Abelian Categories

Kernels and cokernels

A null object in category is an object that is both initial and final. If has a null object, then to any , there exist unique arrows , and . A null object is unique upto isomorphism.

Let have a null object. The kernel of arrow is defined to be an equalizer of arrows . Put more directly, is the kernel of when , and every arrow such that factors uniquely through :

Thus, any category with equalizers, or more generally finite limits, and with zero has kernels for all arrows, and the arrow is unique upto isomorphism of . Like all equalizers, the kernel is necessarily a monic. It is convenient, therefore, to think of the kernel as a subobject of , or the equivalence class of monics .

For example, in with just one element (the identity element ) is a null object, and for any two groups with zero morphism, is the unique morphism which sends all of to the identity in . The kernel of an arbirary morphism is the insertion of the usual kernel all in with . Note that is the normal subgroup, and that in , all kernels are monic, but there are monics which are not kernels.

In the category of pointed sets, the one-point set is a null object, and zero map is the function taking all of to the base in . For any morphism of pointed sets, its kernel is the insertion of the subset of those with , where base point of is identical with base point . The same discussion gives kernels in . In , an epimorphism is determined by its kernel, but this is not the case for or .

In an -category , all equalizers are kernels. Indeed, in such a category, each hom-set is an abelian group. Given parallel arrows and arrow , such that iff . Therefore, the universal can either be described as the equalizer of , or kernel of . The dual notion gives rise to cokernel, as already discussed.