DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Absolute Kan extension

Definition

Let $(L, η)$ be a left Kan extension of $F : C \to E$ along $K : C \to D$ (Left and right Kan extensions). It is absolute when for every functor $H : E \to Z$ (Covariant functor, identity functor, composite functor, and contravariant functor), the pair $(H L, H η)$ is again a left Kan extension of $H F$ along $K$ .

Dually, a right Kan extension $(R, ε)$ of $F$ along $K$ is absolute when for every functor $H : E \to Z$ , the pair $(H R, H ε)$ is again a right Kan extension of $H F$ along $K$ .

Pointwise asks for preservation only by representables of the codomain; absolute asks for preservation by every functor out of the codomain and is therefore the stronger condition.

Depends on

Used by

Adjunctions as absolute Kan extensions, with the preserved converse Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Proposition 6.5.2 (standard reference, not scraped)