CHAPTER 1 Toolbox

1. Categories

Definition 1.1.1. 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 1.1.2. 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 1.1.3. 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 1.1.4. 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 1.1.5. 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 1.1.6. 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 1.1.7. 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 1.1.8. 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 1.1.9. 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 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 1.1.10. 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 1.1.11. 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 1.1.12. 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 1.1.13. 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 1.2.1. 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 1.2.2. 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 1.2.3. 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 1.2.4. 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 1.8).

3. Natural Transformations

Definition 1.3.1. 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 1.3.2. 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 1.3.3. 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 1.3.4. 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 1.3.5. 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 1.2. 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 1.3.6. 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 1.3. 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 1.4.1. 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 1.4.2. 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 1.4.3. 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 1.4.4. (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 1.4.5. 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 1.4.6. 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 1.4.7. 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 1.4.8. 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 1.4.9. 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 1.4.10. 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 1.4.11. 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 1.4.12. 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 1.4.13. 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 1.4.14. 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 1.4.15. 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 (1.4.45.) 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 1.4. 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.