A category is abelian if it is preadditive and

• it has a zero object,
• it has all binary biproducts,
• it has all kernels and cokernels, and
• all monomorphisms and epimorphisms are normal.

Note that the enriched structure on hom-sets is a consequence of the first three axioms. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature.

The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories.

The motivating prototypical example of an abelian category is the category of abelian groups, Ab.

Abelian categories are named after Niels Abel.