CHAPTER 8 Toolbox

39

9. Adjoint Functors and the Yoneda Lemma

Theorem 8.9.1. (Yoneda Lemma) Let C be a category. Given a covariant functor F : C ⟶ Set and an object A ∈ C. Then the map

π : Nat(Mor_{C}(A, -), F) ∋ ϕ ↦ ϕ(A)(1_{A}) ∈ F(A)

is bijective with the inverse map

π^{−1} : F(A) ∋ a ↦ h^{a} ∈ Nat(Mor_{C}(A, -), F),

where h^{a}(B)(f) = F(f)(a).

Proof. For ϕ ∈ Nat(Mor_{C}(A, -), F) we have a map ϕ(A) : Mor_{C}(A, A) ⟶ F(A), hence π with π(ϕ) := ϕ(A)(1_{A}) is a well defined map. For π^{−1} we have to check that h^{a} is a natural transformation. Given f : B ⟶ C in C. Then the diagram

[figure: F(B); F(C); F(f); Mor_{C}(A, B); Mor_{C}(A, C); Mor(A,f); h^{a}(B); h^{a}(C)]

is commutative. In fact if g ∈ Mor_{C}(A, B) then h^{a}(C)Mor_{C}(A, f)(g) = h^{a}(C)(fg) = F(fg)(a) = F(f)F(g)(a) = F(f)h^{a}(B)(a). Thus π^{−1} is well defined.

Let π^{−1}(a) = h^{a}. Then ππ^{−1}(a) = h^{a}(A)(1_{A}) = F(1_{A})(a) = a. Let π(ϕ) = ϕ(A)(1_{A}) = a. Then π^{−1}π(ϕ) = h^{a} and h^{a}(B)(f) = F(f)(a) = F(f)(ϕ(A)(1_{A})) = ϕ(B)Mor_{C}(A, f)(1_{A}) = ϕ(B)(f), also h^{a} = ϕ.

Corollary 8.9.2. Given A, B ∈ C. Then the following hold

1. Mor_{C}(A, B) ∋ f ↦ Mor_{C}(f, -) ∈ Nat(Mor_{C}(B, -), Mor_{C}(A, -)) is a bijective map.

2. With the bijective map from 1. the isomorphisms from Mor_{C}(A, B) correspond to natural isomorphisms from Nat(Mor_{C}(B, -), Mor_{C}(A, -)).

3. For contravariant functors F : C ⟶ Set we have Nat(Mor_{C}(-, A), F) ≅ F(A).

4. Mor_{C}(A, B) ∋ f ↦ Mor_{C}(-, f) ∈ Nat(Mor_{C}(-, A), Mor_{C}(-, B)) is a bijective map that defines a one-to-one correspondence between the isomorphisms from Mor_{C}(A, B) and the natural isomorphisms from Nat(Mor_{C}(-, A), Mor_{C}(-, B)).

Proof. 1. follows from the Yoneda Lemma with F = Mor_{C}(A, -).

2. Observe that h^{f}(C)(g) = Mor_{C}(A, g)(f) = gf = Mor_{C}(f, C)(g) hence h^{f} = Mor_{C}(f, -). Since we have Mor_{C}(f, -)Mor_{C}(g, -) = Mor_{C}(gf, -) and Mor_{C}(f, -) = id_{Mor_{C}(A,-)} if and only if f = 1_{A} we get the one-to-one correspondence between the isomorphisms from 1.

3. and 4. follow by dualizing.

Remark 8.9.3. The map π is a natural transformation in the arguments A and F. More precisely: if f : A ⟶ B and ϕ : F ⟶ G are given then the following diagrams commute

[figure: Nat(Mor_{C}(A, -), G); G(A); π; Nat(Mor_{C}(A, -), F); F(A); π; Nat(Mor(A,-),ϕ); ϕ(A)]

[figure: Nat(Mor_{C}(B, -), F); F(B).; π; Nat(Mor_{C}(A, -), F); F(A); π; Nat(Mor(f,-),F); F(f)]

This can be easily checked. Furthermore we have for ψ : Mor_{C}(A, -) ⟶ F

π Nat(Mor_{C}(A, -), ϕ)(ψ) = π(ϕψ) = (ϕψ)(A)(1_{A}) = ϕ(A)ψ(A)(1_{A}) = ϕ(A)π(ψ)

and π Nat(Mor_{C}(f, -), F)(ψ) = π(ψMor_{C}(f, -)) = (ψMor_{C}(f, -))(B)(1_{B}) = ψ(B)(f) = ψ(B)Mor_{C}(A, f)(1_{A}) = F(f)ψ(A)(1_{A}) = F(f)π(ψ).

Remark 8.9.4. By the previous corollary the representing object A is uniquely determined up to isomorphism by the isomorphism class of the functor Mor_{C}(A, -).

Problem 8.9.1. 1. Determine explicitly all natural endomorphisms from 𝔾_{a} to 𝔾_{a} (as defined in Lemma 2.3.5).

2. Determine all additive natural endomorphisms of 𝔾_{a}.

3. Determine all natural transformations from 𝔾_{a} to 𝔾_{m} (see Lemma 2.3.7).

4. Determine all natural automorphisms of 𝔾_{m}.

Proposition 8.9.5. Let G : C × D ⟶ Set be a covariant bifunctor such that the functor G(C, -) : D ⟶ Set is representable for all C ∈ C. Then there exists a contravariant functor F : C ⟶ D such that G ≅ Mor_{D}(F-, -) holds. Furthermore F is uniquely determined by G up to isomorphism.

Proof. For each C ∈ C choose an object F(C) ∈ D and an isomorphism ξ_{C} : G(C, -) ≅ Mor_{D}(F(C), -). Given f : C ⟶ C′ in C then let F(f) : F(C′) ⟶ F(C) be the uniquely determined morphism (by the Yoneda Lemma) in D such that the diagram

[figure: G(C′, -); Mor_{D}(F(C′), -); ξ_{C′}; G(C, -); Mor_{D}(F(C), -); ξ_{C}; G(f,-); Mor(F(f),-)]

commutes. Because of the uniqueness F(f) and because of the functoriality of G it is easy to see that F(fg) = F(g)F(f) and F(1_{C}) = 1_{F(C)} hold and that F is a contravariant functor.

If F′ : C ⟶ D is given with G ≅ Mor_{D}(F′-, -) then ϕ : Mor_{D}(F-, -) ≅ Mor_{D}(F′-, -). Hence by the Yoneda Lemma ψ(C) : F(C) ≅ F′(C) is an isomorphism for all C ∈ C. With these isomorphisms induced by ϕ the diagram

[figure: Mor_{D}(F′(C′), -); Mor_{D}(F(C′), -); Mor(ψ(C′),-); Mor_{D}(F′(C), -); Mor_{D}(F(C), -); Mor(ψ(C),-); Mor(F′(f),-); Mor(F(f),-)]

commutes. Hence the diagram

[figure: F(C); F′(C); ψ(C); F(C′); F′(C′); ψ(C′); F′(f); F(f)]

commutes. Thus ψ : F ⟶ F′ is a natural isomorphism.

Definition 8.9.6. Let C and D be categories and F : C ⟶ D and G : D ⟶ C be covariant functors. F is called leftadjoint to G and G rightadjoint to F if there is a natural isomorphism of bifunctors ϕ : Mor_{D}(F-, -) ⟶ Mor_{C}(-, G-) from C^{op} × D to Set.

Lemma 8.9.7. If F : C ⟶ D is leftadjoint to G : D ⟶ C then F is uniquely determined by G up to isomorphism. Similarly G is uniquely determined by F up to isomorphism.

Proof. Now we prove the first claim. Assume that also F′ is leftadjoint to G with ϕ′ : Mor_{D}(F′-, -) ⟶ Mor_{C}(-, G-). Then we have a natural isomorphism ϕ′^{−1}ϕ : Mor_{D}(F-, -) ⟶ Mor_{D}(F′-, -). By Proposition 8.9.5 we get F ≅ F′.

Lemma 8.9.8. A functor G : D ⟶ C has a leftadjoint functor iff all functors Mor_{C}(C, G-) are representable.

Proof. follows from 8.9.5.

Lemma 8.9.9. Let F : C ⟶ D and G : D ⟶ C be covariant functors. Then

Nat(Id_{C}, GF) ∋ Φ ↦ G-Φ- ∈ Nat(Mor_{D}(F-, -), Mor_{C}(-, G-))

is a bijective map with inverse map

Nat(Mor_{D}(F-, -), Mor_{C}(-, G-)) ∋ ϕ ↦ ϕ(-, F-)(1_{F-}) ∈ Nat(Id_{C}, GF).

Furthermore

Nat(FG, Id_{C}) ∋ Ψ ↦ Ψ-F- ∈ Nat(Mor_{C}(-, G-), Mor_{D}(F-, -))

is a bijective map with inverse map

Nat(Mor_{C}(-, G-), Mor_{D}(F-, -)) ∋ ψ ↦ ψ(G-, -)(1_{G-}) ∈ Nat(FG, Id_{C}).

Proof. The natural transformation G-Φ- is defined as follows. Given C ∈ C, D ∈ D and f ∈ Mor_{D}(F(C), D) then let (G-Φ-)(C, D)(f) := G(f)Φ(C) : C ⟶ GF(C) ⟶ G(D). It is easy to check the properties of a natural transformation.

Given Φ then one obtains by composition of the two maps G(1_{F(C)})Φ(C) = GF(1_{C})Φ(C) = Φ(C). Given ϕ one obtains

\begin{matrix}G(f)(ϕ(C, F(C))(1_{F(C)}) = Mor_{C}(C, G(f))ϕ(C, F(C))(1_{F(C)}) \\ = ϕ(C, D)Mor_{D}(F(C), f)(1_{F(C)}) = ϕ(C, D)(f). \end{matrix}

The second part of the lemma is proved similarly.

Proposition 8.9.10. Let

ϕ : Mor_{D}(F-, -) ⟶ Mor_{C}(-, G-) and ψ : Mor_{C}(-, G-) ⟶ Mor_{D}(F-, -)

be natural transformations with associated natural transformations (by Lemma 8.9.9) Φ : Id_{C} ⟶ GF resp. Ψ : FG ⟶ Id_{D}.

1) Then we have ϕψ = id_{Mor(-,G-)} if and only if (G ⟶^{ΦG} GFG ⟶^{GΨ} G) = id_{G}.

2) We also have ψϕ = id_{Mor(F-,-)} if and only if (F ⟶^{FΦ} FGF ⟶^{ΨF} F) = id_{F}.

Proof. We get

GΨ(D)ΦG(D) = GΨ(D)ϕ(G(D), FG(D))(1_{FG(D)})
= Mor_{C}(G(D), GΨ(D))ϕ(G(D), FG(D))(1_{FG(D)})
= ϕ(G(D), D)Mor_{D}(FG(D), Ψ(D))(1_{FG(D)})
= ϕ(G(D), D)(Ψ(D))
= ϕ(G(D), D)ψ(G(D), D)(1_{G(D)})
= ϕψ(G(D), D)(1_{G(D)}).

Similarly we get

\begin{matrix}ϕψ(C, D)(f) = ϕ(C, D)ψ(C, D)(f) = G(Ψ(D)F(f))Φ(C) \\ = GΨ(D)GF(f)Φ(C) = GΨ(D)ΦG(D)f. \end{matrix}

Corollary 8.9.11. Let F : C ⟶ D and G : D ⟶ C be functors. F is leftadjoint to G if and only if there are natural transformations Φ : Id_{C} ⟶ GF and Ψ : FG ⟶ Id_{D} such that (GΨ)(ΦG) = id_{G} and (ΨF)(FΦ) = id_{F}.

Definition 8.9.12. The natural transformations Φ : Id_{C} ⟶ GF and Ψ : FG ⟶ Id_{D} given in 8.9.11 are called unit and counit resp. for the adjoint functors F and G.

Problem 8.9.2. 1. Let _{R}M_{S} be a bimodule. Show that the functor M ⊗_{S} - : SM ⟶ _{R}M is leftadjoint to Hom_{R}(M, -) : _{R}M ⟶ _{S}M. Determine the associated unit and counit.

b) Show that there is a natural isomorphism Map(A×B, C) ≅ Map(B, Map(A, C)). Determine the associated unit and counit.

c) Show that there is a natural isomorphism 𝕂-Alg(𝕂G, A) ≅ Gr(G, U(A)). Determine the associated unit and counit.

d) Show that there is a natural isomorphism 𝕂-Alg(U(𝔤), A) ≅ Lie-Alg(𝔤, A^{L}). Determine the corresponding leftadjoint functor and the associated unit and counit.

Definition 8.9.13. Let G : D ⟶ C be a covariant functor. G generates a (co- )universal problem a follows:

Given C ∈ C. Find an object F(C) ∈ D and a morphism ι : C ⟶ G(F(C)) in C such that there is a unique morphism g : F(C) ⟶ D in D for each object D ∈ D and for each morphism f : C ⟶ G(D) in C such that the diagram

[figure: C; G(F(C)); ι; f; G(D); G(g)]

commutes.

A pair (F(C), ι) that satisfies the above conditions is called a universal solution of the (co-)universal problem defined by G and C.

Let F : C ⟶ D be a covariant functor. F generates a universal problem a follows: Given D ∈ D. Find an object G(D) ∈ C and a morphism ν : F(G(D)) ⟶ D in D

such that there is a unique morphism g : C ⟶ G(D) in C for each object C ∈ C and for each morphism f : F(C) ⟶ D in D such that the diagram

[figure: FG(D); D; ν; f; F(C); F(g)]

commutes.

A pair (G(D), ν) that satisfies the above conditions is called a universal solution of the (co-)universal problem defined by F and D.

Proposition 8.9.14. Let F : C ⟶ D be leftadjoint to G : D ⟶ C. Then F(C) and the unit ι = Φ(C) : C ⟶ GF(C) form a (co-)universal solution for the (co- )universal problem defined by G and C.

Furthermore G(D) and the counit ν = Ψ(D) : FG(D) ⟶ D form a universal solution for the universal problem defined by F and D.

Proof. By Theorem 8.9.10 the morphisms ϕ : Mor_{D}(F-, -) ⟶ Mor_{C}(-, G-) and ψ : Mor_{C}(-, G-) ⟶ Mor_{D}(F-, -) are inverses of each other. They are defined with unit and counit as ϕ(C, D)(g) = G(g)Φ(C) resp. ψ(C, D)(f) = Ψ(D)F(f). Hence for each f : C ⟶ G(D) there is a unique g : F(C) ⟶ D such that f = ϕ(C, D)(g) = G(g)Φ(C) = G(g)ι.

The second statement follows analogously.

Remark 8.9.15. If G : D ⟶ C and C ∈ C are given then the (co-)universal solution (F(C), ι : C ⟶ G(D)) can be considered as the best (co-)approximation of the object C in C by an object D in D with the help of a functor G. The object D ∈ D turns out to be F(C).

If F : C ⟶ D and D ∈ D are given then the universal solution (G(D), ν : FG(D) ⟶ D) can be considered as the best approximation of the object D in D by an object C in C with the help of a functor F. The object C ∈ C turns out to be G(D).

Proposition 8.9.16. Given G : D ⟶ C. Assume that for each C ∈ C the universal problem defined by G and C is solvable. Then there is a leftadjoint functor F : C ⟶ D to G.

Given F : C ⟶ D. Assume that for each D ∈ D the universal problem defined by F and D is solvable. Then there is a leftadjoint functor G : D ⟶ C to F.

Proof. Assume that the (co-)universal problem defined by G and C is solved by ι : C ⟶ F(C). Then the map Mor_{C}(C, G(D)) ∋ f ↦ g ∈ Mor_{D}(F(C), D) with G(g)ι = f is bijective. The inverse map is given by g ↦ G(g)ι. This is a natural transformation since the diagram

[figure: Mor_{D}(F(C), D′); Mor_{C}(C, G(D′)); G(-)ι; Mor_{D}(F(C), D); Mor_{C}(C, G(D)); G(-)ι; Mor_{D}(F(C),h); Mor_{C}(C,G(h))]

commutes for each h ∈ Mor_{D}(D, D′). In fact we have

Mor_{C}(C, G(h))(G(g)ι) = G(h)G(g)ι = G(hg)ι = G(Mor_{C}(F(C), h)(g))ι.

Hence for all C ∈ C the functor Mor_{C}(C, G(-)) : D ⟶ Set induced by the bifunctor Mor_{C}(-, G(-)) : C^{op} × D ⟶ Set is representable. By Theorem 8.9.5 there is a functor F : C ⟶ D such that Mor_{C}(-, G(-)) ≅ Mor_{D}(F(-), -).

The second statement follows analogously.

Remark 8.9.17. One can characterize the properties that G : D ⟶ C (resp. F : C ⟶ D) must have in order to possess a left-(right-)adjoint functor. One of the essential properties for this is that G preserves limits (hence direct products and difference kernels).