Abelian category

The theory of abelian categories was developed simultaneously (by Alexander Grothendieck and someone else) during the 1950s to develop a language with the aim of defining cohomology in a general framework. The foundations of abelian categories and homological algebra were outlined in the paper Sur quelques points d'algèbre homologique (often referred to as "Tohoku," owing to the name of the journal in which it was published) by Grothendieck.

A surprising result due to Freyd and Mitchell states that every small abelian category is equivalent to a category of modules. This has one important consequence of allowing one to often prove propositions regarding (small) diagrams in any abelian category by assuming the diagram is in the category of modules, a procedure usually referred to as "diagram chasing."

Definition and Examples

An abelian category is an additive category satisfying the additional requirement that every morphism has a kernel and a cokernel, and that every monomorphism is a kernel of some morphism and every epimorphism is a cokernel of some morphism.

1. The category of modules over a ring $R$. In particular, the category of modules over $\mathbb {Z}$ (which is equivalent to the category of abelian groups), is also an abelian category.
2. The category of sheaves on a topological space $X$ with values in an abelian category is again an abelian category.