nLab
coequalizer

Contents

Contents

Idea

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.

Definition

Definition

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) }.

Equivalently:

Definition

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 }
Remark

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

Properties

Relation to pushouts

Coequalizers are closely related to pushouts:

Proposition

A diagram

XgfYpZX \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} 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 \\ \downarrow && \downarrow^{\mathrlap{p}} \\ X &\underset{}{\longrightarrow}& Z }

is a pushout diagram.

Conversely:

Proposition

A diagram

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

is a pushout square, precisely if

Aι 2f 2ι 1f 1BC(p 1,p 2)DA \stackrel{\overset{\iota_1 \circ f_1}{\longrightarrow}}{\underset{\iota_2 \circ f_2}{\longrightarrow}} B \sqcup C \overset{(p_1,p_2)}{\longrightarrow} D

is a coequalizer diagram.

Examples

Example

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.

Example

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.

References

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.

Last revised on March 27, 2021 at 11:59:58. See the history of this page for a list of all contributions to it.