ZETA FUNCTIONS. NO.8

YOSHIFUMI TSUCHIMOTO

Definition 8.1. A category 𝒞 is a collection of the following data (1) A collection Ob(𝒞) of objects of 𝒞.

(2) For each pair of objects X, Y ∈ Ob(𝒞), a set

Hom_{𝒞}(X, Y)

of morphisms.

(3) For each triple of objects X, Y, Z ∈ Ob(𝒞), a map(“composition (rule)”)

Hom_{𝒞}(X, Y) × Hom_{𝒞}(Y, Z) → Hom_{𝒞}(X, Z)

satisfying the following axioms

(1) Hom(X, Y) ∩ Hom(Z, W) = ∅ unless (X, Y) = (Z, W).

(2) (Existence of an identity) For any X ∈ Ob(𝒞), there exists an element id_{X} ∈ Hom(X, X) such that

id_{X} ∘f = f, g ∘ id_{X} = g

holds for any f ∈ Hom(S, X), g ∈ Hom(X, T) (∀S, T ∈ Ob(𝒞)).

(3) (Associativity) For any objects X, Y, Z, W ∈ Ob(𝒞), and for any morphisms f ∈ Hom(X, Y), g ∈ Hom(Y, Z), h ∈ Hom(Z, W), we have

(f ∘ g) ∘ h = f ∘ (g ∘ h).

Definition 8.2. (1) A morphism f : X → Y in a category is said to be monic if for any object Z of 𝒞 and for any morphism g_{1}, g_{2} : Z → X, we have

f ∘ g_{1} = f ∘ g_{2} ⟹ g_{1} = g_{2}

(2) A morphism f : X → Y in a category is said to be epic if for any object Z of 𝒞 and for any morphism g_{1}, g_{2} : Y → Z, we have

g_{1} ∘ f = g_{2} ∘ f ⟹ g_{1} = g_{2}

Definition 8.3. An object X is called initial (resp. terminal) if Hom(X, Y)(resp. 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 𝒞 are isomorphic.

YOSHIFUMI TSUCHIMOTO

Definition 8.4. Let 𝒞 be a category with a zero object. We say that an object X ∈ Ob(𝒞) is simple when 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) = # End(X) = #Hom(X, X)

where # 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] ∈ P (C) the norm N(P) = N(X) is well-defined, We define the zeta function (s, 𝒞) of 𝒞 as

ζ(s, C) = ∏_{p∈P (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

ζ(s, Ab) = ζ(s).

References

[1] N. Kurokawa, Zeta functions of categories, proc.japan.acad 72 (1996), no. 10, 221–222.