CHAPTER 8 Toolbox

36

8. Representable Functors

Definition 8.8.1. Let F : C ⟶ Set be a covariant functor. A pair (A, x) with A ∈ C, x ∈ F(A) is called a representing (generic, universal) object for F and F is called a representable functor, if for each B ∈ C and y ∈ F(B) there exists a unique f ∈ Mor_{C}(A, B) such that F(f)(x) = y:

[figure: A; B; f; F(A); F(B); F(f); ∋ x; ∋ y]

Proposition 8.8.2. Let (A, x) and (B, y) be representing objects for F. Then there exists a unique isomorphism f : A ⟶ B such that F(f)(x) = y.

[figure: A; B; h; A; k; B; h; 1_{A}; 1_{B}; F(A); F(B); F(h); F(A); F(k); F(B); F(h); 1_{F(A)}; 1_{F(B)}; x; y; x; y]

Examples 8.8.3. 1. Let X ∈ Set and let R be a ring. F : R-Mod ⟶ Set, F(M) := Map(X, M) is a covariant functor. A representing object for F is given by (RX, x : X ⟶ RX) with the property, that for all (M, y : X ⟶ M) there exists a unique f ∈ Hom_{R}(RX, M) such that F(f)(x) = Map(X, f)(x) = fx = y

[figure: X; RX; x; y; M.; f]

2. Given modules M_{R} and _{R}N. Define F : Ab ⟶ Set by F(A) := Bil_{R}(M, N; A). Then F is a covariant functor. A representing object for F is given by (M ⊗_{R} N, ⊗ : M × N ⟶ M ⊗_{R} N) with the property that for all (A, f : M × N ⟶ A) there exists a unique g ∈ Hom(M ⊗_{R} N, A) such that F(g)(⊗) = Bil_{R}(M, N; g)(⊗) = g⊗ = f

[figure: M × N; M ⊗_{R} N; ⊗; f; A.; g]

3. Given a 𝕂-module V. Define F : Alg ⟶ Set by F(A) := Hom(V, A). Then F is a covariant functor. A representing object for F is given by (T (V), ι : V ⟶ T (V)) with the property that for all (A, f : V ⟶ A) the exists a unique g ∈ Mor_{Alg}(T (V), A) such that F(g)(ι) = Hom(V, g)(ι) = gι = f

[figure: V; T (V); ι; f; A.; g]

4. Given a 𝕂-module V. Define F : cAlg ⟶ Set by F(A) := Hom(V, A). Then F is a covariant functor. A representing object for F is given by (S(V), ι : V ⟶ S(V)) with the property that for all (A, f : V ⟶ A) the exists a unique g ∈ Mor_{Alg}(S(V), A) such that F(g)(ι) = Hom(V, g)(ι) = gι = f

[figure: V; S(V); ι; f; A.; g]

Proposition 8.8.4. F has a representing object (A, a) if and only if there is a natural isomorphism φ : F ≅ Mor_{C}(A, −) (with a = φ(A)^{−1}(1_{A})).

Proof. ⟹ : The map

φ(B) : F(B) ∋ y ↦ f ∈ Mor_{C}(A, B) with F(f)(a) = y

is bijective with the inverse map

ψ(B) : Mor_{C}(A, B) ∋ f ↦ F(f)(a) ∈ F(B).

In fact we have y ↦ f ↦ F(f)(a) = y and f ↦ y := F(f)(a) ↦ g : F(g)(a) = y = F(f)(a). By uniqueness we get f = g. Hence all φ(B) are bijective with inverse map ψ(B). It is sufficient to show that ψ is a natural transformation.

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

[figure: Mor_{C}(A, C); F(C); ψ(C); Mor_{C}(A, B); F(B); ψ(B); Mor_{C}(A,g); F(g)]

since ψ(C)Mor_{C}(A, g)(f) = ψ(C)(gf) = F(gf)(a) = F(g)F(f)(a) = F(g)ψ(B)(f).

⇐: Let A be given. Let a := φ(A)^{−1}(1_{A}). For y ∈ F(B) we get y = φ(B)^{−1}(f) = φ(B)^{−1}(f1_{A}) = φ(B)^{−1}Mor_{C}(A, f)(1_{A}) = F(f)φ(A)^{−1}(1_{A}) = F(f)(a) for a uniquely determined f ∈ Mor_{C}(A, B).

Proposition 8.8.5. Given a representable functor F_{X} : C ⟶ Set for each X ∈ D. Given a natural transformation F_{g} : F_{Y} ⟶ F_{X} for each g : X ⟶ Y (contravariant!) such that F depends functorially on X, i.e. F_{1_{X}} = 1_{F_{X}}, F_{hg} = F_{g}F_{h}. Then the representing objects (A_{X}, a_{X}) for F_{X} depend functorially on X, i.e. for each g : X ⟶ Y there is a unique homomorphism A_{g} : A_{X} ⟶ A_{Y} (with F_{X}(A_{g})(a_{X}) = F_{g}(A_{Y})(a_{Y})) and the following identities hold A_{1_{X}} = 1_{A_{X}}, A_{hg} = A_{h}A_{g}.

Proof. Choose a representing object (A_{X}, a_{X}) for F_{X} for each X ∈ C (by the axiom of choice). Then there is a unique homomorphism A_{g} : A_{X} ⟶ A_{Y} with

F_{X}(A_{g})(a_{X}) = F_{g}(A_{Y})(a_{Y}) ∈ F_{X}(A_{Y}),

for each g : X ⟶ Y because F_{g}(A_{Y}) : F_{Y} (A_{Y}) ⟶ F_{X}(A_{Y}) is given. We have F_{X}(A_{1})(a_{X}) = F_{1}(A_{X})(a_{X}) = a_{X} = F_{X}(1)(a_{X}) hence A_{1} = 1, and F_{X}(A_{hg})(a_{X}) = F_{hg}(A_{Z})(a_{Z}) = F_{g}(A_{Z})F_{h}(A_{Z})(a_{Z}) = F_{g}(A_{Z})F_{Y} (A_{h})(a_{Y}) = F_{X}(A_{h})F_{g}(A_{Y})(a_{Y}) = F_{X}(A_{h})F_{X}(A_{g})(a_{X}) = F_{X}(A_{h}A_{g})(a_{X}) hence A_{h}A_{g} = A_{hg} for g : X ⟶ Y and h : Y ⟶ Z in D.

Corollary 8.8.6. 1. Map(X, M) ≅ Hom_{R}(RX, M) is a natural transformation in M (and in X!). In particular Set ∋ X ↦ RX ∈ R-Mod is a functor.

2. Bil_{R}(M, N; A) ≅ Hom(M ⊗_{R} N, A) is a natural transformation in A (and in (M, N) ∈ Mod-R × R-Mod). In particular Mod-R × R-Mod ∋ M, N ↦ M ⊗_{r} N ∈ Ab is a functor.

3. R-Mod-S × S-Mod-T ∋ (M, N) ↦ M ⊗_{S} N ∈ R-Mod-T is a functor.