In the following, let be a functor of abelian categories.

To define exact couples, we begin again with an abelian category.

Let be an abelian category (for example or ).

In particular, every abelian category is normal (and conormal as well).

Exact categories come from abelian categories in the following way.

K-Vect is an important example of an abelian category.

Abelian categories are the most general setting for homological algebra.

Indeed, in an abelian category, every morphism has a kernel.

If, for example, the abelian category is small, i.e. has only a set of objects, then this issue will be no problem.

The result is valid in every abelian category.

More generally, let C be an abelian category.

Consider the category of left -modules is an abelian category.

A left (or right) exact functor between abelian categories is additive.

He is best known for his contributions to Algebra and the theory of Abelian categories.

Chain complexes are easily defined in abelian categories, also.

The name "Ext" comes from the connection between the functors and extensions in abelian categories.

Let be an abelian category with enough projectives.

The G-modules form an abelian category with enough injectives.

So C naturally sits inside an abelian category.

In fact, abelian groups serve as the prototype of abelian categories.

Abelian categories, in particular, are always normal.

Every equivalence or duality of abelian categories is exact.

Say for example that we are working over an abelian category and have only one nonzero term in degree 0:

Fix an abelian category, such as a category of modules over a ring.

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about sub- and factor objects.