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An abelian category is a category
 satisfying the following axioms. Because the later axioms rely on terms whose definitions involve the earlier axioms, we will intersperse the statements of the axioms with such auxiliary definitions as needed.
Axiom 1. For any two objects  in
 , the set of morphisms
 admits an abelian group structure, with group operation denoted by  , satisfying the following naturality requirement: given any diagram of morphisms
we have
 and
 . That is, composition of morphisms must distribute over addition in
 .
The identity element in the group
 will be denoted by 0.
Axiom 2.
 has a zero object.
Axiom 3. For any two objects  in
 , the categorical direct product
 exists in
 .
Given a morphism
 in
 , a kernel of  is a morphism
 such that:
 - For any other morphism
 such that  , there exists a unique morphism
 such that the diagram
commutes.
Likewise, a cokernel of  is a morphism
 such that:
 - For any other morphism
 such that  , there exists a unique morphism
 such that the diagram
commutes.
Axiom 4. Every morphism in
 has a kernel and a cokernel.
The kernel and cokernel of a morphism  in
 will be denoted  and
 , respectively. (Some texts use the notation
 for cokernel.) By the universal properties above, the kernel and cokernel of  are only unique up to isomorphism, but by abuse of notation we write  for a representative element of this isomorphism class.
A morphism
 in
 is called a monomorphism if, for every morphism
 such that  , we have  . Similarly, the morphism  is called an epimorphism if, for every morphism
 such that  , we have  .
Axiom 5.
 for every monomorphism  in
 .
Axiom 6.
 for every epimorphism  in
 .
Remark. Equivalently, an abelian category is an additive category such that Axioms 4-6 are satisfied.
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