TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Adjunctions as absolute Kan extensions, with the preserved converse

Statement

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

If $F ⊣ G$ with unit $η : 1_{C} \Rightarrow G F$ and counit $ε : F G \Rightarrow 1_{D}$ , then:

$(G, η)$ is a left Kan extension of $1_{C}$ along $F$ , and it is absolute;
$(F, ε)$ is a right Kan extension of $1_{D}$ along $G$ , and it is absolute.

Conversely, if $(G, η)$ is a left Kan extension of $1_{C}$ along $F$ and is preserved by $F$ , then $F ⊣ G$ . Dually, if $(F, ε)$ is a right Kan extension of $1_{D}$ along $G$ and is preserved by $G$ , then $F ⊣ G$ .

The preservation clause in the converse is load-bearing.

Facts & Assumptions

Given: Functors $F : C \to D$ and $G : D \to C$ .

[F1]

An adjunction $F ⊣ G$ consists of a unit $η : 1_{C} \Rightarrow G F$ and a counit $ε : F G \Rightarrow 1_{D}$ satisfying $ε F \circ F η = 1_{F}$ and $G ε \circ η G = 1_{G}$ (Adjunction by unit, counit, and the triangle identities).

[F2]

A left Kan extension of $1_{C}$ along $F$ is initial among natural transformations $1_{C} \Rightarrow H F$ , and a right Kan extension of $1_{D}$ along $G$ is terminal among natural transformations $H G \Rightarrow 1_{D}$ (Left and right Kan extensions).

[F3]

An absolute Kan extension is one preserved by every functor out of its codomain (Absolute Kan extension, Covariant functor, identity functor, composite functor, and contravariant functor).

Proof

technique · direct

1.1

Assume $F ⊣ G$ with unit $η$ and counit $ε$ . Let $H : D \to C$ and $α : 1_{C} \Rightarrow H F$ be given. Define ${\overline{α}}_{d} : = H (ε_{d}) \circ α_{G d} : G d \to H d$ . Naturality of $α$ at $η_{c} : c \to G F c$ and the triangle identity $ε_{F c} \circ F (η_{c}) = 1_{F c}$ give $(\overline{α} F) \circ η = α$ . If $γ : G \Rightarrow H$ also satisfies $(γ F) \circ η = α$ , then for each $d$ naturality of $γ$ at $ε_{d} : F G d \to d$ and the triangle identity $G (ε_{d}) \circ η_{G d} = 1_{G d}$ force $γ_{d} = H (ε_{d}) \circ α_{G d} = {\overline{α}}_{d}$ . So $(G, η)$ is a left Kan extension of $1_{C}$ along $F$ . Dually, for $H : C \to D$ and $β : H G \Rightarrow 1_{D}$ , the formula ${\overline{β}}_{c} : = β_{F c} \circ H (η_{c}) : H c \to F c$ gives the unique factorization $ε \circ (\overline{β} G) = β$ , so $(F, ε)$ is a right Kan extension of $1_{D}$ along $G$ .

F1F2

2.1

The same formulas prove absoluteness. Let $S : C \to Z$ and $α : S \Rightarrow H F$ with $H : D \to Z$ . Then ${\overline{α}}_{d} : = H (ε_{d}) \circ α_{G d} : S G d \to H d$ gives the unique factorization $(\overline{α} F) \circ S η = α$ , so $(S G, S η)$ is a left Kan extension of $S$ along $F$ . Thus $(G, η)$ is absolute by [F3]. The right-handed argument is dual.

F1F3step 1.1

3.1

Conversely, suppose $(G, η)$ is a left Kan extension of $1_{C}$ along $F$ and is preserved by $F$ . Applying the preserved Kan-extension property to the identity transformation $1_{F} : F \Rightarrow F$ yields a unique natural transformation $ε : F G \Rightarrow 1_{D}$ with $ε F \circ F η = 1_{F}$ , the first triangle identity. To obtain the second, note that both $1_{G}$ and $G ε \circ η G$ are natural transformations $G \Rightarrow G$ whose composites with $η$ agree: componentwise, naturality of $η$ at $η_{c}$ and the first triangle identity give $G (ε_{F c}) \circ η_{G F c} \circ η_{c} = G (ε_{F c} \circ F η_{c}) \circ η_{c} = η_{c} .$ By the uniqueness clause in the left Kan universal property, $G ε \circ η G = 1_{G}$ . Hence $F ⊣ G$ by [F1]. The right-handed converse is dual.

F1F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)
B. Richter, From Categories to Homotopy Theory, Proposition 4.7.3 (standard reference, not scraped)