CHAPTER 3 Representation Theory, Reconstruction and Tannaka Duality

Introduction

One of the most interesting properties of quantum groups is their representation theory. It has deep applications in theoretical physics. The mathematical side has to distinguish between the representation theory of quantum groups and the representation theory of Hopf algebras. In both cases the particular structure allows to form tensor products of representations such that the category of representations becomes a monoidal (or tensor) category.

The problem we want to study in this chapter is, how much structure of the quantum group or Hopf algebra can be found in the category of representations. We will show that a quantum monoid can be uniquely reconstructed (up to isomorphism) from its representations. The additional structure given by the antipode is itimitely connected with a certain duality of representations. We will also generalize this process of reconstruction.

On the other hand we will show that the process of reconstruction can also be used to obtain the Tambara construction of the universal quantum monoid of a noncommutative geometrical space (from chapter 1.). Thus we will get another perspective for this theorem.

At the end of the chapter you should

• understand representations of Hopf algebras and of quantum groups,

• know the definition and first fundamental properties of monoidal or tensor categories,

• be familiar with the monoidal structure on the category of representations of Hopf algebras and of quantum groups,

• understand why the category of representations contains the full information about the quantum group resp. the Hopf algebra (Theorem of Tannaka-Krein),

• know the process of reconstruction and examples of bialgebras reconstructed from certain diagrams of finite dimensional vector spaces,

• understand better the Tambara construction of a universal algebra for a finite dimensional algebra.

1. Representations of Hopf Algebras

Let A be an algebra over a commutative ring 𝕂. Let A-Mod be the category of A-modules. An A-module is also called a representation of A.

Observe that the action A⊗M ⟶ M satisfying the module axioms and an algebra homomorphism A ⟶ End(M) are equivalent descriptions of an A-module structure on the 𝕂-module M.

The functor U : A-Mod ⟶ 𝕂-Mod with U(_{A}M) = M and U(f) = f is called the forgetful functor or the underlying functor.

If B is a bialgebra then a representation of B is also defined to be a B-module. It will turn out that the property of being a bialgebra leads to the possibility of building tensor products of representations in a canonical way.

Let C be a coalgebra over a commutative ring 𝕂. Let C-Comod be the category of C-comodules. A C-comodule is also called a corepresentation of C.

The functor U : C-Comod ⟶ 𝕂-Mod with U(^{C}M) = M and U(f) = f is called the forgetful functor or the underlying functor.

If B is a bialgebra then a corepresentation of B is also defined to be a B-comodule. It will turn out that the property of being a bialgebra leads to the possibility of building tensor products of corepresentations in a canonical way.

Usually representations of a ring are considered to be modules over the given ring. The role of comodules certainly arises in the context of coalgebras. But it is not quite clear what the good definition of a representation of a quantum group or its representing Hopf algebra is.

For this purpose consider representations M of an ordinary group G. Assume for the simplicity of the argument that G is finite. Representations of G are vector spaces together with a group action G × M ⟶ M. Equivalently they are vector spaces together with a group homomorphism G ⟶ Aut(M) or modules over the group algebra: 𝕂[G] ⊗ M ⟶ M. In the situation of quantum groups we consider the representing Hopf algebra H as algebra of functions on the quantum group G.

Then the algebra of functions on G is the Hopf algebra 𝕂^{G}, the dual of the group algebra 𝕂[G]. An easy exercise shows that the module structure 𝕂[G] ⊗ M ⟶ M translates to the structure of a comodule M ⟶ 𝕂^{G}⊗M and conversely. (Observe that G is finite.) So we should define representations of a quantum group as comodules over the representing Hopf algebra.

Definition 3.1.1. Let G be a quantum group with representing Hopf algebra H. A representation of G is a comodule over the representing Hopf algebra H.

From this definition we obtain immediately that we may form tensor products of representations of quantum groups since the representing algebra is a bialgebra.

We come now to the canonical construction of tensor products of (co-)represen-tations.

Lemma 3.1.2. Let B be a bialgebra. Let M, N ∈ B-Mod be two B-modules. Then M ⊗N is a B-module by the action b(m⊗n) = ∑ b_{(1)}m⊗b_{(2)}n. If f : M ⟶ M′ and g : N ⟶ N′ are homomorphisms of B-modules in B-Mod then f ⊗ g : M ⊗ N ⟶ M′ ⊗ N′ is a homomorphism of B-modules.

Proof. We have homomorphisms of 𝕂-algebras α : B ⟶ End(M) and β : B ⟶ End(N) defining the B-module structure on M and N. Thus we get a homomorphism of algebras can(α ⊗ β)Δ : B ⟶ B ⊗ B ⟶ End(M) ⊗ End(N) ⟶ End(M ⊗ N). Thus M ⊗N is a B-module. The structure is b(m⊗n) = can(α⊗β)(∑ b_{(1)})⊗b_{(2)})(m⊗n) = can(∑ α(b_{(1)}) ⊗ β(b_{(2)}))(m ⊗ n) = ∑ α(b_{(1)})(m) ⊗ β(b_{(2)})(n) = ∑ b_{(1)}m ⊗ b_{(2)}n.

Furthermore we have 1(m ⊗ n) = 1m ⊗ 1m = m ⊗ n.

If f, g are homomorphisms of B-modules, then we have (f ⊗ g)(b(m ⊗ n)) = (f ⊗ g)(∑ b_{(1)}m ⊗ b_{(2)}n) = ∑ f(b_{(1)}m) ⊗ g(b_{(2)}n) = ∑ b_{(1)}f(m) ⊗ b_{(2)}g(n) = b(f(m) ⊗ g(n)) = b(f ⊗ g)(m ⊗ n). Thus f ⊗ g is a homomorphism of B-modules.

Corollary 3.1.3. Let B be a bialgebra. Then ⊗ : B-Mod × B-Mod ⟶ B-Mod with ⊗(M, N) = M ⊗ N and ⊗(f, g) = f ⊗ g is a functor.

Proof. The following are obvious from the ordinary properties of the tensor product over 𝕂. 1_{M} ⊗1_{N} = 1_{M⊗N} and (f ⊗g)(f′⊗g′) = ff′⊗gg′ for M, N, f, f′, g, g′ ∈ B-Mod.

Lemma 3.1.4. Let B be a bialgebra. Let M, N ∈ B-Comod be two B-comodules. Then M⊗N is a B-comodule by the coaction δ_{M⊗N}(m⊗n) = ∑ m_{(1)}n_{(1)}⊗m_{(M)}⊗n_{(N)}.

If f : M ⟶ M′ and g : N ⟶ N′ are homomorphisms of B-comodules in B-Comod then f ⊗ g : M ⊗ N ⟶ M′ ⊗ N′ is a homomorphism of B-comodules.

Proof. The coaction on M ⊗ N may also be described by (∇_{B} ⊗ 1_{M} ⊗ 1_{N})(1_{B} ⊗ τ ⊗ 1_{N})(δ_{M} ⊗ δ_{N}) : M ⊗ N ⟶ B ⊗ M ⊗ B ⊗ N ⟶ B ⊗ B ⊗ M ⊗ N ⟶ B ⊗ M ⊗ N. Although a diagrammatic proof of the coassociativity of the coaction and the law of the counit is quite involved it allows generalization so we give it here.

Consider the next diagram.

Square (1) commutes since M and N are comodules.

Squares (2) and (3) commute since τ : M ⊗ N ⟶ N ⊗ M for 𝕂-modules M and N is a natural transformation.

Square (4) represents an interesting property of τ namely

\begin{matrix}(1 ⊗ 1 ⊗ τ)(τ_{B⊗M,B} ⊗ 1) = (1 ⊗ 1 ⊗ τ)(τ ⊗ 1 ⊗ 1)(1 ⊗ τ ⊗ 1) = \\ (τ ⊗ 1 ⊗ 1)(1 ⊗ 1 ⊗ τ)(1 ⊗ τ ⊗ 1) = (τ ⊗ 1 ⊗ 1)(1 ⊗ τ_{M,B⊗B}) \end{matrix}

that uses the fact that (1 ⊗ g)(f ⊗ 1) = (f ⊗ 1)(1 ⊗ g) holds and that τ_{B⊗M,B} = (τ ⊗ 1)(1 ⊗ τ) and τ_{M,B⊗B} = (1 ⊗ τ)(τ ⊗ 1).

Square (5) and (6) commute by the properties of the tensor product.

Square (7) commutes since B is a bialgebra.

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; δ⊗δ; Δ⊗1⊗Δ⊗1; Δ⊗Δ⊗1⊗1; Δ⊗1⊗1; B ⊗ M ⊗; B ⊗ B ⊗ M⊗; B ⊗ N; B ⊗ B ⊗ N; 1⊗δ⊗1⊗δ; B ⊗ B ⊗ B⊗; B ⊗ M ⊗ N; 1^{2}⊗τ_{M,B⊗B}⊗1; 1⊗τ⊗1; 1⊗τ_{B⊗M,B}⊗1⊗1; 1⊗τ⊗1⊗1⊗1; B ⊗ B ⊗; B ⊗ B ⊗ B⊗; M ⊗ N; M ⊗ B ⊗ N; 1⊗1⊗δ⊗δ; B ⊗ B ⊗ B⊗; B ⊗ M ⊗ N; 1⊗1⊗1⊗τ⊗1; ∇⊗1⊗1; ∇⊗1⊗1⊗1⊗1; ∇⊗1⊗1⊗1⊗1; B ⊗ B⊗; M ⊗ B ⊗ N; B ⊗ M ⊗ N; 1⊗δ⊗δ; B ⊗ B⊗; B ⊗ M ⊗ N; 1⊗1⊗τ⊗1; B ⊗ B ⊗ M ⊗ N; 1⊗∇⊗1⊗1; (1); (2); (3); (4); (5); (6); (7)]

The law of the counit is

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; 1; M ⊗ N; M ⊗ N; 1; M ⊗ N; 1; ε⊗1⊗ε⊗1; ε⊗ε⊗1⊗1; ε⊗1⊗1]

where the last square commutes since ε is a homomorphism of algebras. Now let f and g be homomorphisms of B-comodules. Then the diagram

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; f⊗g; 1⊗f⊗1⊗g; 1⊗1⊗f⊗g; 1⊗f⊗g; B ⊗ M′⊗; B ⊗ N′; M′ ⊗ N′; δ⊗δ; B ⊗ B⊗; M′ ⊗ N′; 1⊗τ⊗1; B ⊗ M′ ⊗ N′; ∇⊗1⊗1]

commutes. Thus f ⊗ g is a homomorphism of B-comodules.

Corollary 3.1.5. Let B be a bialgebra. Then ⊗ : B-Comod × B-Comod ⟶ B-Comod with ⊗(M, N) = M ⊗ N and ⊗(f, g) = f ⊗ g is a functor.

Proposition 3.1.6. Let B be a bialgebra. Then the tensor product ⊗ : B-Mod× B-Mod ⟶ B-Mod satisfies the following properties:

1. The associativity isomorphism α : (M_{1} ⊗ M_{2}) ⊗ M_{3} ⟶ M_{1} ⊗ (M_{2} ⊗ M_{3}) with α((m ⊗ n) ⊗ p) = m ⊗ (n ⊗ p) is a natural transformation from the functor ⊗ ∘ (⊗ × Id) to the functor ⊗ ∘ (Id ×⊗) in the variables M_{1}, M_{2}, and M_{3} in B-Mod.

2. The counit isomorphisms λ : 𝕂 ⊗ M ⟶ M with λ(κ ⊗ m) = κm and ρ : M ⊗ 𝕂 ⟶ M with ρ(m ⊗ κ) = κm are natural transformations in the variable M in B-Mod from the functor 𝕂 ⊗ - resp. - ⊗ 𝕂 to the identity functor Id.

3. The following diagrams of natural transformations are commutative

[figure: ((M_{1} ⊗ M_{2}) ⊗ M_{3}) ⊗ M_{4}; (M_{1} ⊗ (M_{2} ⊗ M_{3})) ⊗ M_{4}; α(M_{1},M_{2},M_{3})⊗1; M_{1} ⊗ ((M_{2} ⊗ M_{3}) ⊗ M_{4}); α(M_{1},M_{2}⊗M_{3},M_{4}); α(M_{1}⊗M_{2},M_{3},M_{4}); 1⊗α(M_{2},M_{3},M_{4}); (M_{1} ⊗ M_{2}) ⊗ (M_{3} ⊗ M_{4}); M_{1} ⊗ (M_{2} ⊗ (M_{3} ⊗ M_{4})); α(M_{1},M_{2},M_{3}⊗M_{4})]

[figure: (M_{1} ⊗ 𝕂) ⊗ M_{2}; M_{1} ⊗ (𝕂 ⊗ M_{2}); α(M_{1},𝕂,M_{2}); M_{1} ⊗ M_{2}; ρ(M_{1})⊗1; 1⊗λ(M_{2})]

Proof. The homomorphisms α, λ, and ρ are already defined in the category 𝕂-Mod and satisfy the claimed properties. So we have to show, that these are homomorphisms in B-Mod and that 𝕂 is a B-module. 𝕂 is a B-module by ε ⊗ 1_{𝕂} : B ⊗ 𝕂 ⟶ 𝕂. The easy verification uses the coassociativity and the counital property of B.

Similarly we get

Proposition 3.1.7. Let B be a bialgebra. Then the tensor product

⊗ : B-Comod × B-Comod ⟶ B-Comod

satisfies the following properties:

1. The associativity isomorphism α : (M_{1} ⊗ M_{2}) ⊗ M_{3} ⟶ M_{1} ⊗ (M_{2} ⊗ M_{3}) with α((m ⊗ n) ⊗ p) = m ⊗ (n ⊗ p) is a natural transformation from the functor ⊗ ∘ (⊗ × Id) to the functor ⊗ ∘ (Id ×⊗) in the variables M_{1}, M_{2}, and M_{3} in B-Comod.

2. The counit isomorphisms λ : 𝕂 ⊗ M ⟶ M with λ(κ ⊗ m) = κm and ρ : M ⊗ 𝕂 ⟶ M with ρ(m ⊗ κ) = κm are natural transformations in the variable M in B-Comod from the functor 𝕂 ⊗ - resp. - ⊗ 𝕂 to the identity functor Id.

3. The following diagrams of natural transformations are commutative

[figure: ((M_{1} ⊗ M_{2}) ⊗ M_{3}) ⊗ M_{4}; (M_{1} ⊗ (M_{2} ⊗ M_{3})) ⊗ M_{4}; α(M_{1},M_{2},M_{3})⊗1; M_{1} ⊗ ((M_{2} ⊗ M_{3}) ⊗ M_{4}); α(M_{1},M_{2}⊗M_{3},M_{4}); α(M_{1}⊗M_{2},M_{3},M_{4}); 1⊗α(M_{2},M_{3},M_{4}); (M_{1} ⊗ M_{2}) ⊗ (M_{3} ⊗ M_{4}); M_{1} ⊗ (M_{2} ⊗ (M_{3} ⊗ M_{4})); α(M_{1},M_{2},M_{3}⊗M_{4})]

[figure: (M_{1} ⊗ 𝕂) ⊗ M_{2}; M_{1} ⊗ (𝕂 ⊗ M_{2}); α(M_{1},𝕂,M_{2}); M_{1} ⊗ M_{2}; ρ(M_{1})⊗1; 1⊗λ(M_{2})]

Remark 3.1.8. We now get some simple properties of the underlying functors U : B-Mod ⟶ 𝕂-Mod resp. U : B-Comod ⟶ 𝕂-Mod that are easily verified.

U(M ⊗ N) = U(M) ⊗ U(N),
U(f ⊗ g) = f ⊗ g,
U(𝕂) = 𝕂,
U(α) = α, U(λ) = λ, U(ρ) = ρ.

Problem 3.1.1. We have seen that in representation theory and in corepresentation theory of quantum groups such as 𝕂G, U(𝔤), SL_{q}(2), U_{q}(sl(2)) the ordinary tensor product (in 𝕂-Mod) of two (co-)reprensentations is in a canonical way again a (co- )reprensentation. For two 𝕂G-modules M and N the structure is g(m⊗n) = gm⊗gn for g ∈ G. For U(𝔤)-modules it is g(m⊗n) = gm⊗n+m⊗gn for g ∈ 𝔤. For U_{q}(sl(2))- modules it is E(m ⊗ n) = m ⊗ En + Em ⊗ Kn, F (m ⊗ n) = K^{−1}m ⊗ F n + F m ⊗ n, K(m ⊗ n) = Km ⊗ Kn.

Remark 3.1.9. Let A and B be algebras over a commutative ring 𝕂. Let f : A ⟶ B be a homomorphism of algebras. Then we have a functor U_{f} : B-Mod ⟶ A-Mod with U_{f}(_{B}M) = _{A}M and U_{f}(g) = g where am := f(a)m for a ∈ A and m ∈ M. The functor U_{f} is also called forgetful or underlying functor.

The action of A on a B-module M can also be seen as the homomorphism A ⟶ B ⟶ End(M).

We denote the underlying functors previously discussed by

U_{A} : A-Mod ⟶ 𝕂-Mod resp. U_{B} : B-Mod ⟶ 𝕂-Mod.

Proposition 3.1.10. Let f : B ⟶ C be a homomorphism of bialgebras. Then U_{f} satisfies the following properties:

U_{f}(M ⊗ N) = U_{f}(M) ⊗ U_{f}(N),
U_{f}(g ⊗ h) = g ⊗ h,
U_{f}(𝕂) = 𝕂,
U_{f}(α) = α, U_{f}(λ) = λ, U_{f}(ρ) = ρ,
U_{B}U_{f}(M) = U_{C}(M),
U_{B}U_{f}(g) = U_{C}(g).

Proof. This is clear since the underlying 𝕂-modules and the 𝕂-linear maps stay unchanged. The only thing to check is that U_{f} generates the correct B-module structure on the tensor product. For U_{f}(M ⊗ N) = M ⊗ N we have b(m ⊗ n) = f(b)(m ⊗ n) = ∑ f(b)_{(1)}m ⊗ f(b)_{(2)}n = ∑ f(b_{(1)})m ⊗ f(b_{(2)})n = ∑ b_{(1)}m ⊗ b_{(2)}n.

Remark 3.1.11. Let C and D be coalgebras over a commutative ring 𝕂. Let f : C ⟶ D be a homomorphism of coalgebras. Then we have a functor U_{f} : C-Comod ⟶ D-Comod with U_{f}(^{C}M) = ^{D}M and U_{f}(g) = g where δ_{D} = (f ⊗ 1)δ_{C} : M ⟶ C ⊗ M ⟶ D ⊗ M. Again the functor U_{f} is called forgetful or underlying functor.

We denote the underlying functors previously discussed by

U_{C} : C-Comod ⟶ 𝕂-Mod resp. U_{D} : D-Comod ⟶ 𝕂-Mod.

Proposition 3.1.12. Let f : B ⟶ C be a homomorphism of bialgebras. Then U_{f} : C-Comod ⟶ D-Comod satisfies the following properties:

U_{f}(M ⊗ N) = U_{f}(M) ⊗ U_{f}(N),
U_{f}(g ⊗ h) = g ⊗ h,
U_{f}(𝕂) = 𝕂,
U_{f}(α) = α, U_{f}(λ) = λ, U_{f}(ρ) = ρ,
U_{C}U_{f}(M) = U_{B}(M),
U_{C}U_{f}(g) = U_{B}(g).

Proof. We leave the proof to the reader.

Proposition 3.1.13. Let H be a Hopf algebra. Let M and N be be H-modules. Then Hom(M, N), the set 𝕂-linear maps from M to N, becomes an H-module by (hf)(m) = ∑ h_{(1)}f(S(h_{(2)}m). This structure makes

Hom : H-Mod × H-Mod ⟶ H-Mod

a functor contravariant in the first variable and covariant in the second variable.

Proof. The main part to be proved is that the action H ⊗ Hom(M, N) ⟶ Hom(M, N) satisfies the associativity law. Let f ∈ Hom(M, N), h, k ∈ H, and m ∈ M. Then ((hk)f)(m) = ∑(hk)_{(1)}f(S((hk)_{(2)}) = ∑ h_{(1)}k_{(1)}f(S(k_{(2)})S(h_{(2)})m) = ∑ h_{(1)}(kf)(S(h_{(2)})m) = (h(kf))(m).

We leave the proof of the other properties, in particular the functorial properties, to the reader.

Corollary 3.1.14. Let M be an H-module. Then the dual 𝕂-module M^{∗} = Hom(M, 𝕂) becomes an H-module by (hf)(m) = f(S(h)m).

Proof. The space 𝕂 is an H-module via ε : H ⟶ 𝕂. Hence we have (hf)(m) = ∑ h_{(1)}f(S(h_{(2)}m) = ∑ ε(h_{(1)})f(S(h_{(2)}m) = f(S(h)m).

2. Monoidal Categories

For our further investigations we need a generalized version of the tensor product that we are going to introduce in this section. This will give us the possibility to study more general versions of the notion of algebras and representations.

Definition 3.2.1. A monoidal category (or tensor category) consists of a category C, a covariant functor ⊗ : C × C ⟶ C, called the tensor product, an object I ∈ C, called the unit, natural isomorphisms

α(A, B, C) : (A ⊗ B) ⊗ C ⟶ A ⊗ (B ⊗ C),

λ(A) : I ⊗ A ⟶ A,
ρ(A) : A ⊗ I ⟶ A,

called associativity, left unit and right unit, such that the following diagrams commute:

[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 ⊗ I) ⊗ B; A ⊗ (I ⊗ B); α(A,I,B); A ⊗ B; ρ(A)⊗1; 1⊗λ(B)]

These diagrams are called coherence diagrams or constraints.

A monoidal category is called a strict monoidal category, if the morphisms α, λ, ρ are the identity morphisms.

Remark 3.2.2. We define A_{1} ⊗ … ⊗ A_{n} := (… (A_{1} ⊗ A_{2}) ⊗ …) ⊗ A_{n}.

There is an important theorem of S. MacLane that says that all diagrams whose morphisms are constructed by using copies of α, λ, ρ, identities, inverses, tensor products and compositions of such commute. We will not prove this theorem. It implies that each monoidal category can be replaced by (is monoidally equivalent to) a strict monoidal category. That means that we may omit in diagrams the morphisms α, λ, ρ or replace them by identities. In particular there is only one automorphism of A_{1} ⊗ … ⊗ A_{n} formed by coherence morphisms namely the identity.

Remark 3.2.3. For each monoidal category C we can construct the monoidal category C^{symm} symmetric to C that coincides with C as a category and has tensor product A ⊠ B := B ⊗ A and the coherence morphisms

α(C, B, A)^{−1} : (A ⊠ B) ⊠ C ⟶ A ⊠ (B ⊠ C),

ρ(A) : I ⊠ A ⟶ A,
λ(A) : A ⊠ I ⟶ A.

Then the coherence diagrams are commutative again, so that C^{symm} is a monoidal category.

Example 3.2.4. 1. Let R be an arbitrary ring. The category _{R}M_{R} of R-R-bimodules with the tensor product M ⊗_{R} N is a monoidal category. In particular the 𝕂-modules form a monoidal category. This is our most important example of a monoidal category.

2. Let B be a bialgebra and B-Mod be the category of left B-modules. We define the structure of a B-module on the tensor product M ⊗ N = M ⊗_{𝕂} N by

B ⊗ M ⊗ N ⟶^{Δ⊗1_{M} ⊗1_{N}} B ⊗ B ⊗ M ⊗ N ⟶^{1_{B}⊗τ⊗1_{N}} B ⊗ M ⊗ B ⊗ N ⟶^{ρ_{M} ⊗ρ_{N}} M ⊗ N

as in the previous section. So B-Mod is a monoidal category by 3.1.7

3. Let B be a bialgebra and B-Comod be the category of B-comodules. The tensor product M ⊗ N = M ⊗_{𝕂} N carries the structure of a B-comodule by

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

as in the previous section. So B-Comod is a monoidal category by 3.1.8

4. Let G be a monoid. A 𝕂-module together with a family of submodules (V_{g}|g ∈ G) is called G-graded if V = ⊕_{g∈G}V_{g}.

Let V and W be G-graded 𝕂-modules. A homomorphism of 𝕂-modules f : V ⟶ W is called G-graded if f(V_{g}) ⊆ W_{g} for all g ∈ G.

The G-graded 𝕂-modules and their homomorphisms form the category (𝕂-Mod)^{G} of G-graded 𝕂-modules.

There is a monoidal structure on (𝕂-Mod)^{G} given by the ordinary tensor product V ⊗ W. The submodules on the tensor product V ⊗ W are given by (V ⊗ W)_{g} := ∑_{h∈G} V_{h} ⊗ W_{h^{−1}g} = ∑_{h,k∈G,hk=g} V_{h} ⊗ W_{k}.

5. A chain complex of 𝕂-modules

M = (… ⟶^{∂_{3}} M_{2} ⟶^{∂_{2}} M_{1} ⟶^{∂_{1}} M_{0})

consists of a family of a family of 𝕂-modules M_{i} and a family of homomorphisms ∂_{n} : M_{n} ⟶ M_{n−1} with ∂_{n−1}∂_{n} = 0. This chain complex is indexed by the monoid ℕ_{0}. One may also consider more general chain complexes indexed by an arbitrary cyclic monoid. Chain complexes indexed by ℕ_{0} × ℕ_{0} are called double complexes. So much more general chain complexes may be considered. We restrict ourselves to chain complexes over ℕ_{0}.

Let M and N be chain complexes. A homomorphism of chain complexes f : M ⟶ N consists of a family of homomorphisms of 𝕂-modules f_{n} : M_{n} ⟶ N_{n} such that f_{n}∂_{n+1} = ∂_{n+1}f_{n+1} for all n ∈ ℕ_{0}.

The chain complexes with these homomorphisms form the category of chain complexes 𝕂-Comp.

If M and N are chain complexes then we form a new chain complex M ⊗ N with (M ⊗ N)_{n} := ⨁_{i=0}^{n} M_{i} ⊗ N_{n−i} and ∂ : (M ⊗ N)_{n} ⟶ (M ⊗ N)_{n−1} given by ∂(m_{i} ⊗ n_{n−i}) := (−1)^{i}∂_{M}(m_{i}) ⊗ n_{n−i} + m_{i} ⊗ ∂(n_{n−i}). This is often called the total complex associated with the double complex of the tensor product of M and N. Then it is easily checked that 𝕂-Comp is a monoidal category with this tensor product.

Problem 3.2.2. 1. Prove that the category (𝕂-Mod)^{G} of G-graded 𝕂-modules is equivalent to the category 𝕂G-Comod of 𝕂G-comodules by the following construction. If V is a G-graded 𝕂-module the V becomes a 𝕂G-comodule by the map δ : V ⟶ 𝕂G⊗V, δ(v) := g⊗v for all v ∈ V_{g} and all g ∈ G. Conversely if V, δ : V ⟶ 𝕂G⊗V is a 𝕂G-comodule then V together with the submodules V_{g} := {v ∈ V |δ(v) = g ⊗ v} is a G-graded 𝕂-module.

Since 𝕂G is a bialgebra the category of 𝕂G-comodules is a monoidal category. Show that the equivalence defined above between (𝕂-Mod)^{G} and 𝕂G-Comod preserves the tensor products, hence that it is a monoidal equivalence.

2. Let B = 𝕂⟨x, y⟩/I where I is generated by x^{2}, xy + yx. Then B is a bialgebra with the diagonal Δ(y) = y⊗y, Δ(x) = x⊗1+y⊗x. The counit is ε(y) = 1, ε(x) = 0. We introduced (the coopposite bialgebra of) this bialgebra in A.7 2.

Show that the category 𝕂-Comp of chain complexes is equivalent to the category B-Comod of B-comodules by the following construction. If M is a chain complex then define a B-comodule on M = ⊕_{i∈ℕ}M_{i} with the structure map δ : M ⟶ B ⊗ M, δ(m) := y^{i} ⊗ m + xy^{i−1} ⊗ ∂_{i}(m) for all m ∈ M_{i} and for all i ∈ ℕ resp. δ(m) := 1 ⊗ m for m ∈ M_{0}. Conversely if M, δ : M ⟶ B ⊗ M is a B-comodule then we define 𝕂-modules M_{i} := {m ∈ M|∃m′ ∈ M[δ(m) = y^{i} ⊗m+xy^{i−1} ⊗m′]} and 𝕂-linear maps ∂_{i} : M_{i} ⟶ M_{i−1} by ∂_{i}(m) := m′ for δ(m) = y^{i} ⊗ m + xy^{i−1} ⊗ m′. Check that this defines an equivalence of categories.

(Hint: Let m ∈ M ∈ B-Comod. Since y^{i}, xy^{i} form a basis of B we have δ(m) = ∑_{i} y^{i} ⊗ m_{i} + ∑_{i} xy^{i} ⊗ m′_{i}. We apply to this the equation (1 ⊗ δ)δ = (Δ ⊗ 1)δ and compare coefficients to get

δ(m_{i}) = y^{i} ⊗ m_{i} + xy^{i−1} ⊗ m′_{i−1}, δ(m′_{i}) = y^{i} ⊗ m′_{i}

for all i ∈ ℕ_{0} (with m′_{−1} = 0). Consequently for each m_{i} ∈ M_{i} there is exactly one ∂(m_{i}) = m′_{i−1} ∈ M such that

δ(m_{i}) = y^{i} ⊗ m_{i} + xy^{i−1} ⊗ ∂(m_{i}).

Since δ(m′_{i−1}) = y^{i−1} ⊗ m′_{i−1} for all i ∈ ℕ we see that ∂(m_{i}) ∈ M_{i−1}. So we have defined ∂ : M_{i} ⟶ M_{i−1}. Furthermore we see from this equation that ∂^{2}(m_{i}) = 0 for all i ∈ ℕ. Hence we have obtained a chain complex from (M, δ).

If we apply (ϵ ⊗ 1)δ(m) = m then we get m = ∑ m_{i} with m_{i} ∈ M_{i} hence M = ⨁_{i∈ℕ} M_{i}. This together with the inverse construction leads to the required equivalence.)

3. A cochain complex has the form

M = (M_{0} ⟶^{∂_{0}} M_{1} ⟶^{∂_{1}} M_{2} ⟶^{∂_{2}} …)

with ∂_{i+1}∂_{i} = 0. Show that the category 𝕂-Cocomp of cochain complexes is equivalent to Comod-B where B is chosen as in example 5.

Lemma 3.2.5. Let C be a monoidal category. Then the following diagrams commute

[figure: (I ⊗ A) ⊗ B; I ⊗ (A ⊗ B); α; A ⊗ B; λ(A)⊗1_{B}; λ(A⊗B); (A ⊗ B) ⊗ I; A ⊗ (B ⊗ I); α; A ⊗ B; ρ(A⊗B); 1_{A}⊗ρ(B)]

and we have λ(I) = ρ(I).

Proof. First we observe that the identity functor Id_{C} and the functor I ⊗ - are isomorphic by the natural isomorphism λ. Thus we have I ⊗ f = I ⊗ g ⟹ f = g. In the following diagram

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

all subdiagrams commute except for the right hand trapezoid. Since all morphisms are isomorphisms the right hand trapezoid must commute also. Hence the first diagram of the Lemma commutes.

In a similar way one shows the commutativity of the second diagram.

Furthermore the following diagram commutes

[figure: I ⊗ (I ⊗ I); (I ⊗ I) ⊗ I; α; I ⊗ (I ⊗ I); α; I ⊗ I; 1⊗ρ; ρ; I ⊗ I; ρ⊗1; 1⊗λ; I; ρ; ρ]

Here the left hand triangle commutes by the previous property. The commutativity of the right hand diagram is given by the axiom. The lower square commutes since ρ is a natural transformation. In particular ρ(1 ⊗ ρ) = ρ(1 ⊗ λ). Since ρ is an isomorphism and I ⊗ - ≅ Id_{C} we get ρ = λ.

Problem 3.2.3. For morphisms f : I ⟶ M and g : I ⟶ N in a monoidal category we define (f ⊗ 1 : N ⟶ M ⊗ N) := (f ⊗ 1_{I})ρ(I)^{−1} and (1 ⊗ g : M ⟶ M ⊗ N) := (1 ⊗ g)λ(I)^{−1}. Show that the diagram

[figure: N; M ⊗ N; f⊗1; I; M; f; g; 1⊗g]

commutes.

Definition 3.2.6. Let (C, ⊗) and (D, ⊗) be monoidal categories. A functor

F : C ⟶ D

together with a natural transformation

ξ(M, N) : F(M) ⊗ F(N) ⟶ F(M ⊗ N)

and a morphism

ξ_{0} : I_{D} ⟶ F(I_{C})

is called weakly monoidal if the following diagrams commute

[figure: (F(M) ⊗ F(N)) ⊗ F(P); F(M ⊗ N) ⊗ F(P); ξ⊗1; F((M ⊗ N) ⊗ P); ξ; α; F(α); F(M) ⊗ (F(N) ⊗ F(P)); F(M) ⊗ F(N ⊗ P); 1⊗ξ; F(M ⊗ (N ⊗ P)); ξ]

[figure: I ⊗ F(M); F(I) ⊗ F(M); ξ_{0}⊗1; F(I ⊗ M); ξ; F(M); λ; F(λ)]

[figure: F(M) ⊗ I; F(M) ⊗ F(I); 1⊗ξ_{0}; F(M ⊗ I); ξ; F(M).; ρ; F(ρ)]

If, in addition, the morphisms ξ and ξ_{0} are isomorphisms then the functor is called a monoidal functor. The functor is called a strict monoidal functor if ξ and ξ_{0} are the identity morphisms.

A natural transformation ζ : F ⟶ F′ between weakly monoidal functors is called a monoidal natural transformation if the diagrams

[figure: F′(M) ⊗ F′(N); F′(M ⊗ N); ξ′; F(M) ⊗ F(N); F(M ⊗ N); ξ; ζ⊗ζ; ζ; F(I); I; ξ_{0}; F′(I); ξ′_{0}; ζ]

commute.

We can generalize the notions of an algebra or of a coalgebra in the context of a monoidal category. We define

Definition 3.2.7. Let C be a monoidal category. An algebra or a monoid in C consists of an object A together with a multiplication ∇ : A ⊗ A ⟶ A that is associative

[figure: A ⊗ A; A; ∇; A ⊗ A ⊗ A; A ⊗ A; id ⊗∇; ∇⊗1; ∇]

or more precisely

and has a unit η : I ⟶ A such that the following diagram commutes

[figure: (A ⊗ A) ⊗ A; A ⊗ (A ⊗ A); α; A ⊗ A; id ⊗∇; A ⊗ A; A; ∇; ∇⊗id; ∇]

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

Let A and B be algebras in C. A morphism of algebras f : A ⟶ B is a morphism in C such that

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

commute.

Remark 3.2.8. It is obvious that the composition of two morphisms of algebras is again a morphism of algebras. The identity also is a morphism of algebras. Thus we obtain the category Alg(C) of algebras in C.

Definition 3.2.9. Let C be a monoidal category. A coalgebra or a comonoid in C consists of an object C together with a comultiplication Δ : A ⟶ A ⊗ A that is coassociative

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

or more precisely

[figure: (C ⊗ C) ⊗ C; C ⊗ (C ⊗ C); α; C ⊗ C; Δ⊗id; C; C ⊗ C; Δ; id ⊗Δ; Δ]

and has a counit ε : C ⟶ I such that the following diagram commutes

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

Let C and D be coalgebras in C. A morphism of coalgebras f : C ⟶ D is a morphism in C such that

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

commute.

Remark 3.2.10. It is obvious that the composition of two morphisms of coalgebras is again a morphism of coalgebras. The identity also is a morphism of coalgebras. Thus we obtain the category Coalg(C) of coalgebras in C.

Remark 3.2.11. Observe that the notions of bialgebra, Hopf algebra, and comodule algebra cannot be generalized to an arbitrary monoidal category since we need to have an algebra structure on the tensor product of two algebras and this requires us to interchange the middle tensor factors. These interchanges or flips are known under the name symmetry, quasisymmetry or braiding and will be discussed later on.

3. Dual Objects

At the end of the first section in Corollary 3.1.15 we saw that the dual of an Hmodule can be constructed. We did not show the corresponding result for comodules. In fact such a construction for comodules needs some finiteness conditions. With this restriction the notion of a dual object can be introduced in an arbitrary monoidal category.

Definition 3.3.1. Let (C, ⊗) be a monoidal category M ∈ C be an object. An object M^{∗} ∈ C together with a morphism ev : M^{∗} ⊗ M ⟶ I is called a left dual for M if there exists a morphism db : I ⟶ M ⊗ M^{∗} in C such that

(M ⟶^{db ⊗1} M ⊗ M^{∗} ⊗ M ⟶^{1⊗ev} M) = 1_{M}

(M^{∗} ⟶^{1⊗db} M^{∗} ⊗ M ⊗ M^{∗} ⟶^{ev ⊗1} M^{∗}) = 1_{M^{∗}}.

A monoidal category is called left rigid if each object M ∈ C has a left dual.

Symmetrically we define: an object^{∗}M ∈ C together with a morphism ev : M⊗^{∗}M ⟶ I is called a right dual for M if there exists a morphism db : I ⟶ ^{∗}M ⊗ M in C such that

(M ⟶^{1⊗db} M ⊗^{∗}M ⊗ M ⟶^{ev ⊗1} M) = 1_{M}

(^{∗}M ⟶^{db ⊗1} ^{∗}M ⊗ M ⊗^{∗}M ⟶^{1⊗ev} ^{∗}M) = 1_{^{∗}M}.

A monoidal category is called right rigid if each object M ∈ C has a left dual. The morphisms ev and db are called the evaluation respectively the dual basis.

Remark 3.3.2. If (M^{∗}, ev) is a left dual for M then obviously (M, ev) is a right dual for M^{∗} and conversely. One uses the same morphism db : I ⟶ M ⊗ M^{∗}.

Lemma 3.3.3. Let (M^{∗}, ev) be a left dual for M. Then there is a natural isomorphism

Mor_{C}(- ⊗ M, -) ≅ Mor_{C}(-, - ⊗ M^{∗}),

i. e. the functor - ⊗ M : C ⟶ C is left adjoint to the functor - ⊗ M^{∗} : C ⟶ C.

Proof. We give the unit and the counit of the pair of adjoint functors. We define Φ(A) := 1_{A} ⊗ db : A ⟶ A ⊗ M ⊗ M^{∗} and Ψ(B) := 1_{B} ⊗ ev : B ⊗ M^{∗} ⊗ M ⟶ B. These are obviously natural transformations. We have commutative diagrams

[figure: (A ⊗ M; A ⊗ M ⊗ M^{∗} ⊗ M; FΦ(A)=; 1_{A}⊗db ⊗1_{M}; A ⊗ M) = 1_{A⊗M}; ΨF(A)=; 1_{A}⊗1_{M} ⊗ev]

and

[figure: (B ⊗ M^{∗}; B ⊗ M^{∗} ⊗ M ⊗ M^{∗}; ΦG(B)=; 1_{B}⊗1_{M^{∗}}⊗db; B ⊗ M^{∗}) = 1_{B⊗M^{∗}}; GΨ(B)=; 1_{B}⊗ev ⊗1_{M^{∗}}]

thus the Lemma has been proved by Corollary A.9.11.

The converse holds as well. If - ⊗ M is left adjoint to - ⊗ M^{∗} then the unit Φ gives a morphism db := Φ(I) : I ⟶ M ⊗ M^{∗} and the counit Ψ gives a morphism ev := Ψ(I) : M^{∗} ⊗ M ⟶ I satisfying the required properties. Thus we have

Corollary 3.3.4. If - ⊗ M : C ⟶ C is left adjoint to - ⊗ M^{∗} : C ⟶ C then M^{∗} is a left dual for M.

Corollary 3.3.5. (M^{∗}, ev) is a left dual for M if and only if there is a natural isomorphism

Mor_{C}(M^{∗} ⊗ -, -) ≅ Mor_{C}(-, M ⊗ -),

i. e. the functor M^{∗} ⊗ - : C ⟶ C is left adjoint to the functor M ⊗ - : C ⟶ C. The natural isomorphism if given by

(f : M^{∗} ⊗ N ⟶ P) ↦ ((1_{M} ⊗ f)(db ⊗1_{N}) : N ⟶ M ⊗ M^{∗} ⊗ N ⟶ M ⊗ P)

and

(g : N ⟶ M ⊗ P) ↦ ((ev ⊗1_{P})(1_{M^{∗}} ⊗ g) : M^{∗} ⊗ N ⟶ M^{∗} ⊗ M ⊗ P ⟶ P).

Proof. We have a natural isomorphism

Mor_{C}(M^{∗} ⊗ -, -) ≅ Mor_{C}(-, M ⊗ -),

iff (M, ev) is a right dual for M^{∗} (as a symmetric statement to Lemma 3.3.3) iff (M^{∗}, ev) is a left dual for M.

Corollary 3.3.6. If M has a left dual then this is unique up to isomorphism.

Proof. Let (M^{∗}, ev) and (M^{!}, ev^{!}) be left duals for M. Then the functors -⊗M^{∗} and - ⊗ M^{!} are isomorphic by Lemma A.9.7. In particular we have M^{∗} ≅ I ⊗ M^{∗} ≅ I ⊗ M^{!} ≅ M^{!}. If we consider the construction of the isomorphism then we get in particular that (ev^{!} ⊗1)(1 ⊗ db) : M^{!} ⟶ M^{!} ⊗ M ⊗ M^{∗} ⟶ M^{∗} is the given isomorphism.

Problem 3.3.4. Let (M^{∗}, ev) be a left dual for M. Then there is a unique morphism db : I ⟶ M ⊗ M^{∗} satisfying the conditions of Definition 3.3.1.

Definition 3.3.7. Let (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) be left duals for M resp. N. For each morphism f : M ⟶ N we define the transposed morphism

(f^{∗} : N^{∗} ⟶ M^{∗}) := (N^{∗} ⟶^{1⊗db_{M}} N^{∗} ⊗ M ⊗ M^{∗} ⟶^{1⊗f⊗1} N^{∗} ⊗ N ⊗ M^{∗} ⟶^{ev_{N} ⊗1} M^{∗}).

With this definition we get that the left dual is a contravariant functor, since we have

Lemma 3.3.8. Let (M^{∗}, ev_{M}), (N^{∗}, ev_{N}), and (P^{∗}, ev_{P}) be left duals for M, N and P respectively.

1. We have (1_{M})^{∗} = 1_{M^{∗}}.

2. If f : M ⟶ N and g : N ⟶ P are given then (gf)^{∗} = f^{∗}g^{∗} holds.

Proof. 1. (1_{M})^{∗} = (ev ⊗1)(1 ⊗ 1 ⊗ 1)(1 ⊗ db) = 1_{M^{∗}}.

2. The following diagram commutes

[figure: M; N ⊗ N^{∗} ⊗ M; db_{N} ⊗1; f; 1⊗1⊗f; N; N ⊗ N^{∗} ⊗ N; db_{N} ⊗1; N; 1⊗ev_{N}; g⊗1⊗1; g; P ⊗ N^{∗} ⊗ N; P; 1⊗ev_{N}]

Hence we have gf = (1 ⊗ ev_{N})(g ⊗ 1 ⊗ f)(db_{N} ⊗1). Thus the following diagram commutes

[figure: P^{∗}; P^{∗} ⊗ N ⊗ N^{∗}; 1⊗db; P^{∗} ⊗ P ⊗ N^{∗}; 1⊗g⊗1; 1⊗db; 1⊗db; 1⊗db; P^{∗} ⊗ M ⊗ M^{∗}; P^{∗} ⊗ N ⊗ N^{∗} ⊗ M ⊗ M^{∗}; 1⊗db ⊗1; P^{∗} ⊗ P ⊗ N^{∗} ⊗ M ⊗ M^{∗}; 1⊗g⊗1; 1⊗gf⊗1; P^{∗} ⊗ P ⊗ M^{∗}; P^{∗} ⊗ P ⊗ N^{∗} ⊗ N ⊗ M^{∗}; 1⊗ev ⊗1; 1⊗f⊗1; ev ⊗1; ev ⊗1; ev ⊗1; M^{∗}; N^{∗} ⊗ N ⊗ M^{∗}; 1⊗ev; N^{∗} ⊗ M ⊗ M^{∗}; 1⊗f⊗1; 1⊗g⊗1⊗f⊗1; N^{∗}; ev ⊗1; 1⊗db]

Problem 3.3.5. 1. In the category of ℕ-graded vector spaces determine all objects M that have a left dual.

2. In the category of chain complexes 𝕂-Comp determine all objects M that have a left dual.

3. In the category of cochain complexes 𝕂-Cocomp determine all objects M that have a left dual.

4. Let (M^{∗}, ev) be a left dual for M. Show that db : I ⟶ M ⊗ M^{∗} is uniquely determined by M, M^{∗}, and ev. (Uniqueness of the dual basis.)

5. Let (M^{∗}, ev) be a left dual for M. Show that ev : M^{∗} ⊗ M ⟶ I is uniquely determined by M, M^{∗}, and db.

Corollary 3.3.9. Let M, N have the left duals (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) and let f : M ⟶ N be a morphism in C. Then the following diagram commutes

[figure: N ⊗ N^{∗}; N ⊗ M^{∗}.; 1⊗f^{∗}; I; M ⊗ M^{∗}; db_{M}; db_{N}; f⊗1]

Proof. The following diagram commutes

[figure: M; N ⊗ N^{∗} ⊗ M; db ⊗1; N; N ⊗ N^{∗} ⊗ N; db ⊗1; f; 1⊗1⊗f; N; 1⊗ev; 1]

This implies (f ⊗ 1_{M^{∗}}) db_{M} = ((1_{N} ⊗ ev_{N})(1_{N} ⊗ 1_{N^{∗}} ⊗ f)(db_{N} ⊗1_{M}) ⊗ 1_{M^{∗}}) db_{M} = (1_{N} ⊗ev_{N} ⊗1_{M^{∗}})(1_{N} ⊗1_{N^{∗}} ⊗f ⊗1_{M^{∗}})(db_{N} ⊗1_{M} ⊗1_{M^{∗}}) db_{M} = (1_{N} ⊗ev_{N} ⊗1_{M^{∗}})(1_{N} ⊗ 1_{N^{∗}} ⊗ f ⊗ 1_{M^{∗}})(1_{N} ⊗ 1_{N^{∗}} ⊗ db_{M}) db_{N} = (1_{N} ⊗ (ev_{N} ⊗1_{M^{∗}})(1_{N^{∗}} ⊗ f ⊗ 1_{M^{∗}})(1_{N^{∗}} ⊗ db_{M})) db_{N} = (1_{N} ⊗ f^{∗}) db_{N}.

Corollary 3.3.10. Let M, N have the left duals (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) and let f : M ⟶ N be a morphism in C. Then the following diagram commutes

[figure: N^{∗} ⊗ N; I.; ev_{N}; N^{∗} ⊗ M; M^{∗} ⊗ M; f^{∗}⊗1; 1⊗f; ev_{M}]

Proof. This statement follows immediately from the symmetry of the definition of a left dual.

Example 3.3.11. Let M ∈ _{R}M_{R} be an R-R-bimodule. Then Hom_{R}(M., R.) is an R-R-bimodule by (rfs)(x) = rf(sx). Furthermore we have the morphism ev : Hom_{R}(M., R.) ⊗_{R} M ⟶ R defined by ev(f ⊗_{R} m) = f(m).

(Dual Basis Lemma:) A module M ∈ M_{R} is called finitely generated and projective if there are elements m_{1}, …, m_{n} ∈ M und m^{1}, …, m^{n} ∈ Hom_{R}(M., R.) such that

∀m ∈ M : ∑_{i=1}^{n} m_{i}m^{i}(m) = m.

Actually this is a consequence of the dual basis lemma. But this definition is equivalent to the usual definition.

Let M ∈ _{R}M_{R}. M as a right R-module is finitely generated and projective iff M has a left dual. The left dual is isomorphic to Hom_{R}(M., R.).

If M_{R} is finitely generated projective then we use db : R ⟶ M ⊗_{R} Hom_{R}(M., R.) with db(1) = ∑_{i=1}^{n} m_{i} ⊗_{R} m^{i}. In fact we have (1 ⊗_{R} ev)(db ⊗_{R}1)(m) = (1 ⊗_{R} ev)(∑ m_{i} ⊗_{R} m^{i} ⊗_{R} m) = ∑ m_{i}m^{i}(m) = m. We have furthermore (ev ⊗_{R}1)(1 ⊗_{R} db)(f)(m) = (ev ⊗_{R}1)(∑_{i=1}^{n} f ⊗_{R} m_{i} ⊗_{R} m^{i})(m) = ∑ f(m_{i})m^{i}(m) = f(∑ m_{i}m^{i}(m)) = f(m) for all m ∈ M hence (ev ⊗_{R}1)(1 ⊗_{R} db)(f) = f.

Conversely if M has a left dual M^{∗} then ev : M^{∗} ⊗_{R} M ⟶ R defines a homomorphism ι : M^{∗} ⟶ Hom_{R}(M., R.) in _{R}M_{R} by ι(m^{∗})(m) = ev(m^{∗} ⊗_{R} m). We define ∑_{i=1}^{n} m_{i} ⊗ m^{i} := db(1) ∈ M ⊗ M^{∗}, then m = (1 ⊗ ev)(db ⊗1)(m) = (1 ⊗ ev)(∑ m_{i} ⊗ m^{i} ⊗ m) = ∑ m_{i}ι(m^{i})(m) so that m_{1}, …, m_{n} ∈ M and ι(m^{1}), …, ι(m^{n}) ∈ Hom_{R}(M., R.) form a dual basis for M, i. e. M is finitely generated and projective as an R-module. Thus M^{∗} and Hom_{R}(M., R.) are isomorphic by the map ι.

Analogously Hom_{R}(.M, .R) is a right dual for M iff M is finitely generated and projective as a left R-module.

Problem 3.3.6. Find an example of an object M in a monoidal category C that has a left dual but no right dual.

Definition 3.3.12. Given objects M, N in C. An object [M, N] is called a left inner Hom of M and N if there is a natural isomorphism Mor_{C}(- ⊗ M, N) ≅ Mor_{C}(-, [M, N]), i. e. if it represents the functor Mor_{C}(- ⊗ M, N).

If there is an isomorphism Mor_{C}(P ⊗ M, N) ≅ Mor_{C}(P, [M, N]) natural in the three variable M, N, P then the category C is called monoidal and left closed.

If there is an isomorphism Mor_{C}(M ⊗ P, N) ≅ Mor_{C}(P, [M, N]) natural in the three variable M, N, P then the category C is called monoidal and right closed.

If M has a left dual M^{∗} in C then there are inner Homs [M, - ] defined by [M, N] := N ⊗ M^{∗}. In particular left rigid monoidal categories are left closed.

Example 3.3.13. 1. The category of finite dimensional vector spaces is a monoidal category where each object has a (left and right) dual. Hence it is (left and right) closed and rigid.

2. Let Ban be the category of (complex) Banach spaces where the morphisms satisfy ∥ f(m) ∥≤∥ m ∥ i. e. the maps are bounded by 1 or contracting. Ban is a monoidal category by the complete tensor product M ˆ⊗N. In Ban exists an inner Hom functor [M, N] that consists of the set of bounded linear maps from M to N made into a Banach space by an appropriate topology. Thus Ban is a monoidal closed category.

3. Let H be a Hopf algebra. The category H-Mod of left H-modules is a monoidal category (see Example 3.2.4 2.). Then Hom_{𝕂}(M, N) is an object in H-Mod by the multiplication

(hf)(m) := ∑ h_{(1)}f(mS(h_{(2)})

as in Proposition 3.1.14.

Hom_{𝕂}(M, N) is an inner Hom functor in the monoidal category H-Mod. The isomorphism ϕ : Hom_{𝕂}(P, Hom_{𝕂}(M, N)) ≅ Hom_{𝕂}(P ⊗ M, N) can be restricted to an isomorphism

Hom_{H}(P, Hom_{𝕂}(M, N)) ≅ Hom_{H}(P ⊗ M, N),

because ϕ(f)(h(p ⊗ m)) = ϕ(f)(∑ h_{(1)}p ⊗ h_{(2)}m) = ∑ f(h_{(1)}p)(h_{(2)}m) = ∑(h_{(1)}(f(p)))(h_{(2)}m) = ∑ h_{(1)}(f(p)(S(h_{(2)})h_{(3)}m)) = h(f(p)(m)) = h(ϕ(f) (p ⊗ m)) and conversely (h(f(p)))(m) = ∑ h_{(1)}(f(p)(S(h_{(2)})m)) = ∑ h_{(1)}(ϕ(f) (p ⊗ S(h_{(2)})m)) = ∑ ϕ(f)(h_{(1)}p ⊗ h_{(2)}S(h_{(3)})m) = ϕ(f)(hp ⊗ m) = f(hp)(m). Thus H-Mod is left closed.

If M ∈ H-Mod is a finite dimensional vector space then the dual vector space M^{∗} := Hom_{𝕂}(M, 𝕂) again is an H-module by (hf)(m) := f(S(h)m). Furthermore M^{∗} is a left dual for M with the morphisms

db : 𝕂 ∋ 1 ↦ ∑_{i} m_{i} ⊗ m^{i} ∈ M ⊗ M^{∗}

and

ev : M^{∗} ⊗ M ∋ f ⊗ m ↦ f(m) ∈ 𝕂

where m_{i} and m^{i} are a dual basis of the vector space M. Clearly we have (1⊗ev)(db ⊗1) = 1_{M} and (ev ⊗1)(1⊗db) = 1_{M^{∗}} since M is a finite dimensional vector space. We have to show that db and ev are H-module homomorphisms. We have

(h db(1))(m) = (h(∑ m_{i} ⊗ m^{i}))(m) = (∑ h_{(1)}m_{i} ⊗ h_{(2)}m^{i})(m) = ∑(h_{(1)}m_{i})((h_{(2)}m^{i})(m)) = ∑(h_{(1)}m_{i})(m^{i}(S(h_{(2)})m)) = ∑ h_{(1)}S(h_{(2)})m = ε(h)m = ε(h)(∑ m_{i} ⊗ m^{i})(m) = ε(h) db(1)(m) = db(ε(h)1)(m) = db(h1)(m),

hence h db(1) = db(h1). Furthermore we have

\begin{matrix}h ev(f ⊗ m) = hf(m) = ∑ h_{(1)}f(S(h_{(2)})h_{(3)}m) = ∑(h_{(1)}f)(h_{(2)}m) = \\ ∑ ev(h_{(1)}f ⊗ h_{(2)}m) = ev(h(f ⊗ m)). \end{matrix}

4. Let H be a Hopf algebra. Then the category of left H-comodules (see Example 3.2.4 3.) is a monoidal category. Let M ∈ H-Comod be a finite dimensional vector space. Let m_{i} be a basis for M and let the comultiplication of the comodule be δ(m_{i}) = ∑ h_{ij} ⊗ m_{j}. Then we have Δ(h_{ik}) = ∑ h_{ij} ⊗ h_{jk}. M^{∗} := Hom_{𝕂}(M, 𝕂) becomes a left H-comodule δ(m^{j}) := ∑ S(h_{ij}) ⊗ m^{i}. One verifies that M^{∗} is a left dual for M.

Lemma 3.3.14. Let M ∈ C be an object with left dual (M^{∗}, ev). Then

1. M ⊗ M^{∗} is an algebra with multiplication

∇ := 1_{M} ⊗ ev ⊗1_{M^{∗}} : M ⊗ M^{∗} ⊗ M ⊗ M^{∗} ⟶ M ⊗ M^{∗}

and unit

u := db : I ⟶ M ⊗ M^{∗};

2. M^{∗} ⊗ M is a coalgebra with comultiplication

Δ := 1_{M^{∗}} ⊗ db ⊗1_{M} : M^{∗} ⊗ M ⟶ M^{∗} ⊗ M ⊗ M^{∗} ⊗ M

and counit

ε := ev : M^{∗} ⊗ M ⟶ I.

Proof. 1. The associativity is given by (∇ ⊗ 1)∇ = (1_{M} ⊗ ev ⊗1_{M^{∗}} ⊗ 1_{M} ⊗ 1_{M^{∗}})(1_{M} ⊗ ev ⊗1_{M^{∗}}) = 1_{M} ⊗ ev ⊗ ev ⊗1_{M^{∗}} = (1_{M} ⊗ 1_{M^{∗}} ⊗ 1_{M} ⊗ ev ⊗1_{M^{∗}})(1_{M} ⊗ ev ⊗1_{M^{∗}}) = (1⊗∇)∇. The axiom for the left unit is ∇(u⊗1) = (1_{M} ⊗ev ⊗1_{M^{∗}})(db ⊗ 1_{M} ⊗ 1_{M^{∗}}) = 1_{M} ⊗ 1_{M^{∗}}.

2. is dual to the statement for algebras.

Lemma 3.3.15. 1. Let A be an algebra in C and left M ∈ C be a left rigid object with left dual (M^{∗}, ev). There is a bijection between the set of morphisms f : A ⊗ M ⟶ M making M a left A-module and the set of algebra morphisms f̃: A ⟶ M ⊗M^{∗}.

2. Let C be a coalgebra in C and left M ∈ C be a left rigid object with left dual (M^{∗}, ev). There is a bijection between the set of morphisms f : M ⟶ M ⊗ C making M a right C-comodule and the set of coalgebra morphisms f̃: M^{∗} ⊗ M ⟶ C.

Proof. 1. By Lemma 3.3.14 the object M ⊗ M^{∗} is an algebra. Given f : A ⊗ M ⟶ M such that M becomes an A-module. By Lemma 3.3.3 we associate f̃ := (f ⊗ 1)(1 ⊗ db) : A ⟶ A ⊗ M ⊗ M^{∗} ⟶ M ⊗ M^{∗}. The compatibility of f̃ with the multiplication is given by the commutative diagram

The unit axiom is given by Conversely let g : A ⟶ M ⊗ M^{∗} be an algebra homomorphism and consider g̃ := (1 ⊗ ev)(g ⊗ 1) : A ⊗ M ⟶ M ⊗ M^{∗} ⊗ M ⟶ M. Then M becomes a left A-module since

[figure: A ⊗ A; A; ∇; A ⊗ A ⊗ M ⊗ M^{∗}; A ⊗ M ⊗ M^{∗}; ∇⊗1⊗1; A ⊗ M ⊗ M^{∗}; M ⊗ M^{∗}; f⊗1; A ⊗ M ⊗ M^{∗} ⊗ M ⊗ M^{∗}; A ⊗ M ⊗ M^{∗}; 1⊗1⊗ev ⊗1; M ⊗ M^{∗} ⊗ M ⊗ M^{∗}; M ⊗ M^{∗}; 1⊗ev ⊗1; f̃⊗f̃; 1⊗f⊗1; 1⊗db ⊗1⊗1; f⊗1; f̃; 1⊗1⊗db; 1⊗f̃; f̃⊗1⊗1; f⊗1⊗1; 1; 1⊗db; 1; f⊗1]

[figure: I; M ⊗ M^{∗}; db; A; A ⊗ M ⊗ M^{∗}; 1⊗db; M ⊗ M^{∗}; f⊗1; u; u⊗1; 1⊗1]

[figure: A ⊗ A ⊗ M; A ⊗ M; ∇⊗1; A ⊗ M ⊗ M^{∗} ⊗ M; M ⊗ M^{∗} ⊗ M ⊗ M^{∗} ⊗ M; g⊗1⊗1⊗1; M ⊗ M^{∗} ⊗ M; 1⊗ev ⊗1⊗1; A ⊗ M; M; g̃; 1⊗g̃; M ⊗ M^{∗} ⊗ M; 1⊗1⊗1⊗ev; g̃; 1⊗g⊗1; 1⊗1⊗ev; g⊗g⊗1; g⊗1; 1⊗ev; 1⊗ev; g⊗1]

and

[figure: A ⊗ M; M ⊗ M^{∗} ⊗ M; g⊗1; M; 1⊗ev; M; u⊗1; db ⊗1; 1]

commute.

2. (M^{∗}, ev) is a left dual for M in the category C if and only if (M^{∗}, db) is the right dual for M in the dual category C^{op}. So if we dualize the result of part 1. we have to change sides, hence 2.

4. Finite reconstruction

The endomorphism ring of a vector space enjoys the following universal property. It is a vector space itself and allows a homomorphism ρ : End(V) ⊗ V ⟶ V. It is universal with respect to this property, i. e. if Z is a vector space and f : Z ⊗ V ⟶ V is a homomorphism, then there is a unique homomorphism g : Z ⟶ End(V) such that

[figure: End(V) ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes.

The algebra structure of End(V) comes for free from this universal property. If we replace the vector space V by a diagram of vector spaces ω : D ⟶ Vec we

get a similar universal object End(ω). Again the universal property induces a unique algebra structure on End(ω).

Problem 3.4.7. 1. Let V be a vector space. Show that there is a universal vector space E and homomorphism ρ : E ⊗ V ⟶ V (such that for each vector space Z and each homomorphism f : Z ⊗ V ⟶ V there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes). We call E and ρ : E ⊗ V ⟶ V a vector space acting universally on V.

2. Let E and ρ : E ⊗ V ⟶ V be a vector space acting universally on V. Show that E uniquely has the structure of an algebra such that V becomes a left E-module.

3. Let ω : D ⟶ Vec be a diagram of vector spaces. Show that there is a universal vector space E and natural transformation ρ : E ⊗ ω ⟶ ω (such that for each vector space Z and each natural transformation f : Z ⊗ω ⟶ ω there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ ω; ω; ρ; f; Z ⊗ ω; g⊗1]

commutes). We call E and ρ : E ⊗ ω ⟶ ω a vector space acting universally on ω.

4. Let E and ρ : E ⊗ ω ⟶ ω be a vector space acting universally on ω. Show that E uniquely has the structure of an algebra such that ω becomes a diagram of left E-modules.

Similar considerations can be carried out for coactions V ⟶ V ⊗C or ω ⟶ ω ⊗C and a coalgebra structure on C. There is one restriction, however. We can only use finite dimensional vector spaces V or diagrams of finite dimensional vector spaces. This will be done further down.

As we have seen, ???

We want to find a universal natural transformation δ : ω ⟶ ω ⊗ coend(ω). For this purpose we consider the isomorphisms

Mor_{C}(ω(X), ω(X) ⊗ M) ≅ Mor_{C}(ω(X)^{∗} ⊗ ω(X), M)

that are given by f ↦ (ev ⊗1)(1 ⊗ f) and as inverse g ↦ (1 ⊗ g)(db ⊗1). We first develop techniques to describe the properties of a natural transformation ϕ : ω ⟶ ω ⊗ M as properties of the associated family g(X) : ω(X)^{∗} ⊗ ω(X) ⟶ M. We will see that g : ω^{∗} ⊗ ω ⟶ M will be a cone. Then we will show that ϕ is a universal natural transformation if and only if its associated cone is universal. In the literature this is called a coend.

Throughout this section assume the following. Let D be an arbitrary diagram scheme. Let C be a cocomplete monoidal category such that the tensor product preserves colimits in both arguments. Let C_{0} be the full subcategory of those objects in C that have a left dual. Let ω : D ⟶ C be a diagram in C such that ω(X) ∈ C_{0} for all X ∈ D, i. e. ω is given by a functor ω_{0} : D ⟶ C_{0}. We call such a diagram a finite diagram in C. Finally for an object M ∈ C let ω ⊗ M : D ⟶ C be the functor with (ω ⊗ M)(X) = ω(X) ⊗ M.

Remark 3.4.1. Consider the following category D̃. For each morphism f : X ⟶ Y there is an object f̃ ∈ D̃. The object corresponding to the identity 1_{X} : X ⟶ X is denoted by X̃ ∈ D̃. For each morphism f : X ⟶ Y in D there are two morphisms f_{1} : f̃ ⟶ X̃ and f_{2} : f̃ ⟶ Ỹ in D̃. Furthermore there are the identities 1_{f} : f̃ ⟶ f̃ in D̃.

Since there are no morphisms with X̃ as domain other than (1_{X})_{i} : X̃ ⟶ X̃ and 1_{f} : f̃ ⟶ f̃ we only have to define the following compositions (1_{X})_{i} ∘ f_{j} := f_{j}. Then D̃ becomes a category and we have 1X̃ = (1_{X})_{1} = (1_{X})_{2}.

We define a diagram ω^{∗} ⊗ ω : D̃ ⟶ C as follows. If f : X ⟶ Y is given then

(ω^{∗} ⊗ ω)(f̃) := ω(Y)^{∗} ⊗ ω(X)

and

\begin{matrix}ω(f_{1}) := ω(f)^{∗} ⊗ ω(1_{X}), \\ ω(f_{2}) := ω(1_{Y})^{∗} ⊗ ω(f). \end{matrix}

The colimit of ω^{∗} ⊗ ω consists of an object coend(ω) ∈ C together with a family of morphisms ι(X, X) : ω(X)^{∗} ⊗ ω(X) ⟶ coend(ω) such that the diagrams

[figure: ω(X)^{∗} ⊗ ω(X); (ω^{∗} ⊗ ω)(f̃) = ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; coend(ω); ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y)]

commute for all f : X ⟶ Y in D. Indeed, such a family ι(X̃) := ι(X, X) can be uniquely extended to a natural transformation by defining ι(f̃) := ι(X, X)(ω(f)^{∗} ⊗ ω(X)) = ι(Y, Y)(ω(Y)^{∗} ⊗ ω(f)). In addition the pair (coend(ω), ι) is universal with respect to this property.

In the literature such a universal object is called a coend of the bifunctor ω^{∗} ⊗ ω : D^{op} × D ⟶ C.

Corollary 3.4.2. The following is a coequalizer

[figure: ∐_{f∈MorD} ω(Zi(f))^{∗} ⊗ ω(Qu(f)); ∐_{X∈Ob D} ω(X)^{∗} ⊗ ω(X); p; q; coend(ω)]

Proof. This is just a reformulation of Remark A.10.11, since the colimit may also be built from the commutative squares given above.

Observe that for the construction of the colimit not all objects of the diagram have to be used but only those of the form ω(X)^{∗} ⊗ ω(X).

Theorem 3.4.3. (Tannaka-Krein)

Let ω : D ⟶ C_{0} ⊆ C be a finite diagram. Then there exists an object coend(ω) ∈ C and a natural transformation δ : ω ⟶ ω ⊗ coend(ω) such that for each object M ∈ C and each natural transformation φ : ω ⟶ ω ⊗ M there exists a unique morphism φ̃: coend(ω) ⟶ M such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ M; 1⊗φ̃]

commutes.

Proof. Let coend(ω) ∈ C together with morphisms ι(f̃) : ω(Y)^{∗} ⊗ ω(X) ⟶ coend(ω) be the colimit of the diagram ω^{∗} ⊗ ω : D̃ ⟶ C. So we get commutative diagrams

[figure: ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); ω(Y)^{∗} ⊗ ω(X); coend(ω); ι(f̃)]

for each f : X ⟶ Y in C.

For X ∈ C we define a morphism δ(X) : ω(X) ⟶ ω(X) ⊗ coend(ω) by (1 ⊗ ι(X, X))(db ⊗1) : ω(X) ⟶ ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⟶ ω(X) ⊗ coend(ω). Then we get as in Corollary 3.3.5 ι(X, X) = (1 ⊗ ev)(1 ⊗ δ(X)).

We show that δ is a natural transformation. For each f : X ⟶ Y the square

[figure: ω(Y) ⊗ ω(Y)^{∗}; ω(Y) ⊗ ω(X)^{∗}.; 1⊗ω(f)^{∗}; I; ω(X) ⊗ ω(X)^{∗}; db_{X}; db_{Y}; ω(f)⊗1]

commutes by Corollary 3.3.9. Thus the following diagram commutes

Now let M ∈ C be an object and φ : ω ⟶ ω ⊗ M a natural transformation. Observe that commutes by Corollary 3.3.10. Thus also the diagram

[figure: ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); db ⊗1; ω(X) ⊗ coend(ω); 1⊗ι(X,X); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(X); ω(Y) ⊗ ω(X)^{∗} ⊗ ω(X); 1⊗ω(f)^{∗}⊗1; ω(Y); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(Y); db ⊗1; ω(Y) ⊗ coend(ω).; 1⊗ι(Y,Y); ω(f); ω(f)⊗1; db ⊗1; 1⊗1⊗ω(f); ω(f)⊗1⊗1; 1⊗ι(X,X)]

[figure: ω(Y)^{∗} ⊗ ω(Y); I; ev; ω(Y)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; 1⊗ω(f); ev]

[figure: ω(X)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); ω(f)^{∗}⊗1; ω(f)^{∗}⊗1⊗1; M; ev ⊗1; ω(Y)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); 1⊗ω(f); 1⊗ω(f)⊗1; ev ⊗1; ω(Y)^{∗} ⊗ ω(Y); ω(Y)^{∗} ⊗ ω(Y) ⊗ M; 1⊗φ(Y)]

commutes. We define φ̃ : coend(ω) ⟶ M from the colimit property as universal factorization

[figure: ω(X)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); (ev ⊗1)(1⊗φ(X)); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); coend(ω); M; φ̃; (ev ⊗1)(1⊗φ(Y))]

Hence the diagram

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M; ω(X) ⊗ M; 1⊗1; φ(X); 1⊗φ̃; ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗1⊗φ(X); db ⊗1; 1⊗ev ⊗1; 1⊗ι(X,X); db ⊗1⊗1]

commutes. The exterior portion of this diagram yields

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M.; 1⊗φ̃; φ(X)]

It remains to show that φ̃ : coend(ω) ⟶ M is uniquely determined. Let φ̃_{0} : coend(ω) ⟶ M be another morphism with φ(X) = (1 ⊗ φ̃_{0})δ(X) for all X ∈ D.

Then the following diagram commutes

[figure: ω(X)^{∗} ⊗ ω(X); coend(ω); ι(X,X); ω(X)^{∗} ⊗ ω(X) ⊗ coend(ω); 1⊗δ(X); 1⊗φ(X); ev ⊗1; 1⊗1⊗φ̃_{0}; φ̃_{0}; ω(X)^{∗} ⊗ ω(X) ⊗ M; M,; ev ⊗1]

hence we have φ̃_{0} = φ̃.

Corollary 3.4.4. The functor Nat(ω, ω ⊗ M) is a representable functor in M represented by coend(ω).

Proof. The universal problem implies the isomorphism

Nat(ω, ω ⊗ M) ≅ Mor_{C}(coend(ω), M)

and the universal natural transformation δ : ω ⟶ ω ⊗ coend(ω) is mapped to the identity under this isomorphism.

It is also possible to construct an isomorphism

Nat(ω, ω′ ⊗ M) ≅ Mor_{C}(cohom(ω′, ω), M)

for different functors ω, ω′ : D ⟶ C and thus define cohomomorphism objects. Observe that only ω′ has to take values in C_{0} since then we can build objects ω′(X)^{∗} ⊗ ω(X).

5. The coalgebra coend

Proposition 3.5.1. Let C be a monoidal category and ω : D ⟶ C be a diagram in C. Assume that there is a universal object coend(ω) and natural transformation δ : ω ⟶ ω ⊗ coend(ω).

Then there is exactly one coalgebra structure on coend(ω) such that the diagrams

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

and

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commute.

Proof. Because of the universal property of coend(ω) there are structure morphisms Δ : coend(ω) ⟶ coend(ω) ⊗ coend(ω) and ϵ : coend(ω) ⟶ I. This implies the coalgebra property similar to the proof of Corollary 3.3.8.

Observe that by this construction all objects and all morphisms of the diagram ω : D ⟶ C_{0} ⊆ C are comodules or morphisms of comodules over the coalgebra coend(ω). In fact C := coend(ω) is the universal coalgebra over which the given diagram becomes a diagram of comodules.

Corollary 3.5.2. Let (D, ω) be a diagram C with objects in C_{0}. Then all objects of the diagram are comodules over the coalgebra C := coend(ω) and all morphisms are morphisms of comodules. If D is another coalgebra and all objects of the diagram are D-comodules by φ(X) : ω(X) ⟶ ω(X) ⊗ D and all morphisms of the diagram are morphisms of D-comodules then there exists a unique morphism of coalgebras φ̃: coend(ω) ⟶ D such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ D; 1⊗φ̃]

commutes.

Proof. The morphisms φ(X) : ω(X) ⟶ ω(X) ⊗ D define a natural transformation since all morphisms of the diagram are morphisms of comodules. So the existence and the uniqueness of a morphism φ̃: coend(ω) ⟶ D is clear. The only thing to show is that this is a morphism of coalgebras. This follows from the universal property of C = coend(ω) and the diagram

[figure: ω; ω ⊗ C; δ; δ; 1⊗Δ; 1; 1⊗φ̃; ω ⊗ C; ω ⊗ C ⊗ C; δ⊗1; 1⊗φ̃; 1⊗φ̃⊗φ̃; ω; ω ⊗ D; φ; φ; 1⊗Δ; ω ⊗ D; ω ⊗ D ⊗ D; φ⊗1]

where the right side of the cube commutes by the universal property. Similarly we get that φ̃ preserves the counit since the following diagram commutes

[figure: ω; ω ⊗ C; δ; ω; ω ⊗ D; φ; 1; 1⊗φ̃; 1; 1⊗ε; 1⊗ε; ω ⊗ 𝕂; 1; ω ⊗ 𝕂; 1]

6. The bialgebra coend

Let ω : D ⟶ C and ω′ : D′ ⟶ C be diagrams in C. We call the diagram (D, ω) ⊗ (D′, ω′) := (D × D′, ω ⊗ ω′) with (ω ⊗ ω′)(X, Y) := ω(X) ⊗ ω′(Y) the tensor product of these two diagrams. The new diagram consists of all possible tensor products of all objects and all morphisms of the original diagrams.

From now on we assume that the category C is the category of vector spaces and we use the symmetry τ : V ⊗ W ⟶ W ⊗ V in Vec.

Proposition 3.6.1. Let (D, ω) and (D′, ω′) be finite diagrams in Vec. Then

coend(ω ⊗ ω′) ≅ coend(ω) ⊗ coend(ω′).

Proof. First observe the following. If two diagrams ω : D ⟶ Vec and ω′ : D′ ⟶ Vec are given then \underrightarrow{lim}_{D} \underrightarrow{lim}_{D′}(ω ⊗ ω′) ≅ \underrightarrow{lim}_{D×D′}(ω ⊗ ω′) ≅ \underrightarrow{lim}_{D}(ω) ⊗ \underrightarrow{lim}_{D′}(ω′) since the tensor product preserves colimits and colimits commute with colimits. For this consider the diagram

[figure: ω(X) ⊗ ω′(Y); ω(X) ⊗ \underrightarrow{lim}_{D′}(ω′); \underrightarrow{lim}_{D}(ω) ⊗ ω′(Y); \underrightarrow{lim}_{D}(ω) ⊗ \underrightarrow{lim}_{D′}(ω′); ≅ \underrightarrow{lim}_{D×D′}(ω ⊗ ω′).]

The maps in the diagram are the injections for the corresponding colimits. In particular we have coend(ω⊗ω′) ≅ \underrightarrow{lim}_{D×D′}((ω⊗ω′)^{∗}⊗(ω⊗ω′)) ≅ \underrightarrow{lim}_{D×D′}(ω^{∗}⊗ω⊗ω′^{∗}⊗ω′) ≅ \underrightarrow{lim}_{D}(ω^{∗} ⊗ ω) ⊗ \underrightarrow{lim}_{D′}(ω′^{∗} ⊗ ω′) ≅ coend(ω) ⊗ coend(ω′).

The (universal) morphism

(ι(X)⊗ι′(Y))(1⊗τ ⊗1) : ω(X)^{∗}⊗ω′(Y)^{∗}⊗ω(X)⊗ω′(Y) ⟶ \underrightarrow{lim}(ω^{∗}⊗ω)⊗\underrightarrow{lim}(ω′^{∗}⊗ω′)

can be identified with the universal morphism

ι(X, Y) : ω(X)^{∗} ⊗ ω′(Y)^{∗} ⊗ ω(X) ⊗ ω′(Y) ⟶ \underrightarrow{lim}((ω ⊗ ω′)^{∗} ⊗ (ω ⊗ ω′)).

Hence the induced morphisms

(1 ⊗ τ ⊗ 1)(δ ⊗ δ′) : ω(X) ⊗ ω′(Y) ⟶ ω(X) ⊗ ω′(Y) ⊗ coend(ω) ⊗ coend(ω′)

and

δ : ω(X) ⊗ ω′(Y) ⟶ ω(X) ⊗ ω′(Y) ⊗ coend(ω ⊗ ω′)

can be identified.

Corollary 3.6.2. For all finite diagrams (D, ω) and (D′, ω′) in D there is a universal natural transformation δ : ω ⊗ ω′ ⟶ ω ⊗ ω′ ⊗ coend(ω) ⊗ coend(ω′) so that for each object M and each natural transformation φ : ω ⊗ ω′ ⟶ ω ⊗ ω′ ⊗ M there exists a unique morphism φ̃: coend(ω) ⊗ coend(ω′) ⟶ M such that

[figure: ω ⊗ ω′; ω ⊗ ω′ ⊗ coend(ω) ⊗ coend(ω′); δ; ω ⊗ ω′ ⊗ M; 1⊗1⊗φ̃; φ]

commutes.

Definition 3.6.3. Let (D, ω) be a diagram in C = Vec. Then ω is called reconstructive

• if there is an object coend(ω) in C and a universal natural transformation δ : ω ⟶ ω ⊗ coend(ω)

• and if (1 ⊗ τ ⊗ 1)(δ ⊗ δ) : ω ⊗ ω ⟶ ω ⊗ ω ⊗ coend(ω) ⊗ coend(ω) is a univesarl natural transformation of bifunctors.

Definition 3.6.4. Let (D, ω) be a diagram in Vec. Let D be a monoidal category and ω be a monoidal functor. Then (D, ω) is called a monoidal diagram.

Let (D, ω) be a monoidal diagram Vec. Let A ∈ Vec be an algebra. A natural transformation φ : ω ⟶ ω ⊗ B is called monoidal monoidal if the diagrams

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; φ(X)⊗φ(Y); ρ; ρ⊗m; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; φ(X⊗Y)]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; ω(I); ω(I) ⊗ B; φ(I)]

commute.

We denote the set of monoidal natural transformations by Nat^{⊗}(ω, ω ⊗ B).

Problem 3.6.8. Show that Nat^{⊗}(ω, ω ⊗ B) is a functor in B.

Theorem 3.6.5. Let (D, ω) be a reconstructive, monoidal diagram in Vec. Then coend(ω) is a bialgebra and δ : ω ⟶ ω ⊗ coend(ω) is a monoidal natural transformation.

If B is a bialgebra and ∂ : ω ⟶ ω ⊗ B is a monoidal natural transformation, then there is a unique homomorphism of bialgebras f : coend(ω) ⟶ B such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; ∂; ω ⊗ B; 1⊗f]

commutes.

Proof. The multiplication of coend(ω) arises from the following diagram

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ coend(ω) ⊗ coend(ω); δ⊗δ; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ coend(ω); δ; ≅ ω(X) ⊗ ω(Y) ⊗ coend(ω)]

For the construction of the unit we consider the diagram D_{0} = ({I}, {id}) together with ω_{0} : D_{0} ⟶ Vec, ω_{0}(I) = 𝕂, the monoidal unit object in the monoidal category of diagrams in Vec. Then (𝕂 ⟶ 𝕂 ⊗ 𝕂) = (ω_{0} ⟶ ω_{0} ⊗ coend(ω_{0})) is the universal map. The following diagram then induced the unit for coend(ω)

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; ω(I); ω(I) ⊗ coend(ω); ≅ 𝕂 ⊗ coend(ω)]

By using the universal property one checks the laws for bialgebras.

The above diagrams show in particular that the natural transformation δ : ω ⟶ ω ⊗ coend(ω) is monoidal.

7. The quantum monoid of a quantum space

Problem 3.7.9. If A is a finite dimensional algebra and δ : A ⟶ M(A) ⊗ A the universal cooperation of the Tambara bialgebra on A from the left then τδ : A ⟶ A ⊗ M(A) (with the same multiplication on M(A)) is a universal cooperation of M(A) on A from the right. The comultiplication defined by this cooperation is τΔ : M(A) ⟶ M(A) ⊗ M(A). Thus we have to distinguish between the left and the right Tambara bialgebra on A and we have M_{r}(A) = M_{l}(A)^{cop}.

Now consider the special monoidal diagram scheme D := D[X; m, u]. To make things simpler we assume that Vec is strict monoidal. The category D[X; m, u] has the objects X ⊗ … ⊗ X = X^{⊗n} for all n ∈ ℕ (and I := X^{⊗0}) and the morphisms m : X ⊗ X ⟶ X, u : I ⟶ X and all morphisms formally constructed from m, u, id by taking tensor products and composition of morphisms.

Let A be an algebra with multiplication m_{A} : A ⊗ A ⟶ A and unit u_{A} : 𝕂 ⟶ A. Then ω_{A} : D ⟶ C defined by ω(X) = A, ω(X^{⊗n}) = A^{⊗n}, ω(m) = m_{A} and ω(u) = u_{A} is a strict monoidal functor. If A is finite dimensional then the diagram is finite. We get

Theorem 3.7.1. Let A be a finite dimensional algebra. Then the algebra M(A) coacting universally from the right on A (the right Tambara bialgebra) M(A) and coend(ω_{A}) are isomorphic as bialgebras.

Proof. We have studied the Tambara bialgebra for left coaction f : A ⟶ M(A)⊗ A but here we need the analogue for universal right coaction f : A ⟶ A ⊗ M(A) (see Problem 3.9).

Let B be an algebra and f : A ⟶ A ⊗ B be a homomorphism of algebras. For ω = ω_{A} we define

φ(X^{⊗n}) : ω(X^{⊗n}) = A^{⊗n} ⟶^{f^{⊗n}} A^{⊗n} ⊗ B^{⊗n} ⟶^{1⊗m_{B}^{n}} A^{⊗n} ⊗ B = ω(X^{⊗n}) ⊗ B,

where m_{B}^{n} : B^{⊗n} ⟶ B is the n-fold multiplication on B. The map φ is a natural transformation since the diagrams

[figure: A; A ⊗ B; φ(X); 𝕂; 𝕂 ⊗ B; φ(I); u; 1⊗u]

and

[figure: A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); A ⊗ A ⊗ B ⊗ B; f⊗f; 1⊗m; m; m⊗m; m⊗1; A ⊗ B; f; 1⊗1; A; A ⊗ B; φ(X)]

commute. Furthermore the following commute

[figure: A^{⊗r} ⊗ A^{⊗s}; A^{⊗r} ⊗ A^{⊗s} ⊗ B ⊗ B; φ(X^{⊗r})⊗φ(X^{⊗s}); A^{⊗r} ⊗ A^{⊗s} ⊗ B^{⊗r} ⊗ B^{⊗s}; A^{⊗(r+s)} ⊗ B^{⊗(r+s)}; A^{⊗(r+s)}; A^{⊗(r+s)} ⊗ B; φ(X^{⊗(r+s)})]

so that φ : ω_{A} ⟶ ω_{A} ⊗ B is a monoidal natural transformation.

Conversely let φ : ω_{A} ⟶ ω_{A} ⊗ B be a natural transformation. Let f := φ(X) : A ⟶ A ⊗ B. Then the following commute

[figure: A ⊗ A; A ⊗ A ⊗ B ⊗ B; f⊗f; =; 1⊗m; A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); m; m⊗1; A; A ⊗ B; f]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; =; 𝕂; 𝕂 ⊗ B; u; u⊗1; A; A ⊗ B.; f]

Hence f : A ⟶ A ⊗ B is a homomorphism of algebras.

Thus we have defined an isomorphism

𝕂Alg(A, A ⊗ B) ≅ Nat^{⊗}(ω_{A}, ω_{A} ⊗ B)

that is natural in B. If A is finite dimensional then the left hand side is represented by the Tambara bialgebra M_{r}(A) and the right hand side by the bialgebra coend(ω_{A}). Thus both bialgebras must be isomorphic.

Corollary 3.7.2. There is a unique isomorphism of bialgebras M_{r}(A) ≅ coend(ω_{A}) such that the diagram

[figure: A; A ⊗ M_{r}(A); A ⊗ coend(ω_{A})]

commutes

Proof. This is a direct consequence of the universal property.

Thus the Tambara bialgebra that represents the universal quantum monoid acting on a finite quantum space may be reconstructed by the Tannaka-Krein reconstruction from representation theory. Similar reconstructions can be given for more complicated quantum spaces such as so called quadratic quantum spaces.

8. Reconstruction and C-categories

Now we show that an arbitrary coalgebra C can be reconstructed by the methods introduced above from its (co-)representations or more precisely from the underlying functor ω : Comod-C ⟶ Vec. In this case one can not use the usual construction of coend(ω) that is restricted to finite dimensional comodules.

The following Theorem is an example that shows that the restriction to finite dimensional comodules in general is too strong for Tannaka reconstruction. There may be universal coendomorphism bialgebras for more general diagrams. On the other hand the following Theorem also holds if one only considers finite dimensional corepresentations of C. However the proof then becomes somewhat more complicated.

Definition 3.8.1. Let C be a monoidal category. A category D together with a bifunctor ⊗ : C ×D ⟶ D and natural isomorphisms β : (A⊗B)⊗M ⟶ A⊗(B ⊗M), η : I ⊗ M ⟶ M is called a C-category if the following diagrams commute

[figure: ((A ⊗ B) ⊗ C) ⊗ M; (A ⊗ (B ⊗ C)) ⊗ M; α(A,B,C)⊗1; A ⊗ ((B ⊗ C) ⊗ M); β(A,B⊗C,M); β(A⊗B,C,M); 1⊗β(B,C,M); (A ⊗ B) ⊗ (C ⊗ M); A ⊗ (B ⊗ (C ⊗ M)); β(A,B,C⊗M)]

[figure: (A ⊗ I) ⊗ M; A ⊗ (I ⊗ M); β(A,I,M); A ⊗ M; ρ(A)⊗1; 1⊗η(M)]

A C-category is called strict if the morphisms β, η are the identities.

Let (D, ⊗) and (D′, ⊗) be C-categories. A functor F : D ⟶ D′ together with a natural transformation ζ(A, M) : A ⊗ F(M) ⟶ F(A ⊗ M) is called a weak C-functor if the following diagrams commute

[figure: (A ⊗ B) ⊗ F(M); F((A ⊗ B) ⊗ M); ζ; β; F(β); A ⊗ (B ⊗ F(M)); A ⊗ F(B ⊗ M); 1⊗ζ; F(A ⊗ (B ⊗ M)); ζ]

[figure: I ⊗ F(M); F(I ⊗ M); ζ; F(M); η; F(η)]

If, in addition, ζ is an isomorphism then we call F a C-functor. The functor is called a strict C-functor if ζ is the identity morphism.

8. RECONSTRUCTION AND C-CATEGORIES 107

A natural transformation φ : F ⟶ F′ between (weak) C-functors is called a C-transformation if

[figure: A ⊗ F′(M); F′(A ⊗ M); ζ′; A ⊗ F(M); F(A ⊗ M); ζ; 1_{A}⊗φ(M); φ(A⊗M)]

commutes.

Example 3.8.2. Let C be a coalgebra and C := Vec. Then the category Comod-C of right C-comodules is a C-category since N ∈ Comod-C and V ∈ C = Vec implies that V ⊗ N is a comodules with the comodule structure of N.

The underlying functor ω : Comod-C ⟶ Vec is a strict C-functor since we have V ⊗ ω(N) = ω(V ⊗ N). Similarly ω ⊗ M : Comod-C ⟶ Vec is a C-functor since V ⊗ (ω(N) ⊗ M) ≅ ω(V ⊗ N) ⊗ M.

Lemma 3.8.3. Let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor. Let φ : ω ⟶ ω ⊗ M be a natural transformation. Then φ is a C-transformation with C = Vec.

Proof. It suffices to show 1_{V} ⊗ φ(N) = φ(V ⊗ N) for an arbitrary comodule N. We show that the diagram

[figure: V ⊗ N; V ⊗ N ⊗ M; 1_{V} ⊗φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); 1; 1]

commutes. Let (v_{i}) be a basis of V. For an arbitrary vector space W let p_{i} : V ⊗ W ⟶ W be the projections defined by p_{i}(t) = p_{i}(∑_{j} v_{j} ⊗ w_{j}) = w_{i} where ∑_{j} v_{j} ⊗ w_{j} is the unique representation of an arbitrary tensor in V ⊗ W. So we get

t = ∑_{i} v_{i} ⊗ p_{i}(t)

for all t ∈ V ⊗W. Now we consider V ⊗N as a comodule by the comodule structure of N. Then the p_{i} : V ⊗ N ⟶ N are homomorphisms of comodules. Hence all diagrams of the form commute. Expressed in formulas this means φ(N)p_{i}(t) = p_{i}φ(V ⊗ N)(t) for all t ∈ V ⊗ N. Hence we have

[figure: N; N ⊗ M.; φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); p_{i}; p_{i}⊗M]

\begin{matrix}(1_{V} ⊗ φ(N))(t) = (1_{V} ⊗ φ(N))(∑ v_{i} ⊗ p_{i}(t)) = ∑ v_{i} ⊗ φ(N)p_{i}(t) \\ = ∑_{i} v_{i} ⊗ p_{i}φ(V ⊗ N)(t) = φ(V ⊗ N)(t) \end{matrix}

So we have 1_{V} ⊗ φ(N) = φ(V ⊗ N) as claimed.

We prove the following Theorem only for the category C = Vec of vector spaces. The Theorem holds in general and says that in an arbitrary symmetric monoidal category C the coalgebra C represents the functor C- Nat(ω, ω ⊗ M) ≅ Mor_{C}(C, M) of natural C-transformations.

Theorem 3.8.4. (Reconstruction of coalgebras) Let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor. Then C ≅ coend(ω).

Proof. Let M in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃: C ⟶ M by φ̃ = (ϵ ⊗ 1)φ(C) using the fact that C is a comodule.

Let N be a C-comodule. Then N is a subcomodule of N ⊗ C by δ : N ⟶ N ⊗ C since the diagram

[figure: N ⊗ C; N ⊗ C ⊗ C; δ⊗1; N; N ⊗ C; δ; δ; 1⊗Δ]

commutes. Thus the following diagram commutes

In particular we have shown that the diagram

commutes.

[figure: N; N ⊗ C; δ; N ⊗ M; N ⊗ C ⊗ M; δ⊗1; φ(N); φ(N⊗C)=1_{N} ⊗φ(C); N ⊗ M; 1; 1⊗ϵ⊗1; 1⊗φ̃]

[figure: ω; ω ⊗ C; δ; φ; ω ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C ⟶ M be another homomorphism with (1 ⊗ g)δ = φ. For c ∈ C we have g(c) = g(ϵ ⊗ 1)Δ(c) = (ϵ ⊗ 1)(1 ⊗ g)Δ(c) = (ϵ ⊗ 1)φ(C)(c) = φ̃(c).

The coalgebra structure from Corollary 3.5.1 is the original coalgebra structure of C. This can be seen as follows. The comultiplication δ : ω ⟶ ω ⊗ C is a natural transformation hence (δ ⊗ 1_{C})δ : ω ⟶ ω ⊗ C ⊗ C is also a natural transformation. As in Corollary 3.5.1 this induced a unique homomorphism Δ : C ⟶ C ⊗ C so that the diagram

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

commutes. In a similar way the natural isomorphism ω ≅ ω ⊗ 𝕂 induces a unique homomorphism ϵ : C ⟶ 𝕂 so that the diagram

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commutes. Because of the uniqueness these must be the structure homomorphisms of C.

We need a more general version of this Theorem in the next chapter. So let C be a coalgebra. Let ω : Comod-C ⟶ Vec be the underlying functor and δ : ω ⟶ ω ⊗C the universal natural transformation for C ≅ coend(ω).

We use the permutation map τ on the tensor product that gives the natural isomorphism

τ : N_{1} ⊗ T_{1} ⊗ N_{2} ⊗ T_{2} ⊗ … ⊗ N_{n} ⊗ T_{n} ≅ N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n} ⊗ T_{1} ⊗ T_{2} … ⊗ ⊗T_{n}

which is uniquely determined by the coherence theorems and is constructed by suitable applications of the flip τ : N ⊗ T ≅ T ⊗ N.

Let ω^{n} : Comod-C × Comod-C × … × Comod-C ⟶ Vec be the functor ω^{n}(N_{1}, N_{2}, …, N_{n}) = ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}). For notational convenience we abbreviate {N}^{n} := N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n}, similarly {C}^{n} = C ⊗ C ⊗ … ⊗ C and {f}^{n} := f_{1} ⊗ f_{2} ⊗ … ⊗ f_{n}. So we get τ : {N ⊗ T }^{n} ≅ {N}^{n} ⊗ {T }^{n}.

Lemma 3.8.5. Let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. Then φ is a C-transformation in the sense that the diagrams

[figure: {V }^{n} ⊗ {N}^{n}; {V }^{n} ⊗ {N}^{n} ⊗ M; {V }^{n}⊗φ(N_{1},... ,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},... ,V_{n}⊗N_{n}); τ; τ⊗M]

commute for all vector spaces V_{i} and C-comodules N_{i}.

Proof. Choose bases {v_{ij}} of the vector spaces V_{i} with corresponding projections p_{ij} : V_{i} ⊗ N_{i} ⟶ N_{i}. Then we have τ(t_{1} ⊗ … ⊗ t_{n}) = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}) so τ = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ {p}^{n}.

The p_{ij_{i}} : V_{i} ⊗N_{i} ⟶ N_{i} are homomorphisms of C-comodules. Hence the diagrams

[figure: {N}^{n}; {N}^{n} ⊗ M; φ(N_{1},... ,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},... ,V_{n}⊗N_{n}); {p}^{n}; {p}^{n}⊗M]

commute for all choices of {p}^{n} = p_{1i_{1}} ⊗ … ⊗ p_{ni_{n}}.

So we get for all t_{i} ∈ V_{i} ⊗ N_{i}

({V }^{n} ⊗ φ(N_{1}, …, N_{n}))τ(t_{1} ⊗ … ⊗ t_{n}) =

= ({V }^{n} ⊗ φ(N_{1}, …, N_{n}))(∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}))

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ φ(N_{1}, …, N_{n}){p}^{n}(t_{1} ⊗ … ⊗ t_{n})

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ ({p}^{n} ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n})

= (τ ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n}).

Theorem 3.8.6. With the notation given above we have

coend(ω^{n}) ≅ C ⊗ C ⊗ … ⊗ C

with the universal natural transformation

δ^{(n)}(N_{1}, N_{2}, …, N_{n}) := τ(δ(N_{1}) ⊗ δ(N_{2}) ⊗ … ⊗ δ(N_{n})) :

ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⟶ ω(N_{1}) ⊗ C ⊗ ω(N_{2}) ⊗ C ⊗ … ⊗ ω(N_{n}) ⊗ C ≅ ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⊗ C ⊗ C ⊗ … ⊗ C.

Proof. We proceed as in the proof of the previous Theorem.

Let M in Vec and let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. We define the homomorphism φ̃: C^{n} = ω(C) ⊗ ω(C) ⊗ … ⊗ ω(C) = C ⊗ C ⊗ … ⊗ C ⟶ M by φ̃ = (ε^{n} ⊗ 1_{M})φ(C, …, C) using the fact that C is a comodule.

As in the preceding proof we get that δ : N_{i} ⟶ N_{i} ⊗ C are homomorphisms of C-comodules. Thus the following diagram commutes

[figure: N_{1} ⊗ … ⊗ N_{n}; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C; δ⊗...⊗δ; N_{1} ⊗ … ⊗ N_{n}⊗; τ; C ⊗ … ⊗ C; N_{1} ⊗ … ⊗ N_{n} ⊗ M; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C ⊗ M; δ⊗...⊗δ⊗M; N_{1} ⊗ … ⊗ N_{n}⊗; τ⊗M; C ⊗ … ⊗ C ⊗ M; φ(N_{1}⊗...⊗N_{n}); φ(N_{1}⊗C,... ,N_{n}⊗C); N_{1}⊗...⊗N_{n}⊗φ(C,... ,C); 1; {1⊗ε}^{n}⊗1; N_{1} ⊗ … ⊗ N_{n} ⊗ M; 1⊗{ϵ}^{n}⊗1]

Hence we get the commutative diagram

[figure: ω^{n}; ω^{n} ⊗ {C}^{n}; δ^{(n)}; φ; ω^{n} ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C^{n} ⟶ M be another homomorphism with (1_{ω^{n}} ⊗ g)δ^{(n)} = φ. We have g = g(ε^{n} ⊗ 1_{C^{n}})τΔ^{n} = g(ε^{n} ⊗ 1_{C^{n}})δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})(1_{C^{n}} ⊗ g)δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})φ(C, …, C) = φ̃.

Now we prove the finite dimensional case of reconstruction of coalgebras.

Proposition 3.8.7. (Reconstruction) Let C be a coalgebra. Let Comod_{0}-C be the category of finite dimensional C-comodules and ω : Comod_{0}-C ⟶ Vec be the underlying functor. Then we have C ≅ coend(ω).

Proof. Let M be in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃ : C ⟶ M as follows. Let c ∈ C. Let N be a finite dimensional C-subcomodule of C containing c. Then we define g(c) := (ϵ|_{N} ⊗ 1)φ(N)(c). If N′ is another finite dimensional subcomodule of C with c ∈ N′ and with N ⊆ N′ then the following commutes

[figure: N; N ⊗ M; φ(N); N′; N′ ⊗ M; φ(N′); C ⊗ M; M; ϵ⊗1]

Thus the definition of φ̃(c) is independent of the choice of N. Furthermore φ̃ : N ⟶ M is obviously a linear map. For any two elements c, c′ ∈ C there is a finite dimensional subcomodule N ⊆ C with c, c′ ∈ N e. g. the sum of the finite dimensional subcomodules containing c and c′ separately. Thus φ̃ can be extended to all of C.

The rest of the proof is essentially the same as the proof of the first reconstruction theorem.

The representations allow to reconstruct further structure of the coalgebra. We prove a reconstruction theorem about bialgebras. Recall that the category of Bcomodules over a bialgebra B is a monoidal category, furthermore that the underlying functor ω : Comod-B ⟶ Vec is a monoidal functor. From this information we can reconstruct the full bialgebra structure of B. We have

Theorem 3.8.8. Let B be a coalgebra. Let Comod-B be a monoidal category such that the underlying functor ω : Comod-B ⟶ Vec is a monoidal functor. Then there is a unique bialgebra structure on B that induces the given monoidal structure on the corepresentations.

Proof. First we prove the uniqueness of the multiplication ∇ : B ⊗ B ⟶ B and of the unit η : 𝕂 ⟶ B. The natural transformation δ : ω ⟶ ω ⊗ B becomes a monoidal natural transformation with ∇ : B ⊗ B ⟶ B and η : 𝕂 ⟶ B We show that ∇ and η are uniquely determined by ω and δ.

Let ∇′ : B ⊗ B ⟶ B and η′ : B ⟶ 𝕂 be morphisms that make δ a monoidal natural transformation. The diagrams

and

commute. In particular the following diagrams commute and

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; δ(X)⊗δ(Y); ρ; ρ⊗∇′; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; δ(X⊗Y)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)] [figure: ω(B) ⊗ ω(B); ω(B) ⊗ ω(B) ⊗ B ⊗ B; δ(B)⊗δ(B); ρ; ρ⊗∇′; ω(B ⊗ B); ω(B ⊗ B) ⊗ B; δ(B⊗B)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)]

Hence we get ∑ b_{(1)}⊗c_{(1)}⊗b_{(2)}c_{(2)} = ∑ b_{(1)}⊗c_{(1)}⊗∇′(b_{(2)}⊗c_{(2)}) and 1⊗1 = 1⊗η′(1). This implies bc = ∑ ϵ(b_{(1)})ϵ(c_{(1)})b_{(2)}c_{(2)} = ∑ ϵ(b_{(1)})ϵ(c_{(1)})∇′(b_{(2)} ⊗ c_{(2)}) = ∇′(b ⊗ c) and 1 = η′(1).

Now we show the existence of a bialgebra structure. Let B be a coalgebra only and let ω : Comod-B ⟶ Vec be a monoidal functor with ξ : ω(M)⊗ω(N) ⟶ ω(M ⊗N) and ξ_{0} : 𝕂 ⟶ ω(𝕂). First we observe that the new tensor product between the comodules M and N coincides with the tensor product of the underlying vector spaces (up to an isomorphism ξ). Because of the coherence theorems for monoidal categories (that also hold in our situation) we may identify along the maps ξ and ξ_{0}.

We define η := (𝕂 ⟶^{δ(𝕂)} 𝕂 ⊗ B ≅ B) and ∇ := (B ⊗ B ⟶^{δ(B⊗B)} B ⊗ B ⊗ B ⟶^{ϵ⊗ϵ⊗1_{B}}

𝕂 ⊗ 𝕂 ⊗ B ≅ B).

Since the structural morphism for the comodule δ : M ⟶ M ⊗ B is a homomorphism of of B comodules where the comodule structure on M ⊗ B is only given by the diagonal of B that is the C-structure on ω : Comod-B ⟶ Vec we get that also δ(M) ⊗ δ(N) : M ⊗ N ⟶ M ⊗ N ⊗ B is a comodule homomorphism. Hence the first square in the following diagram commutes

[figure: M ⊗ N; M ⊗ B ⊗ N ⊗ B; δ(M)⊗δ(N); M ⊗ N ⊗ B; M ⊗ B ⊗ N ⊗ B ⊗ B; δ(M)⊗δ(N)⊗1_{B}; M ⊗ N ⊗ B ⊗ B; 1⊗τ⊗1; M ⊗ N ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1⊗1; δ(M⊗N); δ(M⊗B⊗N⊗B); 1⊗1⊗δ(B⊗B)]

The second square commutes by a similar reasoning since the comodule structure on M ⊗ B resp. N ⊗ B is given by the diagonal on B hence M ⊗ N can be factored out of the natural (C-)transformation. Now we attach

1_{M} ⊗ 1_{N} ⊗ ϵ ⊗ ϵ ⊗ 1_{B} : M ⊗ N ⊗ B ⊗ B ⊗ B ⟶ M ⊗ N ⊗ B

to the commutative rectangle and obtain δ(M ⊗N) = (1_{M} ⊗1_{N} ⊗∇)(1⊗τ ⊗1)(δ(M)⊗ δ(N). Thus the comodule structure on M ⊗ N is induced by the multiplication ∇ : B ⊗ B ⟶ B defined above.

So the following diagrams commute

[figure: B ⊗ B; B ⊗ B ⊗ B ⊗ B; Δ⊗Δ; B ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B; B ⊗ B ⊗ B; δ(B⊗B); B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; 1⊗1⊗Δ; B; B ⊗ B; Δ; 1_{B}⊗1_{B}; 1⊗∇; δ(B⊗B); δ(B⊗B)⊗1; ∇; ∇⊗1; ϵ⊗ϵ⊗1; ϵ⊗ϵ⊗1⊗1]

[figure: B ⊗ B; B; ∇; B ⊗ B; 𝕂; ϵ⊗ϵ; B ⊗ B ⊗ B; δ(B⊗B); ϵ⊗ϵ⊗1; 1; 1⊗1⊗ϵ; ϵ]

[figure: 𝕂; B; η; 𝕂 ⊗ B; 𝕂 ⊗ B ⊗ B; δ(𝕂)⊗1; B ⊗ B; ≅; B; B ⊗ B; η⊗1; 𝕂 ⊗ B; δ(𝕂); δ(𝕂); ≅; ≅; η⊗1; 1_{B}; η; 1⊗Δ; ≅; Δ]

and

[figure: 𝕂; 𝕂; 1; 𝕂 ⊗ B; δ(𝕂); 1⊗ϵ; η; ϵ; B; ≅]

Hence η and ∇ are coalgebra homomorphisms.

To show the associativity of ∇ we identify along the maps α : (M ⊗ N) ⊗ P ≅ M ⊗(N ⊗P) and furthermore simplify the relevant diagram by fixing that σ represents a suitable permutation of the tensor factors. Then the following commute

[figure: B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B)⊗δ(B)⊗δ(B); B ⊗ B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B⊗B)⊗δ(B)); σ(δ(B)⊗δ(B⊗B)); B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; δ(B⊗B⊗B); B; ϵ⊗ϵ⊗ϵ⊗1; 1; 1; 1⊗(∇⊗1); 1⊗(1⊗∇); ∇⊗1; 1⊗∇; 1⊗∇; ∇]

The upper row is the identity hence we get the associative law.

For the proof that η has the properties of a unit we must explicitly consider the coherence morphisms λ and ρ By reasons of symmetry we will only show one half of of the unit axiom. This axiom follows from the commutativity of the following diagram

[figure: B; B ⊗ B; δ(B); B ⊗ B ⊗ 𝕂; ρ^{−1}; B ⊗ B; ρ; B; ϵ⊗1; B ⊗ 𝕂; B ⊗ B ⊗ 𝕂 ⊗ B; δ(B)⊗δ(𝕂); B ⊗ 𝕂 ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B ⊗ B; ρ⊗1⊗1; B ⊗ B; ϵ⊗1⊗1; B ⊗ 𝕂; B ⊗ 𝕂 ⊗ B; δ(B⊗𝕂); B; B ⊗ B; δ(B); B; ϵ⊗1; ρ^{−1}; 1⊗1⊗δ(𝕂); 1⊗1⊗η; 1⊗η; =; 1⊗1⊗∇; 1⊗∇; ∇; ρ; δ(B)⊗1; ρ⊗1]