# Coequalizer

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 : XY.

More explicitly, a coequalizer can be defined as an object Q together with a morphism q : YQ such that qf = qg. 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 : QQ′ such that uq = 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.

## Examples

• In the category of sets, the coequalizer of two functions f, g : XY is the quotient of Y by the smallest equivalence relation $\sim$ such that for every $x\in X$ , we have $f(x)\sim g(x)$ . In particular, if R is an equivalence relation on a set Y, and r1, r2 are the natural projections (RY × 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 category of groups is very similar. Here if f, g : XY 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 Y / im(fg). (This is the cokernel of the morphism fg; 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(gf).

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 : XY in a category C is a coequalizer as defined above, but with the added property that given any functor F: CD, 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.