nLab coequalizer

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:

• If a morphism

  1. is the coequalizer of some pair of parallel morphisms (hence: is a regular epimorphism)

  2. has a kernel pair,

  then it is also the coequalizer of its kernel pair.

• If a kernel pair has a coequalizer, then it is the kernel pair of its coequalizer.

(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.

Related concepts

• equalizer

• regular epimorphism

• propositional truncation object

References

Coequalizers were defined in the paper

• Beno Eckmann, Peter J. Hilton, Group-like structures in general categories II. Equalizers, limits, lengths. Mathematische Annalen 151:2 (1963), 150–186. doi:10.1007/bf01344176.

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:

• Francis Borceux, Section 2.4 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

• Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 (webpage)