Limit and colimit

Limits and colimits generalize many universal constructions in category theory. Since these are cones characterised by universal properties, they are sometimes called universal cones and universal cocones.

As defined below, one takes the (co)limit of a small diagram of a given shape $𝖩$ . When (co)limits exist for all diagrams of a given shape in a category $𝖢$ , we say $𝖢$ has $𝖩$ -(co)limits. If $𝖢$ has $𝖩$ -(co)limits for any small (resp. finite) $𝖩$ , then $𝖢$ is (co)complete (resp. finitely (co)complete).

Definition

The limit of a diagram $𝒟 : 𝖩 \to 𝖢$ is a cone $𝜂$ from $l i m ⟵ 𝒟$ to $𝒟$ such that given any other cone $𝛾$ from $𝐵$ to $𝒟$ , there exists a unique morphism $ℎ \in 𝖢 (𝐵, l i m ⟵ 𝒟)$ such that for all $𝑖, 𝑗 \in 𝖩$ and $𝛼 \in 𝖩 𝑖, 𝑗$ cat

commutes. Informally, the limit of $𝒟$ is the ‘shallowest’ cone over $𝒟$ .¹

Dually, the colimit of $𝒟 : 𝖩 \to 𝖢$ is a cocone $𝜖$ from $l i m ⟶ 𝒟$ to $𝒟$ such that given any other cocone $𝛾$ from $𝐵$ to $𝒟$ there exists a unique morphism $ℎ \in 𝖢 (l i m ⟶ 𝒟, 𝐵)$ such that for all $𝑖, 𝑗 \in 𝖩$ and $𝛼 \in 𝖩 𝑖, 𝑗$

commutes. Informally, the colimit of $𝒟$ is the shallowest cone under $𝒟$ .

Properties

Examples

tidy | en | SemBr | en

2020, Topology: A categorical approach, §4.2, pp. 77–79 ↩

Table of Contents

Limit and colimit

Definition

Properties

Examples

Graph View

Backlinks

Table of Contents

Limit and colimit

Definition

Properties

Examples

Related

Footnotes

Graph View

Backlinks