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# Abelian category

In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernelss and cokernels exist and have the usual properties. The motivating prototype example of an abelian category is the category of abelian groups, whence the name.

 Table of contents 1 Definitions 2 Examples 3 Elementary properties 4 Related concepts 5 History 6 To do

## Definitions

It's possible to define abelian categories in a piecemeal fashion:

It's also possible to define abelian categories all at once. A category is abelian if it has:
Most notably, this latter definition doesn't mention the enrichment over Ab that began the piecemeal definition; that enrichment can be constructed from the assumptions above.

## Elementary properties

Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category.

In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, while the monomorphism is called the image of f.

Subobjects and quotient objects are well behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice.

Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(G,A) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A.

## Related concepts

Abelian categories are the most general concept for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma, the short five lemma, and the snake lemma.

## History

Abelian categories were introduced by Alexander Grothendieck in the middle of the 1950s in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groupss. The two were defined completely differently, but they had formally almost identical properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck managed to unify the two theories: they both arise as derived functors on abelian categories; on the one hand the abelian category of sheaves of abelian groups on a topological space, on the other hand the abelian category of G-modules for a given group G.

## To do

There are still several facts listed in Preadditive category, Additive category, and Preabelian category that should be repeated here when this is the most common context in which they're used.