nLab
coequalizer

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

XgfY X \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=pg p \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

XgfYpZ X \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

XgfYpZ X \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)D A \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.

Last revised on May 21, 2017 at 17:20:28. See the history of this page for a list of all contributions to it.