Preadditive Category - Kernels and Cokernels

Kernels and Cokernels

Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.

When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.

There is a convenient relationship between the kernel and cokernel and the Abelian group structure on the hom-sets. Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers. The alternative term "difference kernel" for binary equalisers derives from this fact.

A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-Abelian. Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-Abelian categories may be found under that subject.