In mathematics, 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 (hence the name).

Definition

The coequalizer is a special kind of colimit in category theory. Specifically it is the colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y.

More explicity, the coequalizer can be defined as an object Q and a morphism q : Y → Q such that q O f = q O 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′ for which the following diagram commutes:

As with all universal constructions, the coequalizer, if it exists, is unique up to a unique isomorphism.

It can be shown that the coequalizer q is an epimorphism in any category.

Examples

• In the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the equivalence relation generated by the relations f(x) = g(x) for all x in X. In particular, if R is an equivalence relation on a set Y, and r_1,2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r₁ and r₂ is the quotient set Y/R.

• The coequalizer in the category of groups 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

S = {f(x)g(x)⁻¹ | for all x ∈ X} ⊂ Y

• 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).

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).

See also

• equalizer
• coproduct
• pushout – a special colimit construction making use a coequalizer and coproduct