Questions: Kan Extensions

Question 1

The universal property of the left Kan extension Lan_K F states that Nat(Lan_K F, G) ≅ Nat(F, G ∘ K) for all G: D → E. A colleague interprets this as meaning (Lan_K F)(K(c)) = F(c) for all c ∈ C. Is this correct?

• A. Yes — the universal property ensures the extension agrees pointwise with F on all objects in the image of K
• B. No — there is a canonical natural transformation η: F ⇒ (Lan_K F) ∘ K (the unit), but the values may differ because (Lan_K F)(K(c)) is computed as a colimit over the comma category (K ↓ K(c)), which may aggregate multiple F-values
• C. Yes — Kan extensions are defined to be exact extensions that preserve all values on the image of K by construction
• D. No — Lan_K F is undefined on objects in the image of K; it is only defined on objects of D outside that image

Answer: (B)

The unit of the Kan extension is a natural transformation η: F ⇒ (Lan_K F) ∘ K, which gives a canonical map from F(c) into (Lan_K F)(K(c)), but they need not be equal. The value (Lan_K F)(K(c)) is computed as the colimit over the comma category (K ↓ K(c)) — all pairs (c', K(c') → K(c)). This category includes the identity K(c) → K(c) but may include other objects too, so the colimit may be 'larger than' F(c). The universal property governs the bijection of natural transformations, not pointwise equality. Only when K is fully faithful does (Lan_K F)(K(c)) recover F(c) exactly.

Question 2

Mac Lane wrote that 'all concepts are Kan extensions.' Which statement correctly describes how left adjoint functors arise as a special case?

• A. Every left adjoint can be factored into a sequential composition of two Kan extensions along intermediate functors
• B. The left adjoint of G: D → C, when it exists, equals the left Kan extension Lan_G Id_C — the extension of the identity functor on C along G
• C. Adjoint functors satisfy a similar universal property to Kan extensions but are defined differently; the connection is a metaphor, not a formal identification
• D. Mac Lane meant that Kan extensions are always adjoint pairs, not that adjoints are themselves Kan extensions

Answer: (B)

If F ⊣ G with G: D → C, the adjoint bijection gives Hom_D(F(c), d) ≅ Hom_C(c, G(d)) for all c ∈ C, d ∈ D. Now consider Lan_G Id_C: C → D, the left Kan extension of the identity Id_C along G. Its universal property is Nat(Lan_G Id_C, H) ≅ Nat(Id_C, H ∘ G) for all H: C → D. Specializing to representable H recovers exactly the adjoint bijection. So F = Lan_G Id_C: the left adjoint is precisely the left Kan extension of the identity along G. This is one concrete sense in which adjunctions are Kan extensions — and why Mac Lane's dictum carries formal weight.

Question 3

When the target category E is cocomplete, the left Kan extension of F: C → E along K: C → D is computed pointwise as (Lan_K F)(d) = colim_{(K ↓ d)} F, where (K ↓ d) is the comma category of objects of C equipped with morphisms into d via K.

• T. True
• F. False

Answer: True

This is the pointwise computation formula for left Kan extensions when E has sufficient colimits. For each d ∈ D, the comma category (K ↓ d) has as objects all pairs (c ∈ C, f: K(c) → d), and the colimit of the functor F (composed with the projection (K ↓ d) → C) gives the value of the Kan extension at d. Intuitively: to extend F to d, you collect all values F(c) for objects c whose image K(c) maps into d, and glue them together in the most general way — a colimit. This pointwise formula exists when E is cocomplete and defines a Kan extension that is also preserved by representable functors.

Question 4

A left Kan extension Lan_K F generally exists for any pair of functors K: C → D and F: C → E, because the universal property uniquely characterizes what the values should be at most object of D.

• T. True
• F. False

Answer: False

The universal property defines what a Kan extension IS if it exists, but does not guarantee existence. Existence of Lan_K F requires sufficient cocompleteness of E (for the colimit formula to work) or other conditions on the functors involved. For small C and D with E = Set (which is cocomplete), left Kan extensions always exist. But for general E lacking necessary colimits, the extension may fail to exist. This is analogous to adjoint functor theorems: the universal property of an adjoint characterizes it precisely, but existence requires additional conditions (solution set condition, completeness). Universal properties characterize; they do not guarantee existence.

Question 5

Explain why Mac Lane's claim that 'all concepts are Kan extensions' is meaningful rather than merely metaphorical. Provide one concrete example showing how a standard categorical construction arises as a Kan extension.

Model answer: The claim is meaningful because Kan extensions subsume the universal properties of the most fundamental constructions in category theory under a single schema. The clearest example is adjoint functors: given G: D → C, its left adjoint F (if it exists) equals the left Kan extension Lan_G Id_C of the identity functor on C along G. The Kan extension universal property Nat(Lan_G Id_C, H) ≅ Nat(Id_C, H ∘ G) specializes to the adjoint bijection Hom(F(c), d) ≅ Hom(c, G(d)) when H is representable. Another example: the colimit of F: J → C is the left Kan extension of F along the unique functor J → 1 (to the terminal category), since the colimit is the universal cocone — exactly what Lan_! F computes at the single object of 1.

The claim is also qualified in the Common Misconceptions: it is a statement about the universality of the construction, not a claim that every theorem in category theory is literally computed as a Kan extension. What it means is that the one concept of Kan extension unifies limits, colimits, adjunctions, the Yoneda embedding, Day convolution, and nerve-realization adjunctions — all fall out of the same abstract pattern. Understanding this unification is what it means to have a mature categorical perspective.