# nLab coequalizer

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## Idea

The concept of coequalizer in a general category is the generalization of the construction where for two functions $f,g$ between sets $X$ and $Y$

$X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} Y$

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

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

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

$p \circ f = p \circ g$

(a map $p$ satisfying this equation is said to “co-equalize” $f$ and $g$) and moreover $p$ is universal with this property.

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

## Definition

###### Definition

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

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

Equivalently:

###### Definition

In a category $\mathcal{C}$ a diagram

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

is called a coequalizer diagram if

1. $p \circ f = p \circ g$;

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

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

###### Remark

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

###### Remark

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

## Properties

### Relation to kernel pairs

###### Proposition

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:

###### Proposition

A diagram

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

is a coequalizer diagram, def. , precisely if

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

is a pushout diagram.

Conversely:

###### Proposition

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

is a pushout square, precisely if

$A \underoverset { q_2 \circ f_2 } { q_1 \circ f_1 } {\rightrightarrows} B \sqcup C \overset{(p_1,p_2)}{\longrightarrow} D$

is a coequalizer diagram.

(Here $B \overset{q_1}{\to} B \sqcup C \overset{q_2}{\leftarrow} C$ denotes the two coprojections into the coproduct.)

## Examples

###### Example

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

###### Example

For $\mathcal{C} =$ Top, the coequalizer of two continuous functions $f$, $g$ 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: