nLab coequalizer




The concept of coequalizer in a general category is the generalization of the construction where for two functions f,gf,g between sets XX and YY

XgfYX \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y

one forms the set Y/ Y/_\sim of equivalence classes induced by the equivalence relation generated by the relation

f(x)g(x)f(x)\sim g(x)

for all xXx \in X. This means that the quotient function p:YY/ p \colon Y \longrightarrow Y/_\sim satisfies

pf=pgp \circ f = p \circ g

(a map pp satisfying this equation is said to “co-equalize” ff and gg) and moreover pp is universal with this property.

In this form this may be phrased generally in any category.



In some category 𝒞\mathcal{C}, the coequalizer coeq(f,g)coeq(f,g) of two parallel morphisms ff and gg between two objects XX and YY is (if it exists), the colimit under the diagram formed by these two morphisms

X gf Y p coeq(f,g).\array{ X && \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} && Y \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && coeq(f,g) }.



In a category 𝒞\mathcal{C} a diagram

XgfYpZX \underoverset{g}{f}{\rightrightarrows} Y \overset{p}{\rightarrow} Z

is called a coequalizer diagram if

  1. pf=pgp \circ f = p \circ g;

  2. pp is universal for this property: i.e. if q:YWq \colon Y \to W is a morphism of 𝒞\mathcal{C} such that qf=qgq \circ f = q \circ g, then there is a unique morphism q:ZWq' \colon Z \to W such that qp=qq' \circ p = q

    X gf Y p Z q !q W \array{ X &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}& Y &\overset{p}{\longrightarrow}& Z \\ && {}^{\mathllap{q}}\downarrow & \swarrow_{\mathrlap{\exists ! \, q'}} \\ && W }


By formal duality, a coequalizer in 𝒞\mathcal{C} is equivalently an equalizer in the opposite category 𝒞 op\mathcal{C}^{op}.


Morphisms that coequalize some pair of parallel morphisms are called regular epimorphisms.


Relation to kernel pairs


In any category:

(e.g. Borceux 1994, Prop. 2.5.7, 2.5.8, Taylor 1999, Lemma 5.6.6

Relation to pushouts

Coequalizers are closely related to pushouts:


A diagram

XfgYpZ X \underoverset {f} {g} {\rightrightarrows} Y \overset{p}{\longrightarrow} Z

is a coequalizer diagram, def. , precisely if

XX (f,g) Y p X Z\array{ X \sqcup X &\overset{(f,g)}{\longrightarrow}& Y \\ \big\downarrow && \big\downarrow{^\mathrlap{p}} \\ X &\underset{}{\longrightarrow}& Z }

is a pushout diagram.



A f 1 B f 2 p 1 C p 2 D \array{ A &\overset{f_1}{\longrightarrow}& B \\ {\mathllap{{}^{f_2}}}\Big\downarrow && \Big\downarrow{{}^\mathrlap{p_1}} \\ C &\underset{p_2}{\longrightarrow}& D }

is a pushout square, precisely if

Aq 2f 2q 1f 1BC(p 1,p 2)D A \underoverset { q_2 \circ f_2 } { q_1 \circ f_1 } {\rightrightarrows} B \sqcup C \overset{(p_1,p_2)}{\longrightarrow} D

is a coequalizer diagram.

(Here Bq 1BCq 2CB \overset{q_1}{\to} B \sqcup C \overset{q_2}{\leftarrow} C denotes the two coprojections into the coproduct.)



For 𝒞=\mathcal{C} = Set, the coequalizer of two functions ff, gg is the quotient set by the equivalence relation generated by the relation f(x)g(x)f(x) \sim g(x) for all xXx \in X.


For 𝒞=\mathcal{C} = Top, the coequalizer of two continuous functions ff, gg is the topological space whose underlying set is the quotient set from example , and whose topology is the corresponding quotient topology.


Coequalizers were defined in the paper

for any finite collection of parallel morphisms. The paper refers to them as right equalizers, whereas equalizers are referred to as left equalizers.

Textbook accounts:

Last revised on May 1, 2023 at 08:39:10. See the history of this page for a list of all contributions to it.