CHAPTER 8 Toolbox

1. Categories

Definition 8.1.1. Let C consist of

1. a class Ob C whose elements A, B, C, … ∈ Ob C are called objects,

2. a family {Mor_{C}(A, B)|A, B ∈ Ob C} of mutually disjoint sets whose elements f, g, … ∈ Mor_{C}(A, B) are called morphisms, and

3. a family {Mor_{C}(A, B) × Mor_{C}(B, C) ∋ (f, g) ↦ gf ∈ Mor_{C}(A, C)|A, B, C ∈

Ob C} of maps called compositions. C is called a category if the following axioms hold for C

1. Associative Law:

∀A, B, C, D ∈ Ob C, f ∈ Mor_{C}(A, B), g ∈ Mor_{C}(B, C), h ∈ Mor_{C}(C, D) :

h(gf) = (hg)f;

2. Identity Law:

∀A ∈ Ob C ∃1_{A} ∈ Mor_{C}(A, A) ∀B, C ∈ Ob C, ∀f ∈ Mor_{C}(A, B), ∀g ∈ Mor_{C}(C, A) :

1_{A}g = g  and  f1_{A} = f.

Examples 8.1.2. 1. The category of sets Set.

2. The categories of R-modules R-Mod, k-vector spaces k-Vec or k-Mod, groups Gr, abelian groups Ab, monoids Mon, commutative monoids cMon, rings Ri, fields Fld, topological spaces Top.

Since modules are highly important for all what follows, we recall the definition and some basic properties.

Definition and Remark 8.1.3. Let R be a ring (always associative with unit). A left R-module _{R}M is an (additively written) abelian group M together with an operation R × M ∋ (r, m) ↦ rm ∈ M such that

1. (rs)m = r(sm),

2. (r + s)m = rm + sm,

3. r(m + m′) = rm + rm′,

4. 1m = m for all r, s ∈ R, m, m′ ∈ M.

Each abelian group is a ℤ-module in a unique way.

A homomorphism of left R-modules f : _{R}M ⟶ _{R}N is a group homomorphism such that f(rm) = rf(m).

Analogously we define right R-modules M_{R} and their homomorphisms.

We denote by Hom_{R}(.M, .N) the set of homomorphisms of left R-modules_{R}M and RN. Similarly Hom_{R}(M., N.) denotes the set of homomorphisms of right R-modules M_{R} and N_{R}. Both sets are abelian groups by (f + g)(m) := f(m) + g(m).

For arbitrary categories we adopt many of the customary notations.

Notation 8.1.4. f ∈ Mor_{C}(A, B) will be written as f : A ⟶ B or A ⟶^{f} B. A is called the domain, B the range of f.

The composition of two morphisms f : A ⟶ B and g : B ⟶ C is written as gf : A ⟶ C or as g ∘ f : A ⟶ C.

Definition and Remark 8.1.5. A morphism f : A ⟶ B is called an isomorphism if there exists a morphism g : B ⟶ A in C such that fg = 1_{B} and gf = 1_{A}. The morphism g is uniquely determined by f since g′ = g′fg = g. We write f^{−1} := g.

An object A is said to be isomorphic to an object B if there exists an isomorphism f : A ⟶ B. If f is an isomorphism the so is f^{−1}. If f : A ⟶ B and g : B ⟶ C are isomorphisms in C then so is gf : A ⟶ C. We have: (f^{−1})^{−1} = f and (gf)^{−1} = f^{−1}g^{−1}. The relation of being isomorphic between objects is an equivalence relation.

Example 8.1.6. In the categories Set, R-Mod, k-Vec, Gr, Ab, Mon, cMon, Ri, Fld the isomorphisms are exactly those morphisms which are bijective as set maps.

In Top the set M = {a, b} with 𝔗_{1} = {∅, {a}, {b}, {a, b}} and with 𝔗_{2} = {∅, M} defines two different topological spaces. The map f = id : (M, 𝔗_{1}) ⟶ (M, 𝔗_{2}) is bijective and continuous. The inverse map, however, is not continuous, hence f is no isomorphism (homeomorphism).

Many well known concepts can be defined for arbitrary categories. We are going to apply some of them. Here are two examples.

Definition 8.1.7. 1. A morphism f : A ⟶ B is called a monomorphism if ∀C ∈ Ob C, ∀g, h ∈ Mor_{C}(C, A) :

fg = fh ⟹ g = h (f is left cancellable).

2. A morphism f : A ⟶ B is called an epimorphism if ∀C ∈ Ob C, ∀g, h ∈ Mor_{C}(B, C) :

gf = hf ⟹ g = h (f is right cancellable).

Definition 8.1.8. Given A, B ∈ C. An object A × B in C together with morphisms p_{A} : A × B ⟶ A and p_{B} : A × B ⟶ B is called a (categorical) product of A and B if for every object T ∈ C and every pair of morphisms f : T ⟶ A and g : T ⟶ B there exists a unique morphism (f, g) : T ⟶ A × B such that the diagram

[figure: T; f; g; A; A × B; p_{A}; B; p_{B}; (f,g)]

commutes.

An object E ∈ C is called a final object if for every object T ∈ C there exists a unique morphism e : T ⟶ E (i.e. Mor_{C}(T, E) consists of exactly one element).

A category C which has a product for any two objects A and B and which has a final object is called a category with finite products.

Remark 8.1.9. If the product (A×B, p_{A}, p_{B}) of two objects A and B in C exists then it is unique up to isomorphism.

If the final object E in C exists then it is unique up to isomorphism.

Problem 8.1.1. Let C be a category with finite products. Give a definition of a product of a family A_{1}, …, A_{n} (n ≥ 0). Show that products of such families exist in C.

Definition and Remark 8.1.10. Let C be a category. Then C^{op} with the following data Ob C^{op} := Ob C, Mor_{C^{op}}(A, B) := Mor_{C}(B, A), and f ∘_{op} g := g ∘ f defines a new category, the dual category to C.

Remark 8.1.11. Any notion expressed in categorical terms (with objects, morphisms, and their composition) has a dual notion, i.e. the given notion in the dual category.

Monomorphisms f in the dual category C^{op} are epimorphisms in the original category C and conversely. A final objects I in the dual category C^{op} is an initial object in the original category C.

Definition 8.1.12. The coproduct of two objects in the category C is defined to be a product of the objects in the dual category C^{op}.

Remark 8.1.13. Equivalent to the preceding definition is the following definition.

Given A, B ∈ C. An object A ⨿ B in C together with morphisms j_{A} : A ⟶ A ⨿ B and j_{B} : B ⟶ A ⨿ B ⟶ B is a (categorical) coproduct of A and B if for every object T ∈ C and every pair of morphisms f : A ⟶ T and g : B ⟶ T there exists a unique morphism [f, g] : A ⨿ B ⟶ T such that the diagram

[figure: T; f; g; A; A ⨿ B; j_{A}; B; j_{B}; [f,g]]

commutes.

The category C is said to have finite coproducts if C^{op} is a category with finite products. In particular coproducts are unique up to isomorphism.

2. Functors

Definition 8.2.1. Let C and D be categories. Let F consist of

1. a map Ob C ∋ A ↦ F(A) ∈ Ob D,

2. a family of maps

{F_{A,B} : Mor_{C}(A, B) ∋ f ↦ F_{A,B}(f) ∈ Mor_{D}(F(A), F(B))|A, B ∈ C}

[or {F_{A,B} : Mor_{C}(A, B) ∋ f ↦ F_{A,B}(f) ∈ Mor_{D}(F(B), F(A))|A, B ∈ C}] F is called a covariant [contravariant] functor if

1. F_{A,A}(1_{A}) = 1_{F(A)} for all A ∈ Ob C,

2. F_{A,C}(gf) = F_{B,C}(g)F_{A,B}(f) for all A, B, C ∈ Ob C.

[F_{A,C}(gf) = F_{A,B}(f)F_{B,C}(g) for all A, B, C ∈ Ob C]. Notation: We write

A ∈ C instead of  A ∈ Ob C
f ∈ C instead of f ∈ Mor_{C}(A, B)
F(f) instead of  F_{A,B}(f).

Examples 8.2.2. 1. Id : Set ⟶ Set

2. Forget : R-Mod ⟶ Set

3. Forget : Ri ⟶ Ab

4. Forget : Ab ⟶ Gr

5. P : Set ⟶ Set, P(M) := power set of M. P(f)(X) := f^{−1}(X) for f : M ⟶ N, X ⊆ N is a contravariant functor.

6. Q : Set ⟶ Set, Q(M) := power set of M. Q(f)(X) := f(X) for f : M ⟶ N, X ⊆ M is a covariant functor.

Lemma 8.2.3. 1. Let X ∈ C. Then

Ob C ∋ A ↦ Mor_{C}(X, A) ∈ Ob Set

Mor_{C}(A, B) ∋ f ↦ Mor_{C}(X, f) ∈ Mor_{Set}(Mor_{C}(X, A), Mor_{C}(X, B)), with Mor_{C}(X, f) : Mor_{C}(X, A) ∋ g ↦ fg ∈ Mor_{C}(X, B) or Mor_{C}(X, f)(g) = fg is a covariant functor Mor_{C}(X, -).

2. Let X ∈ C. Then

Ob C ∋ A ↦ Mor_{C}(A, X) ∈ Ob Set

Mor_{C}(A, B) ∋ f ↦ Mor_{C}(f, X) ∈ Mor_{Set}(Mor_{C}(B, X), Mor_{C}(A, X)) with Mor_{C}(f, X) : Mor_{C}(B, X) ∋ g ↦ gf ∈ Mor_{C}(A, X) or Mor_{C}(f, X)(g) = gf is a contravariant functor Mor_{C}(-, X).

Proof. 1. Mor_{C}(X, 1_{A})(g) = 1_{A}g = g = id(g), Mor_{C}(X, f)Mor_{C}(X, g)(h) = fgh = Mor_{C}(X, fg)(h).

2. analogously.

Remark 8.2.4. The preceding lemma shows that Mor_{C}(-, -) is a functor in both arguments. A functor in two arguments is called a bifunctor. We can regards the bifunctor Mor_{C}(-, -) as a covariant functor

Mor_{C}(-, -) : C^{op} × C ⟶ Set.

The use of the dual category removes the fact that the bifunctor Mor_{C}(-, -) is contravariant in the first variable.

Obviously the composition of two functors is again a functor and this composition is associative. Furthermore for each category C there is an identity functor Id_{C}.

Functors of the form Mor_{C}(X, -) resp. Mor_{C}(-, X) are called representable functors (covariant resp. contravariant) and X is called the representing object (see also section 8.8).

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 8.3.2. 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 8.3.3. 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.

4. Tensor Products

Definition and Remark 8.4.1. Let M_{R} and _{R}N be R-modules, and let A be an abelian group. A map f : M × N ⟶ A is called R-bilinear if

1. f(m + m′, n) = f(m, n) + f(m′, n),

2. f(m, n + n′) = f(m, n) + f(m, n′),

3. f(mr, n) = f(m, rn) for all r ∈ R, m, m′ ∈ M, n, n′ ∈ N.

Let Bil_{R}(M, N; A) denote the set of all R-bilinear maps f : M × N ⟶ A.

Bil_{R}(M, N; A) is an abelian group with (f + g)(m, n) := f(m, n) + g(m, n).

Definition 8.4.2. Let M_{R} and _{R}N be R-modules. An abelian group M ⊗_{R} N together with an R-bilinear map

⊗ : M × N ∋ (m, n) ↦ m ⊗ n ∈ M ⊗_{R} N

is called a tensor product of M and N over R if for each abelian group A and for each R-bilinear map f : M × N ⟶ A there exists a unique group homomorphism g : M ⊗_{R} N ⟶ A such that the diagram

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

commutes. The elements of M ⊗_{R} N are called tensors, the elements of the form m ⊗ n are called decomposable tensors.

Warning: If you want to define a homomorphism f : M ⊗_{R} N ⟶ A with a tensor product as domain you must define it by giving an R-bilinear map defined on M ×N.

Lemma 8.4.3. A tensor product (M ⊗_{R} N, ⊗) defined by M_{R} and _{R}N is unique up to a unique isomorphism.

Proof. Let (M ⊗_{R} N, ⊗) and (M ⊠_{R} N, ⊠) be tensor products. Then

[figure: M × N; ⊗; ⊠; ⊗; ⊠; M ⊗_{R} N; M ⊠_{R} N; h; k; M ⊗_{R} N; M ⊠_{R} N; h]

implies k = h^{−1}.

Because of this fact we will henceforth talk about the tensor product of M and N over R.

Proposition 8.4.4. (Rules of computation in a tensor product) Let (M ⊗_{R} N, ⊗) be the tensor product. Then we have for all r ∈ R, m, m′ ∈ M, n, n′ ∈ N 1. M ⊗_{R} N = {∑_{i} m_{i} ⊗ n_{i} | m_{i} ∈ M, n_{i} ∈ N},

2. (m + m′) ⊗ n = m ⊗ n + m′ ⊗ n,

3. m ⊗ (n + n′) = m ⊗ n + m ⊗ n′,

4. mr ⊗ n = m ⊗ rn (observe in particular, that ⊗ : M × N ⟶ M ⊗ N is not injective in general),

5. if f : M × N ⟶ A is an R-bilinear map and g : M ⊗_{R} N ⟶ A is the induced

homomorphism, then

g(m ⊗ n) = f(m, n).

Proof. 1. Let B := ⟨m ⊗ n⟩ ⊆ M ⊗_{R} N denote the subgroup of M ⊗_{R} N generated by the decomposable tensors m⊗n. Let j : B ⟶ M ⊗_{R}N be the embedding homomorphism. We get an induced map ⊗′ : M × N ⟶ B. In the following diagram

[figure: M × N; B; ⊗′; M ⊗_{R} N; j; B; M ⊗_{R} N; j; ⊗′; id_{B}; jp; p]

we have id_{B} ∘⊗′ = ⊗′, p with p ∘ j ∘ ⊗′ = p ∘ ⊗ = ⊗′ exists since ⊗′ is R-bilinear. Because of jp ∘ ⊗ = j ∘ ⊗′ = ⊗ = id_{M⊗_{R}N} ∘⊗ we get jp = id_{M⊗_{R}N}, hence the embedding j is surjective and thus the identity.

2. (m + m′) ⊗ n = ⊗(m + m′, n) = ⊗(m, n) + ⊗(m′, n) = m ⊗ n + m′ ⊗ n.

3. and 4. analogously.

5. is precisely the definition of the induced homomorphism.

Remark 8.4.5. To construct tensor products, we use the notion of a free module. Let X be a set and R be a ring. An R-module RX together with a map ι : X ⟶

RX is called a free R-module generated by X, if for every R-module M and for every map f : X ⟶ M there exists a unique homomorphism of R-modules g : RX ⟶ M such that the diagram

[figure: X; RX; ι; f; M; g]

commutes.

Free R-modules exist and can be constructed as RX := {α : X ⟶ R| for almost all x ∈ X : α(x) = 0}.

Proposition 8.4.6. Given R-modules M_{R} and _{R}N. Then there exists a tensor product (M ⊗_{R} N, ⊗).

Proof. Define M ⊗_{R} N := ℤ{M × N}/U where ℤ{M × N} is a free ℤ-module over M × N (the free abelian group) and U is generated by

ι(m + m′, n) − ι(m, n) − ι(m′, n)
ι(m, m + n′) − ι(m, n) − ι(m, n′)
ι(mr, n) − ι(m, rn)

for all r ∈ R, m, m′ ∈ M, n, n′ ∈ N. Consider

[figure: M × N; ℤ{M × N}; ι; M ⊗_{R} N; ν; = ℤ{M × N}/U; A; ψ; ρ; g]

Let ψ be given. Then there is a unique ρ ∈ Hom(ℤ{M × N}, A) such that ρι = ψ. Since ψ is R-bilinear we get ρ(ι(m + m′, n) − ι(m, n) − ι(m′n)) = ψ(m + m′, n) − ψ(m, n) − ψ(m′, n) = 0 and similarly ρ(ι(m, n + n′) − ι(m, n) − ι(m, n′)) = 0 and ρ(ι(mr, n) − ι(m, rn)) = 0. So we get ρ(U) = 0. This implies that there is a unique g ∈ Hom(M ⊗_{R} N, A) such that gν = ρ (homomorphism theorem). Let ⊗ := ν ∘ ι. Then ⊗ is bilinear since (m + m′) ⊗ n = ν ∘ ι(m + m′, n) = ν(ι(m + m′, n)) = ν(ι(m + m′, n) − ι(m, n) − ι(m′, n) + ι(m, n) + ι(m′, n)) = ν(ι(m, n) + ι(m′, n)) = ν ∘ ι(m, n) + ν ∘ ι(m′, n) = m ⊗ n + m′ ⊗ n. The other two properties are obtained in an analogous way.

We have to show that (M ⊗_{R} N, ⊗) is a tensor product. The above diagram shows that for each abelian group A and for each R-bilinear map ψ : M × N ⟶ A there is a g ∈ Hom(M ⊗_{R} N, A) such that g ∘ ⊗ = ψ. Given h ∈ Hom(M ⊗_{R} N, A) with h ∘ ⊗ = ψ. Then h ∘ ν ∘ ι = ψ. This implies h ∘ ν = ρ = g ∘ ν hence g = h.

Proposition and Definition 8.4.7. Given two homomorphisms

f ∈ Hom_{R}(M., M′.) and g ∈ Hom_{R}(.N, .N′).

Then there is a unique homomorphism

f ⊗_{R} g ∈ Hom(M ⊗_{R} N, M′ ⊗_{R} N′)

such that f ⊗_{R} g(m ⊗ n) = f(m) ⊗ g(n), i.e. the following diagram commutes

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

Proof. ⊗ ∘ (f × g) is bilinear.

Notation 8.4.8. We often write f ⊗_{R} N := f ⊗_{R} 1_{N} and M ⊗_{R} g := 1_{M} ⊗_{R} g. We have the following rule of computation:

f ⊗_{R} g = (f ⊗_{R} N′) ∘ (M ⊗_{R} g) = (M′ ⊗_{R} g) ∘ (f ⊗_{R} N)

since f × g = (f × N′) ∘ (M × g) = (M′ × g) ∘ (f × N).

Proposition 8.4.9. The following define covariant functors

1. - ⊗ N : Mod-R ⟶ Ab;

2. M ⊗ - : R-Mod ⟶ Ab;

3. - ⊗ - : Mod-R × R-Mod ⟶ Ab.

Proof. (f × g) ∘ (f′ × g′) = ff′ × gg′ implies (f ⊗_{R} g) ∘ (f′ ⊗_{R} g′) = ff′ × gg′. Furthermore 1_{M} × 1_{N} = 1_{M×N} implies 1_{M} ⊗_{R} 1_{N} = 1_{M⊗_{R}N}.

Definition 8.4.10. Let R, S be rings and let M be a left R-module and a right Smodule. M is called an R-S-bimodule if (rm)s = r(ms). We define Hom_{R-S}(.M., .N.) := Hom_{R}(.M, .N) ∩ Hom_{S}(M., N.).

Remark 8.4.11. Let M_{S} be a right S-module and let R × M ⟶ M a map. M is an R-S-bimodule if and only if

1. ∀r ∈ R : (M ∋ m ↦ rm ∈ M) ∈ Hom_{S}(M., M.),

2. ∀r, r′ ∈ R, m ∈ M : (r + r′)m = rm + r′m,

3. ∀r, r′ ∈ R, m ∈ M : (rr′)m = r(r′m),

4. ∀m ∈ M : 1m = m.

Lemma 8.4.12. Let_{R}M_{S} and_{S}N_{T} be bimodules. Then_{R}(M⊗_{S}N)_{T} is a bimodule by r(m ⊗ n) := rm ⊗ n and (m ⊗ n)t := m ⊗ nt.

Proof. Obviously we have 2.-4. Furthermore (r ⊗_{S} id)(m ⊗ n) = rm ⊗ n = r(m ⊗ n) is a homomorphism.

Corollary 8.4.13. Given bimodules _{R}M_{S}, _{S}N_{T}, _{R}M′_{S}, _{S}N′_{T} and homomorphisms f ∈ Hom_{R-S}(.M., .M′.) and g ∈ Hom_{S-T} (.N., .N′.). Then we have f ⊗_{S} g ∈ Hom_{R-T} (.M ⊗_{S} N., .M′ ⊗_{S} N′.).

Proof. f ⊗_{S} g(rm ⊗ nt) = f(rm) ⊗ g(nt) = r(f ⊗_{S} g)(m ⊗ n)t.

Remark 8.4.14. Every module M over a commutative ring 𝕂 and in particular every vector space over a field 𝕂 is a 𝕂-𝕂-bimodule by λm = mλ. So there is an embedding functor ι : 𝕂-Mod ⟶ 𝕂-Mod-𝕂. Observe that there are 𝕂-𝕂-bimodules that do not satisfy λm = mλ. Take for example an automorphism α : 𝕂 ⟶ 𝕂 and a left 𝕂-module M and define mλ := α(λ)m. Then M is such a 𝕂-𝕂-bimodule.

The tensor product M ⊗_{𝕂} N of two 𝕂-𝕂-bimodules M and N is again a 𝕂-𝕂-bimodule. If we have, however, 𝕂-𝕂-bimodules M and N arising from 𝕂-modules as above, i.e. satisfying λm = mλ, then their tensor product M ⊗_{𝕂} N also satisfies this equation, so M ⊗_{𝕂} N comes from a module in 𝕂-Mod. Indeed we have λm ⊗ n = mλ ⊗ n = m ⊗ λn = m ⊗ nλ. Thus the following diagram of functors commutes:

[figure: 𝕂-Mod; 𝕂-Mod-𝕂.; ι; 𝕂-Mod × 𝕂-Mod; 𝕂-Mod-𝕂 × 𝕂-Mod-𝕂; ι × ι; ⊗_{𝕂}; ⊗_{𝕂}]

So we can consider 𝕂-Mod as a (proper) subcategory of 𝕂-Mod-𝕂. The tensor product over 𝕂 can be restricted to 𝕂-Mod.

We write the tensor product of two vector spaces M and N as M ⊗ N.

Theorem 8.4.15. In the category 𝕂-Mod there are natural isomorphisms

1. Associativity Law: α : (M ⊗ N) ⊗ P ≅ M ⊗ (N ⊗ P).

2. Law of the Left Unit: λ : 𝕂 ⊗ M ≅ M.

3. Law of the Right Unit: ρ : M ⊗ 𝕂 ≅ M.

4. Symmetry Law: τ : M ⊗ N ≅ N ⊗ M.

5. Existence of Inner Hom-Functors: Hom(P ⊗ M, N) ≅ Hom(P, Hom(M, N)).

Proof. We only describe the corresponding homomorphisms.

1. Use (8.4.45.) to define α((m ⊗ n) ⊗ p) := m ⊗ (n ⊗ p).

2. Define λ : 𝕂 ⊗ M ⟶ M by λ(r ⊗ m) := rm.

3. Define ρ : M ⊗ 𝕂 ⟶ M by ρ(m ⊗ r) := mr.

4. Define τ(m ⊗ n) := n ⊗ m.

5. For f : P ⊗ M ⟶ N define ϕ(f) : P ⟶ Hom(M, N) by ϕ(f)(p)(m) := f(p ⊗ m).

Usually one identifies threefold tensor products along the map α so that we use M ⊗ N ⊗ P = (M ⊗ N) ⊗ P = M ⊗ (N ⊗ P). For the notion of a monoidal or tensor category, however, this natural transformation is of central importance.

Problem 8.4.4. 1. Give an explicit proof of M ⊗ (X ⊕ Y) ≅ M ⊗ X ⊕ M ⊗ Y.

2. Show that for every finite dimensional vector space V there is a unique element ∑_{i=1}^{n} v_{i} ⊗ v_{i}^{∗} ∈ V ⊗ V^{∗} such that the following holds

∀v ∈ V : ∑_{i} v_{i}^{∗}(v)v_{i} = v.

(Hint: Use an isomorphism End(V) ≅ V ⊗ V^{∗} and dual bases {v_{i}} of V and {v_{i}^{∗}} of V^{∗}.)

3. Show that the following diagrams (coherence diagrams or constraints) commute in 𝕂-Mod:

[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)]

[figure: (A ⊗ 𝕂) ⊗ B; A ⊗ (𝕂 ⊗ B); α(A,𝕂,B); A ⊗ B; ρ(A)⊗1; 1⊗λ(B)]

4. Write τ(A, B) : A ⊗ B ⟶ B ⊗ A for τ(A, B) : a ⊗ b ↦ b ⊗ a. Show that τ is a natural transformation (between which functors?). Show that

[figure: (A ⊗ B) ⊗ C; (B ⊗ A) ⊗ C; τ(X,B)⊗1; B ⊗ (A ⊗ C); α; α; 1⊗τ(A,C); A ⊗ (B ⊗ C); (B ⊗ C) ⊗ A; τ(A,B⊗C); B ⊗ (C ⊗ A); α]

commutes for all A, B, C ∈ 𝕂-Mod and that

τ(B, A)τ(A, B) = id_{A⊗B}

for all A, B in 𝕂-Mod.

5. Find an example of M, N ∈ 𝕂-Mod-𝕂 such that M ⊗_{𝕂} N ≅ N̸ ⊗_{𝕂} M.

5. Algebras

Let 𝕂 be a commutative ring. In most of our applications 𝕂 will be a field. Tensor products of 𝕂-modules will be simply written as M ⊗ N := M ⊗_{K} N. Every such tensor product is again a 𝕂-bimodule since each 𝕂-module M resp. N is a 𝕂-bimodule (see 8.4.14).

Definition 8.5.1. A 𝕂-algebra is a vector space A together with a multiplication ∇ : A ⊗ A ⟶ A that is associative:

[figure: A ⊗ A; A; ∇; A ⊗ A ⊗ A; A ⊗ A; id ⊗∇; ∇⊗id; ∇]

and a unit η : 𝕂 ⟶ A:

[figure: 𝕂 ⊗ A ≅ A ≅ A ⊗ 𝕂; A ⊗ A; id ⊗η; η⊗id; ∇; A ⊗ A; A.; ∇; id]

A 𝕂-algebra A is commutative if the following diagram commutes

[figure: A ⊗ A; A ⊗ A; τ; A.; ∇; ∇]

Let A and B be 𝕂-algebras. A homomorphism of algebras f : A ⟶ B is a 𝕂-linear map such that the following diagrams commute:

[figure: A; B; f; A ⊗ A; B ⊗ B; f⊗f; ∇_{A}; ∇_{B}]

and

[figure: 𝕂; η_{A}; η_{B}; A; B.; f]

Remark 8.5.2. Every 𝕂-algebra A is a ring with the multiplication

A × A ⟶^{⊗} A ⊗ A ⟶^{∇} A.

The unit element is η(1), where 1 is the unit element of 𝕂.

Obviously the composition of two homomorphisms of algebras is again a homomorphism of algebras. Furthermore the identity map is a homomorphism of algebras. Hence the 𝕂-algebras form a category 𝕂-Alg. The category of commutative 𝕂-algebras will be denoted by 𝕂-cAlg.

Problem 8.5.5. 1. Show that End_{K}(V) is a 𝕂-algebra.

2. Show that (A, ∇ : A ⊗ A ⟶ A, η : 𝕂 ⟶ A) is a 𝕂-algebra if and only if

A with the multiplication A × A ⟶^{⊗} A ⊗ A ⟶^{∇} A and the unit η(1) is a ring and η : 𝕂 ⟶ Cent(A) is a ring homomorphism into the center of A.

3. Let V be a 𝕂-module. Show that D(V) := 𝕂 × V with the multiplication (r_{1}, v_{1})(r_{2}, v_{2}) := (r_{1}r_{2}, r_{1}v_{2} + r_{2}v_{1}) is a commutative 𝕂-algebra.

Lemma 8.5.3. Let A and B be algebras. Then A ⊗ B is an algebra with the multiplication (a_{1} ⊗ b_{1})(a_{2} ⊗ b_{2}) := a_{1}a_{2} ⊗ b_{1}b_{2}.

Proof. Certainly the algebra properties can easily be checked by a simple calculation with elements. We prefer for later applications a diagrammatic proof.

Let ∇_{A} : A ⊗ A ⟶ A and ∇_{B} : B ⊗ B ⟶ B denote the multiplications of the two algebras. Then the new multiplication is ∇_{A⊗B} := (∇_{A} ⊗ ∇_{B})(1_{A} ⊗ τ ⊗ 1_{B}) : A ⊗ B ⊗ A ⊗ B ⟶ A ⊗ B where τ : B ⊗ A ⟶ A ⊗ B is the symmetry map from Theorem 8.4.15. Now the following diagrams commute

[figure: A ⊗ B ⊗ A ⊗ B ⊗ A ⊗ B; A ⊗ A ⊗ B ⊗ B ⊗ A ⊗ B; 1⊗τ⊗1^{3}; A ⊗ B ⊗ A ⊗ B; ∇⊗∇⊗1^{2}; A ⊗ B ⊗ A ⊗ A ⊗ B ⊗ B; A ⊗ A ⊗ A ⊗ B ⊗ B ⊗ B; 1⊗τ_{B,A⊗A}⊗1^{3}; A ⊗ A ⊗ B ⊗ B; ∇⊗1⊗∇⊗1; 1^{3}⊗τ⊗1; 1⊗τ_{B⊗B,A}⊗1; 1⊗τ⊗1; A ⊗ A ⊗ B ⊗ A ⊗ B ⊗ B; 1⊗τ⊗1^{3}; 1^{3}⊗τ⊗1; 1^{2}⊗τ⊗1^{2}; A ⊗ B ⊗ A ⊗ B; A ⊗ A ⊗ B ⊗ B; 1⊗τ⊗1; A ⊗ B; ∇⊗∇; 1⊗∇⊗∇; 1⊗∇⊗1⊗∇; ∇⊗∇]

In the left upper rectangle of the diagram the quadrangle commutes by the properties of the tensor product and the two triangles commute by inner properties of τ. The right upper and left lower rectangles commute since τ is a natural transformation and the right lower rectangle commutes by the associativity of the algebras A and B.

Furthermore we use the homomorphism η = η_{A⊗B} : 𝕂 ⟶ 𝕂 ⊗ K ⟶ A ⊗ B in the following commutative diagram

[figure: 𝕂 ⊗ A ⊗ B ≅ A ⊗ B ≅ A ⊗ B ⊗ 𝕂; A ⊗ B ⊗ 𝕂 ⊗ 𝕂; A ⊗ B ⊗ A ⊗ B; 1^{2}⊗η⊗η; 1; A ⊗ 𝕂 ⊗ B ⊗ 𝕂; A ⊗ A ⊗ B ⊗ B; 1⊗η⊗1⊗η; 1⊗τ⊗1; 1⊗τ⊗1; ∇⊗∇; 𝕂 ⊗ 𝕂 ⊗ A ⊗ B; 𝕂 ⊗ A ⊗ 𝕂 ⊗ B; 1⊗τ⊗1; A ⊗ B ⊗ A ⊗ B; A ⊗ A ⊗ B ⊗ B; 1⊗τ⊗1; A ⊗ B.; ∇⊗∇; η⊗η⊗1^{2}; η⊗1⊗η⊗1]

Definition 8.5.4. Let 𝕂 be a commutative ring. Let V be a 𝕂-module. A 𝕂algebra T (V) together with a homomorphism of 𝕂-modules ι : V ⟶ T (V) is called a tensor algebra over V if for each 𝕂-algebra A and for each homomorphism of 𝕂modules f : V ⟶ A there exists a unique homomorphism of 𝕂-algebras g : T (V) ⟶ A such that the diagram

[figure: V; T (V); ι; f; A; g]

commutes.

Note: If you want to define a homomorphism g : T (V) ⟶ A with a tensor algebra as domain you should define it by giving a homomorphism of 𝕂-modules defined on V.

Lemma 8.5.5. A tensor algebra (T (V), ι) defined by V is unique up to a unique isomorphism.

Proof. Let (T (V), ι) and (T′(V), ι′) be tensor algebras over V. Then

[figure: V; ι; ι′; ι; ι′; T (V); T′(V); h; k; T (V); T′(V); h]

implies k = h^{−1}.

Proposition 8.5.6. (Rules of computation in a tensor algebra) Let (T (V), ι) be the tensor algebra over V. Then we have

1. ι : V ⟶ T (V) is injective (so we may identify the elements ι(v) and v for all v ∈ V),

2. T (V) = {∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n},

3. if f : V ⟶ A is a homomorphism of 𝕂-modules, A is a 𝕂-algebra, and g : T (V) ⟶ A is the induced homomorphism of 𝕂-algebras, then

g(∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}) = ∑_{n,ī} f(v_{i_{1}}) · . . . · f(v_{i_{n}}).

Proof. 1. Use the embedding homomorphism j : V ⟶ D(V), where D(V) is defined as in 8.5.3. to construct g : T (V) ⟶ D(V) such that g ∘ ι = j. Since j is injective so is ι.

2. Let B := {∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n}. Obviously B is the subalgebra of T (V) generated by the elements of V. Let j : B ⟶ T (V) be the embedding homomorphism. Then ι : V ⟶ T (V) factors through a linear map ι′ : V ⟶ B. In the following diagram

[figure: V; B; ι′; T (V); j; B; T (V); j; ι′; id_{B}; jp; p]

we have id_{B} ∘ι′ = ι′. p with p ∘ j ∘ ι′ = p ∘ ι = ι′ exists since ι′ is a homomorphism of 𝕂-modules. Because of jp ∘ ι = j ∘ ι′ = ι = id_{T (V)} ∘ι we get jp = id_{T (V)}, hence the embedding j is surjective and thus j is the identity.

3. is precisely the definition of the induced homomorphism.

Proposition 8.5.7. Given a 𝕂-module V. Then there exists a tensor algebra (T (V), ι).

Proof. Define T^{n}(V) := V ⊗ … ⊗ V = V^{⊗n} to be the n-fold tensor product of V. Define T^{0}(V) := 𝕂 and T^{1}(V) := V. We define

T (V) := ⨁_{i≥0} T^{i}(V) = 𝕂 ⊕ V ⊕ (V ⊗ V) ⊕ (V ⊗ V ⊗ V) ⊕ ….

The components T^{n}(V) of T (V) are called homogeneous components.

The canonical isomorphisms T^{m}(V) ⊗ T^{n}(V) ≅ T^{m+n}(V) taken as multiplication

\begin{matrix}∇ : T^{m}(V) ⊗ T^{n}(V) ⟶ T^{m+n}(V) \\ ∇ : T (V) ⊗ T (V) ⟶ T (V) \end{matrix}

and the embedding η : 𝕂 = T^{0}(V) ⟶ T (V) induce the structure of a 𝕂-algebra on T (V). Furthermore we have the embedding ι : V ⟶ T^{1}(V) ⊆ T (V).

We have to show that (T (V), ι) is a tensor algebra. Let f : V ⟶ A be a homomorphism of 𝕂-modules. Each element in T (V) is a sum of decomposable tensors v_{1} ⊗ … ⊗ v_{n}. Define g : T (V) ⟶ A by g(v_{1} ⊗ … ⊗ v_{n}) := f(v_{1}) … f(v_{n}) (and (g : T^{0}(V) ⟶ A) = (η : 𝕂 ⟶ A)). By induction one sees that g is a homomorphism of algebras. Since (g : T^{1}(V) ⟶ A) = (f : V ⟶ A) we get g ∘ ι = f. If h : T (V) ⟶ A is a homomorphism of algebras with h ∘ ι = f we get h(v_{1} ⊗ … ⊗ v_{n}) = h(v_{1}) … h(v_{n}) = f(v_{1}) … f(v_{n}) hence h = g.

Proposition 8.5.8. The construction of tensor algebras T (V) defines a functor T : 𝕂-Mod ⟶ 𝕂-Alg that is left adjoint to the underlying functor U : 𝕂-Alg ⟶ 𝕂-Mod.

Proof. Follows from the universal property and 8.9.16.

Problem 8.5.6. 1. Let X be a set and V := 𝕂X be the free 𝕂-module over X. Show that X ⟶ V ⟶ T (V) defines a free algebra over X, i.e. for every 𝕂algebra A and every map f : X ⟶ A there is a unique homomorphism of 𝕂-algebras g : T (V) ⟶ A such that the diagram

[figure: X; T (V); f; A; g]

commutes.

We write 𝕂⟨X⟩ := T (𝕂X) and call it the polynomial ring over 𝕂 in the noncommuting variables X.

2. Let T (V) and ι : V ⟶ T (V) be a tensor algebra. Regard V as a subset of T (V) by ι. Show that there is a unique homomorphism Δ : T (V) ⟶ T (V) ⊗ T (V) with Δ(v) = v ⊗ 1 + 1 ⊗ v for all v ∈ V.

3. Show that (Δ ⊗ 1)Δ = (1 ⊗ Δ)Δ : T (V) ⟶ T (V) ⊗ T (V) ⊗ T (V).

4. Show that there is a unique homomorphism of algebras ε : T (V) ⟶ 𝕂 with ε(v) = 0 for all v ∈ V.

5. Show that (ε ⊗ 1)Δ = (1 ⊗ ε)Δ = id_{T (V)}.

6. Show that there is a unique homomorphism of algebras S : T (V) ⟶ T (V)^{op} with S(v) = −v. (T (V)^{op} is the opposite algebra of T (V) with multiplication s∗t := ts for all s, t ∈ T (V) = T (V)^{op} and where st denotes the product in T (V).)

7. Show that the diagrams

[figure: T (V); 𝕂; ε; T (V); η; T (V) ⊗ T (V); T (V) ⊗ T (V); 1⊗S; S⊗1; Δ; ∇]

commute.

Definition 8.5.9. Let 𝕂 be a commutative ring. Let V be a 𝕂-module. A 𝕂algebra S(V) together with a homomorphism of 𝕂-modules ι : V ⟶ S(V), such that ι(v) · ι(v′) = ι(v′) · ι(v) for all v, v′ ∈ V, is called a symmetric algebra over V if for each 𝕂-algebra A and for each homomorphism of 𝕂-modules f : V ⟶ A, such that f(v) · f(v′) = f(v′) · f(v) for all v, v′ ∈ V, there exists a unique homomorphism of 𝕂-algebras g : S(V) ⟶ A such that the diagram

[figure: V; S(V); ι; f; A; g]

commutes.

Note: If you want to define a homomorphism g : S(V) ⟶ A with a symmetric algebra as domain you should define it by giving a homomorphism of 𝕂-modules f : V ⟶ A satisfying f(v) · f(v′) = f(v′) · f(v) for all v, v′ ∈ V.

Lemma 8.5.10. A symmetric algebra (S(V), ι) defined by V is unique up to a unique isomorphism.

Proof. Let (S(V), ι) and (S′(V), ι′) be symmetric algebras over V. Then

[figure: V; ι; ι′; ι; ι′; S(V); S′(V); h; k; S(V); S′(V); h]

implies k = h^{−1}.

Proposition 8.5.11. (Rules of computation in a symmetric algebra) Let (S(V), ι) be the symmetric algebra over V. Then we have

1. ι : V ⟶ S(V) is injective (we will identify the elements ι(v) and v for all v ∈ V),

2. S(V) = {∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n},

3. if f : V ⟶ A is a homomorphism of 𝕂-modules satisfying f(v) · f(v′) = f(v′)·f(v) for all v, v′ ∈ V, A is a 𝕂-algebra, and g : S(V) ⟶ A is the induced homomorphism 𝕂-algebras, then

g(∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}) = ∑_{n,ī} f(v_{i_{1}}) · . . . · f(v_{i_{n}}).

Proof. 1. Use the embedding homomorphism j : V ⟶ D(V), where D(V) is the commutative algebra defined in 8.5.3. to construct g : S(V) ⟶ D(V) such that g ∘ ι = j. Since j is injective so is ι.

2. Let B := {∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n}. Obviously B is the subalgebra of S(V) generated by the elements of V. Let j : B ⟶ S(V) be the embedding homomorphism. Then ι : V ⟶ S(V) factors through a linear map ι′ : V ⟶ B. In the following diagram

[figure: V; B; ι′; S(V); j; B; S(V); j; ι′; id_{B}; jp; p]

we have id_{B} ∘ι′ = ι′, p with p ∘ j ∘ ι′ = p ∘ ι = ι′ exists since ι′ is a homomorphism of 𝕂-modules satisfying ι′(v) · ι′(v′) = ι′(v′) · ι′(v) for all v, v′ ∈ V. Because of jp ∘ ι = j ∘ ι′ = ι = id_{S(V)} ∘ι we get jp = id_{S(V)}, hence the embedding j is surjective and thus the identity.

3. is precisely the definition of the induced homomorphism.

Proposition 8.5.12. Let V be a 𝕂-module. The symmetric algebra (S(V), ι) is commutative and satisfies the following universal property:

for each commutative 𝕂-algebra A and for each homomorphism of 𝕂-modules f : V ⟶ A there exists a unique homomorphism of 𝕂-algebras g : S(V) ⟶ A such that the diagram

[figure: V; S(V); ι; f; A; g]

commutes.

Proof. Commutativity follows from the commutativity of the generators: vv′ = v′v which carries over to the elements of the form ∑_{n,ī} v_{i_{1}} · . . . · v_{i_{n}}. The universal property follows since the defining condition f(v) · f(v′) = f(v′) · f(v) for all v, v′ ∈ V is automatically satisfied.

Proposition 8.5.13. Given a 𝕂-module V. Then there exists a symmetric algebra (S(V), ι).

Proof. Define S(V) := T (V)/I where I = ⟨vv′ − v′v|v, v′ ∈ V ⟩ is the two-sided ideal generated by the elements vv′ − v′v. Let ι be the canonical map V ⟶ T (V) ⟶ S(V). Then the universal property is easily verified by the homomorphism theorem for algebras.

Proposition 8.5.14. The construction of symmetric algebras S(V) defines a functor S : 𝕂-Mod ⟶ 𝕂-cAlg that is left adjoint to the underlying functor U : 𝕂-cAlg ⟶ 𝕂-Mod.

Proof. Follows from the universal property and 8.9.16.

Problem 8.5.7. Let X be a set and V := 𝕂X be the free 𝕂-module over X. Show that X ⟶ V ⟶ S(V) defines a free commutative algebra over X, i.e. for every commutative 𝕂-algebra A and every map f : X ⟶ A there is a unique homomorphism of 𝕂-algebras g : S(V) ⟶ A such that the diagram

[figure: X; S(V); f; A; g]

commutes.

The algebra 𝕂[X] := S(𝕂X) is called the polynomial ring over 𝕂 in the (commuting) variables X.

2. Let S(V) and ι : V ⟶ S(V) be a symmetric algebra. Show that there is a unique homomorphism Δ : S(V) ⟶ S(V) ⊗ S(V) with Δ(v) = v ⊗ 1 + 1 ⊗ v for all v ∈ V.

3. Show that (Δ ⊗ 1)Δ = (1 ⊗ Δ)Δ : S(V) ⟶ S(V) ⊗ S(V) ⊗ S(V).

4. Show that there is a unique homomorphism of algebras ε : S(V) ⟶ 𝕂 with ε(v) = 0 for all v ∈ V.

5. Show that (ε ⊗ 1)Δ = (1 ⊗ ε)Δ = id_{S(V)}.

6. Show that there is a unique homomorphism of algebras S : S(V) ⟶ S(V) with S(v) = −v.

7. Show that the diagrams

[figure: S(V); 𝕂; ε; S(V); η; S(V) ⊗ S(V); S(V) ⊗ S(V); 1⊗S; S⊗1; Δ; ∇]

commute.

Definition 8.5.15. Let 𝕂 be a commutative ring. Let V be a 𝕂-module. A 𝕂-algebra E(V) together with a homomorphism of 𝕂-modules ι : V ⟶ E(V), such that ι(v)^{2} = 0 for all v ∈ V, is called an exterior algebra or Grassmann algebra over V if for each 𝕂-algebra A and for each homomorphism of 𝕂-modules f : V ⟶ A, such that f(v)^{2} = 0 for all v ∈ V, there exists a unique homomorphism of 𝕂-algebras g : E(V) ⟶ A such that the diagram

[figure: V; E(V); ι; f; A; g]

commutes.

The multiplication in E(V) is usually denoted by u ∧ v.

Note: If you want to define a homomorphism g : E(V) ⟶ A with an exterior algebra as domain you should define it by giving a homomorphism of 𝕂-modules defined on V satisfying f(v)^{2} = 0 for all v, v′ ∈ V.

Problem 8.5.8. 1. Let f : V ⟶ A be a linear map satisfying f(v)^{2} = 0 for all v ∈ V. Then f(v)f(v′) = −f(v′)f(v) for all v, v′ ∈ V.

2. Let 2 be invertible in 𝕂 (e.g. 𝕂 a field of characteristic ≠ 2). Let f : V ⟶ A be a linear map satisfying f(v)f(v′) = −f(v′)f(v) for all v, v′ ∈ V. Then f(v)^{2} = 0 for all v ∈ V.

Lemma 8.5.16. An exterior algebra (E(V), ι) defined by V is unique up to a unique isomorphism.

Proof. Let (E(V), ι) and (E′(V), ι′) be exterior algebras over V. Then

[figure: V; ι; ι′; ι; ι′; E(V); E′(V); h; k; E(V); E′(V); h]

implies k = h^{−1}.

Proposition 8.5.17. (Rules of computation in an exterior algebra) Let (E(V), ι) be the exterior algebra over V. Then we have

1. ι : V ⟶ E(V) is injective (we will identify the elements ι(v) and v for all v ∈ V),

2. E(V) = {∑_{n,ī} v_{i_{1}} ∧ … ∧ v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n},

3. if f : V ⟶ A is a homomorphism of 𝕂-modules satisfying f(v) · f(v′) = −f(v′) · f(v) for all v, v′ ∈ V, A is a 𝕂-algebra, and g : E(V) ⟶ A is the induced homomorphism 𝕂-algebras, then

g(∑_{n,ī} v_{i_{1}} ∧ … ∧ v_{i_{n}}) = ∑_{n,ī} f(v_{i_{1}}) · . . . · f(v_{i_{n}}).

Proof. 1. Use the embedding homomorphism j : V ⟶ D(V), where D(V) is the algebra defined in 8.5.3. to construct g : E(V) ⟶ D(V) such that g ∘ι = j. Since j is injective so is ι.

2. Let B := {∑_{n,ī} v_{i_{1}}∧…∧v_{i_{n}}|ī = (i_{1}, …, i_{n}) multiindex of length n}. Obviously B is the subalgebra of E(V) generated by the elements of V. Let j : B ⟶ E(V) be the embedding homomorphism. Then ι : V ⟶ E(V) factors through a linear map ι′ : V ⟶ B. In the following diagram

[figure: V; B; ι′; E(V); j; B; E(V); j; ι′; id_{B}; jp; p]

we have id_{B} ∘ι′ = ι′, p with p ∘ j ∘ ι′ = p ∘ ι = ι′ exists since ι′ is a homomorphism of 𝕂-modules satisfying ι′(v) · ι′(v′) = −ι′(v′) · ι′(v) for all v, v′ ∈ V. Because of jp ∘ ι = j ∘ ι′ = ι = id_{E(V)} ∘ι we get jp = id_{E(V)}, hence the embedding j is surjective and thus j is the identity.

3. is precisely the definition of the induced homomorphism.

Proposition 8.5.18. Given a 𝕂-module V. Then there exists an exterior algebra (E(V), ι).

Proof. Define E(V) := T (V)/I where I = ⟨v^{2}|v ∈ V ⟩ is the two-sided ideal generated by the elements v^{2}. Let ι be the canonical map V ⟶ T (V) ⟶ E(V). Then the universal property is easily verified by the homomorphism theorem for algebras.

Problem 8.5.9. 1. Let V be a finite dimensional vector space of dimension n. Show that E(V) is finite dimensional of dimension 2^{n}. (Hint: The homogeneous components E^{i}(V) have dimension ( \begin{matrix}n \\ i \end{matrix} ).

2. Show that the symmetric group S_{n} operates (from the left) on T^{n}(V) by σ(v_{1} ⊗ … ⊗ v_{n}) = v_{σ^{−1}(1)} ⊗ … ⊗ v_{σ^{−1}(n)} with σ ∈ S_{n} and v_{i} ∈ V.

3. A tensor a ∈ T^{n}(V) is called a symmetric tensor if σ(a) = a for all σ ∈ S_{n}. Let Ŝ^{n}(V) be the subspace of symmetric tensors in T^{n}(V).

a) Show that S : T^{n}(V) ∋ a ↦ ∑_{σ∈S_{n}} σ(a) ∈ T^{n}(V) is a linear map.

b) Show that S has its image in Ŝ^{n}(V).

c) Show that Im(S) = Ŝ^{n}(V) if n! is invertible in 𝕂.

d) Show that Ŝ^{n}(V) ↪ T^{n}(V) →^{ν} S^{n}(V) is an isomorphism if n! is invertible in 𝕂 and ν : T^{n}(V) ⟶ S^{n}(V) is the restriction of ν : T (V) ⟶ S(V), the symmetric algebra.

4. A tensor a ∈ T^{n}(V) is called an antisymmetric tensor if σ(a) = ε(σ)a for all σ ∈ S_{n} where ε(σ) is the sign of the permutation σ. Let Ê^{n}(V) be the subspace of antisymmetric tensors in T^{n}(V).

a) Show that E : T^{n}(V) ∋ a ↦ ∑_{σ∈S_{n}} ε(σ)σ(a) ∈ T^{n}(V) is a linear map.

b) Show that E has its image in Ê^{n}(V).

c) Show that Im(E) = Ê^{n}(V) if n! is invertible in 𝕂.

d) Show that Ê^{n}(V) ↪ T^{n}(V) →^{ν} E^{n}(V) is an isomorphism if n! is invertible in 𝕂 and ν : T^{n}(V) ⟶ E^{n}(V) is the restriction of ν : T (V) ⟶ E(V), the exterior algebra.

Definition 8.5.19. Let A be a 𝕂-algebra. A left A-module is a 𝕂-module M together with a homomorphism μ_{M} : A ⊗ M ⟶ M, such that the diagrams

[figure: A ⊗ M; M; μ; A ⊗ A ⊗ M; A ⊗ M; id ⊗μ; ∇⊗id; μ]

and

[figure: M ≅ 𝕂 ⊗ M; A ⊗ M; η⊗id; M; μ; id]

commute.

Let _{A}M and _{A}N be A-modules and let f : M ⟶ N be a 𝕂-linear map. The map f is called a homomorphism of modules if the diagram

[figure: A ⊗ N; N; μ_{N}; A ⊗ M; M; μ_{M}; 1⊗f; f]

commutes.

The left A-modules and their homomorphisms form the category_{A}M of A-modules.

Problem 8.5.10. Show that an abelian group M is a left module over the ring A if and only if M is a 𝕂-module and an A-module in the sense of Definition 8.5.19.

6. Coalgebras

Definition 8.6.1. A 𝕂-coalgebra is a 𝕂-module C together with a comultiplication or diagonal Δ : C ⟶ C ⊗ C that is coassociative:

[figure: C ⊗ C; C ⊗ C ⊗ C; id ⊗Δ; C; C ⊗ C; Δ; Δ; Δ⊗id]

and a counit or augmentation ϵ : C ⟶ 𝕂:

[figure: C; C ⊗ C; Δ; Δ; id ⊗ϵ; C ⊗ C; 𝕂 ⊗ C ≅ C ≅ C ⊗ 𝕂.; ϵ⊗id; id]

A 𝕂-coalgebra C is cocommutative if the following diagram commutes

[figure: C ⊗ C; C ⊗ C; τ; C; Δ; Δ]

Let C and D be 𝕂-coalgebras. A homomorphism of coalgebras f : C ⟶ D is a 𝕂-linear map such that the following diagrams commute:

[figure: C ⊗ C; D ⊗ D; f⊗f; C; D; f; Δ_{C}; Δ_{D}]

and

[figure: 𝕂; ϵ_{C}; ϵ_{D}; C; D.; f]

Remark 8.6.2. Obviously the composition of two homomorphisms of coalgebras is again a homomorphism of coalgebras. Furthermore the identity map is a homomorphism of coalgebras. Hence the 𝕂-coalgebras form a category 𝕂-Coalg. The category of cocommutative 𝕂-coalgebras will be denoted by 𝕂-cCoalg.

Problem 8.6.11. 1. Show that V ⊗V^{∗} is a coalgebra for every finite dimensional vector space V over a field 𝕂 if the comultiplication is defined by Δ(v ⊗ v^{∗}) := ∑_{i=1}^{n} v ⊗ v_{i}^{∗} ⊗ v_{i} ⊗ v^{∗} where {v_{i}} and {v_{i}^{∗}} are dual bases of V resp. V^{∗}.

2. Show that the free 𝕂-modules 𝕂X with the basis X and the comultiplication Δ(x) = x ⊗ x is a coalgebra. What is the counit? Is the counit unique?

3. Show that 𝕂 ⊕ V with Δ(1) = 1 ⊗ 1, Δ(v) = v ⊗ 1 + 1 ⊗ v defines a coalgebra.

4. Let C and D be coalgebras. Then C⊗D is a coalgebra with the comultiplication Δ_{C⊗D} := (1_{C} ⊗ τ ⊗ 1_{D})(Δ_{C} ⊗ Δ_{D}) : C ⊗ D ⊗ C ⊗ D ⟶ C ⊗ D and counit ε = ε_{C⊗D} : C ⊗ D ⟶ 𝕂 ⊗ K ⟶ 𝕂. (The proof is analogous to the proof of Lemma 8.5.3.)

To describe the comultiplication of a 𝕂-coalgebra in terms of elements we introduce a notation first introduced by Sweedler similar to the notation ∇(a ⊗ b) = ab used for algebras. Instead of Δ(c) = ∑ c_{i} ⊗ c′_{i} we write

Δ(c) = ∑ c_{(1)} ⊗ c_{(2)}.

Observe that only the complete expression on the right hand side makes sense, not the components c_{(1)} or c_{(2)} which are not considered as families of elements of C. This notation alone does not help much in the calculations we have to perform later on. So we introduce a more general notation.

Definition 8.6.3. (Sweedler Notation) Let M be an arbitrary 𝕂-module and C be a 𝕂-coalgebra. Then there is a bijection between all multilinear maps

f : C × … × C ⟶ M

and all linear maps

f′ : C ⊗ … ⊗ C ⟶ M.

These maps are associated to each other by the formula

f(c_{1}, …, c_{n}) = f′(c_{1} ⊗ … ⊗ c_{n}).

For c ∈ C we define

∑ f(c_{(1)}, …, c_{(n)}) := f′(Δ^{n−1}(c)),

where Δ^{n−1} denotes the n − 1-fold application of Δ, for example Δ^{n−1} = (Δ ⊗ 1 ⊗ … ⊗ 1) ∘ (Δ ⊗ 1) ∘ Δ.

In particular we obtain for the bilinear map ⊗ : C × C ∋ (c, d) ↦ c ⊗ d ∈ C ⊗ C

∑ c_{(1)} ⊗ c_{(2)} = Δ(c),

and for the multilinear map ⊗^{2} : C × C × C ⟶ C ⊗ C ⊗ C

∑ c_{(1)} ⊗ c_{(2)} ⊗ c_{(3)} = (Δ ⊗ 1)Δ(c) = (1 ⊗ Δ)Δ(c).

With this notation one verifies easily

∑ c_{(1)} ⊗ … ⊗ Δ(c_{(i)}) ⊗ … ⊗ c_{(n)} = ∑ c_{(1)} ⊗ … ⊗ c_{(n+1)}

and

\begin{aligned}∑ c_{(1)} ⊗ … ⊗ ϵ(c_{(i)}) ⊗ … ⊗ c_{(n)} & = ∑ c_{(1)} ⊗ … ⊗ 1 ⊗ … ⊗ c_{(n−1)} \\ & = ∑ c_{(1)} ⊗ … ⊗ c_{(n−1)} \end{aligned}

This notation and its application to multilinear maps will also be used in more general contexts like comodules.

Proposition 8.6.4. Let C be a coalgebra and A an algebra. Then the composition f ∗ g := ∇_{A}(f ⊗ g)Δ_{C} defines a multiplication

Hom(C, A) ⊗ Hom(C, A) ∋ f ⊗ g ↦ f ∗ g ∈ Hom(C, A),

such that Hom(C, A) becomes an algebra. The unit element is given by 𝕂 ∋ α ↦ (c ↦ η(αϵ(c))) ∈ Hom(C, A).

Proof. The multiplication of Hom(C, A) obviously is a bilinear map. The multiplication is associative since (f ∗ g) ∗ h = ∇_{A}((∇_{A}(f ⊗ g)Δ_{C}) ⊗ h)Δ_{C} = ∇_{A}(∇_{A} ⊗ 1)((f ⊗g)⊗h)(Δ_{C} ⊗1)Δ_{C} = ∇_{A}(1⊗∇_{A})(f ⊗(g ⊗h))(1⊗Δ_{C})Δ_{C} = ∇_{A}(f ⊗(∇_{A}(g ⊗ h)Δ_{C}))Δ_{C} = f ∗ (g ∗ h). Furthermore it is unitary with unit 1_{Hom(C,A)} = η_{A}ϵ_{C} since η_{A}ϵ_{C} ∗ f = ∇_{A}(η_{A}ϵ_{C} ⊗ f)Δ_{C} = ∇_{A}(η_{A} ⊗ 1_{A})(1_{K} ⊗ f)(ϵ_{C} ⊗ 1_{C})Δ_{C} = f and similarly f ∗ η_{A}ϵ_{C} = f.

Definition 8.6.5. The multiplication ∗ : Hom(C, A)⊗Hom(C, A) ⟶ Hom(C, A) is called convolution.

Corollary 8.6.6. Let C be a 𝕂-coalgebra. Then C^{∗} = Hom_{K}(C, 𝕂) is an 𝕂algebra.

Proof. Use that 𝕂 itself is a 𝕂-algebra.

Remark 8.6.7. If we write the evaluation as C^{∗} ⊗ C ∋ a ⊗ c ↦ ⟨a, c⟩ ∈ 𝕂 then an element a ∈ C^{∗} is completely determined by the values of ⟨a, c⟩ for all c ∈ C. So the product of a and b in C^{∗} is uniquely determined by the formula

⟨a ∗ b, c⟩ = ⟨a ⊗ b, Δ(c)⟩ = ∑ a(c_{(1)})b(c_{(2)}).

The unit element of C^{∗} is ϵ ∈ C^{∗}.

Lemma 8.6.8. Let 𝕂 be a field and A be a finite dimensional 𝕂-algebra. Then A^{∗} = Hom_{K}(A, 𝕂) is a 𝕂-coalgebra.

Proof. Define the comultiplication on C^{∗} by

Δ : A^{∗} ⟶^{∇^{∗}} (A ⊗ A)^{∗} ⟶^{can^{−1}} A^{∗} ⊗ A^{∗}.

The canonical map can : A^{∗}⊗A^{∗} ⟶ (A⊗A)^{∗} is invertible, since A is finite dimensional. By a diagrammatic proof or by calculation with elements it is easy to show that A^{∗} becomes a 𝕂-coalgebra.

Remark 8.6.9. If 𝕂 is an arbitrary commutative ring, then A^{∗} = Hom_{K}(A, 𝕂) is a 𝕂-coalgebra if A is a finitely generated projective 𝕂-module.

Problem 8.6.12. Find sufficient conditions for an algebra A resp. a coalgebra C such that Hom(A, C) becomes a coalgebra with co-convolution as comultiplication.

Definition 8.6.10. Let C be a 𝕂-coalgebra. A left C-comodule is a 𝕂-module M together with a homomorphism δ_{M} : M ⟶ C ⊗ M, such that the diagrams

[figure: C ⊗ M; C ⊗ C ⊗ M; id ⊗δ; M; C ⊗ M; δ; δ; Δ⊗id]

and

[figure: M; δ; C ⊗ M; 𝕂 ⊗ M ≅ M.; ϵ⊗id; id]

commute.

Let ^{C}M and ^{C}N be C-comodules and let f : M ⟶ N be a 𝕂-linear map. The map f is called a homomorphism of comodules if the diagram

[figure: N; C ⊗ N; δ_{N}; M; C ⊗ M; δ_{M}; f; 1⊗f]

commutes.

The left C-comodules and their homomorphisms form the category ^{C}M of comodules.

Let N be an arbitrary 𝕂-module and M be a C-comodule. Then there is a bijection between all multilinear maps

f : C × … × M ⟶ N

and all linear maps

f′ : C ⊗ … ⊗ M ⟶ N.

These maps are associated to each other by the formula

f(c_{1}, …, c_{n}, m) = f′(c_{1} ⊗ … ⊗ c_{n} ⊗ m).

For m ∈ M we define

∑ f(m_{(1)}, …, m_{(n)}, m_{(M)}) := f′(δ^{n}(m)),

where δ^{n} denotes the n-fold application of δ, i.e. δ^{n} = (1 ⊗ … ⊗ 1 ⊗ δ) ∘ (1 ⊗ δ) ∘ δ.

In particular we obtain for the bilinear map ⊗ : C × M ⟶ C ⊗ M

∑ m_{(1)} ⊗ m_{(M)} = δ(m),

and for the multilinear map ⊗^{2} : C × C × M ⟶ C ⊗ C ⊗ M

∑ m_{(1)} ⊗ m_{(2)} ⊗ m_{(M)} = (1 ⊗ δ)δ(c) = (Δ ⊗ 1)δ(m).

Problem 8.6.13. Show that a finite dimensional vector space V is a comodule over the coalgebra V ⊗ V^{∗} as defined in problem 8.11.1 with the coaction δ(v) := ∑ v ⊗ v_{i}^{∗} ⊗ v_{i} ∈ (V ⊗ V^{∗}) ⊗ V where ∑ v_{i}^{∗} ⊗ v_{i} is the dual basis of V in V^{∗} ⊗ V.

Theorem 8.6.11. (Fundamental Theorem for Comodules) Let 𝕂 be a field. Let M be a left C-comodule and let m ∈ M be given. Then there exists a finite dimensional subcoalgebra C′ ⊆ C and a finite dimensional C′-comodule M′ with m ∈ M′ ⊆ M where M′ ⊆ M is a 𝕂-submodule, such that the diagram

[figure: M; C ⊗ M; δ; M′; C′ ⊗ M′; δ′]

commutes.

Corollary 8.6.12. 1. Each element c ∈ C of a coalgebra is contained in a finite dimensional subcoalgebra of C.

2. Each element m ∈ M of a comodule is contained in a finite dimensional subcomodule of M.

Corollary 8.6.13. 1. Each finite dimensional subspace V of a coalgebra C is contained in a finite dimensional subcoalgebra C′ of C.

2. Each finite dimensional subspace V of a comodule M is contained in a finite dimensional subcomodule M′ of M.

Corollary 8.6.14. 1. Each coalgebra is a union of finite dimensional subcoalgebras.

2. Each comodule is a union of finite dimensional subcomodules.

Proof. (of the Theorem) We can assume that m ≠ 0 for else we can use M′ = 0 and C′ = 0.

Under the representations of δ(m) ∈ C ⊗M as finite sums of decomposable tensors pick one

δ(m) = ∑_{i=1}^{s} c_{i} ⊗ m_{i}

of shortest length s. Then the families (c_{i}|i = 1, …, s) and (m_{i}|i = 1, …, s) are linearly independent. Choose coefficients c_{ij} ∈ C such that

Δ(c_{j}) = ∑_{i=1}^{t} c_{i} ⊗ c_{ij}, ∀j = 1, …, s,

by suitably extending the linearly independent family (c_{i}|i = 1, …, s) to a linearly independent family (c_{i}|i = 1, …, t) and t ≥ s.

We first show that we can choose t = s. By coassociativity we have ∑_{i=1}^{s} c_{i} ⊗ δ(m_{i}) = ∑_{j=1}^{s} Δ(c_{j}) ⊗ m_{j} = ∑_{j=1}^{s} ∑_{i=1}^{t} c_{i} ⊗ c_{ij} ⊗ m_{j}. Since the c_{i} and the m_{j} are linearly independent we can compare coefficients and get

δ(m_{i}) = ∑_{j=1}^{s} c_{ij} ⊗ m_{j}, ∀i = 1, …, s (1)

and 0 = ∑_{j=1}^{s} c_{ij} ⊗ m_{j} for i > s. The last statement implies

c_{ij} = 0, ∀i > s, j = 1, …, s.

Hence we get t = s and

Δ(c_{j}) = ∑_{i=1}^{s} c_{i} ⊗ c_{ij}, ∀j = 1, …, s.

Define finite dimensional subspaces C′ = ⟨c_{ij}|i, j = 1, …, s⟩ ⊆ C and M′ = ⟨m_{i}|i = 1, …, s⟩ ⊆ M. Then by (1) we get δ : M′ ⟶ C′ ⊗ M′. We show that m ∈ M′ and that the restriction of Δ to C′ gives a linear map Δ : C′ ⟶ C′ ⊗ C′ so that the required properties of the theorem are satisfied. First observe that m = ∑ ε(c_{i})m_{i} ∈ M′ and c_{j} = ∑ ε(c_{i})c_{ij} ∈ C′. Using coassociativity we get

\begin{aligned}∑_{i,j=1}^{n} c_{i} ⊗ Δ(c_{ij}) ⊗ m_{j} & = ∑_{k,j=1}^{s} Δ(c_{k}) ⊗ c_{kj} ⊗ m_{j} \\ & = ∑_{i,j,k=1}^{s} c_{i} ⊗ c_{ik} ⊗ c_{kj} ⊗ m_{j} \end{aligned}

hence

Δ(c_{ij}) = ∑_{k=1}^{s} c_{ik} ⊗ c_{kj}. (2)

Remark 8.6.15. We give a sketch of a second proof which is somewhat more technical. Since C is a 𝕂-coalgebra, the dual C^{∗} is an algebra. The comodule structure δ : M ⟶ C ⊗ M leads to a module structure by ρ = (ev ⊗1)(1 ⊗ δ) : C^{∗} ⊗ M ⟶ C^{∗}⊗C⊗M ⟶ M. Consider the submodule N := C^{∗}m. Then N is finite dimensional, since c^{∗}m = ∑_{i=1}^{n}⟨c^{∗}, c_{i}⟩m_{i} for all c^{∗} ∈ C^{∗} where ∑_{i=1}^{n} c_{i} ⊗ m_{i} = δ(m). Observe that C^{∗}m is a subspace of the space generated by the m_{i}. But it does not depend on the choice of the m_{i}. Furthermore if we take δ(m) = ∑ c_{i} ⊗ m_{i} with a shortest representation then the m_{i} are in C^{∗}m since c^{∗}m = ∑⟨c^{∗}, c_{i}⟩m_{i} = m_{i} for c^{∗} an element of a dual basis of the c_{i}.

N is a C-comodule since δ(c^{∗}m) = ∑⟨c^{∗}, c_{i}⟩δ(m_{i}) = ∑⟨c^{∗}, c_{i(1)}⟩c_{i(2)} ⊗ m_{i} ∈ C ⊗ C^{∗}m.

Now we construct a subcoalgebra D of C such that N is a D-comodule with the induced coaction. Let D := N ⊗ N^{∗}. By 8.13 N is a comodule over the coalgebra N ⊗ N^{∗}. Construct a linear map ϕ : D ⟶ C by n ⊗ n^{∗} ↦ ∑ n_{(1)}⟨n^{∗}, n_{(N)}⟩. By definition of the dual basis we have n = ∑ n_{i}⟨n_{i}^{∗}, n⟩. Thus we get

(ϕ ⊗ ϕ)Δ_{D}(n ⊗ n^{∗}) = (ϕ ⊗ ϕ)(∑ n ⊗ n_{i}^{∗} ⊗ n_{i} ⊗ n^{∗})
= ∑ n_{(1)}⟨n_{i}^{∗}, n_{(N)}⟩ ⊗ n_{i(1)}⟨n^{∗}, n_{i(N)}⟩
= ∑ n_{(1)} ⊗ n_{i(1)}⟨n^{∗}, n_{i(N)}⟩⟨n_{i}^{∗}, n_{(N)}⟩
= ∑ n_{(1)} ⊗ n_{(2)}⟨n^{∗}, n_{(N)}⟩ = ∑ Δ_{C}(n_{(1)})⟨n^{∗}, n_{(N)}⟩
= Δ_{C}ϕ(n ⊗ n^{∗}).

Furthermore ε_{C}ϕ(n ⊗ n^{∗}) = ε(∑ n_{(1)}⟨n^{∗}, n_{(N)}⟩ = ⟨n^{∗}, ∑ ε(n_{(1)})n_{(N)}⟩ = ⟨n^{∗}, n⟩ = ε(n⊗n^{∗}). Hence ϕ : D ⟶ C is a homomorphism of coalgebras, D is finite dimensional and the image C′ := ϕ(D) is a finite dimensional subcoalgebra of C. Clearly N is also a C′-comodule, since it is a D-comodule.

Finally we show that the D-comodule structure on N if lifted to the C-comodule structure coincides with the one defined on M. We have

δ_{C}(c^{∗}m) = δ_{C}(∑⟨c^{∗}, m_{(1)}⟩m_{(M)}) = ∑⟨c^{∗}, m_{(1)}⟩m_{(2)} ⊗ m_{(M)}

= ∑⟨c^{∗}, m_{(1)}⟩m_{(2)} ⊗ m_{i}⟨m_{i}^{∗}, m_{(M)}⟩ = ∑⟨c^{∗}, m_{(1)}⟩m_{(2)}⟨m_{i}^{∗}, m_{(M)}⟩ ⊗ m_{i} = (ϕ ⊗ 1)(∑⟨c^{∗}, m_{(1)}⟩m_{(M)} ⊗ m_{i}^{∗} ⊗ m_{i}) = (ϕ ⊗ 1)(∑ c^{∗}m ⊗ m_{i}^{∗} ⊗ m_{i})

= (ϕ ⊗ 1)δ_{D}(c^{∗}m).

7. Bialgebras

Definition 8.7.1. 1. A bialgebra (B, ∇, η, Δ, ϵ) consists of an algebra (B, ∇, η) and a coalgebra (B, Δ, ϵ) such that the diagrams

[figure: B ⊗ B; B ⊗ B ⊗ B ⊗ B; Δ⊗Δ; 1⊗τ⊗1; ∇; B; B ⊗ B; Δ; B ⊗ B ⊗ B ⊗ B; ∇⊗∇]

and

[figure: 𝕂; B; η; η⊗η; B; B ⊗ B; Δ; B ⊗ B; B; ∇; 𝕂; ϵ⊗ϵ; ϵ; 𝕂; 𝕂; id; B; η; ϵ]

commute, i.e. Δ and ϵ are homomorphisms of algebras resp. ∇ and η are homomorphisms of coalgebras.

2. Given bialgebras A and B. A map f : A ⟶ B is called a homomorphism of bialgebras if it is a homomorphism of algebras and a homomorphism of coalgebras.

3. The category of bialgebras is denoted by 𝕂-Bialg.

Problem 8.7.14. 1. Let (B, ∇, η) be an algebra and (B, Δ, ε) be a coalgebra. The following are equivalent:

a) (B, ∇, η, Δ, ε) is a bialgebra.

b) Δ : B ⟶ B ⊗ B and ε : B ⟶ 𝕂 are homomorphisms of 𝕂-algebras.

c) ∇ : B ⊗ B ⟶ B and η : 𝕂 ⟶ B are homomorphisms of 𝕂-coalgebras.

2. Let B be a finite dimensional bialgebra over field 𝕂. Show that the dual space B^{∗} is a bialgebra.

One of the most important properties of bialgebras B is that the tensor product over 𝕂 of two B-modules or two B-comodules is again a B-module.

Proposition 8.7.2. 1. Let B be a bialgebra. Let M and N be left B-modules. Then M ⊗_{𝕂} N is a B-module by the map

B ⊗ M ⊗ N ⟶^{Δ⊗1} B ⊗ B ⊗ M ⊗ N ⟶^{1⊗τ⊗1} B ⊗ M ⊗ B ⊗ N ⟶^{μ⊗μ} M ⊗ N.

2. Let B be a bialgebra. Let M and N be left B-comodules. Then M ⊗_{𝕂} N is a B-comodule by the map

M ⊗ N ⟶^{δ⊗δ} B ⊗ M ⊗ B ⊗ N ⟶^{1⊗τ⊗1} B ⊗ B ⊗ M ⊗ N ⟶^{∇⊗1} B ⊗ M ⊗ N.

3. 𝕂 is a B-module by the map B ⊗ 𝕂 ≅ B ⟶^{ε} 𝕂.

4. 𝕂 is a B-comodule by the map 𝕂 ⟶^{η} B ≅ B ⊗ 𝕂.

Proof. We give a diagrammatic proof for 1. The associativity law is given by

[figure: B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ B ⊗ M ⊗ N; 1⊗Δ⊗1⊗1; B ⊗ B ⊗ M ⊗ B ⊗ N; 1⊗1⊗τ⊗1; B ⊗ M ⊗ N; 1⊗μ⊗μ; B ⊗ B ⊗ B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ B ⊗ M ⊗ B ⊗ N; 1⊗1⊗1⊗τ⊗1; B ⊗ B ⊗ M ⊗ N; 1⊗1⊗μ⊗μ; B ⊗ B ⊗ B ⊗ B ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ B ⊗ B ⊗ N; 1⊗1⊗τ(B⊗B,M)⊗1; B ⊗ M ⊗ B ⊗ N; 1⊗μ⊗1⊗μ; B ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ N; Δ⊗1⊗1; B ⊗ M ⊗ B ⊗ N; 1⊗τ⊗1; M ⊗ N; μ⊗μ; ∇⊗1⊗1; Δ⊗1⊗1⊗1⊗1; Δ⊗1⊗1⊗1⊗1; Δ⊗1⊗1; 1⊗τ⊗1⊗1⊗1; 1⊗τ(B,B⊗M)⊗1⊗1; 1⊗τ⊗1; ∇⊗∇⊗1⊗1; ∇⊗1⊗∇⊗1; μ⊗μ]

The unit law is the commutativity of

[figure: M ⊗ N ≅ 𝕂 ⊗ M ⊗ N; B ⊗ M ⊗ N; η⊗1⊗1; 𝕂 ⊗ 𝕂 ⊗ M ⊗ N; B ⊗ B ⊗ M ⊗ N; η⊗η⊗1⊗1; M ⊗ N ≅ 𝕂 ⊗ M ⊗ 𝕂 ⊗ N; B ⊗ M ⊗ B ⊗ N; η⊗1⊗η⊗1; ≅; 1⊗τ⊗1; Δ⊗1⊗1; 1⊗τ⊗1; μ⊗μ; =; M ⊗ N; 1]

The corresponding properties for comodules follows from the dualized diagrams. The module and comodule properties of 𝕂 are easily checked.

Definition 8.7.3. 1. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let A be a left B-module with structure map μ : B ⊗ A ⟶ A. Let furthermore (A, ∇_{A}, η_{A}) be an algebra such that ∇_{A} and η_{A} are homomorphisms of B-modules. Then (A, ∇_{A}, η_{A}, μ) is called a B-module algebra.

2. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let C be a left B-module with structure map μ : B ⊗C ⟶ C. Let furthermore (C, Δ_{C}, ε_{C}) be a coalgebra such that Δ_{C} and ε_{C} are homomorphisms of B- modules. Then (C, Δ_{C}, ε_{C}, μ) is called a B-module coalgebra.

3. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let A be a left B-comodule with structure map δ : A ⟶ B ⊗ A. Let furthermore (A, ∇_{A}, η_{A}) be an algebra such that ∇_{A} and η_{A} are homomorphisms of B-comodules. Then (A, ∇_{A}, η_{A}, δ) is called a B-comodule algebra.

4. Let (B, ∇, η, Δ, ϵ) be a bialgebra. Let C be a left B-comodule with structure map δ : C ⟶ B ⊗ C. Let furthermore (C, Δ_{C}, ε_{C}) be a coalgebra such that Δ_{C} and ε_{C} are homomorphisms of B-comodules. Then (C, Δ_{C}, ε_{C}, δ) is called a B-comodule coalgebra.

Remark 8.7.4. If (C, Δ_{C}, ε_{C}) is a 𝕂-coalgebra and (C, μ) is a B-module, then (C, Δ_{C}, ε_{C}, μ) is a B-module coalgebra iff μ is a homomorphism of 𝕂-coalgebras.

If (A, ∇_{A}, η_{A}) is a 𝕂-algebra and (A, δ) is a B-comodule, then (A, ∇_{A}, η_{A}, δ) is a B-comodule algebra iff δ is a homomorphism of 𝕂-algebras.

Similar statement for module algebras or comodule coalgebras do not hold.

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.

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 8.9.15. 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 8.9.16. 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).