Next: Bibliography

# Zeta functions. No.8

Yoshifumi Tsuchimoto

DEFINITION 8.1   A category is a collection of the following data
1. A collection of objects of .
2. For each pair of objects , a set

of morphisms.
3. For each triple of objects , a map(composition (rule)'')

satisfying the following axioms

1. unless .
2. (Existence of an identity) For any , there exists an element such that

holds for any ( ).
3. (Associativity) For any objects , and for any morphisms , we have

DEFINITION 8.2

1. A morphism in a category is said to be monic if for any object of and for any morphism , we have

2. A morphism in a category is said to be epic if for any object of and for any morphism , we have

DEFINITION 8.3   An object is called initial (resp. terminal) if consists of only one element for every object . We say that an object is a zero object if is initial and terminal. It follows that all the zero objects of are isomorphic.

DEFINITION 8.4   Let be a category with a zero object. We say that an object is simple when is consisting of monomorph isms and zero-morphisms for every object . The norm of an object is defined as

where is the cardinality of endomorphisms of . We say that a non-zero object is finite if is finite.

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

DEFINITION 8.5   We denote by the isomorphism classes of all finite simple objects of . Remark that for each the norm is well-defined, We define the zeta function of as

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

Next: Bibliography
2013-06-13