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The comma-category and representable-preservation notions of pointwise Kan extension agree

Statement

Let $K : C \to D$ and $F : C \to E$ be functors, with $C$ small and $D, E$ locally small (Small, locally small, and large categories).

For a right Kan extension of $F$ along $K$ , the two definitions

pointwise by the comma-category limit formula, and
pointwise by preservation by all representables,

are equivalent (Pointwise Kan extensions by the comma-category formula, Pointwise Kan extensions as those preserved by representables).

By passage to opposite categories, the same equivalence holds for left Kan extensions.

Facts & Assumptions

Given: Functors $K : C \to D$ and $F : C \to E$ with $C$ small and $D, E$ locally small, and a right Kan extension $(R, ε)$ of $F$ along $K$ .

[F1]

A right Kan extension is pointwise by the comma-category formula when, for every $d$ , the canonical cone with vertex $R (d)$ and legs $ε_{c} \circ R (u)$ indexed by $(c, u : d \to K c)$ is a limit cone; it is pointwise by representable preservation when every covariant representable $E (e, -)$ carries it to a right Kan extension in $S e t$ (Pointwise Kan extensions by the comma-category formula, Pointwise Kan extensions as those preserved by representables).

[L1]

The comma-category formulas compute pointwise Kan extension values (Comma-category limit and colimit formulae compute Kan extensions).

[L2]

Every covariantly representable functor to $S e t$ preserves all existing small limits (Every covariantly representable functor to Set preserves all existing small limits).

[F2]

For fixed $e \in E$ , the functor $E (e, -) : E \to S e t$ is covariantly representable (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

[L3]

For a locally small category, evaluation at the identity gives a bijection $Nat (D (d, -), H) ≅ H (d)$ for every Set-valued functor $H$ (Evaluation at the identity gives $Nat (C (a, -), F) ≅ F (a)$ and proves that the natural-transformation collection is a set).

Proof

technique · direct

1.1F1F2L1L2

Suppose $(R, ε)$ is pointwise by the comma-category formula. Then for each $d$ the value $R (d)$ is the limit of the diagram on $(d ↓ K)$ by [F1]. Because $C$ is small and $D$ locally small, this comma category is small, so [L2] applies to every representable [F2]: $E (e, R (d))$ is the limit in $S e t$ of the Set-valued diagram obtained by applying $E (e, -)$ to that cone. By [L1], this says $E (e, -)$ carries $(R, ε)$ to a right Kan extension of $E (e, F -)$ along $K$ . So the comma-category notion implies the representable-preservation notion.

1.2F1F2algebra

Conversely, suppose every representable $E (e, -)$ carries $(R, ε)$ to a right Kan extension. Fix $d \in D$ and $e \in E$ . A cone from $e$ to the diagram on $(d ↓ K)$ is equivalently a natural transformation $D (d, K -) \Rightarrow E (e, F -) : C \to S e t,$ because its component at $c$ assigns to each $u : d \to K c$ the corresponding leg $e \to F (c)$ .

2.1L3F1step 1.2

The preserved right Kan universal property gives a bijection from the natural transformations in step 1.2 to $Nat (D (d, -), E (e, R -)) .$ By [L3], evaluation at $1_{d}$ identifies the latter set with $E (e, R (d))$ . Under these two bijections a morphism $h : e \to R (d)$ is sent to the canonical cone with legs $ε_{c} \circ R (u) \circ h$ , so the correspondence is natural in $e$ . Hence the canonical cone with vertex $R (d)$ represents the cone functor and is a limit cone.

3.1F1step 2.1∎

Since $d$ was arbitrary, $(R, ε)$ is pointwise by the comma-category formula. The left-handed equivalence is the same argument in opposite categories, exactly as encoded in the left definition of [F1].

Depends on

Used by

FALSE: every Kan extension is pointwise False statement

Dependency tree · two levels

23 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., Theorem 6.3.7 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, §4.3 (standard reference, not scraped)