Coequalizer

A coequalizer is the colimit of a 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. == Examples ==

• In the category of sets, the coequalizer of two functions is the quotient of Y by the smallest equivalence relation ~ such that for every , we have . In particular, if R is an equivalence relation on a set Y, and r1, r2 are the natural projections then the coequalizer of r1 and r2 is the quotient set . (See also: quotient by an equivalence relation.) • The coequalizer in the category of groups is very similar. Here if are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set • : 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 S1 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. == Properties ==

• Every coequalizer is an epimorphism. • In a topos, every epimorphism is the coequalizer of its kernel pair. == Special cases ==

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. == See also ==