CHAPTER 8 Toolbox

6

3. Natural Transformations

Definition 8.3.1. Let F : C ⟶ D and G : C ⟶ D be two functors. A natural transformation or a functorial morphism φ : F ⟶ G is a family of morphisms {φ(A) : F(A) ⟶ G(A)|A ∈ C} such that the diagram

[figure: F(B); G(B); φ(B); F(A); G(A); φ(A); F(f); G(f)]

commutes for all f : A ⟶ B in C, i.e. G(f)φ(A) = φ(B)F(f).

Lemma 8.3.2. Given covariant functors F = Id_{Set} : Set ⟶ Set and G = Mor_{Set}(Mor_{Set}(−, A), A) : Set ⟶ Set for a set A. Then φ : F ⟶ G with

φ(B) : B ∋ b ↦ (Mor_{Set}(B, A) ∋ f ↦ f(b) ∈ A) ∈ G(B)

is a natural transformation.

Proof. Given g : B ⟶ C. Then the following diagram commutes

[figure: B; Mor_{Set}(Mor_{Set}(B, A), A); φ(B); C; Mor_{Set}(Mor_{Set}(C, A), A); φ(C); g; Mor_{Set}(Mor_{Set}(g, A), A)]

since

φ(C)F(g)(b)(f) = φ(C)g(b)(f) = fg(b) = φ(B)(b)(fg)
= [φ(B)(b)Mor_{Set}(g, A)](f) = [Mor_{Set}(Mor_{Set}(g, A), A)φ(A)(b)](f).

Lemma 8.3.3. Let f : A ⟶ B be a morphism in C. Then Mor_{C}(f, -) : Mor_{C}(B, -) ⟶ Mor_{C}(A, -) given by Mor_{C}(f, C) : Mor_{C}(B, C) ∋ g ↦ gf ∈ Mor_{C}(A, C) is a natural transformation of covariant functors.

Let f : A ⟶ B be a morphism in C. Then Mor_{C}(-, f) : Mor_{C}(-, A) ⟶ Mor_{C}(-, B) given by Mor_{C}(C, f) : Mor_{C}(C, A) ∋ g ↦ fg ∈ Mor_{C}(C, B) is a natural transformation of contravariant functors.

Proof. Let h : C ⟶ C′ be a morphism in C. Then the diagrams

[figure: Mor_{C}(B, C′); Mor_{C}(A, C′); Mor_{C}(f,C′); Mor_{C}(B, C); Mor_{C}(A, C); Mor_{C}(f,C); Mor_{C}(B,h); Mor_{C}(A,h)]

and

[figure: Mor_{C}(C, A); Mor_{C}(C, B); Mor_{C}(C,f); Mor_{C}(C′, A); Mor_{C}(C′, B); Mor_{C}(C′,f); Mor_{C}(h,A); Mor_{C}(h,B)]

commute.

Remark 8.3.4. The composition of two natural transformations is again a natural transformation. The identity id_{F}(A) := 1_{F(A)} is also a natural transformation.

Definition 8.3.5. A natural transformation φ : F ⟶ G is called a natural isomorphism if there exists a natural transformation ψ : G ⟶ F such that φ ∘ ψ = id_{G} and ψ ∘ φ = id_{F}. The natural transformation ψ is uniquely determined by φ. We write φ^{−1} := ψ.

A functor F is said to be isomorphic to a functor G if there exists a natural isomorphism φ : F ⟶ G.

Problem 8.3.1. 1. Let F, G : C ⟶ D be functors. Show that a natural transformation φ : F ⟶ G is a natural isomorphism if and only if φ(A) is an isomorphism for all objects A ∈ C.

2. Let (A × B, p_{A}, p_{B}) be the product of A and B in C. Then there is a natural isomorphism

Mor(-, A × B) ≅ Mor_{C}(-, A) × Mor_{C}(-, B).

3. Let C be a category with finite products. For each object A in C show that there exists a morphism Δ_{A} : A ⟶ A × A satisfying p_{1}Δ_{A} = 1_{A} = p_{2}Δ_{A}. Show that this defines a natural transformation. What are the functors?

4. Let C be a category with finite products. Show that there is a bifunctor - × - : C × C ⟶ C such that (- × -)(A, B) is the object of a product of A and B. We denote elements in the image of this functor by A × B := (- × -)(A, B) and similarly f × g.

5. With the notation of the preceding problem show that there is a natural transformation α(A, B, C) : (A × B) × C ≅ A × (B × C). Show that the diagram (coherence or constraints)

[figure: ((A × B) × C) × D; (A × (B × C)) × D; α(A,B,C)×1; A × ((B × C) × D); α(A,B×C,D); α(A×B,C,D); 1×α(B,C,D); (A × B) × (C × D); A × (B × (C × D)); α(A,B,C×D)]

commutes.

6. With the notation of the preceding problem show that there are a natural transformations λ(A) : E × A ⟶ A and ρ(A) : A × E ⟶ A such that the diagram (coherence or constraints)

[figure: (A × E) × B; A × (E × B); α(A,E,B); A × B; ρ(A)×1; 1×λ(B)]

Definition 8.3.6. Let C and D be categories. A covariant functor F : C ⟶ D is called an equivalence of categories if there exists a covariant functor G : D ⟶ C and natural isomorphisms φ : GF ≅ Id_{C} and ψ : FG ≅ Id_{D}.

A contravariant functor F : C ⟶ D is called a duality of categories if there exists a contravariant functor G : D ⟶ C and natural isomorphisms φ : GF ≅ Id_{C} and ψ : FG ≅ Id_{D}.

A category C is said to be equivalent to a category D if there exists an equivalence F : C ⟶ D. A category C is said to be dual to a category D if there exists a duality F : C ⟶ D.

Problem 8.3.2. 1. Show that the dual category C^{op} is dual to the category C.

2. Let D be a category dual to the category C. Show that D is equivalent to the dual category C^{op}.

3. Let F : C ⟶ D be an equivalence with respect to G : D ⟶ C, φ : GF ≅ Id_{C}, and ψ : FG ≅ Id_{D}. Show that G : D ⟶ C is an equivalence. Show that G is uniquely determined by F up to a natural isomorphism.