# nLab coequalizer

### Context

#### Limits and colimits

limits and colimits

# Contents

## Definition

In a category $C$ a diagram of morphisms of $C$

$U \underoverset{b}{a}{\rightrightarrows} V \overset{c}{\rightarrow} X$

is called a coequalizer diagram if

1. $c a=c b$; and
2. $c$ is universal for this property: i.e. if $f: V \to Y$ is a morphism of $C$ such that $f a=f b$, then there is a unique morphism $f': X \to Y$ such that $f'c=f$.

This concept is a special case of that of colimit; specifically, it’s the colimit of the diagram

$U \underoverset{b}{a}{\rightrightarrows} V .$

A coequalizer in $C$ is an equalizer in the opposite category $C^{op}$.

Revised on August 22, 2015 13:26:37 by Rod Mc Guire (108.32.28.201)