RECONSTRUCTION OF HIDDEN SYMMETRIES

BODO PAREIGIS

revised Vers. September 16, 1995

Contents

1. Introduction 1

2. Braided categories and C-categories 4 2.1. C-categories 2.2. Braided categories 2.3. C-monoidal categories 2.4. Rigid categories 2.5. Coadjoint coactions 2.6. C_{0}-generated coalgebras

3. Reconstruction properties 15 3.1. Reconstruction of coalgebras 3.2. Reconstruction of bialgebras 3.3. Reconstruction of morphisms 3.4. Applications

4. Existence theorems in reconstruction theory 22 4.1. Restricted reconstruction 4.2. Finite reconstruction

5. Hidden symmetries 32 5.1. Functors of control categories 5.2. Hidden symmetries of the base category

6. Appendix on 𝕂-additive categories and C-categories 39

References 42

1. Introduction

Groups G are often obtained as groups of symmetries (or automorphisms) of mathematical structures like a vector space (over a fixed field 𝕂) or two vector spaces together with a linear map between them or a whole diagram of vector spaces, where a symmetry of such a diagram is a family of automorphisms one for each vector space which are compatible with the linear maps of the diagram (a natural automorphism). This process of constructing the group of symmetries is a special case of the notion of (Tannaka-Krein) reconstruction.

Conversely given a group G one considers its representations G ⟶ GL(V) in vector spaces V over the field 𝕂. All representations of G form the category _{𝕂G}M of modules, which we may consider as a huge diagram of vector spaces. The category 𝕂GM has an additional interesting structure – the tensor product V ⊗ W of two representations is again a representation in a canonical way, _{𝕂G}M is a monoidal category. A special consequence of reconstruction theory is the fact that G may be recovered as the full group of those symmetries of this huge diagram which are compatible with the tensor product. This process seems to be the inverse of the first one. In a more general setting there are, however, subtle deviations. One may reconstruct much larger groups of symmetries than what one started out with.

More generally we know that algebras A, Lie algebras 𝔤 and Hopf algebras H can be reconstructed from their categories of modules. For the reconstruction of an algebra A one actually needs not only the category of A-modules A-Mod but also the underlying functor ω : A-Mod ⟶ Vec. Then A (as an algebra) can be reconstructed (up to isomorphism) as end(ω), the categorical end of the underlying functor. For the reconstruction of a Hopf algebra H one needs in addition the monoidal structure of H-Mod. Then the full Hopf algebra structure can be reconstructed [DM82, Pa81, Ul89].

This stands in pointed contrast to another similar result, the Morita theorems [Ba68], which show that the knowledge of the category of modules A-Mod of an algebra A does not determine A up to isomorphism.

As we remarked before, the forgetful functor ω : A-Mod ⟶ Vec is essential in the process of reconstruction. In particular one has to consider representations of the given objects (algebras, groups, Lie algebras, Hopf algebras) in vector spaces. Representations in categories of objects with a richer structure like super vector spaces, ⋆-spaces, graded vector spaces, comodules over Hopf algebras have a different behavior.

Instead of the base category Vec we wish to use an arbitrary braided monoidal category A. There are many examples for such a base category A such as L-Mod, the category of modules over a quasitriangular Hopf algebra L, L-Comod, the category of comodules over a coquasitriangular Hopf algebra L, _{L}^{L}YD, the category of Yetter-Drinfeld modules over a Hopf algebra with bijective antipode, or dys C_{A}, the category of dyslectic modules over a commutative algebra in a braided monoidal category C [Pa95]. We study the following problem: given an algebra, a bialgebra, or a Hopf algebra H in A, can it be reconstructed from _{H}A, the underlying functor ω : _{H}A ⟶ A and the monoidal structure? A special case is the reconstruction of a super algebra from its super representations.

The surprising solution of this problem shows that one usually reconstructs an object end(ω) from the underlying functor ω : _{H}A ⟶ A in A that is much bigger than H. In the group case (for H = kG) this amounts to additional symmetries which we call hidden symmetries. A concrete case of such a hidden symmetry is given in 5.1.8. In the (Hopf) algebra case the situation is more complex but we also talk about hidden symmetries. In certain cases we describe precisely the additional hidden symmetries by a smash product decomposition of the reconstructed object.

We control the process of reconstruction by a control category C which operates on ω : _{H}A ⟶ A. With different choices of the control category C we obtain different reconstructed objects end_{C}(ω) from one and the same underlying functor ω : _{H}A ⟶ A. We study the properties of end_{C}(ω).

The second section of this paper is devoted to some basic notions from the theory of braided monoidal categories C and the notion of C-categories. The most interesting examples for C are the categories of modules resp. comodules over Hopf algebras with an additional structure known as a quasitriangular structure resp. braiding, one example is the category of super vector spaces, furthermore the category of Yetter-Drinfeld modules.

In the third section we study the general algebraic structure of reconstructed objects in a braided monoidal category. We separate the discussion of the properties of reconstructed objects strictly from the existence theorems that are treated in the fourth section. We have decided to base our investigations on coalgebras and (right) comodules instead of algebras and (left) modules, because the fundamental structure theorem for comodules makes certain constructions in vector spaces in this case much easier. So we study the coend of a functor ω : B ⟶ A as the universal natural transformation ω ⟶ ω ⊗ U and show that such a universal coend(ω) = U ∈ A carries the structure of a coalgebra, a bialgebra, or even a Hopf algebra depending on the properties of ω. These structures on the coend of a functor have already been studied in various papers [Mj93b, Mj94a, Mj94b, Pa93, Sch92a]. Our techniques allow us to restrict the class of natural transformations (by the notion of C-morphisms and by choosing different control categories C), which gives us a family of different universal transformations ω ⟶ ω ⊗ U_{C} parametrized by the choice of control category C and thus different coalgebras, bialgebras, or Hopf algebras U_{C}. This technique was first used in [Pa78, Pa81]. Now we expand this technique to the case of braided monoidal categories to study the structure of hidden symmetries. It turns out that some of the structure is connected with coadjoint coactions, cosmash products and transmutation.

In the fourth section we show under which conditions a coalgebra C in A can be reconstructed from the category of C-comodules A^{C} in A and the functor ω : A^{C} ⟶ A. If C is a coalgebra in the C-monoidal category A = C and ω : A^{C} ⟶ A is the underlying functor then C = coend_{C}(ω) ([Pa81] and Cor. 4.3 with C_{0} = C). One would like to have a more general theorem of the form: if C is a coalgebra in an arbitrary C-monoidal category A and ω : A^{C} ⟶ A is the underlying functor then C = coend_{C}(ω). This, however, is not true because of hidden symmetries, which live in part in A (or more precisely in coend_{C}(id : A ⟶ A), see section 5). Our use of control categories allows us to reconstruct a coalgebra from its representations (comodules) in an arbitrary monoidal category A, namely C = coend_{A}(ω : A^{C} ⟶ A) (no braiding of A is needed). This fact allows us now to wonder about and study the additional hidden symmetries appearing in coend_{C}(ω). Various refinements of this reconstruction can be found in Theorems 4.1 and 4.2.

The second existence theorem deals with the (re-)construction of C in the case of an arbitrary C-functor ω : B ⟶ A. Conditions for such a reconstruction have been developed in many papers, such as [Ul89, Ul90] in the case A = Vec. In our situation we can prove the following: if ω : B ⟶ C is a C_{0}-functor which factors through the category C_{0} of rigid objects in C and if C is cocomplete then U_{C_{0}} = coend_{C_{0}}(ω) exists. A weak variant of this theorem was proved in [Mj93b] (Theorem 2.2) where B is essentially a category with only a finite number of objects and morphisms, the control category is trivial, and only rigid objects are reconstructed (for more details see the remarks in 4.1.2).

In section five we show as one of the main results of this paper that the universal object coend_{C}(ω) = U_{C} for a functor ω : B ⟶ A tends to decompose into a cosmash product of a Hopf algebra with a coalgebra. In particular we show the following. If H is a braided Hopf algebra in a braided monoidal category D, C is a coalgebra in D^{H}, and ω : (D^{H})^{C} ⟶ D^{H} is the underlying functor. Then there is a canonical isomorphism

f : coend_{D}(ω) ⟶ coend_{D}(id_{D^{H}}) ˜#^{c}coend_{D^{H}}(ω),

where we use a cosmash product with respect to the transmutation multiplication in H. As a special example one obtains for a braided Hopf algebra H over a field 𝕂, an H-comodule coalgebra C, and the functor ω : (Vec^{H})^{C} ⟶ Vec^{H}, that the coend of ω is the cosmash product H ˜#^{c}C ≅ H#^{c}C.

In the Appendix we study certain connections between 𝕂-additive categories and our notion of C-categories and show in particular (Theorem 6.4) why there are no hidden symmetries in the case of representations in ordinary vector spaces.

2. Braided categories and C-categories

Throughout this paper let A be a monoidal category, i.e. a category together with a bifunctor ⊗ : A × A ⟶ A, a neutral object I ∈ A, and natural isomorphisms α : (P ⊗Q)⊗R ⟶ P ⊗(Q⊗R), λ : I ⊗P ⟶ P, and ρ : P ⊗I ⟶ P, satisfying the well-known coherence (constraint) conditions. Without loss of generality (by Mac Lane’s coherence theorem [ML71] Theorem 15.1) we shall assume that A is a strict monoidal category, so that all associativity and unit isomorphisms are identities. Similarly C will be a monoidal category throughout.

We are mainly interested in the case where A is the category of (right) modules M_{B} or comodules M^{B} over a bialgebra B (over a field 𝕂) with the canonical monoidal structure. Further examples are Vec the category of vector spaces over a field 𝕂, vec the category of finite-dimensional vector spaces, mod-B, the category of finite-dimensional (over 𝕂) B-modules, and comod-B, the category of finitedimensional B-comodules. Some interesting topological examples may be found in [Ye90].

2.1. C-categories. We will need the notion of categories, functors, and natural transformations “over” a monoidal category C, which we call C-categories, Cfunctors, and C-morphisms. Many of their properties have been investigated in [Pa77, Pa81]. They are built in analogy to G-sets and their morphisms or R-modules and their morphisms.

Definition 2.1. A category B together with a bifunctor ⊗ : C × B ⟶ B and coherent^{1} natural isomorphisms β : (X ⊗ Y) ⊗ P ⟶ X ⊗ (Y ⊗ P) (for X, Y ∈ C, P ∈ B) and π : I ⊗ P ⟶ P will be called a (left) C-category. (For the coherence conditions see [Se79].) In such a context we will call C a control category.

Some of our main examples are:

2.1.1. A monoidal category A is an A-category.

1Whenever we use the term “coherent” we mean that the given natural transformation is coherent also with respect to the already existing coherent natural transformations, in this case with α, λ, and ρ. The minimal requirements for coherence are obvious in most cases. We do not further investigate them.

2.1.2. Let A (with m_{A} : A ⊗ A ⟶ A and u_{A} : I ⟶ A) be an algebra (a monoid) in A, i.e. the multiplication m_{A} is associative and unital (with unit morphism u_{A}). In the situation A = M^{B}, such an algebra A is called a B-comodule algebra. In the case A = M_{B}, such an algebra A is called a B-module algebra [Sw69].

2.1.3. The category B = A_{A} of (right) A-modules (P, κ : P ⊗ A ⟶ P) in A is a (left) A-category, since X ⊗ P carries the structure of a right A-module by (X ⊗ P) ⊗ A ≅ X ⊗ (P ⊗ A) ⟶ X ⊗ P.

2.1.4. A vector space P is in (M^{B})_{A} if and only if P is a right B-comodule and a right A-module such that δ(pa) = ∑ p_{(0)}a_{(0)} ⊗ p_{(1)}a_{(1)}, a B-A-Hopf module. A vector space P is in (M_{B})_{A} iff P is a right B-module and a right A-module such that (pa)b = ∑(pb_{(1)})(ab_{(2)}), i.e. a B#A-module.

2.1.5. Furthermore let C (with Δ_{C} : C ⟶ C ⊗C and ε_{C} : C ⟶ I) be a coalgebra in A. In the situation A = M^{B}, such a coalgebra C is called a B-comodule coalgebra. In the case A = M_{B}, such a coalgebra C is called a B-module coalgebra.

2.1.6. The category B = A^{C} of (right) C-comodules (P, δ : P ⟶ P ⊗ C) in A is a (left) A-category, since X ⊗ P carries the structure of a right C-comodule by X ⊗ P ⟶ X ⊗ (P ⊗ C) ≅ (X ⊗ P) ⊗ C.

2.1.7. A vector space P is in (M^{B})^{C} if and only if P is a right B-comodule and a right C-comodule such that

[figure: P; P ⊗ B; δ_{B}; P ⊗ C; (P ⊗ C) ⊗ B; δ_{B}; δ_{C}; δ_{C}⊗id_{B}]

commutes, i.e. P is a B#^{c}C-comodule, a comodule over the cosmash product. A vector space P is in (M_{B})^{C} iff P is a right B-module and a right C-comodule such that δ_{C}(pb) = ∑ p_{(0)}b_{(1)} ⊗ p_{(C,1)}b_{(2)}.

Definition 2.2. Let B and B′ be C-categories. A functor ω : B ⟶ B′ together with a coherent natural isomorphism ξ : ω(X ⊗ P) ⟶ X ⊗ ω(P) is called a C-functor.

Observe that our assumption on coherence implies in particular that π′ξ(I, P) = ω(π).

2.1.8. The identity functor id : A ⟶ A is an A-functor. Furthermore the forgetful (underlying) functors ω : A_{A} ⟶ A resp. ω : A^{C} ⟶ A are easily seen to be Afunctors. If f : A ⟶ A′ is an algebra morphism in A, then the induced functor ω : A_{A′} ⟶ A_{A} is an A-functor. Similarly if f : C ⟶ C′ is a coalgebra morphism in A, then the induced functor ω : A^{C} ⟶ A^{C′} is an A-functor.

2.1.9. We will use additional C-functors. Let ω : B ⟶ A be an C-functor and let M ∈ A. Then ω ⊗ M : B ∋ P ↦ ω(P) ⊗ M ∈ A is again a C-functor.

Definition 2.3. Let B and B′ be C-categories and ω : B ⟶ B′ and ω′ : B ⟶ B′ be C-functors. A natural transformation φ : ω ⟶ ω′ is a C-morphism if the following diagram commutes

[figure: ω(X ⊗ P); ω′(X ⊗ P); φ(X⊗P); X ⊗ ω(P); X ⊗ ω′(P).; X⊗φ(P); ξ(X,P); ξ′(X,P)]

We will denote the set^{2} of natural transformation from ω to ω′ by Nat(ω, ω′) and the subset of C-morphisms by Nat_{C}(ω, ω′).

2.1.10. These C-morphisms will be of central importance for reconstruction, so we will give an example. Let B be a bialgebra in Vec and C = A := M_{B}. Let A be an algebra in Vec. It can be considered as a B-module algebra by the trivial action ab := aε(b). Let B := (M_{B})_{A}. Consider the C-functor ω : (M_{B})_{A} ⟶ M_{B} with ω(P) = P, the forgetful functor. Then for any a ∈ A the morphism φ_{a} : ω ⟶ ω, φ_{a}(P) : ω(P) ⟶ ω(P), φ_{a}(p) = pa is a natural transformation and in fact a Cmorphism. For any b ∈ center(B) the morphism φ_{b} : ω ⟶ ω, φ_{b}(P) : ω(P) ⟶ ω(P), φ_{b}(p) = pb is a natural transformation, but in general it is not a C-morphism. If φ_{b} is a C-morphism then for the special choice X = B, P = B ⊗ A, x = 1_{B}, and p = 1_{B} ⊗ 1_{A} we have ∑ b_{(1)} ⊗ b_{(2)} ⊗ 1_{A} = ∑ b_{(1)} ⊗ b_{(2)} ⊗ 1_{A}b_{(3)} = ∑ xb_{(1)} ⊗ pb_{(2)} = φ_{b}(x ⊗ p) = x ⊗ φ_{b}(p) = x ⊗ pb = 1_{B} ⊗ b ⊗ 1_{A}, and hence Δ(b) = 1 ⊗ b which implies b = α · 1_{B} (α ∈ 𝕂). Conversely for b = α · 1_{B} it is easy to see that φ_{b} is a C-morphism.

2.2. Braided categories. For the definition and study of more complicated objects, like bialgebras and Hopf algebras in C, we assume that the monoidal category C is braided (or a quasitensor category) with a natural isomorphism of bifunctors σ_{X,Y} : X ⊗ Y ≅ Y ⊗ X, the braiding, such that (1_{Y} ⊗ σ_{X,Z})(σ_{X,Y} ⊗ 1_{Z}) = σ_{X,Y ⊗Z} and (σ_{X,Z} ⊗ 1_{Y})(1_{X} ⊗ σ_{Y,Z}) = σ_{X⊗Y,Z}.

2.2.1. A quasitriangular structure or universal R-matrix [Dr86] for a bialgebra B = (B, m, u, Δ, ε) in Vec is an invertible element R = ∑ R_{1} ⊗ R_{2} ∈ B ⊗ B such that

(1) ∀b ∈ B : τΔ(b) = RΔ(b)R^{−1},

(2) (Δ ⊗ 1)(R) = R_{13}R_{23},

(3) (1 ⊗ Δ)(R) = R_{13}R_{12} where R_{12} = R ⊗ 1_{B}, R_{13} = ∑ R_{1} ⊗ 1_{B} ⊗ R_{2}, and R_{23} = 1_{B} ⊗ R.

2.2.2. A coquasitriangular structure ([Sch92b] Definition 2.4.4 and [LT91]) or braiding is a convolution-invertible homomorphism r : B ⊗ B ⟶ 𝕂 such that

(1) mτ = r ∗ m ∗ r^{−1},

(2) r(m ⊗ 1) = r^{13}r^{23},

(3) r(1 ⊗ m) = r^{13}r^{12}.

2.2.3. If B is quasitriangular then M_{B} is a braided monoidal category with σ_{X,Y} (x⊗ y) = ∑(yR_{2} ⊗ xR_{1}) [Dr86].

2.2.4. If B is coquasitriangular then M^{B} is a braided monoidal category with σ_{X,Y} (x ⊗ y) = ∑(y_{(0)} ⊗ x_{(0)})r(x_{1} ⊗ y_{1}) ([Sch92b] Remark 2.4.6; see also the last paragraph in [Pa81]).

2There are well known standard methods to handle the set theoretic difficulties of this construction.

2.2.5. Here are some observations from [Mj94a] about algebras, bialgebras and Hopf algebras in braided monoidal categories C. If A and B are algebras in C then so is A ⊗ B. We use the graphical calculus [Ye90] to describe the algebra multiplication as

[figure: A; B; A; B; A; B]

which represents the morphism (m_{A} ⊗ m_{B})(1_{A} ⊗ σ_{B,A} ⊗ 1_{B}). One checks that A ⊗ B becomes an algebra with this multiplication.

This allows us to define a bialgebra in C which is an algebra (B, m, u) and a coalgebra (B, Δ, ε) such that

[figure: B; B; B; B; =; B; B; B; B; ε; ε; ε; u; u; u; u; ε; 1; B; B; B; B; =; =; =; B; B; B; B]

i.e. Δm = (m ⊗ m)(1 ⊗ σ ⊗ 1)(Δ ⊗ Δ), εm = ε ⊗ ε, Δu = u ⊗ u, and εu = 1_{𝕂}.

A bialgebra H in C is a Hopf algebra with antipode S : H ⟶ H in C if it also satisfies

[figure: S; ε; u; S; H; H; H; =; =; H; H; H]

A bialgebra H in C has a twisted antipode S in C if it satisfies

[figure: S; H; H; =; ε; H; u; H; =; H; S; H]

The notion of a braided bialgebra in C is somewhat more subtle and has been studied in [Mj93a].

2.3. C-monoidal categories. Now let C be a braided monoidal category.

Definition 2.4. Let B, B′, and B′′ be C-categories. A bifunctor ω : B × B′ ⟶ B′′ together with natural isomorphisms coherent with the C-structures on B, B′, and B′′,

(1) ξ_{X,P,Q} : ω(X ⊗ P, Q) ⟶ X ⊗ ω(P, Q),

(2) τ_{P,X,Q} : ω(P, X ⊗ Q) ⟶ X ⊗ ω(P, Q), and

(3) τ̃_{X,P,Q} : X ⊗ ω(P, Q) ⟶ ω(P, X ⊗ Q) is called a C-bifunctor, if the following diagrams commute

[figure: ω(X ⊗ P, Y ⊗ Q); Y ⊗ ω(X ⊗ P, Q); τ_{X⊗P,Y,Q}; Y ⊗ X ⊗ ω(P, Q); 1⊗ξ; X ⊗ ω(P, Y ⊗ Q); X ⊗ Y ⊗ ω(P, Q); 1⊗τ_{P,Y,Q}; ξ; σ_{X,Y} ⊗1_{ω(P,Q)}]

[figure: ω(P, X ⊗ Y ⊗ Q); X ⊗ Y ⊗ ω(P, Q); τ_{P,X⊗Y,Q}; X ⊗ ω(P, Y ⊗ Q).; τ_{P,X,Y ⊗Q}; 1_{X}⊗τ_{P,Y,Q}]

[figure: X ⊗ Y ⊗ ω(P, Q); ω(P, X ⊗ Y ⊗ Q); _{τ_{X⊗Y,P,Q}}; X ⊗ ω(P, Y ⊗ Q).; 1_{X}⊗_{τ_{Y,P,Q}}; _{τ_{X,P,Y ⊗Q}}]

and

[figure: Y ⊗ X ⊗ ω(P, Q); Y ⊗ ω(X ⊗ P, Q); 1⊗ξ^{−1}; ω(X ⊗ P, Y ⊗ Q); _{τ_{Y,X⊗P,Q}}; X ⊗ Y ⊗ ω(P, Q); X ⊗ ω(P, Y ⊗ Q); 1⊗_{τ_{Y,P,Q}}; ξ^{−1}; σ_{Y,X}⊗1_{ω(P,Q)}]

(suppressing the coherence isomorphisms α and β from Definition 2.1, i.e. going to the strict case.) The corresponding braid diagrams are

X P Y Q X P Y Q Y X P Q Y X P Q =  P X Y Q P X Y Q = X Y P Q X Y P Q X Y P Q X Y P Q P X Y Q P X Y Q =  Y X P Q Y X P Q = X P Y Q X P Y Q

Observe that τ̃_{X,P,Q} is not the inverse of τ_{P,X,Q}. Both morphisms are associated with σ in the control category, so in braid diagrams they will be represented by a braid with the same orientation as σ.

2.3.1. If A = C is a braided monoidal category, B is an A-category and ω : B ⟶ A is an A-functor, then the bifunctor ω ⊗ ω : B × B ⟶ A given by (ω ⊗ ω)(P, Q) = ω(P)⊗ω(Q) is an A-bifunctor. The bifunctor (ω⊗ω⊗M)(P, Q) = ω(P)⊗ω(Q)⊗M for M ∈ A is also an A-bifunctor.

2.3.2. In a similar way define a C-multifunctor property for multifunctors ω : B_{1} × …×B_{n} ⟶ B. In particular functors of the form ω⊗…⊗ω = ω^{n} : B×…×B ⟶ A and ω⊗…⊗ω⊗M = ω^{n}⊗M : B×…×B ⟶ A are A-multifunctors, if ω : B ⟶ A is an A-functor and M ∈ A.

Definition 2.5. Let B, B′, and B′′ be C-categories and ω, ω′ : B × B′ ⟶ B′′ be C-bifunctors. A natural transformation φ : ω ⟶ ω′ is a C-bimorphism, if the following diagrams commute

[figure: ω(X ⊗ P, Q); ω′(X ⊗ P, Q); φ(X⊗P,Q); X ⊗ ω(P, Q); X ⊗ ω′(P, Q); X⊗φ(P,Q); ξ; ξ′] [figure: ω(P, Y ⊗ Q); ω′(P, Y ⊗ Q); φ(P,Y ⊗Q); Y ⊗ ω(P, Q); Y ⊗ ω′(P, Q); Y ⊗φ(P,Q); τ_{P,Y,Q}; τ′_{P,Y,Q}]

[figure: ω(P, Y ⊗ Q); ω′(P, Y ⊗ Q); φ(P,Y ⊗Q); Y ⊗ ω(P, Q); Y ⊗ ω′(P, Q); Y ⊗φ(P,Q); _{τ_{Y,P,Q}}; τ′_{Y,P,Q}]

Let Nat_{C}(ω, ω′) denote the set of C-bimorphisms.

For multifunctors ω, ω′ : B_{1} × … × B_{n} ⟶ B we proceed in a similar way. A natural transformation of multifunctors φ : ω ⟶ ω′ is called a C-multimorphism, if commutative diagrams as above hold for all variables.

Definition 2.6. Let A be a C-category with a coherent structure of a monoidal category with tensor product P ˆ⊗ Q. If ˆ⊗ : A×A ⟶ A is a coherent C-bifunctor, then A is called a C-monoidal category. In particular the structural morphisms α, λ, and ρ for A are C-morphisms in each variable from A.

If we go to the strict case we assume α_{C}, λ_{C}, ρ_{C}, β, π, α_{A}, λ_{A}, ρ_{A}, and ξ to be identities. Then the necessary equalities for the strict case are

τ(I, X, P) = id,
τ̃((X, I, P) = id,
τ(P, X, Q) ˆ⊗ 1_{R} = τ(P, X, Q ˆ⊗ R),
τ̃(X, P, Q ˆ⊗ R) = τ̃(X, P, Q) ˆ⊗ 1_{R},
(τ(P, X, Q) ˆ⊗ 1_{R})(1_{P} ˆ⊗ τ(Q, X, R)) = τ(P ˆ⊗ Q, X, R),
(1_{P} ˆ⊗ τ̃(X, Q, R))(τ̃(X, P, Q) ˆ⊗ 1_{R}) = τ̃(X, P ˆ⊗ Q, R).

Observe that C is a C-monoidal category (since C is braided).

Definition 2.7. Let A and B be monoidal categories. A monoidal functor is a functor ω : A ⟶ B together with coherent natural isomorphisms υ : ω(P ˆ⊗ Q) ≅ ω(P) ˆ⊗ ω(Q) and ς : ω(I_{A}) ≅ I_{B}.

If A and B are C-monoidal categories, ω : A ⟶ B is a monoidal functor and a C-functor, and υ : ω(P ˆ⊗ Q) ≅ ω(P) ˆ⊗ ω(Q) is a C-bimorphism, then ω is called a C-monoidal functor.

Let ω, ω′ : A ⟶ B be C-monoidal functors. A natural transformation φ : ω ⟶ ω′ is a C-monoidal morphism, if φ is a C-morphism and monoidal.

Definition 2.8. Let A be a C-monoidal category together with a braiding σ = σ_{A} : P ˆ⊗ Q ⟶ Q ˆ⊗ P. We call A a C-braided C-monoidal category, if the braid morphisms in both categories are coherent w.r.t. the braid group, in particular if

[figure: P ˆ⊗ (X ⊗ Q); (X ⊗ Q) ˆ⊗ P; σ; X ⊗ (P ˆ⊗ Q); X ⊗ (Q ˆ⊗ P); 1⊗σ; τ; ξ]

and

[figure: (X ⊗ P) ˆ⊗ Q; Q ˆ⊗ (X ⊗ P); σ; X ⊗ (P ˆ⊗ Q); X ⊗ (Q ˆ⊗ P); 1⊗σ; ξ^{−1}; _{τ}]

commute. Observe, however, that the diagrams

[figure: (X ⊗ P) ˆ⊗ Q; Q ˆ⊗ (X ⊗ P); σ; X ⊗ (P ˆ⊗ Q); X ⊗ (Q ˆ⊗ P); 1⊗σ; ξ; τ]

and

[figure: P ˆ⊗ (X ⊗ Q); (X ⊗ Q) ˆ⊗ P; σ; X ⊗ (P ˆ⊗ Q); X ⊗ (Q ˆ⊗ P); 1⊗σ; _{τ}; ξ^{−1}]

do not necessarily commute since their braid diagrams are

X P Q X P Q X Q P X Q P = ?  and X P Q X P Q X Q P X Q P = ?

In principle arbitrary tensor products of objects from C and from A can be formed and twisted by elements of the braid group with the exception of tensor factors from C appearing on the far right of a tensor product containing tensor factors from A.

Definition 2.9. An object P in a C-monoidal category A is called C-central, if

(X ⊗ P ⊗ Q ⟶^{_{τ}(X,P,Q)} P ⊗ X ⊗ Q ⟶^{τ(P,X,Q)} X ⊗ P ⊗ Q) = id

holds for all X ∈ C and Q ∈ A.

Theorem 2.10. Let A be a C-braided C-monoidal category. Let B be a C-central bialgebra in A, C be a coalgebra in A and z : C ⟶ B be a coalgebra morphism. Then

(1) A^{C} is a C-category;

(2) A^{B} is a C-monoidal category;

(3) ω := A^{z} : A^{C} ⟶ A^{B} is a C-functor;

(4) the forgetful functor ω : A^{C} ⟶ A is a C-functor;

(5) if C is a C-central bialgebra and z : C ⟶ B is a bialgebra morphism then ω := A^{z} : A^{C} ⟶ A^{B} is a C-monoidal functor;

(6) the forgetful functor ω : A^{B} ⟶ A is a C-monoidal functor.

Proof. (1) similar to 2.1.6.

(2) A little calculation shows that A^{B} is a monoidal category ([Mj94a] Prop. 2.5) with the comultiplication on the tensor product given by (1_{P} ⊗ 1_{Q} ⊗ m_{B})(1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q}) : P ⊗ Q ⟶ P ⊗ Q ⊗ B. A^{B} is also a C-category by (1). The natural transformation ξ : (X ⊗ P) ˆ⊗ Q ⟶ X ⊗ (P ˆ⊗ Q) is compatible with the comultiplication with B from the right. So it is in A^{B}. The natural transformation τ : P ˆ⊗ (X ⊗ Q) ≅ X ⊗ (P ˆ⊗ Q) satisfies

[figure: δ; δ; δ; δ; P; X; Q; P; X; Q; =; X; P; Q; B; X; P; Q; B]

hence it is in A^{B}, too. Finally the natural transformation τ̃ : X ⊗ (P ˆ⊗ Q) ≅ P ˆ⊗ (X ⊗ Q) satisfies

[figure: δ; δ; δ; δ; δ; δ; X; P; Q; X; P; Q; X; P; Q; =; =; P; X; Q; B; P; X; Q; B; P; X; Q; B.]

Since the diagrams defining the structure of a C-monoidal category on A^{B} commute in A and consist of morphisms of B-comodules they also commute as diagrams in A^{B}. Thus A^{B} is a C-monoidal category.

(3) and (4) similar to 2.1.8

(5) Since the tensor products in A^{C} and A^{B} are induced by the tensor product in A the natural transformation υ : ω(P ˆ⊗ Q) ⟶ ω(P) ⊗ ω(Q) is the identity which makes ω a C-monoidal functor.

(6) is a special case of (5).

A corresponding result holds by duality for the category A_{B} of modules over a bialgebra B in A.

2.4. Rigid categories.

2.4.1. Another important categorical notion is that of a (right) dual object. This is a generalization of finite-dimensional vector spaces. An object X ∈ C is rigid or has a dual (X^{∗}, ev) where X^{∗} ∈ C and ev : X^{∗} ⊗ X ⟶ I is called the evaluation, if there is a morphism db : I ⟶ X ⊗ X^{∗}, the dual basis, such that

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

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

The monoidal category C is rigid or a tensor category if every object of C has a dual. The full subcategory of objects in C having duals is denoted by C_{0}. An adjoint functor argument shows that the dual of an object is unique up to isomorphism if it exists.

2.4.2. If ω : B ⟶ A is a monoidal functor and P ∈ B is rigid then ω(P) ∈ A is rigid with dual object ω(P^{∗}), evaluation ω(P^{∗})⊗ω(P) ≅ ω(P^{∗}⊗P) ⟶ ω(I_{B}) ≅ I_{A}, and dual basis I_{A} ≅ ω(I_{B}) ⟶ ω(P ⊗ P^{∗}) ≅ ω(P) ⊗ ω(P^{∗}).

Proposition 2.11. Let C be a braided monoidal category. Then the full subcategory C_{0} of rigid objects in C is a rigid braided monoidal category.

Proof. If the evaluation resp. the dual basis are morphisms represented by

[figure: X^{∗}; X; X; X^{∗}]

then the conditions are

[figure: X; X; X^{∗}; X^{∗}; =; =; X; X; X^{∗}; X^{∗}]

If X ∈ C has a dual (X^{∗}, ev) then X^{∗} has the dual (X, ev ∘ σ_{X,X^{∗}}) with the dual basis σ_{X,X^{∗}}^{−1} ∘ db. The corresponding morphisms for X^{∗} are

[figure: X; X^{∗}; X^{∗}; X]

and the relations are

[figure: X^{∗}; X^{∗}; X^{∗}; X^{∗}; X; X; X; X; =; =; =; =; =; =; X^{∗}; X^{∗}; X^{∗}; X^{∗}; X; X; X; X]

If X and Y are in C_{0} then X ⊗ Y has the dual (Y^{∗} ⊗ X^{∗}, ev_{Y} (1_{Y^{∗}} ⊗ ev_{X} ⊗ 1_{Y})). The reader may try the easy graphic and the diagrammatic proofs. Thus C_{0} is a full monoidal subcategory of C which inherits the braiding and contains the duals for every object.

2.5. Coadjoint coactions.

2.5.1. Let A be a braided monoidal category. Let C be a coalgebra in A, H be a Hopf algebra in A and z : C ⟶ H be a coalgebra homomorphism. We define a right coadjoint coaction of H on C by

\begin{aligned}ad := & (1_{C} ⊗ m_{H})(1_{C} ⊗ S ⊗ 1_{H})(σ_{H,C} ⊗ 1_{H}) \\ & (z ⊗ 1_{C} ⊗ z)(1_{C} ⊗ Δ_{C})Δ_{C} : C ⟶ C ⊗ H. \end{aligned}

Proposition 2.12. C with the right coadjoint coaction is an H-comodule coalgebra.

Proof. In graphical notation the right coadjoint coaction is

ad  [figure: z; S; z] C  C = C H  C H

The right coadjoint action is a counary action by

[figure: z; S; z; ε; ε; ε; ε; ad; C; C; C; C; =; =; =; C; C; C; C]

The coaction is coassociative:

[figure: z; S; z; z; S; z; z; S; z; z; z; z; z; z; z; z; z; S; S; S; S; S; ad; ad; ad; C; C; C; C; C; C; C; =; =; =; =; =; =; C; H; H; C; H; H; C; H; H; C; H; H; C; H; H; C; H; H; C; H; H]

The comultiplication is an H-comodule morphism by

[figure: z; S; z; z; S; z; z; S; z; z; z; z; S; S; ad; ad; ad; C; C; C; C; C; =; =; =; =; C; C; H; C; C; H; C; C; H; C; C; H; C; C; H]

and preserves the counit of C

[figure: z; S; z; ε; ε; ε; z; S; μ; ad; C; C; C; C; =; =; =; H; H; H; H]

Now we want to slightly generalize the notion of a right coadjoint coaction to the case where H is only a bialgebra.

Lemma 2.13. Let H be a Hopf algebra, C be a coalgebra, and z : C ⟶ H be a coalgebra morphism. A right coaction ad : C ⟶ C ⊗ H is the right coadjoint coaction iff

[figure: z; ad; z; C; C; =; C; H; C; H]

Proof. If ad is the right coadjoint coaction then the equation of the lemma holds by

[figure: z; ad; z; S; z; z; z; S; z; z; C; C; C; C; =; =; =; C; H; C; H; C; H; C; H]

Conversely if this equality holds then the right coaction ad : C ⟶ C ⊗ H is the right coadjoint coaction since

[figure: z; S; z; z; ad; z; S; z; S; ad; ad; C; C; C; C; =; =; =; C; H; C; H; C; H; C; H]

2.5.2. We say that a coaction ad : C ⟶ C ⊗ B for a given z : C ⟶ B, B a bialgebra, is a right coadjoint coaction if the equation in Lemma 2.13 holds.

More generally if z : C ⟶ B is ⋆-invertible then a coadjoint coaction can be constructed in the same way as above. We don’t know if there are more general conditions for z : C ⟶ B such that a right coadjoint coaction exists nor whether it is unique then.

2.5.3. In the dual situation let f : H ⟶ A be an algebra homomorphism with a Hopf algebra H. The right adjoint action ah = ∑ f(S(h_{1}))·a·f(h_{2}) is characterized by the equation ∑ f(h_{1}) · (ah_{2}) = a · f(h).

2.6. C_{0}-generated coalgebras. We still need another somewhat more general setup. Let C be a monoidal category, C_{0} be a full monoidal subcategory of C. In this situation we consider a special type of coalgebra in C.

Definition 2.14. Let C ∈ C be a coalgebra satisfying the following conditions: (1) C is a colimit in C of a diagram of objects C_{i} in C_{0}.

(2) All morphisms X ⊗ι_{i} ⊗M : X ⊗C_{i} ⊗M ⟶ X ⊗C ⊗M are monomorphisms in C where X ∈ C_{0}, M ∈ C and the ι_{i} : C_{i} ⟶ C are the injections of the colimit diagram.

(3) Every C_{i} is a subcoalgebra of C via ι_{i} : C_{i} ⟶ C.

(4) If (P, δ_{P} : P ⟶ P ⊗ C) is a comodule over C and P ∈ C_{0}, then there exists a C_{i} in the diagram for C and a morphism δ_{P,i} : P ⟶ P ⊗ C_{i} such that

[figure: P; P ⊗ C_{i}; δ_{P,i}; P ⊗ C; δ_{P}; 1⊗ι_{i}]

commutes. Then C is called a C_{0}-generated coalgebra.

2.6.1. If C_{0} = C then the conditions in the previous definition are trivially satisfied. If C = Vec and C_{0} = vec, the category of finite-dimensional vector spaces, then every coalgebra in C is a C_{0}-generated coalgebra by the fundamental theorem for coalgebras ([Sw69] Thm. 2.2.1) and its generalization to the fundamental theorem for comodules.

2.6.2. We denote by C_{0}^{C} the category of C-comodules in C_{0}. Then C_{0}^{C} is a C_{0}- category and the forgetful functor ω : C_{0}^{C} ⟶ C_{0} is a C_{0}-functor.

2.6.3. It is an easy exercise to show for a C_{0}-generated coalgebra, that the (P, δ_{P,i} : P ⟶ P ⊗ C_{i}) are comodules.

3. Reconstruction properties

For the rest of this paper let the control category C be a braided monoidal category and the base category A be a C-monoidal category.

3.1. Reconstruction of coalgebras.

Definition 3.1. We define the category \overline{𝔄}(C) of all C-categories “over” A as follows. The objects are pairs (B, ω) consisting of a C-category B and of a C-functor ω : B ⟶ A. A morphism [χ, ζ] : (B, ω) ⟶ (B′, ω′) is an equivalence class of pairs (χ, ζ) with χ : B ⟶ B′ a C-functor and ζ : ω ⟶ ω′χ a C-isomorphism. Two such pairs (χ, ζ) and (χ′, ζ′) are equivalent if there is a C-isomorphism φ : χ ⟶ χ′ with ζ′ = ω′φ ∘ ζ. Composition is given by [χ′, ζ′] ∘ [χ, ζ] = [χ′χ, ζ′χ ∘ ζ].

Let 𝔄(C) be a full subcategory of \overline{𝔄}(C).

3.1.1. Theorem 2.10 defines a functor A- : A-coalg ⟶ \overline{𝔄}(C) by A-(C) := (A^{C}, ω) where ω is the forgetful functor. Furthermore A-(z) := [A^{z}, id].

3.1.2. If A_{0} is a full C-monoidal subcategory of A, then we define the full subcategory 𝔄_{0}(C) of \overline{𝔄}(C) to consist of those C-categories B over A whose forgetful functor ω : B ⟶ A factors through A_{0}.

3.1.3. In this case we obtain a functor A-_{0} : A-coalg ⟶ 𝔄_{0}(C) by A-_{0}(C) := (A_{0}^{C}, ω) where A_{0}^{C} denotes the full subcategory of A^{C} of those C-comodules whose underlying object is in A_{0}.

Now we address the question which properties of an algebra A or a coalgebra C in a monoidal category A can be recovered from the category of its modules A_{A} resp. comodules A^{C}. Since reconstruction of coalgebras is somewhat simpler (see Theorem 4.7) we will perform the reconstruction of a coalgebra C from A^{C} explicitly and derive the reconstruction of an algebra by duality. As we remarked in the introduction C is not uniquely determined by A^{C}. But if we use additional information about the forgetful C-functor ω : A^{C} ⟶ A we can reconstruct C up to isomorphism with its full structure.

In many cases we can actually “reconstruct” a coalgebra C from an fairly arbitrary C-functor ω : B ⟶ A. We will postpone the discussion of how to obtain the object C ∈ A from a functor ω : B ⟶ A to the next section. In this section we will give the general definition and discuss the structure of such a reconstructed object C.

A different point of view is how to find a left adjoint functor to the functor A- : A-coalg ⟶ 𝔄(C). If such a left adjoint functor does not exist “globally”, it might still exist “locally”, i.e. a certain functor is representable.

Definition 3.2. Let B be a C-category and ω : B ⟶ A be a C-functor. Then the sets Nat_{C}(ω, ω ⊗ M) depend functorially on M ∈ A, i.e. we have a functor

Nat_{C}(ω, ω ⊗ –) : A ⟶ Set.

If this functor is representable then the representing object will be denoted by coend_{C}(ω). It is unique up to isomorphism. (In the dual situation a representing object for Nat_{C}(ω ⊗ –, ω) will be denoted by end_{C}(ω).) So we have

Nat_{C}(ω, ω ⊗ M) ≅ A(coend_{C}(ω), M).

The universal arrow for this functor

δ : ω ⟶ ω ⊗ coend_{C}(ω)

is a C-morphism, the image of the identity in A(coend_{C}(ω), coend_{C}(ω)). It solves the following universal problem

• for every M ∈ A and every C-morphism φ : ω ⟶ ω ⊗ M there is a unique morphism f : C ⟶ M such that

[figure: ω; ω ⊗ C; δ; ω ⊗ M; 1⊗f; φ]

commutes. This universal property is in fact equivalent to the representability of Nat_{C}(ω, ω ⊗–) and induces a universal factorization of ω through the category of comodules A^{C}.

The study of C-functors as conducted here has many properties in common with similar results for general functors. In fact general categories, functors and natural transformations may also be considered as C-categories, C-functors, resp. Cmorphisms for the monoidal category C with one object I and one morphism id_{I}.

Many of the following propositions are well known for the case of a monoidal category C = {I} and can be proved by standard universal abstract nonsense. So we only sketch the idea of the proofs. There are, however, subtle difficulties and restrictions with respect to braidings that do not occur in the case of a symmetric control category C.

Proposition 3.3. If Nat_{C}(ω, ω ⊗ –) is representable, then the representing object C = coend_{C}(ω) is a coalgebra in A. This coalgebra is uniquely determined up to isomorphisms of coalgebras.

Furthermore every object ω(P) ∈ A with P ∈ B is a C-comodule via δ : ω(P) ⟶ ω(P) ⊗ C and every morphism ω(f) is a morphism of C-comodules.

Every object ω(X ⊗P) ∈ A with X ∈ C and P ∈ B is isomorphic as a C-comodule to X ⊗ ω(P) with the structure induced by ω(P).

Proof. The comultiplication Δ and counit ε are uniquely defined by (1_{ω} ⊗ Δ)δ = (δ ⊗ 1_{C})δ and (1_{ω} ⊗ ε)δ = ρ_{ω}^{−1}. The last claim follows since δ is a C-morphism.

We will encounter situations of comodules (P, ϑ : P ⟶ P ⊗ C) in A where we want to know if this comodule comes about as in the previous Proposition. So we define Definition 3.4. Let Nat_{C}(ω, ω ⊗ –) be representable by C ∈ A. Then a comodule (P, ϑ : P ⟶ P ⊗ C) in A can be lifted along ω if there is an object Q ∈ B and a comodule isomorphism (ω(Q), δ) ≅ (P, ϑ). In this case the comodule (P, ϑ) is called liftable along ω and Q ∈ B a lifting.

3.2. Reconstruction of bialgebras. Assume now, that the base category A is a C-braided C-monoidal category. (It will be clear from the context which tensor product is being used, so we simply use ⊗ for the tensor product in A.)

Consider a C-functor ω : B ⟶ A. Then the bifunctors ω ⊗ ω = ω^{2} : B × B ⟶ A and ω^{2} ⊗ M : B × B ⟶ A are C-bifunctors as can be easily checked. The sets Nat_{C}(ω^{2}, ω^{2}⊗M) of C-bimorphisms depend functorially on M, i.e. we have a functor

Nat_{C}(ω^{2}, ω^{2} ⊗ –) : A ⟶ Set.

Let Nat_{C}(ω, ω ⊗ –) be representable with universal C-morphism δ : ω ⟶ ω ⊗ C. In general the morphism

δ_{2} := (1_{ω} ⊗ σ_{coend_{C}(ω),ω} ⊗ 1_{coend_{C}(ω)})(δ ⊗ δ) : ω^{2} ⟶ ω^{2} ⊗ coend_{C}(ω)^{2}

will not be a C-bimorphism. This is, however, the case if C = coend_{C}(ω) is C-central (see Definition 2.9). Similarly δ^{n} is a C-multimorphism if C is C-central.

Definition 3.5. If the functor Nat_{C}(ω, ω ⊗ –) is representable with universal Cmorphism δ : ω ⟶ ω ⊗ C, if C is C-central and if Nat_{C}(ω^{2}, ω^{2} ⊗ –) is also representable with the special universal C-bimorphism

δ_{2} := (1_{ω} ⊗ σ_{C,ω} ⊗ 1_{C})(δ ⊗ δ) : ω ⊗ ω ⟶ ω ⊗ ω ⊗ C ⊗ C

then we say that Nat_{C}(ω, ω ⊗ –) is birepresentable.

In a similar way we proceed for the multifunctor ω⊗…⊗ω = ω^{n}. If C is C-central and the functor Nat_{C}(ω^{n}, ω^{n} ⊗ –) is representable with the universal morphism

δ^{(n)} := τ(δ ⊗ … ⊗ δ) : ω^{n} ⟶ ω^{n} ⊗ coend_{C}(ω)^{n}

with the obvious choice of τ ∈ B_{2n}, the Artin braid group, then we say that Nat_{C}(ω, ω ⊗ –) is n-representable. If this holds for all n ∈ ℕ we say that the functor Nat_{C}(ω, ω ⊗ –) is multirepresentable (fully representable in [Mj93a]).

Proposition 3.6. Let A be C-braided C-monoidal, B be C-monoidal and ω : B ⟶ A be a C-monoidal functor. If Nat_{C}(ω, ω ⊗ –) is multirepresentable, then B := coend_{C}(ω) is a bialgebra in A. This bialgebra is uniquely determined up to isomorphisms of bialgebras.

If in addition B is C-braided (ω will usually not preserve the braiding), then coend_{C}(ω) is coquasitriangular in A.

If ω factors through the full subcategory A_{0} of rigid objects in A then coend_{C}(ω) is a Hopf algebra in A.

Furthermore for any objects P, Q ∈ B the B-comodule structure on ω′(P) ⊗ ω′(Q) is defined by the multiplication on B.

Proof. Similar to [Mj94a] Theorem 3.2. resp. 3.11. We check only that the relevant morphisms that are factorized through the universal morphisms are C-morphisms.

The multiplication of B is defined by the C-bimorphism δ′_{P ⊗Q} : ω(P) ⊗ ω(Q) ≅ ω(P ⊗Q) ⟶ ω(P ⊗Q)⊗B as the uniquely determined morphism m̃_{B} : B⊗B ⟶ B such that

(1_{ω(P)} ⊗ 1_{ω(Q)} ⊗ m̃)(1_{ω(P)} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q}) = δ′_{P ⊗Q}.

The morphism δ′_{P ⊗Q⊗R} : ω(P) ⊗ ω(Q) ⊗ ω(R) ⟶ ω(P ⊗ Q ⊗ R) ⊗ B responsible for associativity is a C-trimorphism.

The coquasitriangular structure r : B ⊗ B ⟶ I is defined by the C-bimorphism σ_{A}^{−1}(ω(Q), ω(P))ω(σ_{B}(P, Q)) : ω(P) ⊗ ω(Q) ⟶ ω(P) ⊗ ω(Q) ⊗ I and the braid equation

δ δr = P Q P Q P Q P Q

where the braid marked with a circle represents the braiding of the category B. This diagram represents the equation

σ_{A}^{−1}(ω(Q), ω(P))ω(σ_{B}(P, Q)) = (1_{ω(P)}⊗1_{ω(Q)}⊗r)(1_{ω(P)}⊗σ_{A}(B, ω(Q))⊗1_{B})(δ_{P} ⊗δ_{Q}).

Observe that the braiding of A^{B} for a braided bialgebra B is described by the equation

δ δ  δ δ r  r P Q  P Q P Q  = Q P  Q P Q P

Finally the antipode is defined by the morphism (ω(ev_{P})⊗σ_{B,ω(P)})(1_{ω(P)}⊗δ_{ω(P)^{∗}} ⊗ 1_{ω(P)})(1_{ω(P)} ⊗ ω(db_{P})) : ω(P) ⟶ ω(P) ⊗ B. To show that this is a C-morphism, we first show the following claim. If P ∈ B defines a trivial B-comodule then ω(P)^{∗} is also trivial. This follows from

[figure: u; u; u; δ; δ; δ; δ; δ; P^{∗}; P^{∗}; P^{∗}; P^{∗}; P^{∗}; P^{∗}; =; =; =; =; =; P^{∗}; B; P^{∗}; B; P^{∗}; B; P^{∗}; B; P^{∗}; B; P^{∗}; B]

Observe that we have X ⊗ P ≅ X ⊗ (I ˆ⊗ P) ≅ (X ⊗ I) ˆ⊗ P = X̃ ˆ⊗ P in B for X̃:= X ⊗ I which gives trivial B-comodules ω(X̃) ≅ X ⊗ ω(I) and ω(X̃)^{∗}. So the diagram (where X denotes ω(X̃) and P denotes ω(P))

[figure: u; δ; δ; δ; δ; δ; δ; X; P; X; P; X; P; X; P; X; P; =; =; =; =; X; P; B; X; P; B; X; P; B; X; P; B; X; P; B]

shows that the morphism (ω(ev_{P})⊗σ_{B,ω(P)})(1_{ω(P)} ⊗δ_{ω(P)^{∗}} ⊗1_{ω(P)})(1_{ω(P)} ⊗ω(db_{P})) : ω(P) ⟶ ω(P) ⊗ B is a C-morphism.

Further interesting properties of A resp. ω for reconstruction may be found in [Dr89, KT92, Ye90]

3.3. Reconstruction of morphisms.

3.3.1. Let (B, ω) and (B′, ω′) be objects in 𝔄(C) and let [χ, ζ] : (B, ω) ⟶ (B′, ω′) be a morphism in 𝔄(C). Then ω : B ⟶ A, ω′ : B′ ⟶ A, and χ : B ⟶ B′ are C-functors, and ζ : ω ≅ ω′χ is a C-isomorphism for the diagram:

[figure: B; B′; χ; A.; ω; ω′]

Let δ : ω ⟶ ω⊗C and ∂ : ω′ ⟶ ω′⊗C′ be universal C-morphisms. Since ζ : ω ⟶ ω′χ is a C-isomorphism there is a unique morphism coend_{C}([χ, ζ]) = z : C ⟶ C′ such that

[figure: ω; ω ⊗ C; δ; ω′χ; ω′χ ⊗ C′; ∂χ; ζ; ζ⊗z]

commutes. If (χ, ζ) and (χ′, ζ′) are equivalent (representatives of [χ, ζ]) by φ : χ ⟶ χ′ with ζ′ = ω′φ ∘ ζ then the diagram

[figure: ω; ω ⊗ C; δ; ω′χ; ω′χ ⊗ C′; ∂χ; ω′χ′; ω′χ′ ⊗ C′; ∂χ′; ζ; ζ′; ω′φ; ζ⊗z; ζ′⊗z′; ω′φ⊗1]

commutes. By the uniqueness of the induced morphism from C to C′ we get z = z′, hence z is uniquely defined by the class [χ, ζ].

It is easy to see that z is a coalgebra morphism.

Observe that ω(P) ⟶^{δ} ω(P) ⊗ C ⟶^{1⊗z} ω(P) ⊗ C′ defines a C′-coaction on ω(P) for every P ∈ B.

3.3.2. Let A be a C-braided C-monoidal category. Let [χ, ζ] : (B, ω) ⟶ (B′, ω′) and δ and ∂ be as before. Furthermore let B′ be a C-monoidal category and ω′ : B′ ⟶ A be a C-monoidal functor. Assume that Nat_{C}(ω′, ω′ ⊗ –) is multirepresentable with the universal morphism ∂ : ω′ ⟶ ω′ ⊗ B. By Proposition 3.6 B is a bialgebra.

We call Nat_{C}(ω, ω ⊗ –) : B ⟶ Set ω′-representable if there is an object C̄ ∈ B′ and a C-morphism d : χ ⟶ χ ⊗ C̄ such that the induced morphism y : C ⟶ ω′C̄ in the commutative diagram

[figure: ω; ω ⊗ C; δ; ω ⊗ ω′C̄; 1⊗y; ω′χ; ω′(χ ⊗ C̄); ω′d; ω′χ ⊗ ω′C̄; υ; ζ; ζ⊗1]

is an isomorphism. Equivalently the morphism

ω ⟶^{ζ} ω′χ ⟶^{ω′d} ω′(χ ⊗ C̄) ⟶^{υ} ω′χ ⊗ ω′C̄ ⟶^{ζ^{−1}⊗1} ω ⊗ ω′C̄

is a universal C-morphism.

3.3.3. Observe that every morphism f in B induces a C-comodule morphism ω(f) in A, and that every morphism g in B′ induces a B-comodule morphism ω′(g) in A. In particular ω′(d) ≅ δ is a B-comodule morphism in A. Furthermore ζ is a B-comodule isomorphism with the B-comodule structure on ω as defined in 3.3.1.

Let ad : C ⟶ C ⊗ B denote the coaction

C ⟶^{y} ω′(C̄) ⟶^{∂(C̄)} ω′(C̄) ⊗ B ⟶^{y^{−1}⊗B} C ⊗ B.

Proposition 3.7. Under the conditions of 3.3.2 [χ, ζ] induces a coalgebra morphism z : C ⟶ B and the coaction of B on C is a right coadjoint coaction.

Proof. The induced coalgebra morphism was defined in 3.3.1. For every P ∈ B and P′ := ω(P) we get

[figure: z; ad; z; z; z; δ; δ; δ; δ; δ; δ; ad; P′; P′; P′; P′; =; =; =; P′; C; B; P′; C; B; P′; C; B; P′; C; B]

where the first and last equalities arise from the coassociativity of the cooperation δ and the middle equality is the fact that (the left lower resp. right upper) δ is a B-comodule morphism, the coaction of B on P′ as in 3.3.1 and on P′ ⊗ C via the multiplication of B. Since the natural transformation ω′χ ⟶^{δ} ω′χ ⊗ C ⟶ ω′χ ⊗ C ⊗ B given by the above graphic diagram induces precisely one morphism C ⟶ C ⊗ B we get (2.5.2) that ad : C ⟶ C ⊗ B is a right coadjoint coaction.

Furthermore we know that C ≅ ω′(C̄) is a B-comodule.

The previous Proposition says in particular that C with the given (coadjoint) B-comodule structure can be lifted along ω′ : B′ ⟶ A.

Corollary 3.8. Under the conditions of Proposition 3.7 if B is a Hopf algebra in A then C is a B-comodule coalgebra under the right coadjoint coaction defined by z : C ⟶ B.

Proof. By Lemma 2.13 ad : C ⟶ C ⊗ B is the uniquely defined coadjoint coaction which by Proposition 2.12 makes C a B-comodule coalgebra.

3.3.4. If coend_{C}(ω) exists for all objects (B, ω) in 𝔄(C) then we have a functor coend_{C} : 𝔄(C) ⟶ A-coalg. If furthermore all full comodule categories A^{C} together with the forgetful functor ω : A^{C} ⟶ A are objects in 𝔄(C) then coend_{C} : 𝔄(C) ⟶ A-coalg is left adjoint to A- : A-coalg ⟶ 𝔄(C) (from 3.1.1).

3.4. Applications.

3.4.1. We specialize C to the case of a one-element category. We call this case full reconstruction. Let ω : B ⟶ A be a functor. Let δ : ω ⟶ ω ⊗ C be a universal morphism. Then the propositions of this section specialize to:

If Nat(ω, ω ⊗ –) is representable, then the representing object C = coend(ω) is a coalgebra in A. This coalgebra is uniquely determined up to isomorphisms of coalgebras.

Furthermore every object ω(P) ∈ A with P ∈ B is a C-comodule via δ : ω(P) ⟶ ω(P) ⊗ C and every morphism ω(f) is a C-comodule morphism.

Let A be braided monoidal, B be monoidal and ω : B ⟶ A be a monoidal functor. If Nat(ω, ω ⊗ –) is multirepresentable, then B = coend(ω) is a bialgebra in A. This bialgebra is uniquely determined up to isomorphisms of bialgebras.

If in addition B is braided, then coend(ω) is coquasitriangular in A.

If B is rigid then coend(ω) is a Hopf algebra in A.

Furthermore for any objects P, Q ∈ B the B-comodule structure on ω(P) ⊗ ω(Q) is defined by the multiplication on coend(ω).

3.4.2. We specialize C = A_{0} with A a braided monoidal category and A_{0} a full (braided) monoidal subcategory. We will call this case restricted reconstruction. Let ω : B ⟶ A_{0} be an A_{0}-functor. Let δ : ω ⟶ ω ⊗ C be a universal A_{0}-morphism. Then the propositions of this section specialize to:

If Nat_{A_{0}}(ω, ω ⊗ –) is representable, then the representing object C = coend_{A_{0}}(ω) is a coalgebra in A. This coalgebra is uniquely determined up to isomorphisms of coalgebras.

Furthermore every object ω(P) ∈ A_{0} with P ∈ B is a C-comodule via δ : ω(P) ⟶ ω(P) ⊗ C and every morphism ω(f) is a C-comodule morphism.

Let B be A_{0}-monoidal and ω : B ⟶ A_{0} be an A_{0}-monoidal functor. If Nat_{A_{0}}(ω, ω⊗–) is multirepresentable, then B = coend_{A_{0}}(ω) is a bialgebra in A. This bialgebra is uniquely determined up to isomorphisms of bialgebras.

If in addition B is A_{0}-braided, then coend_{A_{0}}(ω) is coquasitriangular in A.

If B is rigid then coend_{A_{0}}(ω) is a Hopf algebra in A.

Furthermore for any objects P, Q ∈ B the B-comodule structure on ω(P) ⊗ ω(Q) is defined by the multiplication on coend_{A_{0}}(ω).

4. Existence theorems in reconstruction theory

4.1. Restricted reconstruction. In this section we will study reconstruction of a given coalgebra C with help of the functor Nat_{C_{0}}(ω, ω ⊗ –). We call this restricted reconstruction.

For this purpose let C be a braided monoidal category and C_{0} be a full (braided) monoidal subcategory of C. For a Hopf algebra H in C, a coalgebra C in C and a coalgebra morphism z : C ⟶ H we know that C is a coalgebra in C^{H} with respect to the right coadjoint coaction ad : C ⟶ C ⊗ H by Proposition 2.12. If C is a C_{0}-generated coalgebra then the subcoalgebras C_{i} are also in C^{H} (actually in C_{0}^{H}) by the coadjoint action. We consider the underlying functor ω : C_{0}^{C} ⟶ C^{H}.

Theorem 4.1. Let C be a C_{0}-generated coalgebra in C and H be a Hopf algebra in C. Let z : C ⟶ H be a coalgebra morphism. Let ω := C_{0}^{z} : C_{0}^{C} ⟶ C_{0}^{H} ⊆ C^{H} be the functor induced by z. Then

Nat_{C_{0}}(ω, ω ⊗ –) : C^{H} ⟶ Set

is representable by the coalgebra C = coend_{C_{0}}(ω) in C^{H} with the canonical morphism δ : ω ⟶ ω ⊗ C.

Proof. We define maps

Σ : C^{H}(C, M) ⟶ Nat_{C_{0}}(ω, ω ⊗ M)

and

Π : Nat_{C_{0}}(ω, ω ⊗ M) ⟶ C^{H}(C, M).

The first map is defined by Σ(f)(P, δ_{P}) := (1_{P} ⊗ f)δ_{P} : P ⟶ P ⊗ C ⟶ P ⊗ M. Then Σ(f)(P, δ_{P}) : P ⟶ P ⊗ M is an H-comodule morphism since the following diagram commutes

[figure: P; P ⊗ C; δ_{P}; P ⊗ M; 1⊗f; P ⊗ C ⊗ C ⊗ H; P ⊗ C ⊗ M ⊗ H; 1⊗1⊗f⊗1; P ⊗ H ⊗ C ⊗ H; P ⊗ H ⊗ M ⊗ H; 1⊗1⊗f⊗1; P ⊗ C ⊗ H ⊗ H; P ⊗ M ⊗ H ⊗ H; 1⊗f⊗1⊗1; P ⊗ H; P ⊗ C ⊗ H; δ_{P} ⊗1; P ⊗ M ⊗ H; 1⊗f⊗1; P ⊗ C; δ_{P}; 1⊗z; δ_{P} ⊗ad; 1⊗z⊗1⊗1; 1⊗σ⊗1; 1⊗1⊗m_{H}; δ_{P} ⊗δ_{M}; 1⊗z⊗1⊗1; 1⊗σ⊗1; 1⊗1⊗m_{H}]

where the left hand side commutes by

[figure: δ; δ; δ; δ; z; z; ad; P; P; =; P; C; H; P; C; H]

which follows from Lemma 2.13. Hence Σ(f) : ω ⟶ ω ⊗ M is a natural transformation. Furthermore we have Σ(f)(X ⊗ (P, δ_{P})) = (1_{X⊗P} ⊗ f)δ_{X⊗P} = (1_{X} ⊗ 1_{P} ⊗ f)(1_{X} ⊗ δ_{P}) = 1_{X} ⊗ Σ(f)(P, δ_{P}), so Σ(f) is a C_{0}-morphism.

To define the map Π let φ : ω ⟶ ω ⊗ M be given. Define \underrightarrow{lim} φ(C_{i}) = φ(C) : C ⟶ C ⊗ M as the uniquely determined morphism so that

[figure: C_{i}; C_{i} ⊗ M; φ(C_{i}); C; C ⊗ M; φ(C); ι_{i}; ι_{i}⊗1_{M}]

commutes and let

Π(φ) := (ϵ ⊗ 1_{M})φ(C) : C ⟶ C ⊗ M ⟶ M.

Since C_{i} is a subcoalgebra of C it is an H-subcomodule by the coadjoint coaction induced by the coalgebra morphism zι_{i} : C_{i} ⟶ H. Hence, C is a colimit of H-comodule coalgebras and φ(C) : C ⟶ C ⊗ M is an H-comodule morphism. Consequently Π(φ) is in C^{H}.

We have ΠΣ(f) = (ϵ⊗1_{M}) \underrightarrow{lim}((1_{C_{i}}⊗f)δ_{i}) = (ϵ⊗1_{M})(1_{C}⊗f)Δ = f(ϵ⊗1_{C})Δ_{C} = f. Now observe that δ_{P} : P ⟶ P ⊗ C is a C-comodule morphism with the C-

structure on P ⊗ C coming from the one of C. Thus we get (1 ⊗ δ_{j})δ_{P,j} = (δ_{P,j} ⊗ 1_{C})δ_{P} for the morphism δ_{P,j} : P ⟶ P ⊗ C_{j} which exists by the assumptions about a C_{0}-generated coalgebra. So δ_{P,j} is a C-comodule morphism as well, hence φ(P ⊗ C_{j}, 1_{P} ⊗ δ_{j})δ_{P,j} = (δ_{P,j} ⊗ 1_{M})φ(P, δ_{P}). From this we get

ΣΠ(φ)(P, δ_{P}) = (1_{P} ⊗ (ϵ ⊗ 1_{M})φ(C))δ_{P}
= (1_{P} ⊗ ϵ ⊗ 1_{M})(1_{P} ⊗ \underrightarrow{lim} φ(C_{i}))(1_{P} ⊗ ι_{j})δ_{P,j}
= (1_{P} ⊗ ϵ ⊗ 1_{M})(1_{P} ⊗ ι_{j} ⊗ 1_{M})(1_{P} ⊗ φ(C_{j}))δ_{P,j}
= (1_{P} ⊗ ϵ ⊗ 1_{M})(1_{P} ⊗ ι_{j} ⊗ 1_{M})φ(P ⊗ C_{j})δ_{P,j}
= (1_{P} ⊗ ϵ ⊗ 1_{M})(1_{P} ⊗ ι_{j} ⊗ 1_{M})(δ_{P,j} ⊗ 1_{M})φ(P)
= φ(P, δ_{P}).

So Σ and Π are inverses of each other. The claim about the coalgebra structure is clear from the uniqueness of the reconstructed coalgebra.

4.1.1. Observe that in general the set of all natural transformations – not just the C-morphisms – from ω to ω ⊗ M is too large for the reconstruction of C as we have seen in 2.1.10. By Proposition 6.4, however, this difference does not occur if the base category C is the category of vector spaces Vec over 𝕂. This explains the usual full reconstruction with the functor Nat(ω, ω ⊗ –) like in [DM82, Ul89, Mj94a].

Examples of natural transformations which are not C-morphisms can be derived from 2.1.10.

Now assume that C is a cocomplete braided monoidal category and C_{0} is a (braided) full monoidal subcategory of C. Furthermore assume that the tensor product in C preserves arbitrary colimits in both variables.

Theorem 4.2. Let B be a C_{0}-generated coalgebra in C and a C_{0}-central bialgebra, let H be a braided C_{0}-central Hopf algebra in C. Let z : B ⟶ H be a bialgebra morphism. Let ω := C_{0}^{z} : C_{0}^{B} ⟶ C_{0}^{H} ⊆ C^{H} be the functor induced by z. Then Nat_{C_{0}}(ω, ω ⊗ –) : C^{H} ⟶ Set is multirepresentable by the bialgebra coend_{C_{0}}(ω) in C^{H} which is equal to B as an H-comodule coalgebra under the coadjoint coaction, but carries a different multiplicative structure m̃_{B} : B ⊗ B ⟶ B, the transmuted multiplication.

Proof. We extend the proof of 4.1 to ω^{2}. Define maps

Σ : C^{H}(B ⊗ B, M) ⟶ Nat_{C_{0}}(ω ⊗ ω, ω ⊗ ω ⊗ M)

and

Π : Nat_{C_{0}}(ω ⊗ ω, ω ⊗ ω ⊗ M) ⟶ C^{H}(B ⊗ B, M).

The first map is given by Σ(f)((P, δ_{P}), (Q, δ_{Q})) := (1_{P ⊗Q} ⊗ f)(1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q}) : P ⊗ Q ⟶ P ⊗ B ⊗ Q ⊗ B ⟶ P ⊗ Q ⊗ B ⊗ B ⟶ P ⊗ Q ⊗ M. Then Σ(f)((P, δ_{P}), (Q, δ_{Q})) : P ⊗ Q ⟶ P ⊗ Q ⊗ M is an H-comodule morphism by a similar proof as in 4.1 replacing P by P ⊗ Q. Hence Σ(f) : ω ⊗ ω ⟶ ω ⊗ ω ⊗ M is a natural transformation. To show that Σ(f) is a C_{0}-morphism we observe that ξ : ω(X ⊗ P) ⟶ X ⊗ ω(P) is the identity. So we have

Σ(f)(X ⊗ (P, δ_{P}), (Q, δ_{Q}))
= (1_{X} ⊗ 1_{P} ⊗ 1_{Q} ⊗ f)(1_{X} ⊗ 1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(1_{X} ⊗ δ_{P} ⊗ δ_{Q})
= 1_{X} ⊗ (1_{P} ⊗ 1_{Q} ⊗ f)(1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q})
= 1_{X} ⊗ Σ(f)((P, δ_{P}), (Q, δ_{Q})),
(σ_{P,Y} ⊗ 1_{Q⊗M})Σ(f)((P, δ_{P}), Y ⊗ (Q, δ_{Q}))
= (σ_{P,Y} ⊗ 1_{Q⊗M})(1_{P ⊗Y ⊗Q} ⊗ f)(1_{P} ⊗ σ_{B,Y ⊗Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Y ⊗Q})
= (1_{Y ⊗P ⊗Q} ⊗ f)(1_{Y ⊗P} ⊗ σ_{B,Q} ⊗ 1_{B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})(σ_{P,Y} ⊗ 1_{Q})
= (1_{Y} ⊗ Σ(f)((P, δ_{P}), (Q, δ_{Q})))(σ_{P,Y} ⊗ 1_{Q})

or as a braid diagram

δ  δ  δ δ f  f P Y Q P Y Q = Y P Q M Y P Q M

and

Σ(f)((P, δ_{P}), Y ⊗ (Q, δ_{Q}))(σ_{Y,P} ⊗ 1_{Q})
= (1_{P ⊗Y ⊗Q} ⊗ f)(1_{P} ⊗ σ_{B,Y ⊗Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Y ⊗Q})(σ_{Y,P} ⊗ 1_{Q})
= (1_{P ⊗Y ⊗Q} ⊗ f)(1_{P} ⊗ σ_{B,Y ⊗Q} ⊗ 1_{B})(σ_{Y,P ⊗B} ⊗ 1_{Q⊗B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})
= (1_{P ⊗Y ⊗Q} ⊗ f)(1_{P ⊗Y} ⊗ σ_{B,Q} ⊗ 1_{B})(1_{P} ⊗ (σ_{B,Y} σ_{Y,B} ⊗ 1_{Q⊗B}))
(σ_{Y,P} ⊗ 1_{B⊗Q⊗B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})
= (1_{P ⊗Y ⊗Q} ⊗ f)(1_{P ⊗Y} ⊗ σ_{B,Q} ⊗ 1_{B})(σ_{Y,P} ⊗ 1_{B⊗Q⊗B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})
= (1_{P ⊗Y ⊗Q} ⊗ f)(σ_{Y,P} ⊗ σ_{B,Q} ⊗ 1_{B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})
= (σ_{Y,P} ⊗ 1_{Q⊗M})(1_{Y ⊗P ⊗Q} ⊗ f)(1_{Y ⊗P} σ_{B,Q} ⊗ 1_{B})(1_{Y} ⊗ δ_{P} ⊗ δ_{Q})
= (σ_{Y,P} ⊗ 1_{Q⊗M})(1_{Y} ⊗ Σ(f)((P, δ_{P}), (Q, δ_{Q}))

or as a braid diagram

f  f  f δ  δ  δ δ  δ δ X P Q X P Q X P Q =  = P X Q M P X Q M P X Q M

since B is C_{0}-central in C. Thus Σ(f) is a C_{0}-morphism.

Now we construct the second map Π. Let φ : ω ⊗ ω ⟶ ω ⊗ ω ⊗ M. Define \underrightarrow{lim} φ(B_{i}, B_{j}) = φ(B, B) : B ⊗ B ⟶ B ⊗ B ⊗ M as the uniquely determined morphism so that

[figure: B_{i} ⊗ B_{j}; B_{i} ⊗ B_{j} ⊗ M; φ(B_{i},B_{j}); B ⊗ B; B ⊗ B ⊗ M; φ(B,B); ι_{i}⊗ι_{j}; ι_{i}⊗ι_{j}⊗1_{M}]

commutes. Observe that B ⊗ B is the colimit of the B_{i} ⊗ B_{j} since by assumption the tensor product preserves colimits. Let

Π(φ) := (ϵ ⊗ ϵ ⊗ 1_{M})φ(B, B) : B ⊗ B ⟶ B ⊗ B ⊗ M ⟶ M.

As in the proof of 4.1 Π(φ) is an H-comodule morphism.

We have ΠΣ(f) = (ϵ ⊗ ϵ ⊗ 1_{M})(1_{B⊗B} ⊗ f)(1_{B} ⊗ σ_{B,B} ⊗ 1_{B})(Δ_{B} ⊗ Δ_{B}) = f(ϵ ⊗ ϵ ⊗ 1_{B⊗B})Δ_{B⊗B} = f. Observe that δ_{P} : P ⟶ P ⊗ B is a B-comodule morphism with the B-structure on P ⊗ B coming from the one of B. So we have

ΣΠ(φ)((P, δ_{P}), (Q, δ_{Q}))
= (1_{P ⊗Q} ⊗ ϵ ⊗ ϵ ⊗ 1_{M})(1_{P ⊗Q} ⊗ φ(B, B))(1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q})
= (1_{P ⊗Q} ⊗ ϵ ⊗ ϵ ⊗ 1_{M})(1_{P ⊗Q} ⊗ \underrightarrow{lim} φ(B_{k}, B_{l}))
(1_{P} ⊗ σ_{B,Q} ⊗ 1_{B})(δ_{P} ⊗ δ_{Q})
= (1_{P ⊗Q} ⊗ ϵ ⊗ ϵ ⊗ 1_{M}) \underrightarrow{lim}[(1_{P ⊗Q} ⊗ φ(B_{k}, B_{l}))(1_{P} ⊗ σ_{B_{k},Q} ⊗ 1_{B_{l}})]
(1_{P} ⊗ ι_{i} ⊗ 1_{Q} ⊗ ι_{j})(δ_{P,i} ⊗ δ_{Q,j})
= (1_{P ⊗Q} ⊗ ϵ ⊗ ϵ ⊗ 1_{M})(1_{P ⊗Q} ⊗ ι_{i} ⊗ ι_{j} ⊗ 1_{M})
(1_{P ⊗Q} ⊗ φ(B_{i}, B_{j}))(1_{P} ⊗ σ_{B_{i},Q} ⊗ 1_{B_{j}})](δ_{P,i} ⊗ δ_{Q,j})
= (1_{P ⊗Q} ⊗ ϵι_{i} ⊗ ϵι_{j} ⊗ 1_{M})(1_{P} ⊗ σ_{B_{i},Q} ⊗ 1_{B_{j}} ⊗ 1_{M})
(1_{P} ⊗ φ(B_{i}, Q ⊗ B_{j}))(δ_{P,i} ⊗ δ_{Q,j})
= (1_{P} ⊗ ϵι_{i} ⊗ 1_{Q} ⊗ ϵι_{j} ⊗ 1_{M})φ(P ⊗ B_{i}, Q ⊗ B_{j})(δ_{P,i} ⊗ δ_{Q,j})
= (1_{P} ⊗ ϵι_{i} ⊗ 1_{Q} ⊗ ϵι_{j} ⊗ 1_{M})(δ_{P,i} ⊗ δ_{Q,j} ⊗ 1_{M})φ(P, Q)
= φ((P, δ_{P}), (Q, δ_{Q})).

Thus Σ and Π are inverses of each other.

The multiplicative structure m̃_{B} of B in C^{H} is now the uniquely determined morphism which makes the diagram

[figure: P ⊗ Q; P ⊗ Q ⊗ B ⊗ B; δ_{2}; P ⊗ Q; P ⊗ Q ⊗ B; δ_{P ⊗Q}; 1_{P ⊗Q}; 1_{P ⊗Q}⊗_{m_{B}}]

commutative, since ω⊗ω = ω(-⊗-) ⟶^{δ} ω(-⊗-)⊗B is a C_{0}-bimorphism of bifunctors. The unit is given by λ : I ⟶ I ⊗ B = B. The new multiplication can be described by the braid diagram

[figure: δ; δ; δ; δ; m̃; P; Q; P; Q; =; P; Q; B; P; Q; B]

where the braid morphism on the left is the one in C and the braid morphism one the right is the one in C^{H}.

The proof shows that the universal morphism is (1_{P} ⊗ σ_{B,Q} ⊗ 1_{Q})(δ_{P} ⊗ δ_{Q}) : ω⊗ω ⟶ ω⊗ω⊗B⊗B. An analogous result holds for multifunctors ω⊗…⊗ω = ω^{n} and C_{0}-morphisms φ : ω^{n} ⟶ ω^{n} ⊗ M. This proves that with a suitable element τ ∈ B_{n} in the Artin braid group the morphism

δ_{n} := τδ^{n} : ω^{n} ⟶ ω^{n} ⊗ B^{n}

is the universal C_{0}-morphism for all n ∈ ℕ, in particular coend_{C_{0}}(ω^{n}) = B^{n}.

4.1.2. The proof of 4.1 resp. 4.2 provides a proof for the “representability assumption for modules” (in [Mj90] 3.2) in a very general setting for the functor Nat_{C_{0}}(ω^{n}, ω^{n} ⊗ –) instead of the functor Nat(ω^{n}, ω^{n} ⊗ –). A special case of Theorem 4.2 is [Mj93a] Proposition A.4 if one uses C = Vec.

We have reason to make a few additional comments on the existing literature. In [Mj93b] (Theorem 2.2) the following theorem is given: Let C be a monoidal category and V a rigid braided monoidal category cocomplete over C. Let F : C ⟶ V be a monoidal functor. Then there exists a V-bialgebra, A = Aut(C, F, V), such that F factorizes monoidally as C ⟶ ^{A}V ⟶ V. … A is universal with this property. …

Although Majid considers this theorem “a significant generalization of usual Tannaka-Krein ideas” (hep-th 9504007) it does not even specialize to other known cases of Tannaka-Krein reconstruction such as [Ul90] Proposition 2.4. Observe that the formation of big colimits will usually destroy rigidity. In fact the full subcategory vec of rigid objects in Vec, that is the category of finite dimensional vector spaces, is only cocomplete over finite categories C. There are no nontrivial infinite direct sums in this category.

This is why in reconstruction theorems one usually assumes that the rigid category V is a subcategory of a bigger cocomplete monoidal category in which the universal bialgebra A is then reconstructed as a colimit of a diagram in V.

Furthermore the theorem quoted above only allows to reconstruct rigid bialgebras, e.g. finite dimensional bialgebras in the case V = vec. The proof of this theorem cannot just simply be extended to the more general case because the rigidity of the reconstructed object is explicitly used in the proof of the algebra property, in particular in [Mj93b] Lemma 2.3. A hypothesis on the tensor product of V, namely that it preserves colimits, in order to obtain a bialgebra structure in the more general case, does not become clear in the vague discussion that follows p.205 of [Mj93b].

Theorems 4.1 and 4.2 of [Mj93b] again are only proved under the restriction of finite dimensionality. The discussion of the infinite dimensional situation is not satisfactory (see p. 209). In fact in the dual situation Vec^{op} one reconstructs essentially linearly compact algebras, which are also ordinary algebras, but the bialgebra structure does not live in Vec any more, due to the fact that tensor products do not preserve limits.

So our Theorem 4.2 together with Proposition 3.6 remedy these shortcomings. Through the use of control categories they are clearly much more general.

The following is a generalization of [Pa78], Corollary 6.4.

Corollary 4.3. Let C be a C_{0}-generated coalgebra in the braided monoidal category C and let ω : C_{0}^{C} ⟶ C be the forgetful functor. Then Nat_{C_{0}}(ω, ω ⊗ –) : C ⟶ Set is representable and C = coend_{C_{0}}(ω) as coalgebras in C.

Proof. Use H = 𝕂 in the above theorem.

Corollary 4.4. Let ω : C_{0} ⟶ C be the embedding functor. Then Nat_{C_{0}}(ω, ω ⊗ –) : C ⟶ Set is representable and coend_{C_{0}}(ω) = I.

In particular coend_{C}(Id_{C}) ≅ I for any monoidal category C.

Proof. Use C = 𝕂 in the above Corollary.

As we remarked after the definition of C_{0}-generated coalgebras, the condition that C is C_{0}-generated becomes vacuous in the case C_{0} = C.

Corollary 4.5. Let B be a C_{0}-central bialgebra in C which is C_{0}-generated as a coalgebra, let A := C_{0}^{B}, and let ω : C_{0}^{B} ⟶ C be the forgetful functor. Then Nat_{C_{0}}(ω, ω ⊗ –) is multirepresentable and coend_{C_{0}}(ω) = B as bialgebras.

By dualization of Proposition 4.1 one gets

Corollary 4.6. Let A be a C_{0}-generated algebra in C, and let ω : C_{0A} ⟶ C be the forgetful functor. Then Nat_{C_{0}}(ω ⊗ –, ω) is representable and A = end_{C_{0}}(ω) as algebras in C.

4.1.3. Later on we will need the notion of cosmash products in A. So we define it here and show an important property. If B is a bialgebra and C is a B-comodule-coalgebra in A then we can construct a cosmash product B#^{c}C where the cosmash comultiplication is defined by

\begin{matrix}Δ : B ⊗ C ⟶^{Δ_{B}⊗Δ_{C}} B ⊗ B ⊗ C ⊗ C ⟶^{1⊗δ′_{C}⊗1} B ⊗ B ⊗ C ⊗ B ⊗ C \\ ⟶^{1⊗σ⊗1} B ⊗ C ⊗ B ⊗ B ⊗ C ⟶^{1⊗m_{B}⊗1} B ⊗ C ⊗ B ⊗ C. \end{matrix}

It is easy to see that B#^{c}C is a coalgebra in A.

4.1.4. Let H be a braided Hopf algebra in C, z : B ⟶ H be a bialgebra morphism in C and B̃ be the transmuted bialgebra in C^{H} (as in Theorem 4.2). Let C be a B̃-comodule-coalgebra in A := C^{H}. Then we can form the cosmash product B̃ ˜#^{c}C in C^{H}.

Since B is a bialgebra in C and C is a coalgebra in C, C is also a B-comodule-coalgebra by

[figure: δ; δ; δ; δ; δ; m̃; C; C; C; =; =; C; C; B; C; C; B; C; C; B]

So there is a second way to define a cosmash product B#^{c}C this time in C. These two coalgebra structures on B⊗C, however, coincide as the following diagram shows

[figure: δ; δ; m̃; B; C; B; C; =; B; C; B; C; B; C; B; C.]

4.2. Finite reconstruction. In the previous section we started with an algebra A or a coalgebra C in C and reconstructed them from ω : C_{A} ⟶ C resp. ω : C^{C} ⟶ C. If, however, an arbitrary C-functor ω : B ⟶ A is given it is not clear if Nat_{C}(ω, ω⊗–) is a representable functor or if coend_{C}(ω) exists in C. It is customary to call the construction of coend_{C}(ω) also in this situation “re”construction, although we do not start with an algebra or a coalgebra in C and then reconstruct is from its category of representations.

In one particular situation the (restricted) reconstruction is possible and well known, namely in the case of C = Vec and a functor ω : B ⟶ vec ⊆ Vec into the category of finite-dimensional vector spaces. Various generalizations of this result are known. We will lift this result to braided monoidal categories C.

Let C be a cocomplete braided monoidal category and C_{0} be the full (rigid braided monoidal) subcategory of rigid objects. Furthermore assume that the tensor product in C preserves arbitrary colimits in both variables. Let B be a small category and ω : B ⟶ C_{0} ⊆ C be a functor.

Theorem 4.7. (1) The functor Nat(ω, ω ⊗ M) is multirepresentable:

Nat(ω, ω ⊗ M) ≅ C(coend(ω), M),

(2) If B is a C_{0}-category and ω : B ⟶ C_{0} is a C_{0}-functor, then the functor Nat_{C_{0}}(ω, ω ⊗ M) is representable:

Nat_{C_{0}}(ω, ω ⊗ M) ≅ C(coend_{C_{0}}(ω), M).

If B := coend_{C_{0}}(ω) is C_{0}-central then Nat_{C_{0}}(ω, ω ⊗ M) is multirepresentable.

Proof. (1) The existence of a representing object for the functor Nat(ω, ω ⊗ M) is well known (see [Pa93]). We recall the main steps of its construction, since they play a role in the second part of the proof. The representing object C = coend(ω) is obtained as colimit of the diagram consisting of all wedges for all morphisms f : P ⟶ Q in B:

[figure: ω(P)^{∗} ⊗ ω(P); ω(Q)^{∗} ⊗ ω(P); ω(Q)^{∗} ⊗ ω(Q); ω(f)^{∗}⊗1; 1⊗ω(f)]

Any (cone-) morphism from this diagram to an object M ∈ C

[figure: (1); ω(P)^{∗} ⊗ ω(P); ω(Q)^{∗} ⊗ ω(P); ω(Q)^{∗} ⊗ ω(Q); ω(f)^{∗}⊗1; 1⊗ω(f); M; ψ(P); ψ(Q)]

is given by a natural transformation φ : ω ⟶ ω ⊗ M:

[figure: ω(P); ω(P) ⊗ M; φ(P); ω(Q); ω(Q) ⊗ M; φ(Q); ω(f); ω(f)⊗1_{M}]

The one-to-one correspondence between morphisms ψ and morphisms φ is given, using the evaluation ev_{ω(P)} : ω(P)^{∗} ⊗ ω(P) ⟶ I and the dual basis db_{ω(P)} : I ⟶ ω(P) ⊗ ω(P)^{∗}, as

ψ(P) = (ev_{ω(P)} ⊗ 1_{M})(1_{ω(P)^{∗}} ⊗ φ(P))

and

φ(P) := (1_{ω(P)} ⊗ ψ(P))(db_{ω(P)} ⊗ 1_{ω(P)}).

Given ψ the morphisms φ(P) form a natural transformation since

[figure: ω(P); ω(P) ⊗ ω(P)^{∗} ⊗ ω(P); db_{ω(P)}⊗1_{ω(P)}; ω(P) ⊗ M; 1_{ω(P)}⊗ψ(P); ω(Q) ⊗ ω(Q)^{∗} ⊗ ω(P); ω(Q) ⊗ ω(P)^{∗} ⊗ ω(P); 1_{ω(Q)}⊗ω(f)^{∗}⊗1_{ω(P)}; ω(Q); ω(Q) ⊗ ω(Q)^{∗} ⊗ ω(Q); db_{ω(Q)}⊗1_{ω(Q)}; ω(Q) ⊗ M; 1_{ω(Q)}⊗ψ(Q); ω(f); ω(f)⊗1_{M}; db_{ω(Q)}⊗1_{ω(P)}; ω(f)⊗1_{ω(P)^{∗}}⊗1_{ω(P)}; 1_{ω(Q)}⊗1_{ω(Q)^{∗}}⊗ω(f); 1_{ω(Q)}⊗ψ(P)]

commutes where the left upper part of the diagram commutes by the definition of the adjoint morphism ω(f)^{∗}. Conversely given φ : ω ⟶ ω ⊗ M the diagram

[figure: ω(P)^{∗} ⊗ ω(P); ω(P)^{∗} ⊗ ω(P) ⊗ M; 1_{ω(P)^{∗}}⊗φ(P); ω(Q)^{∗} ⊗ ω(P); ω(Q)^{∗} ⊗ ω(P) ⊗ M; 1_{ω(Q)^{∗}}⊗φ(P); ω(Q)^{∗} ⊗ ω(Q); ω(Q)^{∗} ⊗ ω(Q) ⊗ M; 1_{ω(Q)^{∗}}⊗φ(Q); 1_{ω(Q)^{∗}}⊗ω(f); 1_{ω(Q)^{∗}}⊗ω(f)⊗1_{M}; M; ev_{ω(P)}⊗1_{M}; ω(f)^{∗}⊗1_{ω(P)}; ω(f)^{∗}⊗1_{ω(P)}⊗1_{M}; ev_{ω(Q)}⊗1_{M}]

commutes.

Now we show that the functor Nat(ω, ω ⊗ –) is multirepresentable. We restrict our attention just to the case n = 2. As before there is a bijective correspondence between the natural transformations φ(P, Q) : ω(P) ⊗ ω(Q) ⟶ ω(P) ⊗ ω(Q) ⊗ M and cones ψ(P, Q) : ω(Q)^{∗} ⊗ ω(P)^{∗} ⊗ ω(P) ⊗ ω(Q) ⟶ M.

Let δ : ω ⟶ ω ⊗ coend(ω) be the universal morphism and abbreviate B := coend(ω). Let θ(P) : ω(P)^{∗} ⊗ ω(P) ⟶ B be the induced morphism. In the commutative diagram (colimit of a wedge in the sense used above) induced by morphisms f : P ⟶ R and g : Q ⟶ S in B

[figure: (2); ω(Q)^{∗} ⊗ ω(P)^{∗} ⊗ ω(P) ⊗ ω(Q); ω(P)^{∗} ⊗ ω(P) ⊗ ω(Q)^{∗} ⊗ ω(Q); σ_{ω(Q)^{∗},ω(P)^{∗}⊗ω(P)}^{−1}⊗1_{ω(Q)}; ω(S)^{∗} ⊗ ω(R)^{∗} ⊗ ω(P) ⊗ ω(Q); ω(S)^{∗} ⊗ ω(R)^{∗} ⊗ ω(R) ⊗ ω(S); ω(R)^{∗} ⊗ ω(R) ⊗ ω(S)^{∗} ⊗ ω(S); σ_{ω(S)^{∗},ω(R)^{∗}⊗ω(R)}^{−1}⊗1_{ω(S)}; 1_{ω(S)^{∗}⊗ω(R)^{∗}}⊗ω(f)⊗ω(g); θ(P)⊗θ(Q); ω(g)^{∗}⊗ω(f)^{∗}⊗1_{ω(P)⊗ω(Q)}; B ⊗ B; θ(R)⊗θ(S)]

B ⊗ B is a colimit with

θ(P, Q) = (θ(P) ⊗ θ(Q))(σ_{ω(Q)^{∗},ω(P)^{∗}⊗ω(P)}^{−1} ⊗ 1_{ω(Q)})

since tensor products preserve colimits. We have to show that the induced morphism δ(-, -) : ω⊗ω ⟶ ω⊗ω⊗B⊗B is equal to (1_{ω}⊗σ_{B,ω}⊗1_{B})(δ⊗δ) : ω^{2} ⟶ ω^{2}⊗B⊗B.

We use graphic calculus and observe that the correspondence between the morphisms φ and ψ is given by

[figure: φ; ψ; ψ; φ; ω; ω; ω^{∗}; ω; ω^{∗}; ω; =; =; and; B; B; B; B; ω; ω]

Then we get (writing P and Q instead of ω(P) resp. ω(Q))

[figure: θ; θ; δ; δ; δ; δ; P; P; P; Q; Q; Q; =; =; P; Q; B; B; P; Q; B; B; P; Q; B; B.]

(2) To describe the second isomorphism of the theorem the property of C_{0}-trans-formation for φ : ω ⟶ ω ⊗ M is given by the commutative diagram

[figure: ω(X ⊗ P); ω(X ⊗ P) ⊗ M; φ(X⊗P); X ⊗ ω(P); X ⊗ ω(P) ⊗ M; 1_{X}⊗φ(P); ζ(X,P); ζ(X,P)⊗1_{M}]

which translates into

[figure: (3); ω(P)^{∗} ⊗ X^{∗} ⊗ ω(X ⊗ P); ω(X ⊗ P)^{∗} ⊗ ω(X ⊗ P); ζ(X,P)^{∗}⊗1; ≅; 1_{ω(P)^{∗}}⊗ζ(X,P); ≅; ω(P)^{∗} ⊗ X^{∗} ⊗ X ⊗ ω(P); ω(P)^{∗} ⊗ ω(P); 1_{ω(P)^{∗}}⊗ev_{X}⊗1_{ω(P)}; M; ψ(X⊗P); ψ(P)]

So the colimit coend_{C_{0}}(ω) exists and is described (similar to the way given in [Pa93] Definition 2.2) as ∐_{P ∈Ob(B)} ω(P)^{∗} ⊗ ω(P) modulo the relations given by all f : P ⟶ Q as in the construction of coend(ω) plus the relations for any pair (X, P) given above.

In particular δ : ω ⟶ ω ⊗ B with B := coend_{C_{0}}(ω) is a universal C_{0}-morphism. Now we assume that B = coend_{C_{0}}(ω) is C_{0}-central and show that the functor

Nat_{C_{0}}(ω, ω ⊗ –) is multirepresentable. We restrict our attention again just to the case n = 2. Since δ_{2} := (1_{ω} ⊗ σ_{B,ω} ⊗ 1_{B})(δ ⊗ δ) : ω^{2} ⟶ ω^{2} ⊗ B^{2} is a C_{0}-bimorphism – B is C_{0}-central – we can show that B ⊗ B = coend_{C_{0}}(ω ⊗ ω) with the universal C_{0}-bimorphism δ_{2}. We first show that there is a one-to-one correspondence between C_{0}-bimorphisms φ(P, Q) : ω(P) ⊗ ω(Q) ⟶ ω(P) ⊗ ω(Q) ⊗ M and cones ψ(P, Q) : ω(Q)^{∗} ⊗ ω(P)^{∗} ⊗ ω(P) ⊗ ω(Q) ⟶ M satisfying certain relations given below. We saw earlier for two variables (see diagram (2)) that φ is a natural transformation iff ψ is a cone. So we have to translate the conditions for the C_{0}-structure. It is an easy exercise to show that under the correspondence between φ and ψ the conditions

φ  φ  φ  φ  φ  φ X P Q  X P Q  P X Q  P X Q  X P Q  X P Q =  =  = (a)  (b)  (c) X P QM X P QM X P QM X P QM  P X QM P X QM

are equivalent to the following conditions

[figure: ψ; ψ; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; =; (a); M; M]

[figure: ψ; ψ; ψ; ψ; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; =; =; (b); (c); M; M; M; M]

In particular the induced cone θ_{2} : ω(Q)^{∗} ⊗ω(P)^{∗} ⊗ω(P)⊗ω(Q) ⟶ B ⊗B satisfies the conditions (a), (b), and (c). Observe that condition (b) for φ = δ_{2} implies condition (c) for δ_{2} since B is C_{0}-central. In fact (b) implies

δ_{2}  δ_{2}  δ_{2}  δ  δ P X Q  P X Q  P X Q  P X Q =  =  = P X Q B B P X Q B B P X Q B B P X Q B B

hence

δ_{2}  δ_{2} δ  δ  δ  δ  δ δ X P Q  X P Q  X P Q  X P Q  X P Q =  =  =  = P X Q B B P X Q B B P X Q B B P X Q B B P X Q B B

So condition (b) for ψ = θ_{2} implies condition (c) for θ_{2}. Now we show that θ_{2} : ω(Q)^{∗} ⊗ ω(P)^{∗} ⊗ ω(P) ⊗ ω(Q) ⟶ B ⊗ B is the injection morphism of a colimit. Then δ_{2} : ω ⊗ ω ⟶ ω ⊗ ω ⊗ B ⊗ B is a universal C_{0}-morphism.

We have already seen that diagram (2) is a tensor product of wedges and cone morphisms of the type of diagram (1). Now we show that the relations (a) and (b) come about as tensor products of relations for B. We don’t have to consider condition (c) which is automatically satisfied. The diagrams

and

[figure: θ_{2}; θ_{2}; θ; θ; θ; θ; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; Q^{∗}; P^{∗}; X^{∗}; X; P; Q; =; =; =; B; B; B; B; B; B; B; B]

[figure: θ_{2}; θ_{2}; θ; θ; θ; θ; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; Q^{∗}; X^{∗}; P^{∗}; P; X; Q; =; =; =; B; B; B; B; B; B; B; B]

show that relations (a) and (b) are tensor products with the relation

[figure: θ; θ; P^{∗}; X^{∗}; X; P; P^{∗}; X^{∗}; X; P; =; B; B]

used in the construction of B (up to a preceding isomorphism). Since we are now considering a tensor product of two diagrams and the colimit thereof and since tensor products preserve colimits we have proved the claimed result.

4.2.1. Using the results of section 3 we obtain uniquely determined coalgebra, bialgebra, and Hopf algebra structures (depending on the given functor) on coend(ω) and coend_{C_{0}}(ω).

5. Hidden symmetries

In section 4.1 we studied under which circumstances coalgebras and bialgebras (possibly with a transmuted multiplication) can be reconstructed from their categories of comodules and the functor ω : C^{C} ⟶ C. We saw that they are obtained as the representing object coend_{C}(ω) of Nat_{C}(ω, ω ⊗ –) : C ⟶ Set. In this section we will see that this reconstruction depends strongly on the choice of the control category C. If C is decreased to a category D then the representing (reconstructed) object coend_{D}(ω) becomes larger. We will see that under certain conditions the reconstructed object decomposes into a cosmash product where one factor represents the “hidden symmetries”.

5.1. Functors of control categories. We consider of a braided monoidal functor F : D ⟶ C of control categories.

5.1.1. If B is a C-category via ⊗ : C × B ⟶ B, then B becomes a D-category by

⊗ : D × B ⟶^{F×1} C × B ⟶^{⊗} B

with associativity morphism β(υ ⊗ 1) : (X ⊗ Y) ⊗ P ⟶ X ⊗ (Y ⊗ P) and unary action π(ς ⊗ 1) : I_{D} ⊗ P ⟶ P for X, Y ∈ D and P ∈ B.

5.1.2. If χ : B ⟶ B′ is a C-functor, then χ becomes also a D-functor. If χ : B × B′ ⟶ B′′ is a C-bifunctor, then χ becomes also a D-bifunctor. In both cases the structure morphisms ζ resp. τ remain unchanged.

If χ, χ′ : B ⟶ B′ are C-functors and ζ : χ ⟶ χ′ is a C-morphism, then ζ is also a D-morphism.

5.1.3. If A is a C-monoidal category, then it becomes also a D-monoidal category. The above observations give immediately

Proposition 5.1. If F : D ⟶ C is a braided monoidal functor, then it induces an “underlying” functor 𝔄(F) : 𝔄(C) ⟶ 𝔄(D).

5.1.4. Let (B, ω) ∈ 𝔄(C) and consider 𝔄(F)(B, ω) = (B, ω) with the induced structure morphisms. Assume that Nat_{C}(ω, ω ⊗ –) and Nat_{D}(ω, ω ⊗ –) are representable by coend_{C}(ω) resp. coend_{D}(ω). Then A(coend_{C}(ω), M) ≅ Nat_{C}(ω, ω ⊗ M) ⊆ Nat_{D}(ω, ω ⊗ M) ≅ A(coend_{D}(ω), M) as functors in M ∈ A hence there is an epimorphism of the representing objects coend_{F}(ω) : coend_{D}(ω) ⟶ coend_{C}(ω).

Theorem 5.2. Let F : D ⟶ C be a braided monoidal functor. Let A be a Cmonoidal category, B a C-category and ω : B ⟶ A a C-functor. Assume that coend_{C}(ω) and coend_{D}(ω) exist. Then there is an induced epimorphism of coalgebras coend_{F}(ω) : coend_{D}(ω) ⟶ coend_{C}(ω) in C.

If in addition the comodule (coend_{C}(ω), Δ) is liftable along ω then coend_{F}(ω) : coend_{D}(ω) ⟶ coend_{C}(ω) is a retraction of objects in C.

Proof. Let C := coend_{C}(ω) and D := coend_{D}(ω). Let δ : ω ⟶ ω ⊗ C and ∂ : ω ⟶ ω ⊗ D be the universal morphisms. Write C^{2} := C ⊗ C, D^{2} = D ⊗ D, and f = coend_{F}(ω). Then the commutativity of

[figure: ω; ω ⊗ D; ∂; ω ⊗ D; ω ⊗ D^{2}; ∂⊗1; ω; ω ⊗ C; δ; ω ⊗ C; ω ⊗ C^{2}; δ⊗1; =; 1⊗f; 1⊗f; 1⊗f⊗f; ∂; δ; 1⊗Δ; 1⊗Δ]

and

[figure: ω; ω ⊗ D; ∂; ω ⊗ C; δ; ω ⊗ I; ≅; 1⊗f; 1⊗ε; 1⊗ε]

show that f : D ⟶ C is a coalgebra morphism.

Let g : ω(C̃) ⟶ C be a C-comodule isomorphism. Then the following diagram commutes

[figure: ω(C̃); ω(C̃) ⊗ D; ∂; I ⊗ D ≅ D; εg⊗1; ω(C̃) ⊗ C; δ; 1⊗f; 1⊗f; f; C; g; g⊗1; Δ; C ⊗ C; I ⊗ C ≅ C.; ε⊗1]

The lower morphism of the diagram is the identity hence f is a retraction in A.

5.1.5. Observe that the morphism coend_{F}(ω) : coend_{D}(ω) ⟶ coend_{C}(ω) is the uniquely defined morphism such that (1_{ω} ⊗ coend_{F}(ω)) ∘ ∂ = δ.

5.1.6. The preceding theorem shows that the reconstructed coalgebra D w.r.t. D is larger than C. We consider the additional part in D as hidden symmetries in the sense described in the introduction. It is responsible for D-morphisms φ : ω ⟶ ω ⊗ M which are not C-morphisms or for certain elements in Nat_{D}(ω, ω ⊗ M) which are not contained in Nat_{C}(ω, ω ⊗ M). As we have seen this part of D tends to split off.

5.1.7. An example of a hidden symmetry can be obtained for superalgebra representations. This is dual to the above considerations. Given an algebra A considered as a superalgebra (A, 0). Consider A = C, the category of super vector spaces (𝕂ℤ_{2}-Comod), the category B = C_{A} of super A-modules, and the forgetful functor ω : C_{A} ⟶ C. Let F : D = Vec ⟶ C be the functor which sends each vector space V to the super vector space (V, 0). Any D-morphism φ : ω ⟶ ω is described by its image under φ ∈ Nat_{D}(ω ⊗ I, ω) ≅ A(I, end_{D}(ω)).

The natural transformation φ : P ⟶ P given by (p_{0}, p_{1}) ↦ (p_{0}, −p_{1}) is a symmetry for all representations of A (a natural automorphism of ω), which is not induced by the multiplication with any element of A. A multiplication with an element a = (a, 0) ∈ A on A-modules (P_{0}, P_{1}) in C_{A} would have to satisfy (p_{0}, p_{1})a = (p_{0}a, p_{1}a) = (p_{0}, −p_{1}) for all choices of (p_{0}, p_{1}) which is not possible. The natural transformation φ is, however, a D-morphism and thus comes from multiplication with an element b ∈ end_{D}(ω), in fact from the element e_{1} − e_{t} ∈ (𝕂ℤ_{2})^{∗} ⊆ end_{D}(ω) where e_{1}, e_{t} is the dual basis to 1, t ∈ 𝕂[t]/(t^{2} − 1) = 𝕂ℤ_{2}.

5.1.8. We apply the preceding example in representation theory of groups. If we consider representations of a group G in vector spaces over a field 𝕂, i.e. the category M_{𝕂G}, then each element g ∈ G induces a C-monoidal automorphism φ_{g} : ω ⟶ ω with φ_{g}(p) := pg, where ω : M_{𝕂G} ⟶ M = Vec = C is the forgetful functor. Observe that any natural transformation of functors into C is a C-morphism by Theorem 6.4. Conversely given any C-monoidal automorphism φ : ω ⟶ ω there is precisely one g ∈ G with φ = φ_{g}. Thus G can be reconstructed from its representations, i.e. from ω : M_{𝕂G} ⟶ M.

Now consider representations of G in super vector spaces over 𝕂, i.e. the category A of two-graded vector spaces. Let C = M^{𝕂ℤ_{2}} = A, F : D = Vec ⟶ C = M^{𝕂ℤ_{2}}, and ω : A_{𝕂G} ⟶ A. We may consider 𝕂G as a Hopf algebra (𝕂G, 0) in A and have (p_{0}, p_{1})g = (p_{0}g, p_{1}g) with a suitable G-structure on P_{0} and P_{1} separately. Then each element g ∈ G induces a C-monoidal automorphism φ_{g} : ω ⟶ ω. For any C-monoidal automorphism there is precisely one g ∈ G with φ = φ_{g}. For the D-monoidal automorphism φ : ω ⟶ ω with φ(P_{0}, P_{1})(p_{0}, p_{1}) := (p_{0}, −p_{1}) there is, however, no g ∈ G with φ = φ_{g}. So in this case the group of symmetries (of monoidal automorphisms of ω) is a bigger group than the one we started out with. The given φ is an example of a hidden symmetry.

5.1.9. We consider now the situation of a morphism [χ, ζ] : (B, ω) ⟶ (B′, ω′) in 𝔄(C) together with a braided monoidal functor F : D ⟶ C. Assume the universal objects and morphisms δ : ω ⟶ ω ⊗ C, ∂ : ω ⟶ ω ⊗ D, δ′ : ω′ ⟶ ω′ ⊗ C′, and ∂′ : ω′ ⟶ ω′ ⊗ D′ exist. Then by 3.3.1 we get induced morphisms z : C ⟶ C′ and y : D ⟶ D′ such that (ζ ⊗ z) ∘ δ = δ′χ ∘ ζ and (ζ ⊗ y) ∘ ∂ = ∂′χ ∘ ζ. Furthermore by Theorem 5.2 there are induced morphisms f := coend_{F}(ω) : D ⟶ C and f′ := coend_{F}(ω′) : D′ ⟶ C′ such that (1_{ω} ⊗ f) ∘ ∂ = δ and (1_{ω′} ⊗ f′) ∘ ∂′ = δ′. Hence by the universal property of ∂ the diagram

[figure: ω; ω ⊗ D; ∂; ω′χ; ω′χ ⊗ D′; ∂′χ; ζ; ζ⊗y; ω ⊗ C; ω′χ ⊗ C′; ζ⊗z; 1⊗f; 1⊗f′; δ; δ′χ]

commutes and from ζ ⊗ zf = (ζ ⊗ z)(1 ⊗ f) = (1 ⊗ f′)(ζ ⊗ y) = ζ ⊗ f′y and the uniqueness of induced morphisms we get a commutative diagram of coalgebra morphisms

[figure: D; C; f; D′; C′; f′; y; z]

or

coend_{C}([χ, ζ])coend_{F}(ω) = coend_{F}(ω′)coend_{D}([χ, ζ]).

In particular we have proved

Theorem 5.3. Let F : D ⟶ C be a braided monoidal functor. Assume that the functors coend_{C} : 𝔄(C) ⟶ A-coalg and coend_{D} : 𝔄(C) ⟶ A-coalg exist. Then coend_{F} : coend_{D} ⟶ coend_{C} is a natural epimorphism of functors from 𝔄(C) to A-coalg.

5.1.10. Assume now that B is C-monoidal, that ω : B ⟶ A is a C-monoidal functor and that Nat_{C}(ω, ω ⊗ –) and Nat_{D}(ω, ω ⊗ –) are multirepresentable. Then z = coend_{C}([χ, ζ]) : D ⟶ C is a bialgebra morphism. The multiplicativity follows from the commutative diagram

[figure: ω ⊗ ω; ω ⊗ ω ⊗ D ⊗ D; ∂_{2}; ω(⊗); ω(⊗) ⊗ D; ∂; υ^{−1}; ω ⊗ ω ⊗ D; 1⊗m_{D}; υ^{−1}⊗1; ω ⊗ ω ⊗ C ⊗ C; ω ⊗ ω ⊗ C; 1⊗m_{C}; ω ⊗ C; υ^{−1}⊗1; 1⊗f⊗f; 1⊗f; 1⊗f; δ_{2}; δ]

(where ω(⊗)(P, Q) := ω(P ⊗ Q)) and the unary property is proved similarly.

If the bialgebras C′ and D′ are Hopf algebras then by Corollary 3.8 C is a C′- comodule coalgebra by the coadjoint coaction w.r.t. the induced coalgebra morphism z : C ⟶ C′ and D is a D′-comodule coalgebra by the coadjoint coaction w.r.t. the induced coalgebra morphism y : D ⟶ D′. Furthermore f′ : D′ ⟶ C′ is an epimorphism and a bialgebra (Hopf algebra) morphism.

5.2. Hidden symmetries of the base category. Consider a braided monoidal functor F : D ⟶ C and a morphism [χ, ζ] : (B, ω) ⟶ (B′, ω′) in 𝔄(C). Assume that B′ is a C-monoidal category and ω′ is a C-monoidal functor. Let δ : ω ⟶ ω⊗D be a universal D-morphism. Let D′ be a bialgebra in A and let δ′ : ω′ ⟶ ω′ ⊗D′ be a D-morphism compatible with the bialgebra structure of D′ (e.g. δ′ is a universal D-morphism). Let E be a coalgebra in B′ and let μ : χ ⟶ χ ⊗ E be a C-morphism compatible with the structure of E (e.g. a universal C-morphism).

Theorem 5.4. In the setup given above D′ ⊗ω′E carries the structure of a cosmash product D′#^{c}ω′E and there is a canonical coalgebra morphism f : D ⟶ D′#^{c}ω′E.

Proof. We first observe that ω′E is a coalgebra in A since ω′ is a monoidal functor. Furthermore ω′E is a D′-comodule coalgebra by the morphism δ′E : ω′E ⟶ ω′E ⊗ D′.

Consider the following induced morphism f : D ⟶ D′#^{c}ω′E defined by

[figure: ω; ω ⊗ D; δ; ω′(χ ⊗ E); ω′χ ⊗ ω′E; υ; ω′χ ⊗ D′ ⊗ ω′E.; δ′χ⊗1; ω′χ; ζ; ω′μ; ω ⊗ D′ ⊗ ω′E; 1⊗f; ζ⊗1]

Since the composition along the lower edge of the square is a D-morphism ω ⟶ ω ⊗D′ ⊗ω′E the morphism f is uniquely determined. We show that f is a coalgebra morphism with D′ ⊗ ω′E = D′#^{c}ω′E the cosmash product. For this purpose define a morphism

τ_{0} : D′ ⊗ ω′E ⟶^{1⊗δ′E} D′ ⊗ ω′E ⊗ D′ ⟶^{σ⊗1} ω′E ⊗ D′ ⊗ D′ ⟶^{1⊗m_{D′}} ω′E ⊗ D′.

Then the following diagram of D-morphisms commutes by the very fact that the bialgebra structure of D′ is compatible with δ′. Hence with suitable identifications we get the commutative diagram

[figure: ω′χ; ω′(χ ⊗ E); ω′μ; ω′χ ⊗ ω′E; υ; ω′χ ⊗ D′ ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ D′; ω′(χ ⊗ E) ⊗ D′; ω′μ⊗1; ω′χ ⊗ ω′E ⊗ D′; υ⊗1; δ′χ; δ′(χ⊗E); ω′χ ⊗ D′ ⊗ ω′E ⊗ D′; 1⊗δ′E; ω′χ ⊗ ω′E ⊗ D′ ⊗ D′; 1⊗σ⊗1; 1⊗m_{D′}]

[figure: ω′χ; ω′χ ⊗ ω′E; ω′μ; ω′χ ⊗ D′; ω′χ ⊗ ω′E ⊗ D′.; ω′μ⊗1; δ′χ; ω′χ ⊗ D′ ⊗ ω′E; δ′χ⊗1; 1⊗τ_{0}]

Now consider the commutative diagram

[figure: ω′χ; ω′χ ⊗ D; δ; ω′χ ⊗ D ⊗ D; δ⊗1; ω′χ ⊗ ω′E; ω′χ ⊗ D′ ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ ω′E ⊗ D; ω′χ ⊗ D′ ⊗ ω′E ⊗ D; δ′χ⊗1; ω′χ ⊗ ω′E ⊗ D′ ⊗ ω′E; ω′χ ⊗ D′ ⊗ ω′E ⊗ D′ ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ D′ ⊗ ω′E ⊗ ω′E; ω′χ ⊗ D′ ⊗ D′ ⊗ ω′E ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ ω′E ⊗ ω′E; δ′χ⊗1; ω′μ; ω′μ⊗1; 1⊗f; 1⊗f; 1⊗1⊗f; ω′μ⊗1; ω′μ⊗1; 1⊗τ_{0}⊗1; 1⊗τ_{0}⊗1; 1⊗f⊗1]

From it we get that all morphisms ω′χ ⟶ ω′χ⊗D′⊗ω′E⊗D′⊗ω′E in the following diagram are equal

[figure: ω′χ; ω′χ ⊗ D; δ; ω′χ ⊗ D ⊗ D; δ⊗1; 1⊗Δ_{D}; ω′χ ⊗ ω′E; ω′χ ⊗ D′ ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ ω′E ⊗ D; ω′χ ⊗ D′ ⊗ ω′E ⊗ D; δ′χ⊗1; ω′χ ⊗ ω′E ⊗ D′ ⊗ ω′E; ω′χ ⊗ D′ ⊗ ω′E ⊗ D′ ⊗ ω′E; δ′χ⊗1; ω′χ ⊗ D′ ⊗ ω′E ⊗ ω′E; ω′χ ⊗ D′ ⊗ D′ ⊗ ω′E ⊗ ω′E; δ′χ⊗1; 1⊗Δ_{D′}⊗1; ω′χ ⊗ ω′E ⊗ ω′E; δ′χ⊗1; ω′μ; ω′μ⊗1; 1⊗Δ_{ω′E}; 1⊗Δ_{ω′E}; 1⊗f; 1⊗f; 1⊗1⊗f; ω′μ⊗1; ω′μ⊗1; 1⊗τ_{0}⊗1; 1⊗τ_{0}⊗1; 1⊗f⊗1]

We define

Δ_{D′⊗ω′E} : D′ ⊗ ω′E ⟶^{Δ_{D′}⊗Δ_{ω′E}} D′ ⊗ D′ ⊗ ω′E ⊗ ω′E ⟶^{1⊗τ_{0}⊗1} D′ ⊗ ω′E ⊗ D′ ⊗ ω′E

and observe that δ : ω′χ ⟶ ω′χ⊗D is a universal D-morphism. Hence the diagram of induced morphisms

[figure: D; D ⊗ D; Δ_{D}; D′ ⊗ ω′E; D′ ⊗ ω′E ⊗ D′ ⊗ ω′E; Δ_{D′⊗ω′E}; f; f⊗f]

commutes. It is now easy to see that D′ ⊗ω′E with Δ_{D′⊗ω′E} and ε : D′ ⊗ω′E ⟶^{ε⊗ω′ε} I is a coalgebra, the cosmash product D′#^{c}ω′E, and that f : D ⟶ D′ ⊗ ω′E is a coalgebra morphism.

A special application of the theorem is the following

Corollary 5.5. Let (B, ω) be in 𝔄(C) and let F : D ⟶ C be a braided monoidal functor. If coend_{D}(ω), coend_{C}(ω), and coend_{D}(id_{A}) (with Nat_{D}(id_{A}, id_{A} ⊗ –) multirepresentable) exist then there is a canonical coalgebra morphism

f : coend_{D}(ω) ⟶ coend_{D}(id_{A})#^{c}coend_{C}(ω).

In certain cases the canonical morphism of the preceding corollary is an isomorphism. If this is the case then we have identified the hidden symmetries of the functor ω : B ⟶ A as the component D′ = coend_{D}(id_{A}) in the cosmash product.

Theorem 5.6. Let D be a braided monoidal category and H be a braided Hopf algebra in D. Let A = C = D^{H}. Let C be a coalgebra in A and ω : A^{C} ⟶ A be the forgetful functor. Then with coend_{D}(id_{A}) ˜#^{c}coend_{C}(ω) the cosmash product in A

f : coend_{D}(ω) ⟶ coend_{D}(id_{A}) ˜#^{c}coend_{C}(ω)

is an isomorphism of coalgebras in A.

Proof. We consider the diagram

[figure: A^{C}; A; ω; A; ω; id]

where ω : A^{C} ⟶ A is the forgetful functor together with the functor F : D ⟶ C induced by u : I ⟶ H. It is easy to check that A^{C} = (D^{H})^{C} and D^{H#^{c}C} are isomorphic D-categories with the cosmash product formed in D (for cosmash products and transmutation see also 4.1.4) by sending each object (P, μ : P ⟶

P ⊗ C) in A^{C} to the object (P, δ : P ⟶^{μ} P ⊗ C ⟶^{δ′⊗1} P ⊗ H ⊗ C) in D^{H#^{c}C} (see

2.1.7), where the cosmash comultiplication is defined by

\begin{matrix}Δ : H ⊗ C ⟶^{Δ_{H}⊗Δ_{C}} H ⊗ H ⊗ C ⊗ C ⟶^{1⊗δ′_{C}⊗1} H ⊗ H ⊗ C ⊗ H ⊗ C \\ ⟶^{1⊗σ⊗1} H ⊗ C ⊗ H ⊗ H ⊗ C ⟶^{1⊗m_{H}⊗1} H ⊗ C ⊗ H ⊗ C. \end{matrix}

The morphism z = 1 ⊗ ε_{C} : H#^{c}C ⟶ H is a coalgebra morphism. It induces a D-functor D^{z} : D^{H#^{c}C} ⟶ D^{H} which can be identified with ω : A^{C} ⟶ A.

By Theorem 4.1 we get coend_{D}(ω) = H#^{c}C, the cosmash product defined in D as above.

From Theorem 4.2 we get coend_{D}(id_{A}) = H as a coalgebra but with the new multiplication m̃_{H} : H ⊗ H ⟶ H as defined in Proposition 3.6 by the braid diagram

[figure: δ; δ; δ; δ; m̃; P; Q; P; Q; =; P; Q; H; P; Q; H]

In particular m̃_{H} : H ⊗ H ⟶ H is a morphism of H-comodules under the coadjoint coaction. (This morphism has been studied in [Mj93b] under the notion of transmutation.)

Furthermore we have coend_{C}(ω) = C as coalgebras in A again by Theorem 4.1. By Theorem 5.4 we thus get a canonical coalgebra morphism f : H#^{c}C ⟶

H ˜#^{c}C where the first cosmash product was described above and the second cosmash product comes from the transmutation multiplication on H and the braiding in A = D^{H}. As observed in 4.1.4 these two cosmash products are the same.

With these coalgebras the canonical morphism f is defined as in the proof of Theorem 5.4 by

[figure: ω; ω ⊗ H#^{c}C; δ; ω ⊗ C; ω ⊗ H ⊗ C; δ′ω⊗1; μ; 1⊗f]

with δ = (δ′ω ⊗ 1)μ as above, so that the uniquely determined morphism f under the given identifications is the identity.

Example 5.7. Let H be a coquasitriangular Hopf algebra over the field 𝕂 and let C be an H-comodule coalgebra. Let ω : (M^{H})^{C} ⟶ M^{H} be the forgetful functor. Then

coend(ω) ≅ H#^{c}C.

Proof. Use Proposition 6.4 to show coend(ω) = coend_{M}(ω). This special case can also be derived from [Mj94b] without the use of control categories.

The last example shows that the hidden symmetries as given in 5.1.8 are represented by the Hopf algebra H (with the transmutation multiplication), i.e. H = coend_{M}(id : M^{H} ⟶ M^{H}).

6. Appendix on 𝕂-additive categories and C-categories

Let 𝕂 be a commutative ring and let C := 𝕂-mod be the category of finitely generated projective 𝕂-modules. Then C is a symmetric monoidal 𝕂-abelian category.

Let A be a category with splitting idempotents, i.e. for a morphism f : X ⟶ X with f^{2} = f there are morphisms q : X ⟶ P and j : P ⟶ X with jq = f and qj = 1_{P}. Then q : X ⟶ P is a coequalizer of (1_{X}, f) and q and j are unique up to isomorphisms of P.

Lemma 6.1. Let f : X ⟶ X and g : X ⟶ X be idempotents with fg = gf and splittings (P, q, j) of f and (P′, q′, j′) of g. Let (R, q′′, j′′) be a splitting of the idempotent jgq. Then (R, q′′q, jj′′) is a splitting of fg : X ⟶ X.

Proof. qgj is idempotent since qgjqgj = qgfgj = qggfj = qgjqj = qgj, so a splitting (R, q′′, j′′) exists with qgj = j′′q′′ and q′′j′′ = 1. Then q′′qjj′′ = q′′j′′ = 1_{R} and jj′′q′′q = jqgjq = fgf = ffg = fg.

The following theorem is a generalization of [Sch92a] Lemma 2.2.2.

Theorem 6.2. Let A be a 𝕂-additive category with splitting idempotents. Then A is a C-category.

Proof. Let A be a 𝕂-additive category. Then the following hold for α ∈ 𝕂 and f, g morphisms in A: α(f + g) = αf + αg, α(fg) = (αf)g = f(αg) and α(f ⊕ g) = (αf) ⊕ (αg).

We define the functor (𝕂⊗– : A ⟶ A) := (Id_{A} : A ⟶ A). This is a 𝕂-additive functor. For n ≥ 0 we define (𝕂^{n} ⊗ – : A ⟶ A) := (Id^{n} : A ⟶ A) (so for an object P ∈ A we have 𝕂^{n} ⊗ P = P^{n}) which again is a 𝕂-additive functor.

Let f : 𝕂^{m} ⟶ 𝕂^{n} be a morphism in C. Then f = ∑_{ij} α_{ij}e_{ij} where the e_{ij} are given by the compositions e_{ij} := 𝕂^{m} ⟶^{p_{i}} 𝕂 ⟶^{ι_{j}} 𝕂^{n}, the canonical basis of the matrix space Hom_{𝕂}(𝕂^{m}, 𝕂^{n}).

Since A is additive, we have corresponding natural transformations

ē_{ij}(P) : P^{m} ⟶^{p̄_{i}} P ⟶^{ῑ_{j}} P^{n}.

For f : 𝕂^{m} ⟶ 𝕂^{n} we define f ⊗ P : P^{m} ⟶ P^{n} by f ⊗ P := ∑_{ij} α_{ij}ē_{ij}. Then it is easy to verify, that f ⊗ - : 𝕂^{m} ⊗ – ⟶ 𝕂^{n} ⊗ – is a natural transformation. Furthermore it is easy to see that fg ⊗ - = (f ⊗ -)(g ⊗ -) and id ⊗ - = id. If C^{−} is the full subcategory of C with the objects 𝕂^{n}, then ⊗ : C^{−} × A ⟶ A is a 𝕂-bilinear bifunctor.

Now let X be a finitely generated projective 𝕂-module. Then there are homomorphisms j : X ⟶ 𝕂^{n} and q : 𝕂^{n} ⟶ X for some n ∈ ℕ with qj = 1_{X}. For P ∈ A the morphism jq ⊗ P : 𝕂^{n} ⊗ P ⟶ 𝕂^{n} ⊗ P is an idempotent (jq ⊗ P)^{2} = jq ⊗ P so there is a splitting q ⊗ P : 𝕂^{n} ⊗ P ⟶ X ⊗ P and j ⊗ P : X ⊗ P ⟶ 𝕂^{n} ⊗ P (thus defining q ⊗ P, j ⊗ P and X ⊗ P) with

(j ⊗ P)(q ⊗ P) = jq ⊗ P and (q ⊗ P)(j ⊗ P) = 1_{X⊗P} .

In particular q⊗P : 𝕂^{n}⊗P ⟶ X⊗P is a cokernel of (1−jq)⊗P : 𝕂^{n}⊗P ⟶ 𝕂^{n}⊗P and we have a commutative diagram

[figure: 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; q⊗P; 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; q⊗P; jq⊗P; jq⊗P; =; j⊗P]

Let j : X ⟶ 𝕂^{n}, q : 𝕂^{n} ⟶ X and j′ : X ⟶ 𝕂^{m} and q′ : 𝕂^{m} ⟶ X be two choices of a representation of X as a direct summand of a free 𝕂-modules. Then the following diagram commutes

[figure: 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; q⊗P; 𝕂^{m} ⊗ P; 𝕂^{m} ⊗ P; (1−j′q′)⊗P; [X ⊗ P]; q′⊗P; 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; q⊗P; jq⊗P; jq⊗P; 1_{X⊗P}; j′q⊗P; j′q⊗P; ≅; jq′⊗P; jq′⊗P; ≅]

and the isomorphisms between X ⊗ P and [X ⊗ P] (the corresponding object for the second representation of X) arise from this and the symmetric diagram with (j, q, 𝕂^{n}) and (j′, q′, 𝕂^{m}) interchanged.

Since the morphism (1 − jq) ⊗ P is a natural transformation it commutes with morphisms f : P ⟶ Q and thus induces uniquely determined morphisms on the cokernels X ⊗ f : X ⊗ P ⟶ X ⊗ Q, so that the following diagram commutes:

[figure: 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; 𝕂^{n} ⊗ Q; 𝕂^{n} ⊗ Q; (1−jq)⊗Q; X ⊗ Q.; 𝕂^{n}⊗f; 𝕂^{n}⊗f; X⊗f]

In particular one sees that X ⊗ – : A ⟶ A is a 𝕂-additive functor.

Now let f : X ⟶ X′ be a homomorphism. Choose j : X ⟶ 𝕂^{n}, q : 𝕂^{n} ⟶ X, j′ : X′ ⟶ 𝕂^{m}, and q′ : 𝕂^{m} ⟶ X′ with qj = 1_{X} and q′j′ = 1_{X′}. Then by the cokernel property there is a unique morphism f ⊗ P which makes the following diagram commutative:

[figure: 𝕂^{n} ⊗ P; 𝕂^{n} ⊗ P; (1−jq)⊗P; X ⊗ P; 𝕂^{m} ⊗ P; 𝕂^{m} ⊗ P; (1−j′q′)⊗P; X′ ⊗ P.; j′fq⊗P; j′fq⊗P; f⊗P]

By the universal property of the cokernel we get, that f ⊗ P : X ⊗ P ⟶ X′ ⊗ P is a natural transformation. Furthermore we get fg ⊗- = (f ⊗-)(g ⊗-) and id⊗- = id. In particular we have that ⊗ : C × A ⟶ A is a 𝕂-bilinear bifunctor.

We sketch the construction of the associativity morphism β : (X ⊗ Y) ⊗ P ⟶ X ⊗ (Y ⊗ P). We first observe that 𝕂^{n} ⊗ (𝕂^{m} ⊗ P) ≅ P^{nm} ≅ (𝕂^{n} ⊗ 𝕂^{m}) ⊗ P, which defines β in the free case. Now consider representations q : 𝕂^{n} ⟶ X, j : X ⟶ 𝕂^{n} and q′ : 𝕂^{m} ⟶ Y, j′ : Y ⟶ 𝕂^{m}. Then

f := jq ⊗ 1 ⊗ 1 : 𝕂^{n} ⊗ 𝕂^{m} ⊗ P ⟶ X ⊗ (𝕂^{n} ⊗ P) ⟶ 𝕂^{n} ⊗ 𝕂^{m} ⊗ P
g := 1 ⊗ j′q′ ⊗ 1 : 𝕂^{n} ⊗ 𝕂^{m} ⊗ P ⟶ 𝕂^{n} ⊗ (Y ⊗ P) ⟶ 𝕂^{n} ⊗ 𝕂^{m} ⊗ P

are idempotents with fg = (jq ⊗ 1 ⊗ 1)(1 ⊗ j′q′ ⊗ 1) = jq ⊗ j′q′ ⊗ 1 = gf. So we can apply Lemma 6.1. Since (1 ⊗ q′ ⊗ 1)(q ⊗ 1 ⊗ 1) : 𝕂^{n} ⊗ 𝕂^{m} ⊗ P ⟶ X ⊗ (𝕂^{m} ⊗ P) ⟶ X ⊗ (Y ⊗ P) and (j ⊗ 1 ⊗ 1)(1 ⊗ j′ ⊗ 1) : X ⊗ (Y ⊗ P) ⟶ X ⊗ (𝕂^{m} ⊗ P) ⟶ 𝕂^{n} ⊗ 𝕂^{m} ⊗ P is a splitting of (j ⊗ 1 ⊗ 1)g(q ⊗ 1 ⊗ 1), we get a splitting 𝕂^{n} ⊗ 𝕂^{m} ⊗ P ⟶ X ⊗ (Y ⊗ P) ⟶ 𝕂^{n} ⊗ 𝕂^{m} ⊗ P which defines up to unique isomorphism the object (X ⊗ Y) ⊗ P, so β : (X ⊗ Y) ⊗ P ≅ X ⊗ (Y ⊗ P). By the uniqueness this isomorphism is a natural transformation and coherent.

Hence A is a C-category.

Proposition 6.3. Let A and B be 𝕂-additive categories with splitting idempotents with the C-structure derived in Theorem 6.2 and let ω : B ⟶ A be a 𝕂-additive functor. Then ω is a C-functor.

If ω and ω′ are 𝕂-additive functors from B to A and φ : ω ⟶ ω′ is a natural transformation then φ is a C-morphism.

Proof. An object in C is given as above by the splitting j : X ⟶ 𝕂^{n} and q : 𝕂^{n} ⟶ X. Then the tensor product with X is given as the splitting j⊗1 : X ⊗P ⟶ 𝕂^{n}⊗P with q ⊗1 : 𝕂^{n} ⊗P ⟶ X ⊗P. We define ξ : ω(X ⊗P) ≅ X ⊗ω(P) as the uniquely defined morphism by the isomorphic splittings

[figure: ω(X ⊗ P); X ⊗ ω(P); ξ; ω(𝕂^{n} ⊗ P); 𝕂^{n} ⊗ ω(P); ≅]

We leave it to the reader to check naturality and coherence of ξ.

The following commutative diagram shows that φ is a C-morphism

[figure: ω(X ⊗ P); X ⊗ ω(P); ξ; ω(𝕂^{n} ⊗ P); 𝕂^{n} ⊗ ω(P); ≅; ω′(X ⊗ P); X ⊗ ω′(P); ω′(𝕂^{n} ⊗ P); 𝕂^{n} ⊗ ω′(P); ω(P^{n}) =; ω′(P^{n}) =; = ω(P)^{n}; = ω′(P)^{n}.; φ(X⊗P); X⊗φ(P); φ(P^{n}); φ(P)^{n}]

Theorem 6.4. Let C be a full monoidal subcategory of M = 𝕂-Mod and A be a C-category. Let ω, ω′ : A ⟶ C be C-functors. Then every natural transformation φ : ω ⟶ ω′ is a C-morphism.

Proof. Since

[figure: ω(𝕂 ⊗ P); ω′(𝕂 ⊗ P); φ(𝕂⊗P); ω(P); ω′(P); φ(P); 𝕂 ⊗ ω(P); 𝕂 ⊗ ω′(P); 1⊗φ(P); ω(π_{P}); λ_{P}^{−1}; ω′(π_{P}); λ_{P}^{−1}]

commutes and the vertical morphisms are by coherence the canonical morphisms ξ : ω(𝕂 ⊗ P) ⟶ 𝕂 ⊗ ω(P) resp. ξ′ : ω′(𝕂 ⊗ P) ⟶ 𝕂 ⊗ ω′(P). For x ∈ X ∈ C let f_{x} : 𝕂 ⟶ X be the homomorphism with f_{x}(1) = x. Then the following diagram commutes, possibly with the exception of the front face

[figure: ω(𝕂 ⊗ P); ω′(𝕂 ⊗ P); φ(𝕂⊗P); 𝕂 ⊗ ω(P); 𝕂 ⊗ ω′(P); 1_{𝕂}⊗φ(P); ω(X ⊗ P); ω′(X ⊗ P); φ(X⊗P); X ⊗ ω(P); X ⊗ ω′(P).; 1_{X}⊗φ(P); ξ; ξ; ξ′; ξ′; ω(f_{x}⊗1); f_{x}⊗1; ω′(f_{x}⊗1); f_{x}⊗1]

So for q ∈ ω(P) we get φ(X ⊗ P)ξ^{−1}(x ⊗ q) = φ(X ⊗ P)ξ^{−1}(f_{x} ⊗ 1)(1 ⊗ q) = ξ′^{−1}(1_{X} ⊗ φ(P))(f_{x} ⊗ 1)(1 ⊗ q) = ξ′^{−1}(1_{X} ⊗ φ(P))(x ⊗ q). This holds for all x ∈ X and all q ∈ ω(P) so that φ(X ⊗ P)ξ^{−1} = (ξ′^{−1} ⊗ 1)(1_{X} ⊗ φ(P)) and φ is a Cmorphism.

References

[Ba68] Higman Bass: Algebraic K-theory. W. A. Benjamin, New York – Amsterdam 1968.

[DM82] P. Deligne, J. S. Milne: Tannakian Categories. In: Hodge Cycles, Motives and Shimura Varieties. Springer LN Math 900 (1982) 101-228.

[Dr86] V. G. Drinfeld: Quantum groups. In ICM Proceedings Berkeley, 1986.

[Dr89] V. G. Drinfeld: Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations. In: A. A. Belavin: Problems of modern QFT, Springer 1989.

[KT92] Christian Kassel, Vladimir Turaev: Double Construction for Monoidal Categories. Publ. de l’Inst. Rech. Math. Av. 507/P-294 (1992).

[LT91] Richard G. Larson, Jacob Towber: Two Dual Classes of Bialgebras Related to ‘Quantum Groups’ and ‘Quantum Lie Algebras’. Communications in Algebra 19, (1991), 3295-3345.

[ML71] Saunders Mac Lane: Categories for the Working Mathematician. Springer-Verlag New York, Heidelberg, Berlin, 1971.

[Mj90] Shahn Majid: Rank of Quantum Groups and Braided Groups in Dual Form. DAMTP/90-44, Euler Inst. Programme on Quantum Groups, Leningrad, 1990.

[Mj93a] Shahn Majid: Transmutation theory and rank for quantum braided groups. Math. Proc. Camb. Phil. Soc. 113 (1993) 45-70.

[Mj93b] Shahn Majid: Braided groups. J. of Pure and Appl. Alg. 86 (1993) 187-221.

[Mj94a] Shahn Majid: Algebras and Hopf algebras in braided categories. In: Advances in Hopf Algebras. LN pure and appl. math. 158 (1994) 55-105.

[Mj94b] Shahn Majid: Cross Products by Braided Groups and Bosonisation. J. of Algebra 163 (1994) 165-190.

[Pa77] Bodo Pareigis: Non-additive Ring and Module Theory II. C-Categories, C-Functors and C-Morphisms. Publ. Math. (Debrecen) 24 (1977) 351-361.

[Pa78] Bodo Pareigis: Non-additive Ring and Module Theory III. Morita equivalences. Publ. Math. (Debrecen) 25 (1978) 177-186.

[Pa81] Bodo Pareigis: A Non-Commutative Non-Cocommutative Hopf Algebra in “Nature”. J. Alg. 70 (1981) 356-374.

[Pa93] Bodo Pareigis: Endomorphism Bialgebras of Diagrams and of Non-Commutative Algebras and Spaces. In: Advances in Hopf Algebras. LN pure and appl. math. 158 (1994) 153-186.

[Pa95] Bodo Pareigis: On Braiding and Dyslexia. J. Alg. 171 (1995) 413-425.

[Sch92a] Peter Schauenburg: Tannaka Duality for Arbitrary Hopf Algebras. Algebra Berichte No 66, Verlag Reinhard Fischer, München, 1992.

[Sch92b] Peter Schauenburg: On Coquasitriangular Hopf Algebras and the Quantum Yang-Baxter Equation. Algebra Berichte No 67, Verlag Reinhard Fischer, München, 1992.

[Se79] Mohammed Seoud: Kombinatorische Behandlung von Kohärenzfragen in monoidalen Kategorien. Dissertation Universität München 1979.

[Sw69] Moss E. Sweedler: Hopf algebras. W. A. Benjamin, New York 1969.

[Ul89] Karl-H. Ulbrich: Determining Bialgebras from k-Valued Monoidal Functors. preprint 1989, 8pp.

[Ul90] Karl-H. Ulbrich: Tannakian Categories for Non-Commutative Hopf Algebras. Israel J. Math. 72 (1990).

[Ye90] David N. Yetter: Quantum groups and representations of monoidal categories. Math. Proc. Camb. Phil. Soc. 108 (1990) 261-290.

Mathematisches Institut der Universität München, Germany E-mail address: pareigis@rz.mathematik.uni-muenchen.de