TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Pointwise Kan extensions exist under smallness and completeness hypotheses

Statement

Let $K : C \to D$ and $F : C \to E$ be functors.

If $C$ is small, $D$ locally small, and $E$ cocomplete, then for every $d \in D$ the comma category $(K ↓ d)$ is small and the pointwise left Kan extension value at $d$ exists.

If $C$ is small, $D$ locally small, and $E$ complete, then for every $d \in D$ the comma category $(d ↓ K)$ is small and the pointwise right Kan extension value at $d$ exists.

These are objectwise existence statements. A global functor ${Lan}_{K} F$ or ${Ran}_{K} F$ is obtained only when the corresponding colimits or limits are supplied, with chosen universal cones, for every $d$ .

Facts & Assumptions

Given: Functors $K : C \to D$ and $F : C \to E$ with $C$ small and $D$ locally small.

[F1]

A category is small when its objects and morphisms form sets, and locally small when every hom-collection is a set (Small, locally small, and large categories).

[F2]

A category is cocomplete when it has all small colimits and complete when it has all small limits (Finite, small, and large limits and colimits; complete and cocomplete categories).

[L1]

For each object $d$ , the comma-category colimit over $(K ↓ d)$ computes the pointwise left Kan extension value, and the comma-category limit over $(d ↓ K)$ computes the pointwise right Kan extension value (Comma-category limit and colimit formulae compute Kan extensions, Pointwise Kan extensions by the comma-category formula).

Proof

technique · direct

1.1F1

Because $C$ is small and $D$ is locally small, the objects of $(K ↓ d)$ form a set: they are pairs $(c, u)$ with $c \in C$ and $u \in D (K c, d)$ , and both pieces are set-sized by [F1]. Its morphisms are arrows of $C$ satisfying one extra equation, so they also form a set. The same argument applies to $(d ↓ K)$ . Thus both comma categories are small.

2.1F2L1step 1.1

If $E$ is cocomplete, then every small diagram in $E$ has a colimit by [F2], so the diagram from $(K ↓ d)$ into $E$ has a colimit for each $d$ ; by [L1] that colimit is the pointwise left Kan extension value at $d$ . Dually, if $E$ is complete, then every diagram from $(d ↓ K)$ into $E$ has a limit, and [L1] makes it the pointwise right Kan extension value at $d$ .

3.1L1step 2.1∎

The values obtained in step 2.1 exist objectwise. A global pointwise Kan extension functor requires, in addition, that these colimits or limits and their universal cones be supplied for every $d$ , because that is the data used in [L1] to assemble the arrow maps of ${Lan}_{K} F$ or ${Ran}_{K} F$ .

Depends on

Used by

The codensity monad of the small skeleton of finite sets is the ultrafilter monad Theorem

Dependency tree · two levels

12 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., Corollary 6.2.9 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, §§4.1-4.2 (standard reference, not scraped)