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:

C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear);

C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts;

given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f).

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 → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.

Examples

The original example of an additive category is the category Ab of abelian groups.

Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.

Other common examples:

The category of (left) modules over a ring R, in particular:

the category of vector spaces over a field K.

The category of (Hausdorff) abelian topological groups.

The category of Banach spaces.

The category of Fréchet spaces.

The category of (Hausdorff) bornological spaces.

These will give you an idea of what to think of; for more examples, see abelian category (every abelian category is pre-abelian).