Abelian Categories

The notion of abelian category is a generalization of , the category of abelian groups.

Abelian Category

A preadditive category is abelian if

• it has a zero object;
• it has all binary products and coproducts;
• it has all kernels and cokernels;
• all monomorphisms and epimorphisms are normal.
  

Kernel

In a preadditive category with a zero object, a kernel of a morphism is a morphism such that and for any morphism with , there exists a unique morphism such that . That is, the following diagram commutes:

Exact Sequence

A sequence of morphisms between objects in an abelian category is exact at if . The sequence is an exact sequence if it is exact at every object.

Short Exact Sequence & Long Exact Sequence

A short exact sequence is an exact sequence of the form . Consequently,

• is monic.
• is epic.
• .
• .

A longer exact sequence is called a long exact sequence.