Weak Hopf Monoids in Braided Monoidal Categories Craig Pastro and Ross Street

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Volume 3 2009 No. 2

Weak Hopf monoids in braided monoidal categories Craig Pastro and Ross Street

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ALGEBRA AND NUMBER THEORY 3:2(2009)

Weak Hopf monoids in braided monoidal categories Craig Pastro and Ross Street

We develop the theory of weak bimonoids in braided monoidal categories and show that they are in one-to-one correspondence with quantum categories with a separable Frobenius object-of-objects. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R ⊗ R.

Introduction 149 1. Weak bimonoids 153 2. Weak Hopf monoids 160 3. The monoidal category of A-comodules 163 4. Frobenius monoid example 175 5. Quantum groupoids 177 6. Weak Hopf monoids and quantum groupoids 182 Appendix A. String diagrams and basic deﬁnitions 196 Appendix B. Proofs of the properties of s, t, and r 203 Acknowledgments 206 References 206

Introduction

Weak Hopf algebras, introduced by Bohm,¨ Nill, and Szlachanyi´ in the papers [Bohm¨ and Szlachonyi´ 1996; Nill 1998; Szlachanyi´ 1997; Bohm¨ et al. 1999], are generalizations of Hopf algebras and were proposed as an alternative to weak quasi- Hopf algebras. A weak bialgebra is both an associative algebra and a coassociative coalgebra, but instead of requiring that the multiplication and unit morphism are

MSC2000: primary 16W30; secondary 18B40, 18D10. Keywords: weak bimonoids (weak bialgebra), weak Hopf monoids (weak Hopf algebra), quantum category, quantum groupoid, monoidal category. Pastro is partially supported by GCOE, Kyoto University. The majority of this work was completed while he was a PhD student at Macquarie University, Australia, and during that time was gratefully supported by an international Macquarie University Research Scholarship and a Scott Russell John- son Memorial Scholarship. Street gratefully acknowledges the support of the Australian Research Council Discovery Grant DP0771252.

149 150 Craig Pastro and Ross Street coalgebra morphisms (or equivalently that the comultiplication and the counit are algebra morphisms), other “weakened” axioms are imposed. The multiplication is still required to be comultiplicative (equivalently, the comultiplication is still required to be multiplicative), but the counit is no longer required to be an algebra morphism and the unit is no longer required to be a coalgebra morphism. Instead, these requirements are replaced by weakened versions (see Equations (v) and (w) below). As the name suggests, any bialgebra satisﬁes these weakened axioms and is therefore a weak bialgebra. For a given a weak bialgebra A, one may deﬁne source and target morphisms s, t : A → A whose images s(A) and t(A) are called the source and target (counital) subalgebras. Nill [1998] has shown that Hayashi’s face algebras [1998] are special cases of weak bialgebras for which the, say, target subalgebra is commutative. A weak Hopf algebra is a weak bialgebra H that is equipped with an antipode ν : H → H satisfying the axioms1

µ(ν ⊗ 1)δ = t, µ(1 ⊗ ν)δ = s, and µ3(ν ⊗ 1 ⊗ ν)δ3 = ν, where µ3 = µ(µ ⊗ 1) and δ3 = (δ ⊗ 1)δ. Again, any Hopf algebra satisfies these weakened axioms and so is a weak Hopf algebra. Nill [1998] has also shown that the (finite-dimensional) generalized Kac algebras of Yamanouchi [1994] are examples of weak Hopf algebras with involutive antipode. Weak Hopf algebras have also been called “quantum groupoids” [Nikshych and Vainerman 2002], but in this paper this is not what we mean by quantum groupoid. Perhaps the simplest examples of weak bialgebras and weak Hopf algebras are category algebras and groupoid algebras, respectively. Suppose that k is a field, and let Ꮿ be a category with set of objects Ꮿ0 and set of morphisms Ꮿ1. The category algebra k[Ꮿ] is the vector space k[Ꮿ1] over k with basis Ꮿ1. Elements are formal linear combinations of the elements of Ꮿ1 with coefficients in k, that is,

α f + βg + · · · with α, β ∈ k and f, g ∈ Ꮿ1. An associative multiplication on k[Ꮿ] is deﬁned by g ◦ f if g ◦ f exists, µ( f, g) = f · g = 0 otherwise and extended by linearity to k[Ꮿ]. This algebra does not have a unit unless Ꮿ0 is ﬁnite, in which case the unit is X η(1) = e = 1A,

A∈Ꮿ0

1There may be some discrepancy with what we call the source and target morphisms and what exists in the literature. This arises from our convention of taking multiplication in the groupoid algebra to be f · g = g ◦ f (whenever g ◦ f is deﬁned). Weak Hopf monoids in braided monoidal categories 151 making k[Ꮿ] into a unital algebra; all algebras (monoids) considered in this paper will be unital. A comultiplication and counit may be deﬁned on k[Ꮿ] as δ( f ) = f ⊗ f, ( f ) = 1, making k[Ꮿ] into a coalgebra. Note that k[Ꮿ] equipped with this algebra and coalgebra structure will not satisfy any of the following usual bialgebra axioms:

µ = ⊗ , δη = η ⊗ η, η = 1k. The one bialgebra axiom that does hold is δµ = (µ⊗µ)(1⊗c⊗1)(δ⊗δ). Equipped with this algebra and coalgebra structure, k[Ꮿ] does, however, satisfy the axioms of a weak bialgebra. Furthermore, if Ꮿ is a groupoid, then k[Ꮿ], which is then called the groupoid algebra, is an example of a weak Hopf algebra with antipode ν : k[Ꮿ] → k[Ꮿ] defined by ν( f ) = f −1 and extended by linearity. If f : A → B ∈ Ꮿ, the source and target morphisms s, t : k[Ꮿ] → k[Ꮿ] are given by s( f ) = 1A and t( f ) = 1B, as one would expect. In this paper we define weak bialgebras and weak Hopf algebras in a braided monoidal category ᐂ, where we prefer to call them “weak bimonoids” and “weak Hopf monoids”. The only difference between our definition of a weak bimonoid in ᐂ and the one given by Bohm,¨ Nill, and Szlachanyi´ [Bohm¨ et al. 1999] is that a choice of “crossing” must be made in the axioms. Our definition is not as general as the one given by J. N. Alonso, J. M. Fernandez,´ and R. Gonzalez´ in [Alonso Alvarez´ et al. 2008a; 2008b], but, in the case that their weak Yang–Baxter operator tA,A is the braiding cA,A and their idempotent ∇A⊗A = 1A⊗A, our choices of crossings are the same. Our difference in defining weak bimonoids occurs in the choice of source and target morphisms. We have chosen s : A → A and t : A → A so that (1) the “globular” identities ts = s and st = t hold, (2) the source subcomonoid and target subcomonoid coincide (up to isomorphism) and are denoted by C; and (3) s : A → C◦ and t : A → C are comonoid morphisms. These properties of the source and target morphisms are essential for our point L R of view of quantum categories. These are s = 5A and t = 5A in the notation 0 of [Alonso Alvarez´ et al. 2008a; 2008b] and s = s and t = s in the notation of [Schauenburg 2003], with the appropriate choice of crossings. We choose to work in the Cauchy completion ᏽᐂ of ᐂ. The category ᏽᐂ is also called the “completion under idempotents” of ᐂ or the “Karoubi envelope” of ᐂ. We do this rather than assume that idempotents split in ᐂ. Suppose A is a weak bimonoid in ᏽᐂ. In this case we find C by splitting either the source or target morphism. As in [Schauenburg 2003, Proposition 4.2], C is a separable Frobenius monoid in ᏽᐂ, meaning that (C, µ, η, δ, ) is a Frobenius monoid with µδ = 1C . 152 Craig Pastro and Ross Street

Our deﬁnition of weak Hopf monoid is the same as the one proposed in [Bohm¨ et al. 1999] for the symmetric case and as in [Alonso Alvarez´ et al. 2008a; 2008b] when restricted to the braided case. A weak bimonoid H is a weak Hopf monoid if it is equipped with an antipode ν : H → H satisfying

µ(ν ⊗ 1)δ = t, µ(1 ⊗ ν)δ = r, and µ3(ν ⊗ 1 ⊗ ν)δ3 = ν, where r = νs. This r : H → H turns out to be the “usual” source morphism, that L ´ is, 5H in the notation of [Alonso Alvarez et al. 2008a; 2008b]. Ignoring crossings r is t in the notation of [Schauenburg 2003], and our r and t correspond respectively to uL and uR in the notation of [Bohm¨ et al. 1999], wherein the morphism s does not appear. Usually, in the second axiom above, µ(1 ⊗ ν)δ = r, the right side is equal to the chosen source map s of the weak bimonoid H. The reason that this r does not work for us as a source morphism is that it does not satisfy all three requirements for the source morphism mentioned above. This choice of r allows us to show that any Frobenius monoid in ᐂ yields a weak Hopf monoid R ⊗ R with bijective antipode; see [Bohm¨ et al. 1999, example in the appendix]. There are a number of generalizations of bialgebras and Hopf algebras to their “many object” versions, for example, Sweedler’s generalized bialgebras [1974], which were later generalized by Takeuchi to ×R-bialgebras [1977]; the quantum groupoids of Lu [1996] and Xu [2001]; Schauenburg’s ×R-Hopf algebras [2000]; the bialgebroids and Hopf algebroids of Bohm¨ and Szlachanyi´ [2004]; the face algebras [Hayashi 1998] and generalized Kac algebras [Yamanouchi 1994]; and the ones of interest in this paper, the quantum categories and quantum groupoids of Day and Street [2004]. Brzezinski´ and Militaru [2002, Theorem 3.1] have shown that the quantum groupoids of Lu and Xu are equivalent to Takeuchi’s ×R-bialgebras. Schauenburg [1998] has shown that face algebras are an example of ×R-bialgebras for which R is commutative and separable. In [2003, Theorem 5.1], Schauenburg shows that weak bialgebras are also examples of ×R-bialgebras for which R is separable Frobenius (there called Frobenius-separable). Schauenburg also shows in [2003, Theorem 6.1] that a weak Hopf algebra may be characterized as a weak bialgebra H for which a certain canonical map H ⊗C H → µ(δ(η(1)), H ⊗ H) is a bijection. As a corollary he shows that the ×R-bialgebra associated to the weak Hopf algebra is actually a ×R-Hopf algebra. While bialgebras are self dual, bialgebroids are not. The dual of a bialgebroid is called a “bicoalgebroid” by Brzezinski´ and Militaru [2002] and further studied by Balint´ [2008b]. In the terminology of [Day and Street 2004], these structures are quantum categories in the monoidal category of vector spaces. According to [Day and Street 2004], a quantum category in ᐂ consists of two comonoids A and C in ᐂ, with A playing the role of the object of morphisms, and C the object-of-objects. There are source and target morphisms s, t : A → C, Weak Hopf monoids in braided monoidal categories 153 a “composition” morphism µ : A ⊗C A → A, and a “unit” morphism η : C → A, all in ᐂ. These data must satisfy a number of axioms. Indeed, ordinary categories are quantum categories in the category of sets. Motivated by the duality present in ∗-autonomous categories [Barr 1995], Day and Street define a quantum groupoid to be a quantum category equipped with a generalized antipode coming from a ∗-autonomous structure. In this paper we show there is a bijection between weak bimonoids and quantum categories for which the object-of-objects is a separable Frobenius monoid. In the case that the weak bimonoid is equipped with an invertible antipode, making it a weak Hopf monoid, we show how to yield a quantum groupoid. The outline of this paper is as follows: In Section 1, we provide the definition of a weak bimonoid A in a braided monoidal category ᐂ and define the source and target morphisms. We then move to the Cauchy completion ᏽᐂ and prove the three required properties of our source and target morphisms. In this section we also prove that C, the object-of-objects of A, is a separable Frobenius monoid. In Section 2, we introduce Weak Hopf monoids in braided monoidal categories. In Section 3, we describe a monoidal structure on the categories Bicomod(C) of C-bicomodules in ᐂ, and Comod(A) of right A-comodules in ᐂ, such that the underlying functor U : Comod(A) → Bicomod(C) is strong monoidal. If H is a weak Hopf monoid, we are then able to show that the category Comod f (H), consisting of the dualizable objects of Comod(H), is left autonomous. In Section 4, we prove that any separable Frobenius monoid R in a braided monoidal category ᐂ yields an example of a weak Hopf monoid R ⊗ R with invertible antipode in ᐂ. In Section 5, we recall the definitions of quantum categories and of quantum groupoids, and in Section 6, we show the correspondence between weak bimonoids and quantum categories with separable Frobenius object-of-objects. In Section 6, we also show that a weak Hopf monoid with invertible antipode yields a quantum groupoid. This paper depends heavily on the string diagrams in braided monoidal categories of Joyal and Street [1993], which were shown to be rigorous in [1991]. The reader unfamiliar with string diagrams may first want to read Appendix A, where we review some preliminary concepts using these diagrams. Many string proofs also appear in Appendix B.

1. Weak bimonoids

A weak bialgebra [Bohm¨ and Szlachonyi´ 1996; Nill 1998; Szlachanyi´ 1997; Bohm¨ et al. 1999] is a generalization of a bialgebra with weakened axioms. These weakened axioms replace the three that follow by requiring that the unit be a coalgebra 154 Craig Pastro and Ross Street morphism and the counit be an algebra morphism. With the appropriate choices of under and over crossings, the deﬁnition of a weak bialgebra carries over rather straightforwardly into braided monoidal categories, where we prefer to call it a “weak bimonoid”.

1.1. Weak bimonoids. Suppose that ᐂ = (ᐂ, ⊗, I, c) is a braided monoidal category. Deﬁnition 1.1. A weak bimonoid A = (A, µ, η, δ, ) in ᐂ is an object A ∈ ᐂ equipped with the structure of a monoid (A, µ, η) and a comonoid (A, δ, ) satisfying the following equations. ? ?? JJ t = JJtJ (b) ?? tt ? 99 Õ 99 ÕÕ wGG Õ = ww G = (v)

ØÙ 9 = ØÙ GG w ØÙ = ØÙ ØÙ (w) ÕÕ 99 Gww ÕÕ ØÙ 9 ØÙ ØÙ ØÙ ØÙ Suppose A is a weak bimonoid, and deﬁne the source and target morphisms s, t : A → A of A as

s = , and t = . ØÙ ØÙ Notice that s : A → A is invariant under rotation by π, while t : A → A is invariant under horizontal reﬂection ØÙ and the inverse ØÙ braiding. Importantly, under either of these transformations • (m) and (c) are interchanged,2 • (b) is invariant, and • (v) and (w) are interchanged. Note that these are not the “usual” source and target morphisms. They were chosen, as mentioned in the introduction, precisely because we need them to satisfy the following three properties: (i) the “globular” identities ts = s and st = t hold; (ii) the source subcomonoid and target subcomonoid coincide (up to isomorphism), and are denoted by C; (iii) s : A → C◦ and t : A → C are comonoid morphisms.

2The (m) and (c) refer to the monoid and comonoid identities found in Appendix A. Weak Hopf monoids in braided monoidal categories 155

These properties will be proved in this section. Note that we will run into the usual source morphism (which we call r) in Deﬁnition 2.1, which deﬁnes weak Hopf monoids. We tabulate properties of the source morphism s in Figure 1, properties of the target morphism t in Figure 2, and properties involving the interaction of s and t in Figure 3. Proofs of these properties may be found in Appendix B. Suppose A and B are weak bimonoids in ᐂ.A morphism of weak bimonoids f : A → B is a morphism f : A → B in ᐂ that is both a monoid morphism and a comonoid morphism.

Under (b) and (w) Under (b) and (v)

× 77 × s s s × 7 × JJ ( ) = = × = J = XX 1 7 JJJ XX s s s ×× 77 × ×× ØÙ ØÙ : s ::Ô ØÙ Ô s ØÙ (2) s = :: s = = Ô = ÔÔÔ : ØÙ ØÙ ØÙ ØÙ s 7 s s 7 ØÙ 7 ×× ØÙ = ØÙ = × ØÙ (3) 7 s s × 7 s ×× 7 77 × 77 × s s s ? s 7×× 7×× ? = ?? ? = = = ?? (4) ?? ?? ? ×77 ×77 s s s s ×× 7 ×× 7 s s s ?? = s s ? = ?? (5) ?? ? s s s s s Under (b)

ooOOO OoO oo (6) oooOO = s G w Gww Under (b) and [(w) or (v)]

s (7) = s s Figure 1. Properties of s. 156 Craig Pastro and Ross Street

Under (b) and (w) Under (b) and (v) × 7 t × 77 ×× + = t = t × = ttt+ = (1) 7 JJ XX t , × 7 J t + t ×× 7 + / ØÙ ØÙ : t :: ÔÔ ØÙ (2) t = :: t = ?? = ØÙ Ô t = ÔÔÔ : ØÙ ØÙ ØÙ ØÙ 7 t 7 × t t ? ØÙ 7 × ØÙ (3) = ? = × ØÙ t ×77 ØÙ t ×× 7 t 7 7 7 × 7 × t t t ? t 7× × 7×× = ?? ? = = = (4) ?? ?? ×77 ×77 t t t t ×× 7 ×× 7 t t t :: Ô = t t : = Ô:: (5) Ô :: Ô Ô : t Ô t t Ô t Ô t Under (b)

×77 × t = (6) 7×7 7×× Under (b) and [(w) or (v)]

t (7) = t t Figure 2. Properties of t.

Lemma 1.2. Suppose A and B are weak bimonoids in ᐂ each with source and target morphisms s and t. If f : A → B is a morphism of weak bimonoids, then f s = s f and f t = t f.

Proof. The proof of the ﬁrst statement is

?? 7 s ?? ?? 7 f f f f = ?? = = f f f = = ?? = . ØÙ 7 ØÙ ?? ?? f f ØÙ f f ØÙ 7 ?? s '&%$ !"# '&%$ !"# ØÙ '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# ØÙ '&%$ !"# '&%$ !"# ØÙ The second statement follows ØÙ from a similar ØÙ proof. ØÙ Weak Hopf monoids in braided monoidal categories 157

Under (w)

s t (8) = s = t t s t s t s (9) = = s ×77 t ×77 ×× 7 ×× 7 Under (v)

zDD (10) ?? = s z t t s AA AA| ~~ A Under (w) and (v)

?? (11) ?? = t s

Figure 3. Interactions of s and t.

In what follows A = (A, µ, η, δ, ) will always denote a weak bimonoid and s, t : A → A will always denote the source and target morphisms. From properties (7) and (7) in Figures 1 and 2 respectively, s and t are idempotents. In the following we will work in the Cauchy completion (= completion under idempotents = Karoubi envelope) ᏽᐂ of ᐂ. We do this rather than assume that idempotents split in ᐂ.

1.2. Cauchy completion. Given a category ᐂ, its Cauchy completion ᏽᐂ is the category whose objects are pairs (X, e) with X ∈ ᐂ and e : X → X ∈ ᐂ an idempotent. A morphism (X, e) → (X 0, e0) in ᏽᐂ is a morphism f : X → X 0 ∈ ᐂ such that e0 f e = f . Note that the identity morphism of (X, e) is e itself. The point of working in the Cauchy completion is that every idempotent f : (X, e) → (X, e) in ᏽᐂ has a splitting, namely, f (X, e)(/ X, e) < @ << ÒÒ << ÒÒ f << ÒÒ f < ÒÒ (X, f ).

If ᐂ is a monoidal category, then ᏽᐂ is a monoidal category via (X, e) ⊗ (X 0, e0) = (X ⊗ X 0, e ⊗ e0). 158 Craig Pastro and Ross Street

The category ᐂ may be fully embedded in ᏽᐂ by sending X ∈ ᐂ to (X, 1) ∈ ᏽᐂ and f : X → Y ∈ ᐂ to f : (X, 1) → (Y, 1), which is obviously a morphism in ᏽᐂ. When working in ᏽᐂ we will often identify an object X ∈ ᐂ with (X, 1) ∈ ᏽᐂ.

1.3. Properties of the source and target morphisms. Let A = (A, 1) be a weak bimonoid in ᏽᐂ. From the deﬁnition of the Cauchy completion, the result of splitting the source morphism s is (A, s), and similarly, the result of splitting the target morphism t is (A, t). The following proposition shows that these two objects are isomorphic. Proposition 1.3. The idempotent t : (A, 1) → (A, 1) has the two splittings t t (A, 1)(/ A, 1) (A, 1)(/ A, 1) < @ < @ << ÒÒ << ÒÒ << ÒÒ and << ÒÒ t << ÒÒ t t << ÒÒ s < ÒÒ < ÒÒ (A, t) (A, s).

In this case s : (A, s) → (A, t) and t : (A, t) → (A, s) are inverse morphisms, and hence (A, t) =∼ (A, s). Proof. This result follows from the identities ts = s and st = t (property (8) in Figure 3). We will denote this object by C = (A, t) and call it the object-of-objects of A. In the propositions next we will show that C is a comonoid and that it is a separable Frobenius monoid; this is similar to what was done in [Schauenburg 2003] (where it was called Frobenius-separable). Proposition 1.4. The object C = (A, t) equipped with δ t⊗t δ = C / C ⊗ C / C ⊗ C and = C / I is a comonoid in ᏽᐂ. If C is furthermore equipped with t⊗t µ η µ = C ⊗ C / C ⊗ C / C and η = I / C, then C is a separable Frobenius monoid in ᏽᐂ (see Deﬁnition A.5). Proof. We ﬁrst observe that (t ⊗ t)δ : C → C ⊗ C and : C → I are in ᏽᐂ, which follows respectively from (5) and (2). The comonoid identities are given as

? ? ? ? ? ? t t (5) ? (c) ?? (5) t t = ×77 t = t ×77 = ?? t × t t × t ?? t t t t Weak Hopf monoids in braided monoidal categories 159 and ?? ?? t t (2) ?? (c) (c) ? (2) t t = t = t = t ? = .

To see that C is a separable ØÙ Frobenius monoid ØÙ we ﬁrst observe that µ and η ØÙ ØÙ are morphisms in ᏽᐂ from (5) and (2), and the monoid identities are dual to the comonoid identities. The following calculation proves that the Frobenius condition holds.

? t t t t ? G G t ?? t ? t (7) G t (5) G t (3) JJ (4) ? = t G = t G = JJJ = t t t G t G t ?? ?? t t t t t t t (4) t (3) t ww (5) t ww (7) t t = ttt = t = t = t t ww t ww t t t t t Finally, that this is a separable Frobenius monoid follows from ? ? 7 t t t t (7) × 7 (5) 7 (3) 7 (6) µδ = = t × t = × 7 = × 7 = t = 1 . 7 t × t ×7× t C t ? t 7×× 7 7 ? 7×× 7×× Corollary 1.5. Every morphism of weak bimonoids induces an isomorphism on the objects-of-objects. That is, if (A, 1) and (B, 1) are weak bimonoids, and f : (A, 1) → (B, 1) is a morphism of weak bimonoids, then the induced morphism t f t : (A, t) → (B, t) is an isomorphism. Proof. If f : A → B is a morphism of weak bimonoids, then by Lemma 1.2 f t = t f and f s = st. The corollary now follows from Propositions 1.4 and A.3. Proposition 1.6. If we write C◦ for the comonoid C with the “opposite” comultiplication deﬁned via

δ t⊗t c zDD C / C ⊗ C / C ⊗ C / C ⊗ C = t z t , AA AA| ~~ A then s : A → C◦ and t : A → C are comonoid morphisms. That is, the diagrams s t A / C A / C

δ c(t⊗t)δ and δ (t⊗t)δ s⊗s t⊗t ⊗ A / C ⊗ CA ⊗ A / C ⊗ CA commute. 160 Craig Pastro and Ross Street

Proof. The second diagram expresses

t ?? ?? = t t , t t which is exactly (5), and the calculation

s s s s s s ? (5) (8) ?? (3) (10) (9) zDD (8) ? = ? = s s = ? = zDD = t z t = zDD s s ? ? s z t t z t s s t s AA | AA | t AA s F w AA ~~ A FFw ~~ A yy F shows that the ﬁrst diagram commutes.

2. Weak Hopf monoids

In this section we introduce weak Hopf monoids. A weak Hopf monoid is a weak bimonoid H equipped with an antipode ν : H → H satisfying the three axioms ν ∗ 1 = t, 1 ∗ ν = r, and ν ∗ 1 ∗ ν = ν, where f ∗g = µ( f ⊗g)δ is the convolution product, and the morphism r : H → H is introduced below. This turns out to be the usual deﬁnition of weak Hopf monoids as found in the literature; in the symmetric case see [Bohm¨ et al. 1999], and in the braided case see [Alonso Alvarez´ et al. 2008a; 2008b]. Note property (15), which says that r = νs.

2.1. The endomorphism r and weak Hopf monoids. Deﬁne an endomorphism r : A → A by rotating the target morphism t : A → A by π, that is,

r = . ØÙ Since r is just t rotated by π, all the identities for t in Figure 2 rotated by π hold ØÙ for r. We list some additional identities of r interacting with s and t.

r s = s = r (12) s r ? ?? zDD t r ?? ?? = r z t ?? = r t (13) t r AA JJtt AA| ~~ A 7 7 r 77 ×× s 77 ×× = × = × s r (14) s r Weak Hopf monoids in braided monoidal categories 161

The proofs of these properties may also be found in Appendix B.

Deﬁnition 2.1. A weak bimonoid H is called a weak Hopf monoid if it is equipped with an endomorphism ν : H → H, called the antipode, satisfying

? ? ?? ?? ν = t , ν = r , ν ν = ν . ?? ?? ? ? The axioms of a weak Hopf monoid immediately imply the identities

? ? ?? ?? ν = t ν = ν r . ?? ?? ? ? The antipode is unique since if ν0 is another, then

ν0 = ν0 ∗ 1 ∗ ν0 = t ∗ ν0 = ν ∗ 1 ∗ ν0 = ν ∗ r = ν ∗ 1 ∗ ν = ν.

Proposition 2.2 [Alonso Alvarez´ et al. 2003, Proposition 1.4]. Suppose H and K are weak Hopf monoids in ᐂ and that f : H → K is both a monoid and comonoid morphism. Then f preserves the antipode, that is, f ν = ν f .

The proof is also due to the authors of [Alonso Alvarez´ et al. 2003], where a similar proof may be found. We include it here for completeness.

Proof. Recall from Lemma 1.2 that t f = f t, from which we may easily conclude that r f = f r. The proposition is then established by the calculation

? G ?? w GG ?? ??? ??? ν f ν w ν ? ν r ν f ν ν r Lemma 1.2 ν 7 (c) = ??? = = = × 7 = f f f f × f f f f r 1 ×77'&%$ !"# ν f ??? ??? 1 7 × ν '&%$ !"# 1 w× G G 1ww '&%$ !"# '&%$ G!"#ww'&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# '&%$ !"# 1 ww1 7w 1 ?? ?? f × 77 1 ? ? (c) ν × f ν t f f f ν f = 7 × = Lemma= 1.2 = ?? = 7×× ? . f ν t ν t ?'&%$ !"# ν ν f ν ?? ?? ?? ?? '&%$ !"# ?'&%$ !"# '&%$ !"# ?'&%$ !"# '&%$ !"# ? '&%$ !"# '&%$ !"# Therefore, if H and K are weak Hopf monoids in ᐂ, then a morphism of weak Hopf monoids f : H → K is a morphism f : H → K in ᐂ that is a monoid and a comonoid morphism. We list some properties of the antipode ν : H → H. 162 Craig Pastro and Ross Street

Proposition 2.3.

s = r (15) ν ν r r ν t t = = = = (16) t ν t r ν r D ν ν zz D = = ν A ν ×77 AA| × 7 ~ AA × ~ (17) ØÙ 7 AA ØÙ 77 ×× AA~~ × | A ν = = ν D ν ν Dzz ØÙ ØÙ The last identity (17) states that ν : H → H is both an anticomonoid morphism and an antimonoid morphism.

Proof. The calculation

s s s s ν ? (9) ? ν (12) = ?? = ?? = = r t ν ν ν ?? r ? veriﬁes the identity (15), and

?? ?? r r ν ? ? r r ν t ? ν (3) ν ν ν (3) ? ν = ?? = = = = ?? = t t ν ν r ν ?? ?? t t ? ? verifies the first identity of (16). The second follows from a similar calculation. We prove the first two properties of (17), which show that ν is an anticomonoid morphism. The remaining two properties of (17), which show that ν is an antimonoid mophism, following from rotating the diagrams by π. The proof of the counit property is easy enough:

? ? ν ? (2) ? ν (2) ν = t ? ν = ν = r = . ? ØÙ ØÙ ØÙ ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 163

The following calculation proves that the antipode is anticomultiplicative.

? ? ?? ?? ? ν ? ν ? ? ? ?? ?? // ?? ? ν r ? ν r (b) ? (3) ν ν (4) ν ν = ν ? = ν r = ? = = ?? O ? ?? OOOoo ?? r ? ? oo OO ?? ?? ?? ? ? r R r ? ?? RRRR ? RR 0 / ? 00 // 1 ?? ?? 0 ?? / 1 ν ?/ ?? 0 ?? / 1 / ν 4 00 ν / / DD 11 ν 44 0 ?? / / DD 1 ν (c) O 4 0 (b) ? // // (c),ν t ν 1 = = OOOoo ν 00 = / / = 11 oo OO ? ν / JJ ν ?? ν JJJ ν GG v ? t w LL ν vv tt ww LL w vv tt w LL w v ww L vv tt w vv ttt ( ) ( 1 ) ? ( ? 11 ( ) ?? ( ? D 1 ( ) ν ? ( ?? DD 1 ( ) w (( ?? ? t D 11 (( )) ww ( ?? ?? (3) ν 1 ν ν ? ( ) (4) ( (c),ν ?? ? = J 1 = ?? ) = ν ( = ? ? JJ ν ν ) ( ν O t r ν JJ JJ )) (( OOO ooo t www JJ GG OO o t J ν t G ν OOoOo ww lll mmm oo O ww ll mm ll mmm ? ? ? ?? ?? ?? ?? (13) ?? ? (c),ν ? = ?? ?? = ν ? ν ν r t ν ? OOO oo ?? OOG ooo ?? GG ww GwG www G

3. The monoidal category of A-comodules

Suppose A = (A, 1) is a weak bimonoid in ᏽᐂ and let C = (A, t), which we recall is a separable Frobenius monoid. In this section we describe a monoidal structure on the categories Bicomod(C) of C-bicomodules in ᏽᐂ, and Comod(A) of right A-comodules in ᏽᐂ, such that the underlying functor

U : Comod(A) → Bicomod(C) is strong monoidal. If A is a weak Hopf monoid then we show that Comod f (A), the subcategory consisting of the dualizable objects, is left autonomous. 164 Craig Pastro and Ross Street

This section is fairly standard in the ᐂ = Vect case — see for example [Bohm¨ and Szlachanyi´ 2000; Nill 1998; Nikshych and Vainerman 2002] — and carries over rather straightforwardly to the general braided ᐂ case; see [Day et al. 2003].

3.1. The monoidal structure on C-bicomodules. Suppose, for this section, that idempotents split in ᐂ, that C ∈ ᐂ is a separable Frobenius monoid, and that M ∈ ᐂ is a C-bicomodule with coaction γ : M → C ⊗ M ⊗ C.

A left C-coaction and a right C-coaction are obtained from γ by involving the counit as follows: γ 1⊗1⊗ γl = M / C ⊗ M ⊗ C / C ⊗ M ,

γ ⊗1⊗1 γr = M / C ⊗ M ⊗ C / M ⊗ C .

Suppose now that N is another C-bicomodule. We now wish to define the tensor product of M and N. Before doing so we will need the following definition. Definition 3.1. Let f, g : X → Y be a parallel pair in ᐂ. This pair is called cosplit when there is an arrow d : Y → X such that

d f = 1X and f dg = gdg. It is not hard to see that, in this case, dg : X → X is an idempotent and a splitting of dg, that is, dg 1 XX/ QQ/ ?? ? ?? ? ?? ?? x ?? y y ?? x ? ? Q X provides an absolute equalizer (Q, y) for f and g. Now M and N are C-bicomodules and we have two morphisms

γr ⊗1 / M ⊗ NM/ ⊗ C ⊗ N. 1⊗γl

Proposition 3.2. The pair γr ⊗ 1 and 1 ⊗ γl are cosplit by

d = (1 ⊗ ⊗ 1)(1 ⊗ µ ⊗ 1)(γr ⊗ 1 ⊗ 1) : M ⊗ C ⊗ N → M ⊗ N. Proof. Here we barely sketch the proof and note that we prove a very similar statement in greater detail in Proposition 3.3. That d(γr ⊗1) = 1 follows from the separable property of the Frobenius monoid, and (γr ⊗ 1)d(1 ⊗ γl ) = (1 ⊗ γl )d(1 ⊗ γl ) follows from the Frobenius property. Weak Hopf monoids in braided monoidal categories 165

Thus, the equalizer of the two morphisms γr ⊗1 and 1⊗γl is found by splitting the idempotent d(1 ⊗ γl ), which is possible from our assumption that idempotents split in ᐂ. So the equalizer exists and is absolute. This equalizer is then defined to be the tensor product of M and N over C, denoted M ⊗C N. That this defines a monoidal structure on the category Bicomod(C) with tensor product ⊗C and unit C is yet to be proved. However, we thought it better to write the next section more explicitly from which the details here may be filled in.

3.2. The tensor product of A-comodules. Let A = (A, 1) be a weak bimonoid in ᏽᐂ. Let C = (A, t). It will be shown that the monoidal structure on the category of right A-comodules is ⊗C , the tensor product over C, with unit C. Suppose that M is a right A-comodule. We know that s : A → C◦ and t : A → C are comonoid morphisms and that property (10) holds, recalling that property (10) expresses the commutativity of the diagram s⊗t A ⊗ A / C ⊗ C δ rr8 rrr A r c LL LLL δ L& t⊗s A ⊗ A / C ⊗ C.

Therefore M may be made into a C-bicomodule via γ 1⊗δ c−1⊗1 s⊗1⊗t γ = M / M ⊗ A / M ⊗ A ⊗ A / A ⊗ M ⊗ A / C ⊗ M ⊗ C, which is M ?? A ?/ γ = ooo/ s oo t

C M C in strings. The left and right C-coactions are

JJ = kJ = GG γl s kkk and γr t .

The tensor product of two A -comodules M and N over C then may be deﬁned as in Section 3.1. We derive an explicit description of M ⊗C N. Suppose M and N are A-comodules. Two morphisms M ⊗ N → M ⊗ C ⊗ N are given as

M N M N A J A GG JJ γr ⊗ 1 = t and 1 ⊗ γl = qq . s qq M CN M CN 166 Craig Pastro and Ross Street

Proposition 3.3. The pair γr ⊗ 1 and 1 ⊗ γl are cosplit by M CN

G t d = GG . M N Proof. That d is a morphism in ᏽᐂ follows ØÙ immediately as t is idempotent. The calculation G G O t GG OO4 (7) t (c) (6) ?? (c) d(γr ⊗ 1) = t = GG = 44 t = = = 1M⊗N GG G 4 G ØÙ ⊗ = ØÙ ØÙ shows that d(γr 1) 1, ØÙ and the identity

(γr ⊗ 1)d(1 ⊗ γl ) = (1 ⊗ γl )d(1 ⊗ γl ) follows from

JJJ mm JJJ s mm m JJ JJJ s mmm G J O ? r GG mmm OOJ ? rrr ⊗ ⊗ = t (8)= GG (2)= mm (c)= JJllll (12)= s t (γr 1)d(1 γl ) GG G 4 G GG t 44 jj G t jj G G G t t ØÙ ØÙ ØÙ ØÙ ØÙ JJJ J JJJ mmm JJ mm s m mmm s mm (2) t/m/m (c) GG (8) t = s = G = GG 44 k s G 4kkk JJJ mm JJ s mm mJ s mmm ØÙ ØÙ = ØÙ ⊗ ⊗ (1 γl )d(1 γl ).

The idempotent d(1 ⊗ γl ) will be denoted by m, which gains a simpler representation from the calculation

JJJ mm s mm JJJ mmm JJJ t (8)= s m (2)= GG mm = GG GG Gmmm m. G G ØÙ ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 167

A splitting of m, that is, m m (M ⊗ N, 1)(/ M ⊗ N, 1) (M ⊗ N, m)(/ M ⊗ N, m) AA }> AA }> AA }} AA }} AA }} AA }} m AA }} m m AA }} m A }} A }} (M ⊗ N, m) (M ⊗ N, 1), provides an absolute equalizer (M ⊗ N, m) of (γr ⊗ 1) and (1 ⊗ γl ). Thus, the tensor product of M and N over C is

M ⊗C N = (M ⊗ N, m).

3.3. The coaction on the tensor product. If Comod(A) is to be a monoidal category with underlying functor U : Comod(A) → Bicomod(C) strong monoidal, then the tensor product of two A-comodules must also be an A-comodule. In this section we show that the obvious coaction on M ⊗C N, namely,

QQQ 77 γ = QQQ7 : M ⊗C N → M ⊗C N ⊗ A, does the job.

Lemma 3.4. The coaction γ : M ⊗C N → M ⊗C N ⊗ A, as deﬁned above, is a morphism in ᏽᐂ. That is,

?? QQ 7 ?? r QQQ77 ?rrr QQ 7 Q = QQQ77 = ?? . Q ?? r QQQ 77 ?rrr QQQ7 ØÙ Proof. The ﬁrst equality is given by ØÙ

?? O O ?? r OOO OOO QQ 7 ?rrr (c) OOJ OO (b) QQQ77 (c) QQ 7 = JJJtt = Q = QQQ77 , tt J ? Q QQQ 77 ? QQQ7 ØÙ ØÙ and the second by the similar calculation: ØÙ

QQ 7 OO OO QQQ77 OO OO RR ?? Q (c) OOJ OO (b) RRRR?? (c) QQ 7 ?? = JJJtt = R = QQQ77 . ?? r tt J Q ?rrr

⊗ ØÙ ØÙ Proposition 3.5. (M ØÙ C N, γ ) is an A-comodule. 168 Craig Pastro and Ross Street

Proof. Coassociativity is proved as usual, by

O O T J RR ?? OOO OOO TTTTJJ RRRR?? (b) OO OO (c) TTJTJ R = JJJtt = QQ 7 , t JJ QQQ77 // t Q / and the counit condition is proved as

? ? ? ? ?? rrr ? ? ?? SS 4 r (c) ? oo (b) G (c) ?? SSS44 Lemma= 3.4 = 99 Õo = GG ooo = ?? r = 9 o ? rr 1M⊗C N . QQQ 77 Õ 9 ? r QQQ7 ? ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ 3.4. Comod(A) is a monoidal ØÙ category. We now set out to prove the claim, at the beginning of this section, that (Comod(A), ⊗C , C) is a monoidal category. It will turn out that associativity is a strict equality (if it is so in ᐂ) and the unit conditions are only up to isomorphism. We state this as a theorem and devote the remainder of this section to its proof.

Theorem 3.6. Comod(A) = (Comod(A), ⊗C , C) is a monoidal category.

First note that C itself is an A-comodule with coaction C t . ×77 ×× 7 C A The following lemma will be useful.

Lemma 3.7. The following identities hold.

t t t O JJ t OOOtt Q 7 = JJJ Q 7 = ?t? O QQQ 7 J QQQ 7 r QQ7 t QQ7 s rr Proof. The ﬁrst identity is proved by

t t J t J t (9)= QQQ 77 (11)= JJJ (c)= JJJ QQQ 77 QQQ7 J J J , QQQ7 t t s Weak Hopf monoids in braided monoidal categories 169 and the second is proved by

t t t OOOtt (3)= (11)= t J t (c)= ?t OO QQQ 77 QQQ 77 JJJtt ? . QQQ7 QQQ7 tt JJ rr t s s r

Proof of Theorem 3.6. Consider (M ⊗C N)⊗ C P and M ⊗C (N ⊗C P) in ᏽᐂ. The former is (M ⊗ N ⊗ P, u) and the latter is (M ⊗ N ⊗ P, v), where

? ?? J ? JJ l JJllll Jlll = / = ?? u WWW // ?? and v JJJ ll . WW// l ?? oll lll ?ooo ØÙ ØÙ

Since, by Lemma 3.4, γ is a morphism ØÙ in ᏽᐂ, both ØÙ u and v may be rewritten as

7 ? V 7 ? VVVV7llll ,

proving the (strict) equality (M ⊗C N) ⊗ ØÙ C P = M ⊗C (N ⊗C P) in ᏽᐂ (since we are writing as if ᐂ were strict). ∼ ∼ It remains to prove M ⊗C C = M = C ⊗C M. By deﬁnition

t t ? ?? M ⊗C C = (M ⊗ C, J ? ) and C ⊗C M = (C ⊗ M, JJ l ). JJllll Jlll

We will show that the morphisms ØÙ ØÙ

G t GG GG : M ⊗C C → M and t : M → M ⊗C C ∼ will establish the isomorphism ØÙ M ⊗C C = M, and

? t ? ?? oo : C ⊗ M → M o : M → C ⊗ M oo C and s oo C ∼ will establish the ØÙ isomorphism M = C ⊗C M. These morphisms are easily seen to be in ᏽᐂ, and the fact that they are mutually inverse pairs is given in one direction by Lemma 3.7, and in the other by an easy string calculation making use of the identity (6) in Figure 1 or (6) in Figure 2. 170 Craig Pastro and Ross Street

It now remains to show that these four morphisms are A-comodules morphisms, that is, that they are in Comod(A). Note that M⊗C C and C⊗C M are A-comodules via the coactions

t t QQQ 77 and QQQ 77 QQQ7 QQQ7 respectively. We then have these facts:

G t • GG : M ⊗C C → M is an A-comodule morphism since

t ØÙ t OOO t O OO QQQ // J OO t t t QQQ/ Lemma 3.7 t (7) J (c),(6) L t (c) t (c) LL (4) LL (c) GG = = t = LL = GG = LL = LL = G . t L L G t JJJ ? ? ?? JJJ ?? ? ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ GG • t : M → M ⊗C C is an A-comodule morphism since

OO t OO OOO OOO (7) t Lemma 3.7 O t (c) t (6) t = = OOO = OO = J . QQ 77 O O t QQ 77 QQQ7 t t QQQ7 t ?? o • ooo : C ⊗C M → M is an A-comodule morphism since

ØÙ t 7 TT 7 t 7 t TTT7 TTT 77 7 TT X 7 ?? t 7 2 XXXXX7 Lemma 3.7 ll (8) ?? (c),(6) SSS77 (c) t M 2 t = s l = ll = S = MM22 ? ? s l ? M // ll ? ?? ll t ? lll llll l ØÙ ØÙ ØÙ ØÙ 2 J J t M 2 t JJ t JJ (c) MM22 (4) RR 77 (c) ?? jjj = M = RRRR7 = j . ? // / ØÙ ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 171

?? • o : M → C ⊗ M A s oo C is an -comodule morphism since

?? ? ?? oo ? oo s o ooo ? s o Lemma 3.7 (8) s M 7 (c),(6) ?? = 77 = MM77 = t VVV7 M ?? . t 7 ?V ?? oo VV 7 ? oo s o VVV7 ooo s o s ∼ ∼ Thus, M ⊗C C = M = C ⊗C M in ᏽᐂ.

Thus, Comod(A) = (Comod(A), ⊗C , C) is a monoidal category.

3.5. The forgetful functor from A-comodules to C-bicomodules. There is a forgetful functor U : Comod(A) → Bicomod(C) that assigns to each A-comodule M a C-bicomodule UM that is M itself with coaction M ??A ?/ ooo/ . s oo t C M C Obviously a morphism of A-comodules f : M → N is automatically a morphism of the underlying C-bicomodules f : UM → UN. Proposition 3.8. The forgetful functor U : Comod(A) → Bicomod(C) is strong monoidal. Proof. We must establish the C-bicomodule isomorphisms ∼ ∼ C = UC and UM ⊗C UN = U(M ⊗C N).

The ﬁrst is obvious. To establish the second we observe that the object UM⊗C UN is (M ⊗C N, m) with coaction ?? ?? r ?rrr ? ? GG s jjj t ØÙ and U(M ⊗C N) is also (M ⊗C N, m) but with coaction

2 TT 22 TTTT2 ' . iiii' s iii t

172 Craig Pastro and Ross Street

The following calculation shows that these two coactions are the same, and hence ∼ the isomorphism U(M ⊗C N) = UM ⊗C UN.

? ? ? ? ? 7 ? 7 ?* ?? ?? rr 7 s ooo 7 s ooo* JJJ ll* rr (2) 7o (c),(10) 7o * (2) ll ** ? = = t = ?? t ? GG ?? G ?? j s jjj t jj G jj s jj s j t s j ØÙ ØÙ ØÙ ØÙ ? OOO Y ?? TTTT OOO 77 YYYYY? TTL XXXXX7 (4)= (c)= LLL (4)= X ?? 4 s jj// s lll t jjj jjjj jjjj t s j t ØÙ This may seem to be a strict equality, but as tensor products are really only deﬁned up to isomorphism, we prefer “strong”.

3.6. Comod f (H) is left autonomous. Let ᐂf denote the subcategory of ᐂ consisting of the objects with a left dual (since ᐂ is braided, left duals are right duals), and suppose that H is a weak Hopf monoid. There is a forgetful functor Ul : Comod(H) → ᐂ deﬁned as the composite of the two forgetful functors Comod(H) → Bicomod(C) and Bicomod(C) → ᐂ. Sometimes this composite Ul : Comod(H) → ᐂ is called the long forgetful functor, as opposed to the short forgetful functor U : Comod(H) → Bicomod(C). Let us say an object M ∈ Comod(H) is dualizable if Ul M has a left dual in ᐂ, that is, Ul M ∈ ᐂ f . Denote by Comod f (H) the subcategory of Comod(H) consisting of the dualizable objects. The goal of this section is to prove the following proposition.

Proposition 3.9. If H is a weak Hopf monoid, then the category Comod f (H) is left autonomous (= left compact = left rigid).

∗ Suppose M ∈ Comod f (H) has a left dual M in ᐂ. Using the antipode of H, a coaction on M∗ is deﬁned as M∗

JJJ . Jν

M∗ A By (17) it is easy to see that this deﬁnes a comodule structure on M∗. We claim ∗ ∗ that M is the left dual of M in Comod f (H). Deﬁne morphisms e : M ⊗C M → C Weak Hopf monoids in braided monoidal categories 173

∗ and n : C → M ⊗C M via

?? OO t e = t and n = O r . OÔÔ ∗ Proposition 3.10. Suppose M ∈ Comod f (H) with underlying ØÙ left dual M . Then M∗ with evaluation and coevaluation morphisms e and n respectively is the left dual of M in Comod f (H). That is, Comod f (H) is left autonomous. Proof. Let M, M∗, e, and n be as above. We will ﬁrst show that e and n are comodule morphisms, and secondly that they satisfy the triangle identities. The following calculation shows that e is a comodule morphism.

H HH ? ?? J J HH J ? t XXX tri J t (c) Jν (4) ν ν t (3),(7) XX ν = = KK = J = = J J t J ν t SSS t t S t ?? ? To show that n is a comodule morphism, we must establish the equality

OO t O r OÔÔ t = OO O r , O OOO TT ν TTTT ØÙ TT ØÙ which is proved by the calculation

OOO t O OO r OOO OO/T OOO OÔÔ ν /TT ν ν tri OO ν,(c) ν (17) J: t = O r t = SSS / t = :: QQQ × S5SS// RR j: OO Q× 55 SSS RRjGjjj : T O ν UUUU 5 S GG TTTT ØÙ UU 55 TTT ØÙ ØÙ ØÙ OOO Q ν OO t (b) QQ 22 ν O r t (4) OO t (4) O = QQQ2 = O = O r = OO . t OO OÔÔ r O ? ? ØÙ e n ØÙ It remains to show that and satisfy ØÙ the triangle identities, that is, that the following composites are the identity: 174 Craig Pastro and Ross Street

∼ n⊗1 ∗ 1⊗e ∼ (i) M = C ⊗C M / M ⊗C M ⊗C M / M ⊗C C = M ;

∗ ∼ ∗ 1⊗n ∗ ∗ e⊗1 ∗ ∼ ∗ (ii) M = M ⊗C C / M ⊗C M ⊗C M / C ⊗C M = M . ∼ ∼ Recall that M = M ⊗C C and M = C ⊗C M via

? G G t ?? t ? G G o oo t , G and s oo , oo respectively. ØÙ ØÙ The calculation

?? O y J OOO s yy s J ?? s r ? T J ? TT r t t J ? ? - tri (c),(13) r J (2),(c) (6) ? - = J = J = O = = 1M , -- r J OOO MMM t ?? MM t OO O M JJJ ? t MM qq OO ?? MMqqq ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ proves (i), and ØÙ (ii) is given by

OO O ν OO TTT TTT t OO ν ν TT t T TTT T r S OO TT r t SS UUUU O r C T UUG J T CC TT r G tri TT t CC (13) TTT (2),ν E (2),(c) ν G = SS C = r E = TT EE = ??www T × T EE T ν ØÙ ?? TT ν ×× TT ν JJJ r w t QQQ OO× JJ ? t GG O Q ν OO J ØÙ ?? t OO ØÙ ?×? t ?? t t ØÙ ? ? OOO ØÙ ØÙ ØÙ ØÙ ØÙ T TTTT T?? TTT (13) ν ? (2) TTT ν T (2),(c) tri = = ?? = TT t = = = ∗ ν ? 1M . ?? t ? r ? ØÙ ∗ ØÙ This completes the proof ØÙ that M is the left dual of M in Comod f (H), and hence that Comod f (H) is left autonomous. Weak Hopf monoids in braided monoidal categories 175

4. Frobenius monoid example

Let R be a separable Frobenius monoid in ᐂ. In this section we prove that R ⊗ R is an example of a weak Hopf monoid with an invertible antipode. In the case ᐂ = Vect, this example is essentially the same as in [Bohm¨ et al. 1999, Appendix]. Let R be a Frobenius monoid in ᐂ. Then R ⊗ R becomes a comonoid via

δ = and = . where, for simplicity, in this section we will adopt the simpler notation

?? = ? = ?? and , ? ØÙ ⊗ and R R becomes a monoid via ØÙ A = AA = t AAt == µ = Ft A == and η = . Fyy ØÙ ØÙ The comonoid structure is via the comonad generated by the adjunction R a R. The monoid structure is the usual monoid structure (viewing R as a monoid) on the tensor product R◦ ⊗ R, where R◦ is the opposite monoid of R.

Proposition 4.1. If R is separable, meaning µδ = 1R, then R ⊗ R is a weak bimonoid. An invertible antipode ν on R ⊗ R is given by

ν = , which makes R ⊗ R into a weak Hopf monoid.

The next three sets of calculations establish the axioms (b), (v), and (w), and hence the ﬁrst claim. Axiom (b) is given by

99 × 9 HH DD oo H D × 99 HHo DD HH DDoo ' 9 9 o H D Ho D ' × 9 9 nat oOOO sep oOo H D ⊗ ⊗ c ⊗ ⊗ = '× 99 9 = = OO = (µ µ)(1 1)(δ δ) × ' 9 99 δµ. × '' 99 9 ?×× ' ? 9 ? ' ? 176 Craig Pastro and Ross Street

Axiom (v) is seen from the diagrams

:: 6 t : 66 ::t 6 99 Õ t?? 6O 9 Õ PP OOO 9ÕÕ : PPP OO PPP E P E : 4 : 6 ØÙ (( : t 4 : 6t ( ::t 44 ::t 6 (( t : 4 t : 66 : 6 / : t?? 44 ?t? OO G : 6 rr / :t (c),tri PPP (c) PPP OO ww GG : :: rr 6 /t :: = PPP = PPP OO w r : 66 // :: PP PP ?r? tJJJ E P E P oo E E ØÙ ØÙ 2 // 22 / : 6 2 / :: 6t . ,, < vv Ò 22 + nn x>x Ô :t 66 .. , vv @@ Ò 2 nn ++ xx > Ô t? : 6 . , vv << ÒÒ ÒÒÒ (c) ?nn 2 + x Ô tri ?Q : . , Ò @@ = ? +x ÔÔ >> = QQQ . vv , ÒÒ< Ò @@ 99 xx + Ô > QO v . ,, Ò << @ 9 + ÔÔ > OOO ?v? ÒÒ ÒJ < @ x?x9 + Ô >> OOO JJ ? Ô :: 6 t ØÙ ØÙ : 66 ::t 6 ?t? 6O (c) PP OOO = PPP OO PPP E P E

For (w), by the naturality of the braiding and the counit property of R, each equation in (w), that is,

9 , GG w , ÕÕ 99 Gww ÕÕ ØÙ 9 ØÙ ØÙ ØÙ ØÙ is easily seen to be equal to the diagram

. ØÙ ØÙ Thus, R⊗ R is a weak bimonoid. We next prove that R⊗ R is a weak Hopf monoid with invertible antipode

ν = .

An inverse to ν is easily seen to be given by

ν−1 = : , Weak Hopf monoids in braided monoidal categories 177 and so the antipode is invertible. We note that (in simpliﬁed form)

? ?? r = and t = . 7 7×× ØÙ ØÙ The following calculations then prove the antipode axioms.

Õ 99 ÕÕ 99 Õ22 z ? 2 z ?? ?? z 2 tri sep ? µ(ν ⊗ 1)δ = 22 = = = t = 27 == 77y ==y 7 yF = 77 ØÙ Fyy

9 ÕÕ 99 ÕÕ % / 9 % //zz %% z / nat sep µ(1 ⊗ ν)δ = % // = = = r % / 7 7 == 77y 7 × 7 × == 7 × × =y 77 yF = 7 ØÙ Fyy

? ?? ?? 5 ?? 5 zz // z % // z 55 /z %% /z z 5 z / % z / 5 6 z // % // 5 66z / % / sep sep z 66 (c) ⊗ ⊗ = == 77y = = 6 = = µ3(ν 1 ν)δ3 = 7 6 ν ==y 7 ØÙ 6 yF = 77 Fyy D N ØÙ DD NNoo DDoo NNN Joo D NN JJttt

Thus, R ⊗ R is a weak Hopf monoid with invertible antipode.

5. Quantum groupoids

In this section we recall the quantum categories and quantum groupoids of Day and Street [2004], where there is a succinct deﬁnition on [page 216] in terms of “basic data” and “Hopf basic data”. Here we give the unpacked deﬁnition of quantum category and quantum groupoid which is essentially found in [page 221]; however, we do make a correction. 178 Craig Pastro and Ross Street

Our setting is a braided monoidal category ᐂ=(ᐂ, ⊗, I, c) in which the functors

A ⊗ − : ᐂ → ᐂ with A ∈ ᐂ, preserve coreﬂexive equalizers, that is, equalizers of pairs of morphisms with a common left inverse.

5.1. Quantum categories. Suppose A and C are comonoids in ᐂ and s : A → C◦ and t : A → C are comonoid morphisms such that the diagram

s⊗t A ⊗ A / C ⊗ C δ rr8 rrr A r c LL LLL δ L& t⊗s A ⊗ A / C ⊗ C commutes. Then A may be viewed as a C-bicomodule with left and right coactions deﬁned respectively via

δ 1⊗s c−1 γl = A / A ⊗ A / A ⊗ C / C ⊗ A ,

δ 1⊗t γr = A / A ⊗ A / A ⊗ C .

Recall that the tensor product P = A ⊗C A of A with itself over C is deﬁned as the equalizer

γr ⊗1 ι / P / A ⊗ A / A ⊗ C ⊗ A . 1⊗γl

The diagrams

⊗ 1⊗γr ⊗1 ι γl 1 / P / A ⊗ A / C ⊗ A ⊗ A / C ⊗ A ⊗ C ⊗ A, 1⊗1⊗γl

⊗ γr ⊗1⊗1 ι 1 γr / P / A ⊗ A / A ⊗ A ⊗ C / A ⊗ C ⊗ A ⊗ C 1⊗γl ⊗1 may be seen to commute and therefore induce respectively a left C- and right C- coaction on P. These coactions make P into a C-bicomodule. The commutativity of the diagram

⊗ ⊗ ⊗ 1⊗1⊗γr ⊗1 ι δ δ ⊗ 1 c 1 ⊗ / P / A ⊗ A / A 4 / A 4 / A ⊗ A ⊗ A ⊗ C ⊗ A 1⊗1⊗1⊗γl Weak Hopf monoids in braided monoidal categories 179 may be seen from

ι ι ι ι y EE y + 77 ?? Eyy E* 2yy + $RR ?? EE * 22 ++ $ R s * :: EE * = + = $$ = ** ØØ :; , E ** t + $ * Ø ;; 33 * ; 2 3 < * Ø s * ; 22ww 3 G < * Ø 6 t * ;; 2 33 GG ; *Ø 6 Ô * ;;y 2 3 GG ;; Ø * ÔÔ5 * yy ; 2 3 G ; ØØ * ÔÔ 5 and since 1 ⊗ 1 ⊗ ι is the equalizer of 1 ⊗ 1 ⊗ γr ⊗ 1 and 1 ⊗ 1 ⊗ 1 ⊗ γl , there is a unique morphism δl : P → A ⊗ A ⊗ P making the diagram ι δ⊗δ P / A ⊗ A / A ⊗ A ⊗ A ⊗ A

δl 1⊗c⊗1

A ⊗ A ⊗ P / A ⊗ A ⊗ A ⊗ A 1⊗1⊗ι commute. In strings,

ι δl 22 = 99 2 ι . : : Ô// * ::Ô / * Ô : / '&%$ !"# * It is easy to see (postcompose with the monomorphism 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ ι) that the morphism δl is the left coaction of the comonoid A ⊗ A on P that makes P into a (left) A ⊗ A-comodule. This means that the diagrams

δl P / A ⊗ A ⊗ P δl P / A ⊗ A ⊗ P δl E EE EE A ⊗ A ⊗ P 1⊗1⊗δl EE ⊗⊗1 1 EE EE δ⊗δ⊗1 E" P 1⊗c⊗1⊗1 A ⊗ A ⊗ A ⊗ A ⊗ P / A ⊗ A ⊗ A ⊗ A ⊗ P commute.

5.2. The deﬁnition. We are now ready to state the deﬁnition. A quantum category in ᐂ consists of the data A = (A, C, s, t, µ, η) where A, C, s, t are as above, and µ : P = A ⊗C A → A and η : C → A are morphisms in ᐂ, called the composition morphism and unit morphism, respectively. This data must satisfy axioms (B1) through (B6) below. (B1) (A, µ, η) is a monoid in Bicomod(C). 180 Craig Pastro and Ross Street

(B2) The following diagram commutes. t⊗⊗1 ⊗ δl / 1 µ P / A ⊗ A ⊗ P / C ⊗ P / C ⊗ A ⊗s⊗1 Before stating (B3), we use (B2) to show that the diagram ⊗ ⊗ γr ⊗1⊗1 δl 1 1 µ / P / A ⊗ A ⊗ P / A ⊗ A ⊗ A / A ⊗ C ⊗ A ⊗ A 1⊗γl ⊗1 commutes, as seen by the calculation

δl δl δl %5 : 9 δl δl3 % 55 :: 9 Õ !! 3 , 5 : δl ÕÕ ! 3 = ,, 5 = δl = # 3 = . ! 3 '&%$ !"# 55 '&%$ !"# + '&%$ !"# # 33 ! µ 5 ÕÕ + # 3 '&%$ !"#s t '&%$ !"#! µ t + s µ µ ! × µ '&%$ !"# ! t × '&%$ !"# ×× ØÙ × ØÙ ØÙ Since ι⊗1 is the equalizer of γr ⊗1⊗1 and 1⊗ γl ⊗1, there is a unique morphism δr : P → P ⊗ A making the square

δl P / A ⊗ A ⊗ P

δr 1⊗1⊗µ ι⊗1 P ⊗ A / A ⊗ A ⊗ A commute. We can now state (B3). (B3) The following diagram commutes. µ PA/

δr δ µ⊗1 P ⊗ A / A ⊗ A (B4) The following diagram commutes. µ PA/

ι ⊗ A ⊗ A / I (B5) The following diagram commutes. C OO OOO OO' η 7 I ooo ooo A o Weak Hopf monoids in braided monoidal categories 181

(B6) The following diagram commutes.

δ s⊗1 A / A ⊗ A / C ⊗ A η |= BB η⊗1 || BB | BB || η δ ! C / A / A ⊗ A BB |= BB || η B |η⊗1 B! δ t⊗1 || A / A ⊗ A / C ⊗ A

A consequence of these axioms is that P becomes a left A ⊗ A-, right A- bicomodule. The axiom (B6) makes C into a right A-comodule via

η δ s⊗1 C / A / A ⊗ A / C ⊗ A .

We refer to A as the object-of-arrows and C as the object-of-objects.

5.3. Quantum groupoids. Suppose we have comonoid isomorphisms =∼ =∼ υ : C◦◦ / C and ν : A◦ / A .

⊗3 Denote by Pl the left A -comodule P with coaction deﬁned by

δ 1⊗1⊗1⊗ν 1⊗1⊗cP,A P / A ⊗ A ⊗ P ⊗ A / A ⊗ A ⊗ P ⊗ A / A ⊗ A ⊗ A ⊗ P ,

⊗3 and by Pr the left A -comodule P with coaction deﬁned by

−1 −1 c δ 1⊗1⊗1⊗ν A⊗A⊗P,A P / A ⊗ A ⊗ P ⊗ A / A ⊗ A ⊗ P ⊗ A / A ⊗ A ⊗ A ⊗ P .

⊗3 Furthermore, suppose that θ : Pl → Pr is a left A -comodule isomorphism. We deﬁne a quantum groupoid in ᐂ to be a quantum category A in ᐂ equipped with an υ, ν, and θ satisfying (G1) through (G3) below.

(G1) sν = t, (G2) tν = υs, and (G3) the diagram3

ς cC,C⊗C P / C ⊗ C ⊗ C / C ⊗ C ⊗ C

θ 1⊗1⊗υ ς P / C ⊗ C ⊗ C

3This corrects [Day and Street 2004, Section 12, page 223]. 182 Craig Pastro and Ross Street

commutes, where the morphism ς : P → C⊗3 is deﬁned by taking either of the equal routes

γr ⊗1 ⊗ ⊗ ι / s 1 t ⊗ P / A ⊗ A / A ⊗ C ⊗ A / C 3. 1⊗γl

6. Weak Hopf monoids and quantum groupoids

The goal of this section is to prove the following theorem. Theorem 6.1. There is a bijection in ޑᐂ between weak bimonoids and quantum categories with separable Frobenius object-of-objects. Also, if the weak bimonoid is equipped with an invertible antipode, making it a weak Hopf monoid, then the quantum category becomes a quantum groupoid. In the case ᐂ = Vect, it has been shown in [Brzezinski´ and Militaru 2002, Proposition 5.2] that a weak Hopf monoid with invertible antipode yields a quantum groupoid. Concerning the converse, we would like to warmly thank Gabriella Bohm¨ for not only suggesting that it may be true, and pointing out that the ᐂ = Vect case appears in the PhD thesis of Imre Balint´ [2008a], but for also helping us with the proof. The theorem has an obvious corollary. Corollary 6.2. Any Frobenius monoid in ᏽᐂ yields a quantum groupoid in ᏽᐂ. Proof. By Proposition 4.1, every Frobenius monoid R in ᏽᐂ leads to a weak Hopf monoid with invertible antipode R ⊗ R. Apply Theorem 6.1 to this weak Hopf monoid with invertible antipode to get a quantum groupoid in ᏽᐂ. Now let us discuss the necessary data for the theorem. The proof involves many string calculations, which may be found in Sections 6.1 and 6.2. Suppose A = (A, 1) is a weak bimonoid in ᏽᐂ with source morphism s and target morphism t, and put C = (A, t). Our claim is that this data along with

77 × µ = 7×× : P → A, η = t : C → A forms a quantum category in ᏽᐂ. If moreover A is a weak Hopf monoid in ᏽᐂ with an invertible antipode ν : A → A, then setting

υ = tννt : C◦◦ → C, ν = ν : A◦ → A, θ = : P → P. ν yields a quantum groupoid in ᏽᐂ. The details will be proved in Section 6.1. Weak Hopf monoids in braided monoidal categories 183

For the other direction, recall from Section 3.1 that, if C is a separable Frobenius monoid and M and N are C-comodules, we may form M ⊗C N, the tensor product over C of M and N. Moreover, the tensor product over C is a retract of the tensor product in ᐂ so that ι M ⊗C N / M ⊗ N has a retraction m : M ⊗ N → M ⊗C N. Again, from Section 3.1, we see that we may explicitly write ιm as

δr ⊗δl 1⊗µ⊗1 1⊗⊗1 ιm = M ⊗ N / M ⊗ C ⊗ C ⊗ NM/ ⊗ C ⊗ N / M ⊗ N.

Graphically, M N ⊗ ⊗ 7 M N M C N M C N 7 ×× m OCC ι M ⊗ N = OOooo = C and M N . (@) ι 7 ×× 7 m M N M N ØÙ Now suppose A = (A, C, s, t, µ, η) is a quantum category in ᐂ, in which C = (C, µ, η, δ, ) is a separable Frobenius monoid. The comonoid A is therefore a C-bicomodule, so that ι : P → A ⊗ A has a retraction m : A ⊗ A → P. This is such that ?? ?? r = t s rr = ιm DDzz and mι 1P . A If we then deﬁne a multiplication and ØÙ unit for as m µ η η µ = (A ⊗ A / P / A) , η = (I / C / A), where η : I → C comes from the fact that C is a Frobenius module, then we have the data for a weak bimonoid A. That this is actually a weak bimonoid will be proved in Section 6.2. Let us see that this correspondence between weak bimonoids and quantum categories with separable Frobenius object-of-objects is one-to-one. Suppose we have a weak bimonoid A = (A, µ, η, δ, ) in ᏽᐂ. It becomes a quantum category in ᏽᐂ by setting

C := (A, t), s := s, t := t, η := t, and µ := µ.

This quantum category then becomes a weak bimonoid by setting

µ := µ ◦ m and η := t ◦ η 184 Craig Pastro and Ross Street where, in this case, ?? ?? r JJJ = t s rr (2)= GG mm m DDzz Gmmm ◦ m = and, as we see from the ﬁrst paragraph ØÙ of Section ØÙ 6.1, µ µ, and moreover, the morphism t ◦ η = η by axiom (2) for weak bimonoids. Thus, we have ended up with the weak bimonoid A that we started with. Let us now go in the other direction. Suppose that we have a quantum category A = (A, C, s, t, µ, η) with (A, δ, ) a comonoid, and C = (C, µC , ηC , δC , C ) a separable Frobenius monoid. Then A becomes a weak bimonoid with

0 0 µ := µ ◦ m, where m : A ⊗ A → P, and η := η ◦ ηC .

We note that the source and target morphisms for the weak bimonoid are given by

s0 := η ◦ s and t0 := η ◦ t.

First, we observe that C =∼ (A, t) and P =∼ (A ⊗ A, ιm) respectively via η m x w 0 C P A ⊗ A m (A, t ) t / ( , 1) and ( , 1)(ι / , ι ).

The ﬁrst isomorphism is established in one direction by deﬁnition and in the other by tη = (1 ⊗ t)( ⊗ 1)δη

= ( ⊗ 1)(η ⊗ 1)δC since η is a C-comodule morphism

= (C ⊗ 1)δC by (B5)

= 1C . The second isomorphism is again established in one direction by deﬁnition and in the other by the fact that m is a retract of ι. This weak bimonoid becomes a quantum category by stripping off the m from µ0 so that we are left with the original µ. Since t0 is a morphism from (A, t) to (A, 1), if we wish to consider it as a morphism from C to A, we must precompose with η : (C, 1) → (A, t). This gives t0 ◦ η = η ◦ t ◦ η = η, and so we are left with the original η. We have already seen that C =∼ (A, t), so we are left with our original quantum category. This establishes the bijection one-to-one correspondence between quantum categories and weak bimonoids, and moreover shows how to construct a quantum groupoid from a weak Hopf monoid. The remainder of this section is devoted to proving the details of the theorem. Weak Hopf monoids in braided monoidal categories 185

6.1. Weak bimonoids yield quantum categories. In this section, we prove that a weak bimonoid in ᏽᐂ yields a quantum category in ᏽᐂ. Suppose that A = (A, 1) is a weak bimonoid with source and target morphisms s and t, respectively. Set C = (A, t) as usual. The morphisms s and t are obviously in ᏽᐂ, hence so is η = t, and ?? JJ DD z 77 × Jllll nat DDz (b) 7 × = zz D = × = µ 77 × 7×× ØÙ shows that µ is as well. Recall that ØÙ

? J ? P = (A ⊗ A, JJllll ).

The morphisms δ : P → A ⊗ A ⊗ P and δ : P → P ⊗ A are given by l r ØÙ

Q ?? QQ JJ l QQQ = Jlll = δl and δr ?? . :: Ô** JJJ lll ::Ô * l Ô : * ØÙ The two calculations ØÙ

?? ? JJJ lll J ? l JJllll ?? 4 4 (c) ? JJJ ll 44 Ô44 = ? = ll 4Ô 4 JJJ lll Ô 4 4? l 44 44 J ? 4 ÔÔ 4 ØÙ JJllll 44 ØÙ Ô44 44 44 44Ô 44 Ô Ô 4 4 ØÙ ØÙ and ØÙ ?? QQQ iii TT i TTT QQQ TDD z QQ QQ (c) DDz (b) Q QQQ = ? zz D = ? Q J ? J ? ? JJllll JJllll J ØÙ ? JJllll ØÙ show that these are morphisms in ᏽᐂ. ØÙ ØÙ To see that (A, µ, η) is ØÙ a comonoid in Bicomod(C), notice that associativity follows from that of µ viewed as a weak bimonoid, and the counit property may be seen from property (6), that is,

−1 µ(1 ⊗ t)δ = 1A and µ(s ⊗ 1)c δ = 1A, 186 Craig Pastro and Ross Street and so (B1) holds. (B2) follows from one application of (12), (B3) follows from (b), (B4) from (c), and (B5) follows from (2). The calculation

t t t (3) t (8) s = = t s t t t veriﬁes (B6). Thus, A = (A, C, s, t, µ, η) is a quantum category in ᏽᐂ. We now wish to show that a weak Hopf monoid with invertible antipode yields a quantum groupoid. So suppose that our weak bimonoid A is equipped with an invertible antipode ν : A → A, and set

υ = tννt : C◦◦ → C, ν = ν : A◦ → A, θ = : P → P. ν

The morphisms υ and ν are obviously morphisms in ᏽᐂ, and the two calculations

? J ? JJllll 44 :R:RR 44 (c) ::RRRR4 (b) = :: = ν ØÙ TT ν ν TTTT ØÙ and

ν (17) (2) ν t (16) t t (4) = t = = = t ? ν J ? JJJ r ν ν JJllll ν JJ JJ J J ν J ν JJ ν ØÙ ØÙ ØÙ ØÙ ØÙ

(16)= ν (2)= (c)= =ν (2),(c)= ν 4 r ν r JJJ 444 ν JJ ν ν ν J ν ØÙ ØÙ show that θ is as well. ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 187

Lemma 6.3. An inverse for θ is given by

O OOO :: ν-1 θ −1 = : , : :? J ? JJl76540123lll

ØÙ Proof. Since

?? JJ JJ l (c) JE (b) Jlll = EEy = JJJ , O y E JJ OOO

ØÙ ØÙ it is clear that θ −1 is a morphism in ᏽᐂ. That θ −1 is an inverse for θ may be seen in one direction from

? 9 ν ? ν 99 ?? 9 ?? KK ? OO ?? 99 vv − OO (17) ? 9 (c) v K 1 -1 -1 -1 ?vv -1 θ θ = = ν ν = / ν = ?? ν -1 4 / ? y :: ν 4 / fffyy : O 4 J / fff ? : OO Ô J Ô Q ? :? 76540123ÔÔOO?76540123 ÔÔJJ?76540123 QQiiii 76540123 J ? ÔJÔ ? ÔJÔ ? JJl76540123lll JJllll JJllll

ØÙ ? ?? ? ØÙ ? ØÙ o J ? ØÙ s jjj ? s o JJ lll ? (†) kk (4) ?oo (2) l J ? = ?? = = ?? = JJllll OO j ?? JJ l Ojjj OOO jjj Jlll j ØÙ ØÙ ØÙ = 1P , ØÙ ØÙ ØÙ where (†) is given by

? ?? ?? r ?? ν ν KK v GG (17) ν (15) vK = wwG = = = s vv -1 ww G , v?? ν ν-1 ν-1 -1 ?? y EE ν-1 ν yy Eyy y 76540123 76540123 76540123 76540123 76540123 188 Craig Pastro and Ross Street and in the other direction by

OO OO O OOO -1 TTT TT :: ν T TT : -1 444 ν −1 :: (‡) Ô (c) D ν-1 (17) -1 θθ = ?? = Ô4 = D = ν J JJ76540123ll ÔÔ 4 zzzDD ll 4 z?? 76540123 ? ν ν 76540123 7776540123× ν ν × ØÙ ν T J TTT (c),ν ?? (4) t ?? (2) ?? = = t = = JJJ l = ? rrr lll 1P , ν ? ? r ? ØÙ for which the ﬁrst step (‡) holds because θ is a ØÙ morphism in ᏽᐂ. That the antipode ν : A◦ → A is a comonoid isomorphism is our assumption. That υ : C◦◦ → C is as well may be seen from the calculation (t ⊗ t)δυ = (t ⊗ t)δtννt = (t ⊗ t)δννt by (3) = (t ⊗ t)c(ν ⊗ ν)δνt by (17) = (t ⊗ t)c(ν ⊗ ν)c(ν ⊗ ν)δt by (17) = (t ⊗ t)(ν ⊗ ν)(ν ⊗ ν)ccδt by nat = (t ⊗ t)(t ⊗ t)(ν ⊗ ν)(ν ⊗ ν)ccδt by (7) = (t ⊗ t)(ν ⊗ ν)(r ⊗ r)(ν ⊗ ν)ccδt by (16) = (t ⊗ t)(ν ⊗ ν)(ν ⊗ ν)(t ⊗ t)ccδt by (16) = (t ⊗ t)(ν ⊗ ν)(ν ⊗ ν)(t ⊗ t)ccδ by (5) = (υ ⊗ υ)ccδ by (5).

An inverse for υ is given by the morphism υ−1 = tν−1ν−1t, as may be seen in one direction by the calculation

υ−1υ = tν−1ν−1ttννt = ttν−1ν−1ννtt by (16) = tν−1ν−1ννt by (7) = tt

= t = 1C by (7). Weak Hopf monoids in braided monoidal categories 189

The other direction is similar. Recall that the left A ⊗ A-, right A-coaction δ on P is deﬁned by taking the diagonal of the commutative square

δl P / A ⊗ A ⊗ P

δr 1⊗1⊗δr

δl ⊗1 P ⊗ A / A ⊗ A ⊗ P ⊗ A.

We note that δ may be written as

U UUUU U UUUU MM ?? UU MMM JJJ ll ? EE y PP ll J ? (c) EyE (b) PPP ?? = JJllll = y = PP = δ. JJJ lll :: Ô** l : * : / ::Ô * :: Ô* :: / : * :: Ô** :Ô * :: // Ô : ØÙ :Ô * Ô : * Ô : * ØÙ ØÙ ØÙ

⊗3 We must show that θ is a left A -comodule isomorphism Pl → Pr . That is, we must prove the commutativity of the square

γ ⊗3 Pl / A ⊗ Pl

θ 1⊗θ γ ⊗3 Pr / A ⊗ Pr ,

⊗3 where the left A -coactions on Pl and Pr were deﬁned using δ (see Section 5.3). The clockwise direction around the square is

P PPP PPP KK PPP PPP KK O P 44 KKK OOO ν B 44 KK OOO << 6 (17) B (c) KK (§) < 66 = BB ν = KK = << 6 BB (( ν < 6 B ØØ ( ν ν << BBØ ( ?? : * < ν ν ( C ν : Ô* < 7 EØØ CC CC v ::Ô * 7 EE CC Cv : * ν 77ØØ CCv CC Ô 7 v C C Ø 7 190 Craig Pastro and Ross Street where the last step (§) is given by the calculation

O OOOoo ?? KK s KK s ooo OOO ?? KKs KKsK Õ9o9 Õ99 ν s Kν ν s ν Õ 9 Õ 9 (17) G (b) (17) ν Õ ν ν Õ ν ν = G w = = M - w KK s MMM - qqq KKsK MM qq - ?? ?? ss K q MM - ?? ? ? qqq MM- OO OOoo < ooo OOO < Õ,o, Õ,, << ÕÕ ,, ÕÕ ,, << ν M ν q < (17) MM --qq nat < = MMMqq - = ν ν . qq MMM-- 77 × q4qq M ×? ν 44 ?? { ??{ ν {{ ?? { The counterclockwise direction is

+ ++ ++ TTT J TTTT JJJttt TTTT ν ?? t JJ LLL HH ν ν DD tt J LJJ M ' HHH DD tGtt J MM ' (17) H ÖÖ (b) ν (c) G zz ν MM', = ÖÖH = DD = zzGG , Ö H D z -1 ν-1 ν-1 C ν-1 JzJJ ν ν-1 CC CC Jtt C{{{ C{{{ rr rrr r 76540123 ## * $ 76540123 765401234 $ 765401234 $ # **76540123ÔÔ$$ 4 $ 4 $ # * $ 44 $ 44 $ # Ô 4 $ 4 $ L R RR LL RRR OO RRR LL RRR OOO RRR LL R OO ?? (c) L (11) (c),(6) = 9 s ν = s = t = ν . 9 s ν ν ss9 zz ww ss % zz ww : * % z % w % ::ÔÔ** % % % :: * % % Ô

⊗3 Thus, θ is a left A -comodule morphism Pl → Pr . The inverse of θ then is a ⊗3 left A -comodule morphism Pr → Pl . We now prove the properties (G1) through (G3) required of a quantum groupoid. The calculation

ν t ν (12) (16) (12) t (8) = r = r = = t s s s s Weak Hopf monoids in braided monoidal categories 191 veriﬁes (G1), and (G2) is established by

s s ν r t ν (7) (16) (15) ν (8) def s = t = ν = = ν = . t ν υ t t ν t t It remains to prove (G3), that is, we must show that θ makes the square ς cC,C⊗C P / C⊗3 / C⊗3

θ 1⊗1⊗υ ς P / C⊗3 commute. The clockwise direction around the square is

? J ? ?? ?? JJllll JJJ ll kk ll kkk Ô ttJJJ 7 7 ÔÔ Mt qq ×× ×× 7 ×× ×× 7 s Ô s qqM t s s t s s t ØÙ × × × × Ó (10) ØÙ ×× (4) ×× = × = × = ν ÓÓ t ×× × t ×× × t Ó ?? s t t ν ν ν ν ν ν t t t for which the last step holds since tννts = tννs by (8) = tνr by (15) = ttν by (16) = tν by (7). The counterclockwise direction is

?? ?? ?? ν ?? ν ν t ν t ν (17)= ν ν (2)= (16)= (4)= ?? JJ r ν Q i J t t ν t QQiii t JJ JJ J 7 7 J J JJ × 7 t × 7 × s × t 7 7 7 s t × 7 × 7 × 7 s × t s × t s × t ØÙ ØÙ ØÙ ØÙ ØÙ 192 Craig Pastro and Ross Street

?? (16) ν ν (2) ?? ν ?? (2),(c) = = ν ν = s ν = ν . r t JJJ ?? JJJ t t s t t ×77 ×77 ×77 s × t s × t s × t ØÙ ØÙ ØÙ This establishes the commutativity of the square.

6.2. Quantum categories are weak bimonoids. In this section we prove that a quantum category with a separable Frobenius object-of-objects yields a weak bimonoid. Thus, suppose that A = (A, C, s, t, µ, η) is a quantum category in ᐂ with C = (C, µ, η, δ, ) a separable Frobenius monoid. The object A is a comonoid in ᐂ, and our goal here is to show that equipping it with a multiplication and unit as m µ η η µ = (A ⊗ A / P / A) , η = (I / C / A) then yields a weak bimonoid A in ᐂ (and hence in ᏽᐂ). Let us begin by establishing that the multiplication and unit deﬁned here give a monoid structure on A. Note that the morphisms

η ⊗C 1 : C ⊗C A → A ⊗C A and 1 ⊗C η : A ⊗C C → A ⊗C A are the unique morphisms such that (η ⊗ 1)ι = ι(η ⊗C 1) : C ⊗C A → A ⊗ A and (1 ⊗ η)ι = ι(1 ⊗C η) : A ⊗C C → A ⊗ A, respectively. Thus, we have

η ⊗C 1 = m(η ⊗ 1)ι and 1 ⊗C η = m(1 ⊗ η)ι, so that

(η ⊗C 1)λ = m(η ⊗ 1)δl and (1 ⊗C η)ρ = m(1 ⊗ η)δr , where λ : A → C ⊗C A and ρ : A → A⊗C C are the left and right unit isomorphisms, respectively. Also, µ⊗C 1 and 1⊗C µ are the unique morphisms such that (µ⊗1)ι = ι(µ⊗C 1) and (1 ⊗ µ)ι = ι(1 ⊗C µ). Thus,

1 ⊗C µ = m(1 ⊗ µ)ι and µ ⊗C 1 = m(µ ⊗ 1)ι.

Given these identities, one of the unit conditions is seen from the calculation Weak Hopf monoids in braided monoidal categories 193

η η ?? ?? Joo?? 7 ?? s ee o ØÙ ?? yy ?? s oo JJ λ λ η m t s y η 7 (@) ØÙ (@) 4 ØÙ (f) m = ι = = = η = η⊗C 1 = = 1A. ØÙ ?? −1 m m λ µ m µ m µ ØÙ µ µ /.() *+-, µ ØÙ 76540123 The third equality uses the fact that η is a C-comodule morphism. The other unit condition may be calculated similarly. To establish associativity, we ﬁrst prove the following lemma. PA AP 7 7 7 ×× 7 ×× m µ m µ 7 × Lemma 6.4. ι = m and ι = m . × 7 µ ι µ ι × 77 × 77 AA × AA × Proof. We prove the ﬁrst equality. The second is similar. 7 7 ×× µ m ? µ ( ) γr γl ? ? ( ) 7 @= 22 = ? rr @= m ι 2 t s r , µ DDzz µ× ι 7 ×× 7 ØÙ where the second step follows since µ is a C-comodule ØÙ morphism. The following calculation then shows that associativity holds. , ,, ,, ,, , , , m m m , ,, 22 ,, µ m 2 m m m 2 µ m m 2 ι ι 2 = = = = m ι µ µ µ⊗ 1 22 22 C µ m m m µ /. -, µ µ µ () *+ , , ,, ,, m ,, m , , ,, m m µ m

m m µ = = ι 2 = = 2 1⊗C µ µ ι m µ µ m m /. -, () *+ µ µ 194 Craig Pastro and Ross Street

This then proves that (A, µ, η) is a monoid in ᐂ. We now prove the remaining axioms for a weak bimonoid. Axiom (b) is given by the calculation

77 × 77 × 7 m× 7 m× 7 × 7 × m× 7 7 m× 7 × 7 ×× m× m δr deﬁnition δl ι δ / G (B3) ( ) of δl l ( ) / w G @ ι / @ m ι Gw µ = δr = = / = = GG www m µ GG m'&%$ !"# '&%$ !"# mw m ×77 µ '&%$ !"# µ m × 7 µ × '&%$ !"# µ µ µ µ ? L KK s J ? LLL ? KKs J t s lll ?? ? m s Km Jt t s lll KK s L m Jt KKsK (@=) LLL rrr (c)= (@=) µ ι (@=) m s m LLrL . m r rr Lm µ m µ µ ØÙ m µ µ µ ØÙ µ Axiom (v) is established with three calculations. The ﬁrst is given as follows, where the third equality below follows from the fact that µ is a C-comodule morphism.

, , ,, ,, ,, , ,, ,, m m m , , m m µ µ ?? 2 (B4) 2 (@) µ r (B4) 2 = 2 = = γr s rr = γr s m m ?? ?? ?? J l ?ww ?×× t s ll µ ι µ ι Jt , ,, '&%$ !"# ,'&%$ !"# , ØÙ ,, ØÙ ØÙ m ØÙ J ØÙ ØÙ ØÙ ØÙ deﬁnition ØÙ ØÙ ØÙ ØÙ tHtt JJ of γr (@) ? vvv ?? = ι = ? vv HH = t s t s ?? t ? s t 4 s ?? ?? s ? 4 t ? ? ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ The second: ØÙ

? GG ? mww m ? m m (B4) (@),(c) ? = = t ? s t ? s ι ι ? ? µ µ , , , , ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 195

The third:

J J Mttt JJ Mttt JJ J M qq M qq tHtt JJ m qqMm (B4) m qqMm (@),(c) vv ?? = = vvvHH = t s t s t ? s t 4 s ?? ?? µ µ ι ι ? 4 , , , , ØÙ ØÙ ØÙ ØÙ To ØÙ prove ØÙ the ﬁnal axiom ØÙ ØÙ ØÙ (w) ØÙ for a weak bimonoid, we need a lemma.

s s s s η η η η η η 2 2 Lemma 6.5. 2 = ? and 2 = ooOO . ØÙ m ? m ØÙ yy t s y ι ι / η / η / / Proof. The ﬁrst property of the lemma may be seen as follows. (The second equality below holds since η is a C-comodule morphism.)

s s s s s s s η η η η η η η η (@) η (f) (B6) (B6) 22 = = = OO = G = ? = ? ØÙ m ooOO ooOO ooy w G ? ? ØÙ J yy yy yy t w s s t t J t s y ? s y s G w J η ?? w ι t t η ww G η η η η // ØÙ η ØÙ ØÙ The second property of the lemma is proven as

s s s s s η η η η η η (@) η (f) 22 = ØÙ OO = = = OO , m ØÙ oo ? 2 ?? oo J yy ? 22 t yy t s y t D s y Jt D ØÙ η ι/ η / η ØÙ ØÙ where the second equality holds again since η is a C-comodule morphism and the last equality follows from the proof of the ﬁrst part. 196 Craig Pastro and Ross Street

The following two calculations prove the axiom (w). In both calculations the last equality follows from the monoid structure on A.

η η ØÙ ØÙ η η s s ØÙ η η η η ØÙ Lemma η s 7 η ?? η ØÙ ?? ØÙ (B6)= (@=),(c) ØÙ m 6.5= (B6)= t = m η ? ØÙ ?? ? ØÙ 4 m ι t η 44 µ 4 7×× - 4 µ 7 -- η m m - m µ µ µ η η η η η ×× ØÙ η yy ØÙ s ØÙ QQ ØÙ M ØÙ s y η η mmQmQm ØÙ mmmMm m?? m?? η Q m m η η Lemma ØÙ Q mmm ØÙ (@) (@),(c) (B6) 7 6.5 ? m??m QQ = = = m× = ? m ι ι ØÙ yy ?? s y %% ι µ m % OO m η m D µ DD µ m µ µ η ØÙ o?o OO ÕÕ η (B6)= s Õ ??? = ØÙ 4 η 44 ?? 44 m

µ This completes the proof. Thus, a quantum category with a separable Frobenius object-of-objects yields a weak bimonoid.

Appendix A. String diagrams and basic deﬁnitions

In this appendix we give a quick introduction to string diagrams in a braided monoidal category ᐂ = (ᐂ, ⊗, I, c) [Joyal and Street 1993] and use these to deﬁne monoid, module, comonoid, comodule, and separable Frobenius monoid in ᐂ. The string calculus was shown to be rigorous in [Joyal and Street 1991]. Weak Hopf monoids in braided monoidal categories 197

A.1. String diagrams. Suppose that ᐂ=(ᐂ, ⊗, I, c) is a braided (strict) monoidal category. In a string diagram, objects label edges and morphisms label nodes. For example, if f : A ⊗ B → C ⊗ D ⊗ E is a morphism in ᐂ, it is represented as

AB / // / f = f , 33 33 3 C '&%$ !"#DE where this diagram is meant to be read top to bottom. The identity morphism on an object will be represented as the object itself, as in

A A = .

A special case is the object I ∈ ᐂ, which is represented as the empty string. If, in ᐂ, there are morphisms f : A⊗B →C⊗D⊗E and g : D⊗E⊗F → G⊗H, then they may be composed as

f ⊗1 1⊗g A ⊗ B ⊗ FC/ ⊗ D ⊗ E ⊗ F / C ⊗ G ⊗ H, which may be represented as vertical concatenation

,, f (1 ⊗ g)( f ⊗ 1) = g '&%$ !"# , , '&%$ !"# (where we have left off the objects). The tensor product of morphisms, say / / // // / / f and g , // // // // '&%$ !"# '&%$ !"# is represented by horizontal juxtaposition:

// // // // f ⊗ g = f g // // // // '&%$ !"# '&%$ !"# (again leaving off the objects). 198 Craig Pastro and Ross Street

The braiding cA,B : A ⊗ B → B ⊗ A is represented as a left-over-right crossing. The inverse braiding is then represented as a right-over-left crossing: AB BA 22 22 2 −1 2 cA,B = 22 , c = . 2 A,B 22 2 2 BA AB Suppose A ∈ ᐂ has a chosen left dual A∗, which we denote by A∗ a A (it would be an adjunction if we were to view ᐂ as a one object bicategory). The evaluation ∗ ∗ and coevaluation morphisms eA : A ⊗ A → I and n A : I → A⊗ A are represented as A∗ A eA = and n A = . A A∗ The triangle equalities become A A∗ A A∗ ∗ A = and A = .

A A∗ To simplify the string diagrams in what follows, we will omit the nodes from certain morphisms (for example, multiplication and comultiplication morphisms) or simplify them (for example, unit and counit morphisms).

A.2. Monoids and modules. A monoid A = (A, µ, η) in ᐂ is an object A ∈ ᐂ equipped with morphisms 2 22 µ = 2 : A ⊗ A → A and η = : I → A,

ØÙ called the multiplication and unit of the monoid respectively, satisfying 9 9 9 9 9 99 ÕÕ 99ÕÕ ÕÕ 99 9ÕÕ 4 ÕÕ 99 9ÕÕ = 9ÕÕ = 9ÕÕ and 4ÕÕ = = 9 . (m) ØÙ ØÙ If A and B are monoids, a monoid morphism f : A → B is a morphism in ᐂ satisfying AA AA 7 77 ×× A × = f f = 22 and f . f 2 B B ØÙ '&%$ !"# '&%$ !"# ØÙ B B '&%$ !"# Monoids make sense'&%$ !"# in any monoidal category, however, in order that the tensor product A ⊗ B of monoids A, B ∈ ᐂ should again be a monoid, there must be a Weak Hopf monoids in braided monoidal categories 199

“switch” morphism cA,B : A ⊗ B → B ⊗ A in ᐂ given by, say, a braiding. In this case, A ⊗ B becomes a monoid in ᐂ via / / // // µ = / / and η = . ØÙ ØÙ Suppose that A is a monoid in ᐂ.A right A-module in ᐂ is an object M ∈ ᐂ equipped with a morphism

MA 2 22 µ = 2 : M ⊗ A → M, M called the action of A on M, satisfying

MAA MAA M 99 Õ Õ 99 99 Õ ,, M 9Õ9 Õ = 99 Õ , = ÕÕ ÕÕ and , A . (m) M M M ØÙ Notice that we use the same label “(m)” as in the monoid axioms (and “(c)” below for the comodule axioms). This should not cause any confusion as the labeling of strings disambiguates a multiplication and an action; however, the labeling will usually be left off. If M and N are modules, a module morphism f : M → N is a morphism in ᐂ satisfying MA MA 7 77 ×× × f = 22 . f 2 N '&%$ !"#N '&%$ !"# A.3. Comonoids and comodules. Comonoids and comodules are dual to monoids and modules. Explicitly, a comonoid C =(C, δ, ) in ᐂ is an object C ∈ᐂ equipped with morphisms

δ = 2 : A → A ⊗ A and = : A → I, 22 2 called the comultiplication and counit of the comonoid ØÙ respectively, satisfying

9 = 9 = 9 and 9 = = 44 . (c) ÕÕ 99 Õ9Õ 99 ÕÕ 99 99 ÕÕ ÕÕ 9 ÕÕ 9 9 ÕÕ ÕÕ 9 9 ÕÕ

ØÙ ØÙ 200 Craig Pastro and Ross Street

If C, D are comonoids, a comonoid morphism f : C → D is a morphism in ᐂ satisfying A A A f 22 2 f A = f f and = . ×77 B ××'&%$ !"#7 BB BB '&%$ !"# '&%$ !"# '&%$ !"# ØÙ Similarly here, ᐂ must contain a switch morphism ØÙ cC,D : C ⊗ D → D ⊗ C in order that the tensor product C ⊗ D of comonoids C, D ∈ ᐂ should again be a comonoid. In this case the comultiplication and counit are given by

δ = / / and = . // // / / Suppose that C is a comonoid in ᐂ.A right C-comodule ØÙ ØÙ in ᐂ is an object M ∈ ᐂ equipped with a morphism M = 2 : M → M ⊗ C γ 22 , 2 M C called the coaction of A on M, satisfying

M M M M 9 = 9 44 = ÕÕ 99 Õ9Õ 99 and C . (c) ÕÕ ÕÕ 9 ÕÕ 9 9 M CC M CC M If M and N are C-comodules, a comodule morphism ØÙ f : M → N is a morphism in ᐂ satisfying M M f * = ** f * . ×77 * ×× 7 * NC'&%$ !"# NC '&%$ !"# In this paper we also make use of C-bicomodules. Suppose that M is both a left C-comodule and a right C-comodule with coactions

γl : M → C ⊗ M and γr : M → M ⊗ C.

If the square γl M / C ⊗ M

γr 1⊗γr

γl ⊗1 M ⊗ CC/ ⊗ M ⊗ C Weak Hopf monoids in braided monoidal categories 201 commutes, meaning C C 9 = 9 ÕÕ 99 Õ9Õ 99 ÕÕ ÕÕ 9 ÕÕ 9 9 M C M M C M in string diagrams, then M is called a C-bicomodule. The diagonal of the square will be denoted by γ : M → C ⊗ M ⊗ C.

A.4. Frobenius monoids. A Frobenius monoid R in ᐂ is both a monoid and a comonoid in ᐂ that additionally satisﬁes the “Frobenius condition”: δ⊗1 R ⊗ RR/ ⊗ R ⊗ R

1⊗δ 1⊗µ µ⊗1 R ⊗ R ⊗ R / R ⊗ R. In strings the Frobenius condition is displayed as ? ?? ?? = . (f)

We will now review some basic facts about Frobenius monoids. Lemma A.1. (1 ⊗ µ)(δ ⊗ 1) = δµ = (µ ⊗ 1)(1 ⊗ δ) : R ⊗ R → R ⊗ R. Proof. The left equality is proved by the string calculation

? ??? ?? (c) ?? ?? (f) JJJ (c) JJJ (f) oo (c) ?? = ?? = JJ = JJ = oo = . ? ?? ?? ?? ? ? ? ØÙ The right equality follows from a similar ØÙ calculation. ØÙ ØÙ Deﬁne morphisms ρ and σ by η δ ρ = I / R / R ⊗ R = 7 , ×× 77 × ØÙ 7 µ 7 × σ = R ⊗ R / R / I = 7×× .

Proposition A.2. The morphisms ρ and σ respectively form the unit and counit of an adjunction R a R. ØÙ Proof. One of the triangle identities is given as

(f) (m),(c) tt = JJJ = , ttt JJ ØÙ ØÙ ØÙ ØÙ 202 Craig Pastro and Ross Street and the other should now be clear.

A morphism of Frobenius monoids f : R → S is a morphism in ᐂ that is both a monoid and comonoid morphism.

Proposition A.3. Any morphism of Frobenius monoids f : R → S is an isomorphism.

Proof. Given f : R → S, deﬁne f −1 : S → R by

⊗ ⊗ ⊗ −1 ρ 1 1 f 1 1⊗σ GG f = S / R ⊗ R ⊗ S / R ⊗ S ⊗ S / R = f G . ØÙ G '&%$ !"# It is then an easy calculation to show that f −1 is the inverse of f , namely, ØÙ

f JJ GG JJ J (f) (m),(c) = f f = J = JJ = tt = ?? JJ tt . f −1 ØÙ ØÙ f ØÙ t '&%$ !"# ØÙ '&%$ !"# '&%$ !"# ?>=<89:; '&%$ !"# ØÙ − ØÙ ØÙ That f f 1 = 1 may be seen by viewing ØÙ the previous calculation upside down.

A similar calculation shows that

1⊗ρ 1⊗ f ⊗1 σ⊗1 w S / S ⊗ R ⊗ R / S ⊗ S ⊗ R / R = f w . ww ØÙ '&%$ !"# is also an inverse of f . Therefore: ØÙ

Corollary A.4. For any morphism of Frobenius monoids f : R → S, we have

G G ww f G = f . ØÙ G ww ØÙ '&%$ !"# '&%$ !"# Deﬁnition A.5. A Frobenius monoid ØÙ R is said ØÙ to be separable if and only if µδ =1, that is, ? ?? = ?? . ? Weak Hopf monoids in braided monoidal categories 203

Appendix B. Proofs of the properties of s, t, and r

As we have noted in Section 1, the source morphism s : A → A is invariant under rotation by π, the target t : A → A is invariant under horizontal reﬂection, and the endomorphism r is t rotated by π. This reduces the number of proofs we present, since the others are derivable.

s =s (c)= (w)= JJJ t =s s 7 ?? tt JJJ (1) ×× 77 ? × ?? ØÙ ØÙ ØÙ ØÙ ØÙ × s s ØÙ (c) ØÙ (w) s s × = = ? = PPP ØÙ n = 7 ? nEnnPPn XX ×× 77 ? Eyy × ?? ØÙ ØÙ ØÙ ØÙ ØÙ t t ØÙ (c) ØÙ (w) t t = = 99 = OOO j ØÙ jj = 7 99 j JJJ ×× 77 ? × ?? ØÙ ØÙ ØÙ ØÙ ØÙ × t t ØÙ (c) (w) t × = = 99 ØÙ = JJJ kk ØÙ = t 7 99 9kkJJk XX ×× 77 ? 9w × ?? ØÙ ØÙ ØÙ w ØÙ ØÙ ØÙ ØÙ ØÙ

s nat J (w) (c) = ×× = JJttt = 9 = : (2) s × ÕÕ 99 ÔÔ:: ØÙ ØÙ ØÙ ÕÕ 9 Ô ØÙ ØÙ ØÙ ØÙ s (m) (c) s = ØÙ = ? ØÙ = ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ PP t nat PnPPnn (w) (c) = = nEEn y = 9 = : t ØÙ y ÕÕ 99 ÔÔ:: ØÙ ØÙ ÕÕ 9 Ô ØÙ ØÙ ØÙ ØÙ ØÙ t ØÙ (m) (c) ØÙ t = = ? = ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ

s (1) s (2) s (1) s = TT = = (3) s JJJ 7 s ×× 77 ØÙ × ØÙ t (1) t (2) t (1) t = TT = = JJJ 7 t t ×× 77 ØÙ × ØÙ 204 Craig Pastro and Ross Street

s s (b) (1) s (m) D (b) s = J = DD = DD z = ?? JJtt Dzz{{{ zD s ?? (4) 7 tt JJ zDD zz D ×× 77 z ØÙ ØÙ {{{ × s s s s s ?? (b) (1) XXX X (m) (b) = JJ t = X = nn = 7 tJtJJ ?? VVVVnVn × 7 t ØÙ jj?j?j nnn VV ×× 7 jjjj ? ØÙ

t t (b) (1) t (m) D (b) t = J = DD = DD z = ?? JJtt Dzz{{{ zD t ?? 7 tt JJ zDD zz D ×× 77 z ØÙ ØÙ {{{ × t t t t t ?? (b) (1) XXX X (m) (b) = JJ t = X = nn = 7 tJtJJ ?? VVVVnVn × 7 t ØÙ jj?j?j nnn VV ×× 7 jjjj ? ØÙ

s s s s s s ?? (4)= GG (3)= G (4)= s s G GG ?? (5) s s t t t t t t ØÙ ØÙ ?? (4)= GG (3)= G (4)= t t G GG ?? t t ØÙ ØÙ + JJ ++ × tJtt J s nat J (b) ? × (m),(c) JJtt = + = JJtt = × = s ttJ ++ t JJ ? (6) ?? + t × ØÙ + ØÙ ×× ØÙ ØÙ ØÙ 77 ØÙ t J (b) 7 (m),(c) t = JJtJt = = tt J ? ØÙ × ×× ØÙ ØÙ ØÙ s s (2) s t t (2) t = = = s = = = t (7) s s t t ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ s s (2) s t t (2) t = = = s = = = t (8) t t s s ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ Weak Hopf monoids in braided monoidal categories 205

t t t (3) (8) (3) t = t = = 7 (9) s t × 7 s ×× 7 s s s (3) (8) (3) s = s = = 7 t s × 7 t ×× 7 ? ?? FFF x ? s t xxF ?? s t G ?? , nat x FF (v) ? , w G s = = xx F = = s w t (10) t x ØÙ ØÙ KKKss ØÙ ØÙ sss KK ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ

?? (m) (4) (m) 44 (2) 44 (2) 44 (4) ?? = EE = 4 = 4 = 4 = 4 = (11) t EE 44 t ØÙ t ØÙ t ØÙ ØÙ s ØÙ s

r r,s nat (v) 99 (c) s = ØÙ = = 99 = = s (12) s ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ ?? F ? ? t,r nat FFxx (v) ?? t,r DD ?? = = xF = ? = r zz t t r xx FF ? A (13) xx F AA|| ØÙ ØÙ ~~ AA ØÙ ØÙ ØÙ ØÙ ~ ØÙ ØÙ ØÙ ØÙ ØÙ ØÙ r 7 r s 77 ×× (3)= s (12)= (3)= × s (14) s s s s 7 s r 77 ×× (3)= r (12)= (3)= × r r r r 206 Craig Pastro and Ross Street

Acknowledgments

We would like to thank Gabriella Bohm¨ for her helpful comments and suggestions, and particularly for her help with Theorem 6.1, both for its statement and also for helping with the proof. We would also like to thank J. N. Alonso, J. M. Fernandez,´ and R. Gonzalez´ for sending us copies of their preprints [Alonso Alvarez´ et al. 2008a; 2008b], and an anonymous referee for helpful comments and suggestions.

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Communicated by Susan Montgomery Received 2008-01-26 Revised 2008-11-14 Accepted 2008-12-23 [email protected] Research Institute for the Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan www.kurims.kyoto-u.ac.jp/~craig/ [email protected] Department of Mathematics, Macquarie University, New South Wales 2109, Australia www.maths.mq.edu.au/~street