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 projection function p:YY/ p \colon Y \longrightarrow Y/_\sim satisfies

pf=pg p \circ f = p \circ g

and in fact pp is universal with this property, hence it “co-equalizes” ff and gg.

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. 2, 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(y)f(x) \sim g(y) 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 1, and whose topology is the corresponding quotient topology.

Revised on April 21, 2016 05:30:24 by Urs Schreiber (131.220.184.222)