The axiomatization of abelian categories was intended to capture some of the useful properties of categories in homological algebra. This entry gives some examples of abelian categories.

Example 1   The category of abelian groups is an abelian category. [Proof]

Since abelian groups are a special case of -modules, one might ask whether categories of -modules are also abelian categories. This is the case.

Example 2   For any ring , the category of left -modules is an abelian category.

Example 3   For any ring , the category of complexes of left -modules is an abelian category.

Example 4   For any topological space , the category of sheaves of abelian groups over is an abelian category.