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.

## Definition

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 coequalizing arrow *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 smallest equivalence relation ~ such that for every*x*∈*X*, we have*f*(*x*) ~*g*(*x*).^{[1]}In particular, if*R*is an equivalence relation on a set*Y*, and*r*_{1},*r*_{2}are the natural projections (*R*⊂*Y*×*Y*) →*Y*then the coequalizer of*r*_{1}and*r*_{2}is the quotient set*Y*/*R*. (See also: quotient by an equivalence relation.) - 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 - 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 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.

## 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 *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.

## See also

## Notes

## References

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.- Coequalizers – page 65
- Absolute coequalizers – page 149

## External links

- Interactive Web page, which generates examples of coequalizers in the category of finite sets. Written by Jocelyn Paine.