that an object $X$ is a zero object if $X$ is initial and terminal. It follows that all the zero objects of $\mathcal{C}$ are isomorphic.

DEFINITION 8.4   Let $\mathcal{C}$ be a category with a zero object. We say that an object $X\in \operatorname{Ob}(\mathcal{C})$ is simple when $\operatorname{Hom}(X, Y)$ is consisting of monomorph isms and zero-morphisms for every object $Y$ . The norm $N(X)$ of an object $X$ is defined as

$$N(X)= \char93 \operatorname{End}(X) = \char93 Hom(X,X)$$

where $\char93 \operatorname{End}(X)$ is the cardinality of endomorphisms of $X$ . We say that a non-zero object $X$ is finite if $N(X)$ is finite.

The treatment here is based on a paper of Kurokawa[1].

DEFINITION 8.5   We denote by $P(C)$ the isomorphism classes of all finite simple objects of $C$ . Remark that for each $P= [X]\in P(C)$ the norm $N(P)= N(X)$ is well-defined, We define the zeta function $(s, \mathcal{C})$ of $\mathcal{C}$ as

$$\zeta(s, C)= \prod_{p \in P(\mathcal{C})} (1-N(p)^{-s} )^{-1}.$$

PROPOSITION 8.6   The zeta function of the category Ab of abelian groups is equal to the Riemann zeta function. In other words, we have

$$\zeta(s,{\rm Ab})=\zeta(s).$$