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Category theory MOC

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 . When (co)limits exist for all diagrams of a given shape in a category , we say has -(co)limits. If has -(co)limits for any small (resp. finite) , then is (co)complete (resp. finitely (co)complete).

Definition

The limit of a diagram is a cone from to such that given any other cone from to , there exists a unique morphism such that for all and cat

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

Dually, the colimit of is a cocone from to such that given any other cocone from to there exists a unique morphism such that for all and

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

Properties

Examples

tidy | en | sembr | en

Footnotes

  1. 2020, Topology: A categorical approach, §4.2, pp. 77–79 ↩

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