Pre-abelian Category

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.

Spelled out in more detail, this means that a category C is pre-abelian if:

  1. C is preadditive, that is enriched over the monoidal category of abelian groups;
  2. C has all biproducts, which are both finite products and finite coproducts;
  3. given any morphism f: AB in C, the equaliser of f and the zero morphism from A to B exists (this is the kernel), as does the coequaliser (this is the cokernel).

Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism AOB, where O is a zero object, guaranteed to exist by item 2.

Read more about Pre-abelian Category:  Examples, Elementary Properties, Exact Functors, Special Cases

Other articles related to "category":

Pre-abelian Category - Special Cases
... An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal ... The pre-abelian categories most commonly studied are in fact abelian categories for example, Ab is an abelian category ...

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