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: A → B 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 A → O → B, where O is a zero object, guaranteed to exist by item 2.