An Abelian category is an additive category with kernels and cokernels in which every monomorphism is the kernel of its cokernel and every epimorphism is the cokernel of its kernel. These axioms support the constructions of homological algebra. The basic examples of such categories are the category of Abelian groups and, more generally, the category of modules over a ring. Abelian categories are widely used in algebra, algebraic geometry, and topology.
Many of the same constructions that are found in categories of modules, such as kernels, exact sequences, and commutative diagrams are available in Abelian categories. A disadvantage that must be overcome is the fact that the objects in a category do not necessarily have elements that can be manipulated directly, so the traditional definitions do not work. As a result, methods must be developed that allow definition and manipulation of objects without the use of elements.
As an example, consider the definition of the kernel of a morphism, which states that given 
, the kernel of 
is defined to be a morphism 
with 
such that all morphisms 
such that 
, factor uniquely through 
. Notice that this definition does not guarantee that the kernel
exists, but only gives properties which uniquely identify it if it does exist. Similar
work must be done to define the cokernel of a morphism.
With these definitions, a category can be defined as Abelian if it satisfies the following five properties:
1. For two objects 
and 
, the set 
of morphisms from 
to 
has the structure of an Abelian
group. This group structure must be arranged so that composition
of morphisms is bilinear.
2. There is an object, denoted 0, that is both an initial object and a terminal object.
3. Products and coproducts of finite collections of objects always exist.
4. Kernels and cokernels always exist.
5. If 
is a morphism whose kernel
is 0, then 
is the kernel of its cokernel.
If 
is a morphism whose cokernel
is 0, then 
is the cokernel of its kernel.
A category that satisfies only the first three properties (the first making it a preadditive category) is called an additive category.
Examples of Abelian categories include:
1. For a commutative ring 
, the category of modules
over 
is an Abelian category. This is the basic example.
2. The category of vector spaces over a fixed field is an Abelian category.
3. The category of sheaves of Abelian groups over a topological space is an Abelian category (Stacks Project, Lemma 18.3.1).
Freyd's theorem states that every small Abelian category is a subcategory of some category of modules over a ring. Mitchell (1964) has strengthened this, saying every small Abelian category is a full subcategory of a category of modules over a ring. Despite this result, the terminology and methods of Abelian categories remain useful and powerful.
