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 f, g : X → Y.
More explicitly, a coequalizer can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g. Moreover, the pair (Q, q) must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ such that u ∘ q = q′. 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 coequalizer q is an epimorphism in any category.
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:
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 f, g : X → Y in a category C is a coequalizer as defined above, but with the added property that given any functor F: C → D, 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.
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