Abelian Category - Definitions

Definitions

A category is abelian if

• it has a zero object,
• it has all binary products and binary coproducts, and
• all monomorphisms and epimorphisms are normal.

This definition is equivalent to the following "piecemeal" definition:

• A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear.
• A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products.
• An additive category is preabelian if every morphism has both a kernel and a cokernel.
• Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism.

Note that the enriched structure on hom-sets is a consequence of the three axioms of the first definition. 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.