An Abelian category is a category for which the constructions and techniques of homological algebra are available. 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 such that all morphisms such that , factor 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.
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 is called an additive category.
Examples of Abelian categories include:
Freyd's theorem states that every Abelian category is a subcategory of some category of modules over a ring. Mitchell (1964) has strengthened this, saying every 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.