**Yoshifumi Tsuchimoto**

- A collection
of
**objects**of . - For each pair of objects
, a set
**morphisms**. - For each triple of objects
,
a map(``composition (rule)'')

satisfying the following axioms

- unless .
- (Existence of an identity) For any
,
there exists an element
such that
- (Associativity)
For any objects
, and for any morphisms
,
we have

- A morphism
in a category is said to be
**monic**if for any object of and for any morphism , we have - A morphism
in a category is said to be
**epic**if for any object of and for any morphism , we have

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