Coequalizer

In category theory, a coequalizer (or coequaliser) is a generalization of the quotient of a set by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

In category theory, a coequalizer (or coequaliser) is a generalization of the quotient of a set by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. 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}\mid x\in X\} For abelian groups the coequalizer is particularly simple. It is just the factor group . (This is the cokernel of the morphism ; see the next section). In the category of topological spaces, the circle object S 1 can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex. Coequalizers can be large: There are exactly two functors from the category 1 having one object and one identity arrow, to the category 2 with two objects and one non-identity arrow going between them. The coequalizer of these two functors is the monoid of natural numbers under addition, considered as a one-object category. In particular, this shows that while every coequalizing arrow is epic, it is not necessarily surjective. 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.

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