Coequalizer

In , a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary . It is the categorical construction 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 of the parallel morphisms f and g 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 the , the coequalizer of two functions f, g : X → Y 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 (R ⊂ Y × Y) → Y then the coequalizer of r1 and r2 is the quotient set Y/R. (See also: quotient by an equivalence relation.) The coequalizer in the is very similar. Here if f, g : X → Y are group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set For abelian groups the coequalizer is particularly simple. It is just the factor group Y / im(f – g). (This is the cokernel of the morphism f – g; see the next section). In the , the circle object 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.