Editing Abelian category (section)

==Definitions==
A category is '''abelian''' if it is ''[[Preadditive category|preadditive]]'' and
*it has a [[zero object]],
*it has all binary [[biproduct]]s,
*it has all [[Kernel (category theory)|kernels]] and [[cokernel]]s, and
*all [[monomorphism]]s and [[epimorphism]]s are [[Normal morphism|normal]].

This definition is equivalent<ref>Peter Freyd, [http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html Abelian Categories]</ref> to the following "piecemeal" definition:
* A category is ''[[Preadditive category|preadditive]]'' if it is [[enriched category|enriched]] over the [[monoidal category]] {{math|1='''Ab'''}} of [[abelian group]]s. This means that all [[hom-set]]s are abelian groups and the composition of morphisms is [[bilinear operator|bilinear]]. 
* A preadditive category is ''[[Additive category|additive]]'' if every [[finite set]] of objects has a [[biproduct]]. This means that we can form finite [[direct sum of modules|direct sum]]s and [[direct product]]s. In <ref>Handbook of categorical algebra, vol. 2, F. Borceux</ref> Def. 1.2.6, it is required that an additive category have a zero object (empty biproduct).
* An additive category is ''[[preabelian category|preabelian]]'' if every morphism has both a [[kernel (category theory)|kernel]] and a [[cokernel]].
* Finally, a preabelian category is '''abelian''' if every [[monomorphism]] and every [[epimorphism]] is [[normal morphism|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-set]]s is a ''consequence'' of the first three [[axiom]]s of the first definition. This highlights the foundational relevance of the category of [[Abelian group]]s in the theory and its canonical nature.

The concept of [[exact sequence]] arises naturally in this setting, and it turns out that [[exact functor]]s, 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 category|exact categories]], forming a very special case of [[regular category|regular categories]].