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.
DEFINITION 8.5
We denote by

the isomorphism classes of all finite simple
objects of

. Remark that for each
![$ P= [X]\in P(C)$](img32.png)
the norm

is well-defined,
We define the zeta function

of

as