In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.
A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms .
More explicitly, a coequalizer of the parallel morphisms f and g can be defined as an object Q together with a morphism such that . Moreover, the pair must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism such that . This information can be captured by the following commutative diagram:
As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).
It can be shown that a coequalizing arrow q is an epimorphism in any category.
S=\{f(x)g(x)-1\midx\inX\}
In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism.
In preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups). In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference:
coeq(f, g) = coker(g – f).
A stronger notion is that of an absolute coequalizer, this is a coequalizer that is preserved under all functors.Formally, an absolute coequalizer of a pair of parallel arrows in a category C is a coequalizer as defined above, but with the added property that given any functor, F(Q) together with F(q) is the coequalizer of F(f) and F(g) in the category D. Split coequalizers are examples of absolute coequalizers.