# Cokernel

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of a morphism in a category

The concept dual to the concept of the kernel of a morphism in a category. In categories of vector spaces, groups, rings, etc. it describes a largest quotient object of an object that annihilates the image of a homomorphism .

Let be a category with null morphisms. A morphism is called a cokernel of a morphism if and if any morphism such that can be expressed in unique way as . A cokernel of a morphism is denoted by .

If and then for a unique isomorphism .

Conversely, if and is an isomorphism, then is a cokernel of . Thus, all cokernels of a morphism form a quotient object of , which is denoted by . If , then is a normal epimorphism. The converse need not be true. The cokernel of the zero morphism is . The cokernel of the unit morphism exists if and only if contains a zero object.

In a category with a zero object, a morphism has a cokernel if and only if contains a co-Cartesian square with respect to the morphisms and . This condition is satisfied, in particular, for any morphism of a right locally small category with a zero object and products.

The co-Cartesian square, or fibred sum or pushout, of two morphisms , is (if it exists) a commutative diagram such that for any two morphisms , such that there exists a unique morphism for which , .