Cokernel

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(Redirected from Cokernel (category theory))

In abstract algebra, the cokernel of a homomorphism f : XY is the quotient of Y by the image of f. In a topological setting, one typically takes the closure of the image before passing to the quotient. For instance, if f : H1H2 is a bounded linear operator between Hilbert spaces, then coker(f) is the quotient of H2 by the closure of the range of f.

In category theory

One can also define the cokernel in the general framework of category theory. In order for the definition to make sense the category in question must have zero morphisms. The cokernel of a morphism f : XY is defined as the coequalizer of f and the zero morphism 0XY : XY.

Explicitly, this means the following. The cokernel of f : XY is an object Q and a morphism q : YQ such that the diagram

Missing image
Cokernel-01.png
Image:Cokernel-01.png

commutes. Moreover the morphism q must be universal for this diagram, i.e. any other such q′: YQ′ can be obtained by composing q with a unique morphism u : Q to Q′:

Missing image
Cokernel-02.png
Image:Cokernel-02.png

As with all universal constructions the cokernel, if it exists, is unique up to a unique isomorphism.

Like all coequalizers, the cokernel q : YQ is necessarily an epimorphism. Conversely an epimorphism is called normal (or conormal) if it is the cokernel of some morphism. A category is called conormal if every epimorphism is normal (e.g. the category of groups is conormal).

This cokernel is dual to the kernels of category theory, hence the name.

Examples

In the category of groups, the cokernel of a group homomorphism f : GH is the quotient of H by the normal closure of the image of f. In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo the image of f:

coker(f) = H / im(f).

Special cases

In a preadditive category, it makes sense to add and subtract morphisms. In such a category, the coequalizer of two morphisms f and g (if it exists) is just the cokernel of their difference:

coeq(f, g) = coker(g - f)

In a pre-abelian category (a special kind of preadditive category) the existence of kernels and cokernels is guaranteed. In such categories the image and coimage of a morphism f are given by

im(f) = ker(coker f)
coim(f) = coker(ker f)

Abelian categories are even better behaved with respect to cokernels. In particular, every abelian category is conormal (and normal as well). That is, every epimorphism e : AB can be written as the cokernel of some morphism. Specifically, e is the cokernel of its own kernel:

e = coker(ker e)
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