coequalizer in nLab

Context

Limits and colimits

• Idea
• Definition
• Properties
• Relation to kernel pairs
• Relation to pushouts
• Examples
• Related concepts
• References

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

Definition

In a category 𝒞\mathcal{C} a diagram

X⇉gfY→pZX \underoverset{g}{f}{\rightrightarrows} Y \overset{p}{\rightarrow} Z

is called a coequalizer diagram if

1. p∘f=p∘gp \circ f = p \circ g;

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

   X ⟶g⟶f 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 }

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.

Proposition

A diagram

X⇉fgY⟶pZ X \underoverset {f} {g} {\rightrightarrows} Y \overset{p}{\longrightarrow} Z

is a coequalizer diagram, def. , precisely if

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

is a pushout diagram.

Proposition

A ⟶f 1 B f 2↓ ↓ p 1 C ⟶p 2 D \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⇉q 2∘f 2q 1∘f 1B⊔C⟶(p 1,p 2)D 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.