Theory and Applications of Categories, Vol. 22, No. 4, 2009, pp. 77–96.

ON ENDOMORPHISM ALGEBRAS OF SEPARABLE MONOIDAL FUNCTORS

BRIAN DAY AND CRAIG PASTRO

Abstract. We show that the (co)endomorphism algebra of a sufficiently separable “fibre” functor into Vect_{k}, for k a field of characteristic 0, has the structure of what we call a “unital” von Neumann core in Vect_{k}. For Vect_{k}, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group.

1. Introduction

Let 𝒞 = (𝒞 , ⊗, I, c) be a braided (or even symmetric) monoidal category. Recall that an algebra in 𝒞 is an object A ∈ 𝒞 equipped with a multiplication μ : A ⊗ A → A and a unit η : I → A satisfying μ_{3} = μ(1 ⊗ μ) = μ(μ ⊗ 1) : A^{⊗3} → A (associativity) and μ(η ⊗ 1) = 1 = μ(1 ⊗ η) : A → A (unit conditions). Dually, a coalgebra in 𝒞 is an object C ∈ 𝒞 equipped with a comultiplication δ : C → C ⊗ C and a counit ϵ : C → I satisfying δ_{3} = (1 ⊗ δ)δ = (δ ⊗ 1)δ : C → C^{⊗3} (coassociativity) and (ϵ ⊗ 1)δ = 1 = (1 ⊗ ϵ)δ : C → C (counit conditions).

A very weak bialgebra in 𝒞 is an object A ∈ 𝒞 with both the structure of an algebra and a coalgebra in 𝒞 related by the axiom

δμ = (μ ⊗ μ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A → A ⊗ A.

For example, when 𝒞 = Vect_{k}, any k-bialgebra or weak k-bialgebra is a very weak bialgebra in this sense.

We note briefly that, if A is such a structure, but has no unit or counit, we simply call A a semibialgebra, or core for short. This minimal structure on A is then called a von Neumann core in 𝒞 if it also is equipped with an endomorphism S : A → A in 𝒞 satisfying the axiom

μ_{3}(1 ⊗ S ⊗ 1)δ_{3} = 1 : A → A.

The first author gratefully acknowledges partial support of an Australian Research Council grant, while the second author is partially supported by GCOE, Kyoto University. Parts of this work were completed while the second author was a Ph.D. student at Macquarie University, Australia, and during that time was gratefully supported by an international Macquarie University Research Scholarship and a Scott Russell Johnson Memorial Scholarship. The authors would like to thank Ross Street for several helpful comments.

Received by the editors 2007-12-21 and, in revised form, 2009-04-17.

Transmitted by Jean-Louis Loday. Published on 2009-04-20. 2000 Mathematics Subject Classification: 18D99, 16B50. Key words and phrases: separable fibre functor, Tannaka reconstruction, bialgebra, von Neumann

core.

\begin{matrix}c \\ ◯ \end{matrix} Brian Day and Craig Pastro, 2009. Permission to copy for private use granted.

A von Neumann regular semigroup is precisely a von Neumann core in Set, while the free k-vector space on it is a special type of von Neumann core in Vect_{k}. However, within this article we shall always suppose that A has both a unit and a counit. For example, when 𝒞 = Vect_{k}, a Hopf k-algebra or a weak Hopf k-algebra is a von Neumann core in this somewhat stronger sense.

Since groups A in Set are characterized by the (stronger) axiom

1 ⊗ η = (1 ⊗ μ)(1 ⊗ S ⊗ 1)δ_{3} : A → A ⊗ A,  (†)

a very weak bialgebra A satisfying (†), in the general 𝒞, will be called a unital von Neumann core in 𝒞. Such a unital von Neumann core A always has a left inverse to the “fusion” operator [9]

(1 ⊗ μ)(δ ⊗ 1) : A ⊗ A → A ⊗ A,

namely

(1 ⊗ μ)(1 ⊗ S ⊗ 1)(δ ⊗ 1) : A ⊗ A → A ⊗ A.

Any Hopf algebra in 𝒞 satisfies the stronger axiom (†), but a weak Hopf algebra does not necessarily do so. In this article we are mainly interested in producing a unital von Neumann core, namely End^{∨}U, associated to a certain type of split monoidal functor U into Vect_{k}. It seems unlikely that all unital von Neumann cores in Vect_{k} may be reproduced as such.

We will tacitly assume throughout the article that the ground category [8] is Vect = Vect_{k}, for k a field of characteristic 0, so that the categories and functors considered here are all k-linear (although any reasonable category [𝒟, Vect] of parameterized vector spaces would suffice). We denote by Vect_{f} the full subcategory of Vect consisting of the finite dimensional vector spaces, and we further suppose that 𝒞 = (𝒞 , ⊗, I, c) is a braided monoidal category with a “fibre” functor

U : 𝒞 → Vect,

with both a monoidal structure (U, r, r_{0}) and a comonoidal structure (U, i, i_{0}), which need not be inverse to one another. We call U separable^{1} if ri = 1 and i_{0}r_{0} = dim(UI) · 1; i.e., for all A, B ∈ 𝒞, the diagrams

[figure: U(A ⊗ B); UA ⊗ UB; U(A ⊗ B); i; r; 1; k; UI; k; r_{0}; i_{0}; dim UI·1]
(r ⊗ 1)(1 ⊗ i) = ir : UA ⊗ U(B ⊗ C) → U(A ⊗ B) ⊗ UC, and
(1 ⊗ r)(i ⊗ 1) = ir : U(A ⊗ B) ⊗ UC → UA ⊗ U(B ⊗ C)

1Strictly, we should also require the conditions (cf. [1, 10])

in order for U to be called “separable”, but we do not need these here.

commute.

First we produce an algebra structure (μ, η) on

End^{∨}U = ∫ ^{C} U(C)^{∗} ⊗ UC

using the monoidal and comonoidal structures on U. Secondly, we suppose that 𝒞 has a suitable small generating set 𝒜 of objects, and produce a coalgebra structure (δ, ϵ) on End^{∨}U when each value UA, A ∈ 𝒜, is finite dimensional. Finally, we assume that each A ∈ 𝒜 has a ⊗-dual A^{∗} which also lies in 𝒜, and that U is equipped with an isomorphism

U(A^{∗}) ≅ U(A)^{∗}

for all A ∈ 𝒜. This isomorphism should be suitably related to the evaluation and coevaluation maps of 𝒞 and Vect_{f} which then allows us to define a natural non-degenerate form

U(A^{∗}) ⊗ UA → k.

This last assumption is sufficient to provide End^{∨}U with an automorphism S so that it becomes a unital von Neumann core in the above sense whenever (U, r, r_{0}) is a braided monoidal functor.

By way of examples, we note that many separable monoidal functors are constructable from separable monoidal categories, i.e., from monoidal categories 𝒞 for which the tensor product map

⊗ : 𝒞 (A, B) ⊗ 𝒞 (C, D) → 𝒞 (A ⊗ C, B ⊗ D)

is a naturally split epimorphism (as is the case for some finite cartesian products such as Vect_{f}^{n}). A closely related source of examples is the notion of a weak dimension functor on 𝒞 (cf. [6]); this is a comonoidal functor

(d, i, i_{0}) : 𝒞 → Set_{f}

for which the comonoidal transformation components

i = i_{C,D} : d(C ⊗ D) → dC × dD

are injective functions, while the unique map i_{0} : dI → 1 is surjective. Various examples are described at the conclusion of the paper.

We suppose the reader is familiar to some extent with the standard Tannaka reconstruction problem when restricted to the case of U strong monoidal (see [7] for example).

2. The very weak bialgebra End^{∨}U

If 𝒞 is a (k-linear) monoidal category and

U : 𝒞 → Vect

has a monoidal structure (U, r, r_{0}) and a comonoidal structure (U, i, i_{0}), then End^{∨}U, when it exists, has an associative and unital k-algebra structure whose multiplication μ is the composite map

[figure: ∫ ^{C}U(C)^{∗} ⊗ UC ⊗ ∫ ^{D}U(D)^{∗} ⊗ UD; ∫ ^{C,D}U(C)^{∗} ⊗ U(D)^{∗} ⊗ UC ⊗ UD; ≅; ∫ ^{C,D}(UC ⊗ UD)^{∗} ⊗ UC ⊗ UD; can; ∫ ^{C,D}U(C ⊗ D)^{∗} ⊗ U(C ⊗ D); ∫ i^{∗}⊗r; ∫ ^{B}U(B)^{∗} ⊗ UB; _{∫ ⊗}; μ]

while the unit η is given by

[figure: k; k^{∗} ⊗ k; ≅; UI^{∗} ⊗ UI.; i_{0}^{∗}⊗r_{0}; ∫ ^{C}U(C)^{∗} ⊗ UC; _{copr_{C=I}}; η]

The associativity and unit axioms for (End^{∨}U, μ, η) now follow directly from the corresponding associativity and unit axioms for (U, r, r_{0}) and (U, i, i_{0}). An augmentation ϵ is given by

[figure: U(D)^{∗} ⊗ UD; ∫ ^{C}U(C)^{∗} ⊗ UC; ^{copr^{C=D}}; k; ϵ; e]

in Vect, where e denotes evaluation in Vect.

We also observe that the coend

End^{∨}U = ∫ ^{C} U(C)^{∗} ⊗ UC

actually exists in Vect if 𝒞 contains a small full subcategory 𝒜 with the property that the family

{Uf : UA → UC | f ∈ 𝒞 (A, C), A ∈ 𝒜 }

is epimorphic in Vect for each object C ∈ 𝒞. In fact, we shall use the stronger condition that the maps

α_{C} : ∫ ^{A∈𝒜} 𝒞 (A, C) ⊗ UA → UC

should be isomorphisms, not just epimorphisms. This stronger condition implies that we can effectively replace ∫^{C∈𝒞} by ∫^{A∈𝒜} since by the Yoneda lemma

∫ ^{C} U(C)^{∗} ⊗ UC ≅ ∫ ^{C} U(C)^{∗} ⊗ (∫ ^{A} 𝒞 (A, C) ⊗ UA)
≅ ∫ ^{A} U(A)^{∗} ⊗ UA.

If we furthermore ask that each value UA be finite dimensional for A in 𝒜, then

End^{∨}U ≅ ∫ ^{A∈𝒜} U(A)^{∗} ⊗ UA

is canonically a k-coalgebra with counit the augmentation ϵ, and comultiplication δ given by

[figure: U(A)^{∗} ⊗ UA; ∫ ^{A}U(A)^{∗} ⊗ UA; copr; ∫ ^{A}U(A)^{∗} ⊗ UA ⊗ ∫ ^{A}U(A)^{∗} ⊗ UA; δ; U(A)^{∗} ⊗ UA ⊗ U(A)^{∗} ⊗ UA,; 1⊗n⊗1; copr ⊗ copr]

where n denotes the coevaluation morphism in Vect_{f}.

2.1. Proposition. If U is separable then End^{∨}U satisfies the k-bialgebra axiom expressed by the commutativity of

[figure: End^{∨}U ⊗ End^{∨}U; End^{∨}U; μ; End^{∨}U ⊗ End^{∨}U.; δ; (End^{∨}U)^{⊗4}; δ⊗δ; (End^{∨}U)^{⊗4}; 1⊗c⊗1; μ⊗μ]

Proof. Let ℬ denote the monoidal full subcategory of 𝒞 generated by 𝒜 (we will essentially replace 𝒞 by this small category ℬ). Then, for all C, D in ℬ, we have, by induction on the tensor lengths of C and D, that U(C ⊗ D) is finite dimensional since it is a retract of UC ⊗ UD. Moreover, we have

∫ ^{A∈𝒜} U(A)^{∗} ⊗ UA ≅ ∫ ^{B∈ℬ} U(B)^{∗} ⊗ UB

by the Yoneda lemma, since the natural transformation

α = α_{B} : ∫ ^{A∈𝒜} 𝒞 (A, B) ⊗ UA → UB

is an isomorphism for all B ∈ ℬ. Since ri = 1, the triangle

[figure: k; (UC ⊗ UD) ⊗ (UC ⊗ UD)^{∗}; U(C ⊗ D) ⊗ U(C ⊗ D)^{∗},; r⊗i^{∗}; n; n]

commutes in Vect_{f}, where n denotes the coevaluation maps. The asserted bialgebra axiom then holds on End^{∨}U since it reduces to the following diagram on filling in the definitions of μ and δ (where, for the moment, we have dropped the symbol “⊗”):

[figure: UC U(C)^{∗} UD U(D)^{∗}; UC UD U(C)^{∗} U(D)^{∗}; UC UD (UC UD)^{∗}; U(C D) U(C D)^{∗}; UC (UC U(C)^{∗}) U(C)^{∗} UD (UD U(D)^{∗}) U(D)^{∗}; UC UD UC UD U(C)^{∗} U(D)^{∗} U(C)^{∗} U(D)^{∗}; UC UD UC UD (UC UD)^{∗} (UC UD)^{∗}; U(C D) U(C D) U(C D)^{∗} U(C D)^{∗}; ≅; ≅; r i^{∗}; ≅; ≅; r r i^{∗} i^{∗}; 1 n 1 1 n 1; 1 n 1; 1 n 1]

for all C, D ∈ ℬ.

Notably the bialgebra axiom expressed by the commutativity of

[figure: End^{∨}U ⊗ End^{∨}U; End^{∨}U; μ; k; ϵ; ϵ⊗ϵ]

does not hold in general, while the form of the axiom expressed by

[figure: k; End^{∨}U; η; End^{∨}U ⊗ End^{∨}U; δ; η⊗η]

holds only if η = r_{0} ⊗ i_{0}^{∗}. Also ϵη = dim UI · 1 for U separable.

The single k-bialgebra axiom established in the above proposition implies that the “fusion” operator (1 ⊗ μ)(δ ⊗ 1) : A ⊗ A → A ⊗ A satisfies the fusion equation (see [9] for details).

The k-linear dual of End^{∨}U is of course

[ ∫ ^{C} U(C)^{∗} ⊗ U(C), k] ≅ ∫_{C}[U(C)^{∗}, U(C)^{∗}]

which is the endomorphism k-algebra of the functor

U(−)^{∗} : 𝒞^{op} → Vect.

If ob 𝒜 is finite, so that

∫ ^{A} U(A)^{∗} ⊗ UA

is finite dimensional, then

∫_{C}[U(C)^{∗}, U(C)^{∗}] ≅ ∫_{A}[U(A)^{∗}, U(A)^{∗}]

is also a k-coalgebra.

3. The unital von Neumann core End^{∨}U

We now take 𝒞 = (𝒞 , ⊗, I, c) to be a braided monoidal category and 𝒜 ⊂ 𝒞 to be a small full subcategory of 𝒞 for which the monoidal and comonoidal functor U : 𝒞 → Vect induces

U : 𝒜 → Vect_{f}

on restriction to 𝒜. We suppose that 𝒜 is such that

• the identity I of ⊗ lies in 𝒜, and each object of A ∈ 𝒜 has a ⊗-dual A^{∗} lying in 𝒜.

With respect to U, we suppose 𝒜 has the properties

• “U-irreducibility”: 𝒜 (A, B) ≠ 0 implies dim UA = dim UB for all A, B ∈ 𝒜,

• “U-density”: the canonical map

α_{C} : ∫ ^{A∈𝒜} 𝒞 (A, C) ⊗ UA → UC

is an isomorphism for all C ∈ 𝒞,

• “U-trace”: each object of 𝒜 has a U-trace in 𝒞 (I, I), where by U-trace of A ∈ 𝒜 we mean an isomorphism d(A) in 𝒞 (I, I) such that the following two diagrams commute.

We shall assume dim UI ≠ 0 so that the latter assumption implies dim UA ≠ 0, for all A ∈ 𝒜.

[figure: I; A ⊗ A^{∗}; n; A^{∗} ⊗ A; c; I; e; d(A); k; UI; r_{0}; UI; dim UI·U(d(A)); k; dim UA; r_{0}]

We require also a natural isomorphism

u = u_{A} : U(A^{∗}) ⟶^{≅} U(A)^{∗}

such that

[figure: k; UI; r_{0}; U(A ⊗ A^{∗}); Un; UA ⊗ U(A)^{∗}; n; UA ⊗ U(A^{∗}); 1⊗u^{−1}; r]  (n, r, r_{0})

commutes, and

[figure: U(A^{∗} ⊗ A); UI; Ue; k; i_{0}; U(A^{∗}) ⊗ UA; i; U(A)^{∗} ⊗ UA; u⊗1; e]  (e, i, i_{0})

commutes. This means that U “preserves duals” when restricted to 𝒜.

An endomorphism

σ : End^{∨}U → End^{∨}U

may be defined by components

[figure: U(A)^{∗} ⊗ UA; U(A^{∗})^{∗} ⊗ U(A^{∗}),; σ_{A}; ∫ ^{A}U(A)^{∗} ⊗ UA; copr; ∫ ^{A}U(A)^{∗} ⊗ UA; copr; σ]

each σ_{A} being given by commutativity of

[figure: U(A)^{∗} ⊗ UA; U(A^{∗})^{∗} ⊗ U(A^{∗}); σ_{A}; U(A)^{∗} ⊗ U(A)^{∗∗}; 1⊗ρ; U(A^{∗}) ⊗ U(A^{∗})^{∗},; u^{−1}⊗u^{∗}; c]

where ρ denotes the canonical isomorphism from a finite dimensional vector space to its double dual. Clearly each component σ_{A} is invertible.

3.1. Theorem. Let 𝒞, 𝒜, and U be as above, and suppose that U is braided and separable as a monoidal functor. Then there is an automorphism S on End^{∨}U such that (End^{∨}U, μ, η, δ, ϵ, S) is a unital von Neumann core in Vect_{k}.

Proof. A family of maps {S_{A} | A ∈ 𝒜 } is defined by

S_{A} = dim UI · (dim UA)^{−1} · σ_{A}.

Then, by the U-irreducibility assumption on the category 𝒜, this family induces an automorphism S on the coend

End^{∨}U ≅ ∑_{n=1}^{∞} ∫ ^{A∈𝒜_{n}} U(A)^{∗} ⊗ UA,

where 𝒜_{n} is the full subcategory of 𝒜 determined by {A | dim UA = n}. We now take S to be the prospective core endomorphism on End^{∨}U and check that

1 ⊗ η = (1 ⊗ μ)(1 ⊗ S ⊗ 1)δ_{3}.

From the definition of μ and δ, we require commutativity of the exterior of the following diagram (where, again, we have dropped the symbol “⊗”):

[figure: U(A)^{∗} UA U(A)^{∗} UA U(A)^{∗} UA; U(A)^{∗} UA U(A)^{∗} UA; U(A)^{∗} UA; U(A)^{∗} UA k; U(A)^{∗} UA U(A^{∗})^{∗} U(A^{∗}) U(A)^{∗} UA; U(A)^{∗} UA U(A)^{∗∗} U(A)^{∗} U(A)^{∗} UA; U(A)^{∗} UA U(A)^{∗∗} U(A)^{∗} U(A)^{∗} UA; U(A)^{∗} UA U(A^{∗})^{∗} U(A^{∗}) U(A)^{∗} UA; ≅; U(A)^{∗} UA (U(A^{∗}) UA)^{∗} U(A^{∗}) UA; ≅; U(A)^{∗} UA U(A^{∗} A)^{∗} U(A^{∗} A); 1 1 i^{∗} r; U(A)^{∗} UA ∫ ^{B}U(B)^{∗} UB; 1 1 copr; U(A)^{∗} UA UA U(A)^{∗} U(A)^{∗} UA; 1 1 n 1 1; U(A)^{∗} UA UA U(A)^{∗} U(A)^{∗} UA; 1 1 1 c 1; 1 1 c 1 1; 1 n 1 1 1; 1 n 1; ≅; 1 1 S_{A} 1 1; dim UI·(dim UA)^{−1}·(1 1 e^{∗} 1 1); 1 1 η; 1 1 u^{∗} u^{−1} 1 1; 1 1 1 c 1; (1); (2); (3)]

The region labelled by (1) commutes on composition with 1 ⊗ n ⊗ 1 since

[figure: k; UA ⊗ U(A)^{∗}; n; UA ⊗ UA ⊗ U(A)^{∗} ⊗ U(A)^{∗}; 1⊗n⊗1; UA ⊗ UA ⊗ U(A)^{∗} ⊗ U(A)^{∗}; 1⊗1⊗c; UA ⊗ U(A)^{∗} ⊗ UA ⊗ U(A)^{∗}; 1⊗c⊗1; UA ⊗ U(A)^{∗}; n; n⊗1⊗1]

commutes (choose a basis for UA). The region labelled by (2) now commutes by inspection of

[figure: UA ⊗ U(A)^{∗}; UA ⊗ UA ⊗ U(A)^{∗} ⊗ U(A)^{∗}; 1⊗n⊗1; UA ⊗ UA ⊗ U(A)^{∗} ⊗ U(A)^{∗}; 1⊗1⊗c; UA ⊗ U(A)^{∗} ⊗ UA ⊗ U(A)^{∗}; 1⊗c⊗1; UA ⊗ U(A)^{∗∗} ⊗ U(A)^{∗} ⊗ U(A)^{∗}.; 1⊗e^{∗}⊗1; UA ⊗ U(A)^{∗∗} ⊗ U(A)^{∗} ⊗ U(A)^{∗}; 1⊗1⊗c; UA ⊗ U(A)^{∗} ⊗ U(A)^{∗∗} ⊗ U(A)^{∗}; 1⊗c⊗1; UA ⊗ U(A^{∗}) ⊗ U(A^{∗})^{∗} ⊗ U(A)^{∗}; UA ⊗ U(A^{∗})^{∗} ⊗ U(A^{∗}) ⊗ U(A)^{∗}; 1⊗u^{−1}⊗u^{∗}⊗1; 1⊗c⊗1; 1⊗σ_{A}⊗1; 1⊗1⊗ρ⊗1; 1⊗ρ⊗1⊗1]

From the definition of the U-trace d(A) of A ∈ 𝒜, we have that

[figure: k; k; dim UI·(dim UA)^{−1}; UI; r_{0}; UI; r_{0}; U(d(A)^{−1})]

commutes, so that the exterior of

[figure: k; k; dim UI·(dim UA)^{−1}; UA ⊗ U(A)^{∗}; n; UA ⊗ U(A)^{∗}; 1⊗u^{−1}; U(A ⊗ A^{∗}); r; UI; r_{0}; UI; U(d(A)^{−1}); Un; r_{0}; (n,r,r_{0})]

commutes.

Finally, the region labelled by (3) commutes on examination of the following diagram

[figure: k^{∗} ⊗ k; k^{∗} ⊗ UA ⊗ U(A)^{∗}; k^{∗} ⊗ U(A)^{∗} ⊗ UA; (U(A)^{∗} ⊗ UA)^{∗} ⊗ U(A)^{∗} ⊗ UA; (U(A^{∗}) ⊗ UA)^{∗} ⊗ U(A^{∗}) ⊗ UA; U(A^{∗} ⊗ A)^{∗} ⊗ U(A^{∗} ⊗ A); ∫ ^{B}U(B)^{∗} ⊗ UB; k^{∗} ⊗ U(A^{∗}) ⊗ UA; 1⊗u^{−1}⊗1; U(I)^{∗} ⊗ U(A^{∗} ⊗ A); i_{0}^{∗}⊗r; U(e)^{∗}⊗1; k^{∗} ⊗ UA ⊗ U(A^{∗}); 1⊗1⊗u^{−1}; U(I)^{∗} ⊗ U(A ⊗ A^{∗}); i_{0}^{∗}⊗r; U(I)^{∗} ⊗ UI; U(I)^{∗} ⊗ UI; U(I)^{∗} ⊗ UI; 1; copr; i_{0}^{∗}⊗r_{0}; copr; 1⊗dim UI·(dim UA)^{−1}·n; 1⊗c; e^{∗}⊗1⊗1; (u⊗1)^{∗}⊗(u^{−1}⊗1); i^{∗}⊗r; copr; 1⊗c; 1⊗Uc; 1⊗Ue; 1⊗U(d(A)^{−1}); 1⊗Un; (n,r,r_{0}); (∗); (e,i,i_{0})]

whose commutativity depends on the hypothesis that (U, r, r_{0}) is braided monoidal in order for

[figure: UA ⊗ U(A^{∗}); U(A^{∗}) ⊗ UA; c; U(A^{∗} ⊗ A); r; U(A ⊗ A^{∗}); r; Uc; (∗)]

to commute.

4. The fusion operator

The unital von Neumann axiom on End^{∨}U implies that the fusion operator

f = (1 ⊗ μ)(δ ⊗ 1) : End^{∨}U ⊗ End^{∨}U → End^{∨}U ⊗ End^{∨}U

has a left inverse, namely g = (1 ⊗ μ)(1 ⊗ S ⊗ 1)(δ ⊗ 1). For this we consider the following diagram.

[figure: End^{∨}U ⊗ End^{∨}U; (End^{∨}U)^{⊗3}; End^{∨}U ⊗ End^{∨}U; (End^{∨}U)^{⊗3}; End^{∨}U ⊗ End^{∨}U; (End^{∨}U)^{⊗4}; (End^{∨}U)^{⊗4}; (End^{∨}U)^{⊗3}; (End^{∨}U)^{⊗3}; δ⊗1; 1⊗η⊗1; δ_{3}⊗1; 1; 1⊗μ; 1⊗δ⊗1; δ⊗1⊗1; δ⊗1; 1⊗1⊗μ; 1⊗S⊗1⊗1; 1⊗S⊗1; 1⊗μ; 1⊗μ⊗1; 1⊗1⊗μ; 1⊗μ]

In particular f = (1 ⊗ μ)(δ ⊗ 1) is a partial isomorphism, i.e., fgf = f and gfg = g.

5. Examples of separable monoidal functors in the present context

Unless otherwise indicated, categories, functors, and natural transformations shall be k-linear, for k a field of characteristic 0.

For these examples we recall that a (small) k-linear promonoidal category (𝒜 , p, j) (previously called “premonoidal” in [2]) consists of a k-linear category 𝒜 and two k-linear functors

p : 𝒜^{op} ⊗ 𝒜^{op} ⊗ 𝒜 → Vect
j : 𝒜 → Vect

equipped with associativity and unit constraints satisfying axioms (as described in [2]) analogous to those used to define a monoidal structure on 𝒜. The notion of a symmetric promonoidal category (also introduced in [2]) was extended in [4] to that of a braided promonoidal category.

The main point is that (braided) promonoidal structures on 𝒜 correspond to cocontinuous (braided) monoidal structures on the functor category [𝒜 , Vect]. This latter monoidal structure is often called the convolution product of 𝒜 and Vect and is given explicity by the coend formula

(f ∗ g)(c) = ∫ ^{a,b} p(a, b, c) ⊗ fa ⊗ gb

in Vect. The unit of this convolution product is given by j.

5.1. Example. Let (𝒜 , p, j) be a small braided promonoidal category with

𝒜 (I, I) ≅ k where j = 𝒜 (I, −),

and suppose that each hom-space 𝒜 (a, b) is finite dimensional. Let f : 𝒜 → Vect_{f} be a very weak bialgebra in the convolution [𝒜 , Vect] so that we have maps

μ : f ∗ f → f and η : j → f

and

δ : f → f ∗ f and ϵ : f → j,

satisfying associativity and unital axioms, plus the very weak bialgebra axiom. Suppose also that 𝒜 ⊂ 𝒞 where 𝒞 is a separable braided monoidal category, with

p(a, b, c) ≅ 𝒞 (a ⊗ b, c) and j(a) ≅ 𝒞 (I, a)

naturally, and suppose the induced maps

∫ ^{c∈𝒜} p(a, b, c) ⊗ 𝒞 (c, C) → 𝒞 (a ⊗ b, C)

are isomorphisms (e.g., 𝒜 monoidal). We also suppose that each a ∈ 𝒜 has a dual a^{∗} ∈ 𝒜.

Define a functor U : 𝒞 → Vect by

UC = ∫ ^{a∈𝒜} fa ⊗ 𝒞 (a, C);

then, by the Yoneda lemma, U(a^{∗}) ≅ U(a)^{∗} if f(a^{∗}) ≅ f(a)^{∗} for a ∈ 𝒜. Furthermore, by the Yoneda lemma,

UI = ∫ ^{a∈𝒜} fa ⊗ 𝒞 (a, I) ≅ fI

so that, by our assumption 𝒜 (I, I) ≅ k the maps η and ϵ induce respectively maps

r_{0} : k → UI and i_{0} : UI → k.

Maps r and i are described in the following diagram.

[figure: UC ⊗ UD; ∫ ^{a,b}fa ⊗ fb ⊗ 𝒞 (a, C) ⊗ 𝒞 (b, D); ≅; ∫ ^{a,b}fa ⊗ fb ⊗ 𝒞 (a ⊗ b, C ⊗ D); ∫ ^{a,b}fa ⊗ fb ⊗ ∫ ^{c}p(a, b, c) ⊗ 𝒞 (c, C ⊗ D); ∫ ^{c}fc ⊗ 𝒞 (c, C ⊗ D); U(C ⊗ D); 1; r; i; 𝒞 separable; ≅; μ; δ]

These then produce a braided monoidal and comonoidal structure on U. Moreover, we have i_{0}r_{0} = dim UI · 1 if and only if ϵ_{I}η_{I} = dim fI · 1, and if f is a separable very weak bialgebra, then U is separable since ri = 1 if μδ = 1.

Therefore, Theorem 3.1 may be applied when 𝒜 and U satisfy the “U-irreducibility” and “U-trace” criteria.

5.2. Example. Suppose that (𝒜^{op}, p, j) is a small braided promonoidal category with I ∈ 𝒜 such that j ≅ 𝒜 (−, I) and with each x ∈ 𝒜 an “atom” in 𝒞 (i.e., an object x ∈ 𝒞 for which 𝒞 (x, −) preserves all colimits) where 𝒞 is a cocomplete and cocontinuous braided monoidal category containing 𝒜 and each x ∈ 𝒜 has a dual x^{∗} ∈ 𝒜. Suppose that the inclusion 𝒜 ⊂ 𝒞 is dense over Vect (that is, the canonical evaluation morphism

∫ ^{a} 𝒞 (a, C) · a → C

is an isomorphism for all C ∈ 𝒞), and

x ⊗ y ≅ ∫ ^{z} p(x, y, z) · z (naturally in x, y ∈ 𝒜)

so that

𝒞 (a, x ⊗ y) = 𝒞 (a, ∫ ^{z} p(x, y, z) · z)

≅ ∫ ^{z} p(x, y, z) ⊗ 𝒞 (a, z) since a ∈ 𝒜 is an atom in 𝒞,

≅ p(x, y, a) by the Yoneda lemma applied to z ∈ 𝒜.

Let W : 𝒜 → Vect be a strong braided promonoidal functor on 𝒜. This means that we have structure isomorphisms

W x ⊗ W y ≅ ∫ ^{z} 𝒞 (z, x ⊗ y) ⊗ W z and
k ≅ W I

satisfying suitable associativity and unital coherence axioms. Define a functor U : 𝒞 → Vect by

UC = ∫ ^{a} 𝒞 (a, C) ⊗ W a.

Then, if we suppose that W (x^{∗}) ≅ W (x)^{∗} for all x ∈ 𝒜, we have

U(x^{∗}) = ∫ ^{a} 𝒞 (a, x^{∗}) ⊗ W a
≅ W (x^{∗})
≅ W (x)^{∗}
≅ ( ∫ ^{a} 𝒞 (a, x) ⊗ W a)^{∗}
= U(x)^{∗},

so that U(x^{∗}) ≅ U(x)^{∗}, and

i_{0} : UI = ∫ ^{a} 𝒞 (a, I) ⊗ W a
≅ W I
≅ k,

so that i_{0}r_{0} = 1 and r_{0}i_{0} = 1. Also there are mutually inverse composite maps r and i given by:

r : UC ⊗ UD ≅ ∫ ^{x,y} 𝒞 (x, C) ⊗ 𝒞 (y, D) ⊗ Ux ⊗ Uy
≅ ∫ ^{x,y} 𝒞 (x, C) ⊗ 𝒞 (y, D) ⊗ W x ⊗ W y
≅ ∫ ^{x,y} 𝒞 (x, C) ⊗ 𝒞 (y, D) ⊗ ∫ ^{z} 𝒞 (z, x ⊗ y) ⊗ W z
≅ ∫ ^{z} 𝒞 (z, C ⊗ D) ⊗ W z
≅ U(C ⊗ D),

which uses the assumptions that 𝒞 is cocontinuous monoidal and 𝒜 ⊂ 𝒞 is dense. Thus, ri = 1 and ir = 1 so that U is a braided strong monoidal functor.

5.3. Example. (See [6] Proposition 3.) Let 𝒞 be a braided compact monoidal category and let 𝒜 ⊂ 𝒞 be a full finite discrete Cauchy generator of 𝒞 which contains I and is closed under dualization in 𝒞. As in the Häring-Oldenburg case [6], we suppose that each hom-space 𝒞 (C, D) is finite dimensional with a chosen natural isomorphism 𝒞 (C^{∗}, D^{∗}) ≅ 𝒞 (C, D)^{∗}.

Then we have a separable monoidal functor

UC = ⨁_{a,b∈𝒜} 𝒞 (a, C ⊗ b),

whose structure maps are given by the composites

UC ⊗ UD ≅ ⨁_{a,b,c,d} 𝒞 (c, C ⊗ b) ⊗ 𝒞 (a, D ⊗ d)

[figure: c=d; adjoint]⨁_{a,b,c} 𝒞 (c, C ⊗ b) ⊗ 𝒞 (a, D ⊗ c)
≅ ⨁_{a,b} 𝒞 (a, D ⊗ (C ⊗ b))
≅ ⨁_{a,b} 𝒞 (a, (D ⊗ C) ⊗ b)
≅ ⨁_{a,b} 𝒞 (a, (C ⊗ D) ⊗ b)
= U(C ⊗ D),

and r_{0} : k → UI the diagonal, with i_{0} its adjoint. Moreover

U(C^{∗}) = ⨁_{a,b} 𝒞 (a, C^{∗} ⊗ b)

≅ ⨁_{a,b} 𝒞 (a^{∗}, C^{∗} ⊗ b^{∗})

≅ ⨁_{a,b} 𝒞 (a, C ⊗ b)^{∗}

≅ UC^{∗}

for all C ∈ 𝒞.

5.4. Example. Let (𝒜 , p, j) be a finite braided promonoidal category over Set_{f} with I ∈ 𝒜 such that j ≅ 𝒜 (I, −) and with a braided promonoidal functor

d : 𝒜^{op} → Set_{f}

for which each structure map

u : ∫ ^{z} p(x, y, z) × dz → dx × dy

is an injection, and u_{0} : dI → 1 is a surjection. Then we have corresponding maps

∫ ^{z}k[p(x, y, z)] ⊗ k[dz] [figure]k[dx] ⊗ k[dy]

and

k[dI] [figure]k[1],

where k[s] denotes the free k-vector space on the (finite) set s, in Vect_{f}. Define the functor U : 𝒞 → Vect_{f} by

Uf = ∫ ^{x} fx ⊗ k[dx]

for f ∈ 𝒞 = [k_{∗}𝒜 , Vect_{f}], with the convolution braided monoidal closed structure, where k_{∗}𝒜 is the free k-linear category on 𝒜 so that

r : Uf ⊗ Ug = (∫ ^{x} fx ⊗ k[dx]) ⊗ (∫ ^{y} gx ⊗ k[dy])
≅ ∫ ^{x,y} fx ⊗ gy ⊗ (k[dx] ⊗ k[dy])
⇄ ∫ ^{x,y} fx ⊗ gy ⊗ (∫ ^{z} k[p(x, y, z)] ⊗ k[dz])
≅ ∫ ^{z} (∫ ^{x,y} fx ⊗ gy ⊗ k[p(x, y, z)]) ⊗ k[dz]
= ∫ ^{z}(f ⊗ g)(z) ⊗ k[dz]
= ∫ ^{z} U(f ⊗ g)

and

i_{0} : UI = ∫ ^{x} k[𝒜 (I, x)] ⊗ k[dx]
≅ k[dI]
⇄ k[1] ≅ k.

Hence i_{0}r_{0} = dim UI ·1 = |dI|·1. Thus, U becomes a braided separable monoidal functor.

5.5. Example. Let 𝒜 be a finite (discrete) set and give the cartesian product 𝒜 × 𝒜 the Set_{f}-promonoidal structure corresponding to bimodule composition (i.e., to matrix multiplication). If

d : 𝒜 × 𝒜 → Set_{f}

is a braided promonoidal functor, then its associated structure maps

∑_{z,z′} p((x, x′), (y, y′), (z, z′)) × d(z, z′) = ∑_{z,z′} 𝒜 (z, x) × 𝒜 (x′, y) × 𝒜 (y′, z′) × d(z, z′)
≅ 𝒜 (x′, y) × d(x, y′)
→ d(x, x′) × d(y, y′),

and

∑_{z,z′} j(z, z′) × d(z, z′) = ∑_{z,z′} 𝒜 (z, z′) × d(z, z′)
≅ ∑_{z} d(z, z)
→ 1,

are determined by components

d(x, y′) ↣ d(x, y) × d(y, y′)
d(z, z) ↠ 1

which give 𝒜 the structure of a discrete cocategory over Set_{f}.

Define the functor U : 𝒞 = [k_{∗}(𝒜 × 𝒜), Vect_{f}] → Vect_{f} by

Uf = ⨁_{x,y}(f(x, y) ⊗ k[d(x, y)]).

Then we obtain monoidal and comonoidal structure maps

U(f ⊗ g) [figure: i; r]Uf ⊗ Ug
UI [figure: i_{0}; r_{0}]k ≅ k[1]

from the canonical maps

⨁_{x,y,z} f(x, z) ⊗ g(z, y) ⊗ k[d(x, y)]

[figure: z=u=v; adjoint]⨁_{x,u} (f(x, u) ⊗ k[d(x, u)]) ⊗ ⨁_{v,y} (g(v, y) ⊗ k[d(v, y)])

and

⨁_{z} k[d(z, z)] ⇄ k ≅ k[1].

These give U the structure of a separable braided monoidal functor on 𝒞.

6. Concluding remarks

If the original “fibre” functor U is faithful and exact then the Tannaka equivalence (duality)

Lex(𝒞^{op}, Vect) ≃ Comod(End^{∨}U)

is available, where Lex(𝒞^{op}, Vect) is the category of k-linear left exact functors from 𝒞^{op} to Vect. (See [3] for example.) Thus, since 𝒞 is braided monoidal, so is Comod(End^{∨}U) with the tensor product and unit induced by the convolution product on Lex(𝒞^{op}, Vect); for convenience we recall [3] that, for 𝒞 compact, this convolution product is given by the restriction to Lex(𝒞^{op}, Vect) of the coend

F ∗ G = ∫ ^{C,D} F C ⊗ GD ⊗ 𝒞 (−, C ⊗ D)
≅ ∫ ^{C} F C ⊗ G(C^{∗} ⊗ −)

computed in the whole functor category [𝒞^{op}, Vect]. Moreover, when U is separable monoidal, the category Co(End^{∨}U) of cofree coactions of End^{∨}U (as constructed in [7] for example) also has a monoidal structure (Co(End^{∨}U), ⊗, k), this time obtained from the algebra structure of End^{∨}U. The forgetful inclusion

Comod(End^{∨}U) ⊂ Co(End^{∨}U)

preserves colimits while Comod(End^{∨}U) has a small generator, namely {UC | C ∈ 𝒞 }, and thus, from the special adjoint functor theorem, this inclusion has a right adjoint. The value of the adjunction’s counit at the functor F ⊗ G in Co(End^{∨}U) is then a split monomorphism and, in particular, the monoidal forgetful functor

Comod(End^{∨}U) → Vect,

which is the composite Comod(End^{∨}U) ⊂ Co(End^{∨}U) → Vect, is a separable braided monoidal functor extension of the given functor U : 𝒞 → Vect.

References

[1] Aurelio Carboni. Matrices, relations, and group representations, J. Algebra 136 no. 2 (1991): 497–529.

[2] Brian Day. On closed categories of functors, in Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics 137 (1970): 1–38.

[3] Brian J. Day. Enriched Tannaka reconstruction, J. Pure Appl. Algebra 108 no. 1 (1996): 17–22.

[4] B. Day, E. Panchadcharam, and R. Street. Lax braidings and the lax centre, in Hopf Algebras and Generalizations, Contemporary Mathematics 441 (2007): 1–17.

[5] Brian Day and Ross Street. Quantum categories, star autonomy, and quantum groupoids, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004): 187–226.

[6] Reinhard Häring-Oldenburg. Reconstruction of weak quasi-Hopf algebras, J. Algebra 194 no. 1 (1997): 14–35.

[7] André Joyal and Ross Street. An Introduction to Tannaka Duality and Quantum Groups, Lecture Notes in Mathematics 1488 (Springer-Verlag, Berlin, 1991): 411– 492.

[8] G. M. Kelly. Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64. Cambridge University Press, Cambridge, 1982. Also at

    http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html

[9] Ross Street. Fusion operators and cocycloids in monoidal categories, Appl. Categ. Struct. 6 (1998): 177–191.

[10] Kornél Szlachányi. Finite quantum groupoids and inclusions of finite type, Fields Inst. Comm. 30 (2001): 393–407.

Department of Mathematics Macquarie University New South Wales 2109 Australia

Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502 Japan

Email: craig@kurims.kyoto-u.ac.jp

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