Colimits

Summary

A colimit is a limit in the opposite category — an initial cocone. Coproducts are dual to products, coequalizers dual to equalizers, pushouts dual to pullbacks. In Set, the colimit is a quotient of a coproduct, identifying elements according to the diagram’s maps. Epimorphisms are the dual of monomorphisms.

Overview

Every concept for limits has a dual for colimits: just reverse all arrows. Colimits are the “join-like” constructions in category theory, capturing quotients, unions, and free constructions where limits capture intersections, subsets, and universal maps into.

Main Content

Definition 5.2.1: Colimit

A cocone over a diagram $D:\mathcal{I}\to \mathcal{A}$ with vertex $A$ is a natural transformation $\lambda :D\Rightarrow \Delta A$.

Concretely: maps ${\lambda }_I:DI\to A$ for each $I$, such that for every $u:I\to J$: ${\lambda }_J\circ Du={\lambda }_I$.

A colimit of $D$ is an initial cocone: a cocone $\lambda :D\Rightarrow \Delta C$ such that for every cocone $\mu :D\Rightarrow \Delta A$, there is a unique $\bar{\mu }:C\to A$ with $\bar{\mu }\circ {\lambda }_I={\mu }_I$ for all $I$.

Write $C=\mathrm{colim}D$ or ${\mathrm{colim}}_{I\in \mathcal{I}}DI$.

Key Special Cases

Diagram Shape	Colimit Name
Empty	Initial object
Discrete (two objects)	Coproduct $A\bigsqcup B$
Parallel pair $\bullet \rightrightarrows \bullet$	Coequalizer
Span $\bullet \leftarrow \bullet \to \bullet$	Pushout

Coproducts

Definition: Coproduct

A coproduct of $A,B\in \mathcal{A}$ is an object $A\bigsqcup B$ with maps $i_1:A\to A\bigsqcup B$ and $i_2:B\to A\bigsqcup B$ (the coprojections or injections) such that for any $C$ and $f:A\to C$, $g:B\to C$, there is a unique $[f,g]:A\bigsqcup B\to C$ with $[f,g]\circ i_1=f$ and $[f,g]\circ i_2=g$.

Examples:

• In Set: $A\bigsqcup B$ = disjoint union.
• In Top: disjoint union of spaces.
• In Grp: coproduct = free product $A\ast B$.
• In Ab: $A\oplus B=A\times B$ (finite coproduct = product in abelian categories).
• In Set${}_{\ast }$ (pointed sets): wedge $A\vee B$.
• In a preorder: $a\bigsqcup b=\sup (a,b)$ (least upper bound).

Coequalizers

Definition: Coequalizer

A coequalizer of $s,t:X\rightrightarrows Y$ is an object $Q$ with a map $q:Y\to Q$ such that $q\circ s=q\circ t$, universal with this property.

In Set: $Q=Y/\sim$ where $\sim$ is the equivalence relation generated by $s(x)\sim t(x)$ for all $x\in X$.

Examples:

• In Grp: coequalizer of $s,t:G\rightrightarrows H$ is $H/⟨s{t}^{-1}(g):g\in G⟩$.
• In Top: coequalizer = quotient space.
• In Ring: quotient by the ideal generated by $\{s(x)-t(x)\}$.

Colimit Formula in Set

Example: Colimit Formula in Set (BCT, Ch. 5.2)

For $D:\mathcal{I}\to \mathbf{Set}$, the colimit is:

$$\mathrm{colim}D=\left(\underset{I\in \mathcal{I}}{⨆}DI\right)/\sim$$

where $\sim$ is the equivalence relation generated by: $x\sim (Du)(x)$ for all $u:I\to J$ in $\mathcal{I}$ and $x\in DI$.

I.e., the colimit is a quotient of a coproduct.

Epimorphisms

Definition 5.2.17: Epimorphism

A map $f:A\to B$ is an epimorphism (or epic) if for all $g,h:B\to C$, $g\circ f=h\circ f$ implies $g=h$.

In Set: epimorphisms are surjections. Coequalizers are always epimorphisms. In Ring: the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is an epimorphism in Ring but not a surjection.

Coproducts + coequalizers generate all colimits, dually to how products + equalizers generate all limits.

Cocompleteness

Definition: Cocomplete Category

A category is cocomplete if it has all small colimits.

Set, Top, Grp, Ab are all cocomplete. Any presheaf category $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ is both complete and cocomplete.

Connections

• Left adjoints preserve colimits (Adjoints and Limits): the dual of “right adjoints preserve limits.”
• Colimits in presheaf categories (Limits in Presheaf Categories): in $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$, colimits are also computed pointwise.
• The density theorem (Limits in Presheaf Categories): every presheaf is a colimit of representables.

See Also

• General Limits — Limits (the dual notion)
• Pullbacks — Pushouts are the colimit dual
• Functors and Limits — Preservation of colimits
• Adjoints and Limits — Left adjoints preserve colimits
• Limits in Presheaf Categories — Colimits pointwise in presheaf categories