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The notion of abelian category is an abstraction of basic properties of the category Ab of abelian groups, more generally of the category RMod of modules over some ring, and still more generally of categories of sheaves of abelian groups and of modules. It is such that much of the homological algebra of chain complexes can be developed inside every abelian category. (from nLab)

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Examples of formalisation of abelian categories

The UniMath library has a formalization on abelian categories and homological algebra (done by Tommi Pannila). See: Abelian categories Short exact sequences Five lemma The documentation can be found …
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