Weak Hopf monoids in braided monoidal categories

WEAK HOPF MONOIDS IN BRAIDED MONOID AL CA TEGORIES CRAIG P ASTRO AND ROSS STREET Abstract. W e dev elop the t heory of w eak bimonoids in braided monoidal categories and show them to b e quant um categories in a certain sense. W eak Hopf m onoids are shown to b e quant um groupoids. Eac h separable F rob en ius monoid R leads to a we ak Hopf monoid R ⊗ R . Contents 1. Int ro duction 1 2. W eak bimonoids 5 3. W eak Hopf monoids 10 4. The monoida l ca tegory of A -co modules 13 5. F rob enius mono id example 23 6. Quantum g roupoids 26 7. W eak Hopf monoids are quantum group oids 30 Appendix A. String dia grams and basic definitions 36 Appendix B. Pro ofs of th e properties of s , t , and r 42 References 45 1. Introduction W eak Hopf algebras were intro duced b y B¨ ohm, Nill, and Szlach´ an yi in a series of paper s [5, 15, 22, 4]. They are generalizatio ns of Hopf a lgebras and were pro- po sed a s an alternative to weak quasi- Hopf alg e bras. A weak bialgebra is bo th an assoc iativ e algebra and a coasso ciativ e co algebra, but instea d of requiring that the multiplication and unit mor phism are co algebra morphisms (or equiv alently that the comultiplication a nd the counit ar e alg ebra morphisms) other “ weakened” axioms a re impos e d. The multiplication is still required to be comultiplicativ e (equiv alently , the comultiplication is s till req uired to be mu ltiplicative), but the counit is no longe r r equired to b e an algebra morphism and the unit is no longer required to b e a co algebra morphism. I ns tead, these requirements a re replaced b y weak ened v ersions (see eq uations (v) and (w) b elow) . As the na me suggests, an y bialgebra satisfies these weak ened ax ioms and is therefore a weak bialg e br a. Given a weak bialgebra A one may define sour c e and targ et morphisms s, t : A / / A whose ima g es s ( A ) and t ( A ) are called the “sourc e and target (counital) Date : Octob er 28, 2018. The first author gratefully ackno wl ed ges supp ort of an i n te rnational Macquarie Univ ersity Researc h Sc holarship while the s ec ond gratefully ac kno wledges supp ort of the Australian Research Council Di s co very Gran t DP0771252. 1 2 CRAIG P ASTR O AND ROSS STREET subalgebra s”. It has be e n shown by Nill [15] tha t Hayashi’s face algebras [11] are sp ecial cas e s of weak bia lgebras for which the, say , target subalgebra is comm uta- tive. A weak Hopf a lgebra is a w eak bialgebra H equipp ed with an antipo de ν : H / / H s atisfying the axioms 1 µ ( ν ⊗ 1) δ = t, µ (1 ⊗ ν ) δ = s, and µ 3 ( ν ⊗ 1 ⊗ ν ) δ 3 = ν , where µ 3 = µ ( µ ⊗ 1) and δ 3 = ( δ ⊗ 1 ) δ . Aga in, any Hopf alg e br a satisfies these weak ened axio ms a nd so is a weak Hopf algebra. Also in [15] Nill has shown that the (finite dimensional) genera lized Kac algebras o f Y amanouchi [25] ar e examples of w eak Hopf alge br as with inv olutive antipo de. W eak Hopf alg ebras have also b een called “quantum group oids” [16] a nd in this pape r this is not what w e mean b y quantum gr oupoid. Perhaps the simplest example of weak bialge br as and w eak Hopf algebras are, resp ectiv ely , categor y alg ebras and gr oupoid a lgebras. Supp ose that k is a field and let C be a categor y with se t of ob ject C 0 and set o f morphism C 1 . T he c ate gory algebr a k [ C ] is the vector space k [ C 1 ] over k with basis C 1 . Elements are formal linear combinations o f the elements of C 1 with co efficients in k , i.e., αf + β g + · · · with α, β ∈ k and f , g ∈ C 1 . An asso ciativ e multiplication on k [ C ] is defined by µ ( f , g ) = f · g = ( g ◦ f if g ◦ f exists 0 otherwise and extended b y linearity to k [ C ]. This algebra do es not ha v e a un it unless C 0 is finite, in which cas e the unit is η (1) = e = X A ∈ ob C 1 A , making k [ C ] in to a unital algebra; all alg ebras (monoids) considered in this pap er will b e unital. A co m ultiplication and counit may b e defined on k [ C ] as δ ( f ) = f ⊗ f ǫ ( f ) = 1 making k [ C ] into a co algebra. Note that k [ C ] equipped with this algebra and coalgebr a structure will not sa tisfy any o f the f ollowing usua l bialgebr a axioms: ǫµ = ǫ ⊗ ǫ δ η = η ⊗ η ǫη = 1 k . The one bialg ebra axiom that do es hold is δ µ = ( µ ⊗ µ )(1 ⊗ c ⊗ 1 )( δ ⊗ δ ). Equipp ed with this algebr a and coa lgebra structure k [ C ] do es, howev er, satisfy the ax io ms of a weak bialgebr a. F urthermo r e, if C is a group oid, then k [ C ], which is then called the gr oup oid algebr a , is an example of a weak Hopf algebra with a n tipo de ν : k [ C ] / / k [ C ] defined by ν ( f ) = f − 1 . 1 There may b e some di screpa ncy wi th what w e call the source and target mor phisms and what exists in the l iterat ure. This arises f rom our conv ention of taking multiplication i n the group oid algebra to b e f · g = g ◦ f (whenev er g ◦ f is defined). WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 3 and extended by linearity . I f f : A / / B ∈ C , the so urce and targe t morphisms s, t : k [ C ] / / k [ C ] are given by s ( f ) = 1 A and t ( f ) = 1 B , as o ne would exp ect. In this pap er w e define weak bialgebras and weak Hopf algebr as in a braided monoidal ca tegory V , where prefer to call them “weak bimonoids” a nd “weak Hopf monoids”. T o define a weak bimonoid in V the only diff erence from the definition given b y B¨ ohm, Nill, and Szla c h´ an yi [4] is that a choice of “crossing ” m ust b e made in the axioms . Our definition is not as genera l as the one given b y J. N. Alonso ´ Alv are z , J. M. F ern´ andez Vilab oa, a nd R. Gonz´ ale z Rodr ´ ıguez in [1, 2], but, in the ca se that their weak Y ang-Ba xter o perator t A,A is the bra iding c A,A and their idemp oten t ∇ A ⊗ A = 1 A ⊗ A , then our choices o f cross ing s are the same. Our difference in defining weak bimonoids o ccurs in the choice of s ource and ta rget morphisms. W e ha v e chosen s : A / / A a nd t : A / / A s o that: (1) the “g lobular” identities ts = s and st = t hold; (2) the source sub comonoid and tar get s ubcomonoid co incide (up to isomor - phism), and is denoted by C ; (3) s : A / / C ◦ and t : A / / C are c omonoid morphisms. These prop erties of the sourc e and target morphisms ar e esse ntial for our view of quantum categor ies. These are s = ¯ Π L A and t = Π R A in the no tation of [1, 2] and s = ǫ ′ s and t = ǫ s in the notation o f [19], with the appr opriate choice o f cross ing s. W e cho ose to work in the Cauc h y completion Q V of V . The catego ry Q V is also ca lled the “completion under idemp oten ts” of V or the “K aroubi env elope” of V . This is done rather than assume that idempotents split in V . Supp ose that A is a weak bimonoid in Q V . In this case we find C b y s plittin g either the sour ce or target morphism. As in [1 9, Pr op. 4.2], C is a separable F rob enius mono id in Q V , meaning tha t ( C, µ, η , δ, ǫ ) is a F rob enius monoid with µδ = 1 C . It tur ns o ut that our definition o f w eak Hopf mo noid is (in the symmetric cas e ) the same a s what is prop osed in [4], and in the braide d case in [1 , 2]. A weak bimonoid H is a weak Hopf mo no id if it is equipped with an a n tipo de ν : H / / H satisfying µ ( ν ⊗ 1) δ = t, µ (1 ⊗ ν ) δ = r , and µ 3 ( ν ⊗ 1 ⊗ ν ) δ 3 = ν, where r = ν s . This r : H / / H he r e turns out to b e the “usual” so urce mor phism; Π L H in the notation of [1, 2]. Ignoring cro ssings r is ǫ t in the notation of [19] and our r and t corresp ond resp ectiv ely to ⊓ L and ⊓ R in the no tation of [4 ]; the morphism s do es not app ear in [4]. Usua lly , in the second axio m ab ov e, µ (1 ⊗ nu ) δ = r , the right-hand side is equal to the chosen source map s of the weak bimonoid H . The rea son that this r do es not w ork as a source mor phism for us is that it do es not satisfy all three requirements for the source morphism mentioned abov e. This choice of r allows us to show that any F r o benius monoid in V yields a weak Hopf monoid R ⊗ R with bijective a n tipo de (cf. the example in the App endix of [4]). There are a num ber of generaliz ations of bia lg ebras and Hopf algebra s to their “many ob ject” versions. F or example, Sweedler’s g eneralized bia lg ebras [21], which were later gener alized b y T a keuchi to × R -bialgebra s [23], the quantum group oids of Lu [14] and Xu [24], Schauenburg’s × R -Hopf alg ebras [18], the bialge br oids a nd Hopf algebroids of B¨ ohm and Szlach´ anyi [7], the ea rlier mentioned fa c e alge br as [11] 4 CRAIG P ASTR O AND ROSS STREET and genera lized Kac algebr as [25], and, the o nes of interest in this pap er, the quan- tum ca tegories and quantum group oids of Da y and Street [9]. It has b een shown by Brzezi ´ nski and Militaru that the qua ntum g r oupoids o f Lu and Xu are equiv- alent to T a k euc hi’s × R -bialgebra s [8, Thm. 3.1]. Schauenburg has shown in [1 7 ] that face algebras are an example of × R -bialgebra s for whic h R is commut ative and sepa rable. In [19, Thm. 5.1] Sc hauen burg has shown that w eak bialg ebras are also examples of × R -bialgebra s for which R is se pa rable F r obenius (there ca lled F rob enius-separa ble). Schauen burg also shows in [19, Thm. 6 .1] that a w eak Hopf algebra ma y b e c haracterized as a w eak bialgebra H for which a certain canonical map H ⊗ C H / / µ ( δ ( η (1)) , H ⊗ H ) is a bijection. As a corollar y he sho ws that a weak Hopf algebra is a × R -Hopf alg ebra. Quantum group oids w ere introduced in [9]. They firs t in troduce quant um cat- egories. A quantum ca tegory in V consists of tw o co monoids A a nd C in V , with A playing the role of the ob ject-of-mo rphisms and C the ob ject-of-o b jects. There are source and target morphisms s, t : A / / C , a “comp osition” morphism µ : A ⊗ C A / / A , and a “ unit” morphis m η : C / / A a ll in V . This data must satisfy a num b er of axioms. Indeed, ordinary categorie s are examples of quan- tum categories . Motiv a ted by the dualit y found in ∗ - autonomous categ ories [3], they then define a quantum group oid to be a qua n tum category equipp ed with a generalized a n tipo de coming f rom a ∗ -auto no mous structure. In this pap er we show that weak bimonoids are examples o f quantum catego ries for whic h t he ob ject-o f-ob jects C is a separable F rob enius monoid, a nd that weak Hopf mo noids with inv ertible antipo de are quan tum g roupoids . An o utline of this pa per is as follo ws: In § 2 w e provide the definition of weak bimonoid A in a bra ided mo noidal ca te- gory V and define the s ource and tar get morphisms. W e then mov e to the Cauc h y completion Q V a nd prove the three required prop erties of o ur so urce and target morphisms mentioned ab ov e. In this section we also prov e that C , the o b ject-of- ob jects of A , is a se parable F rob enius mono id. W eak Hopf monoids in braided monoidal categ o ries a re intro duced in § 3 . In § 4 we describ e a mo noidal structur e on the categor ies Bicomo d ( C ) of C - bicomo dules in V , and Como d ( A ) of r igh t A -c omodules in V , such that the un- derlying functor U : Como d ( A ) / / Bicomo d ( C ) is str ong mo noidal. If H is a weak Hopf monoid, then we a re able to s ho w that the category Com od f ( H ), consisting o f the dualiza ble ob jects of Como d ( H ), is left autonomous. In § 5 w e pr o v e that any sepa rable F r obenius monoid R in a braided monoidal category V yields an example o f a weak Hopf monoid R ⊗ R w ith inv ertible antipo de in V . The definitions of q uan tum catego ries and quantum group oids are recalled in § 6, and in § 7 we show th at an y weak bimonoid is a quantum category and a n y w eak Hopf mo noid w ith in vertible a n tipo de is a quantum group oid. This pap er depends heavily o n o f t he string diagrams in braided monoidal cat- egories of Joy al and Street [13], which were shown to b e rigor ous in [12]. The reader unfamiliar with str ing dia grams ma y first w ant t o read App endix A where we re view s ome preliminary co ncepts using these diagrams . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 5 W e would like to thank J. N. Alonso ´ Alv are z , J. M. F ern´ andez Vilab oa, a nd R. Gonz´ alez Ro dr ´ ıguez for sending us copies of their preprints [1, 2]. 2. Weak bimonoids A w eak bia lgebra [5, 15, 22, 4] is a generalization of a bialgebr a with weak ened axioms. These weak ened a xioms repla c e the three axioms that s a y that the unit is a coalgebr a morphism and the counit is a n algebr a morphism. With the a ppropriate choices of under and o ver c rossings the definition of a weak bialgebra carr ies ov er rather straightforwardly into braided monoidal categor ies, where we prefer to ca ll it a “weak bimonoid” . 2.1. W eak bim onoids. Supp ose that V = ( V , ⊗ , I , c ) is a braided monoidal cat- egory . Definition 2. 1. A we ak bimonoid A = ( A, µ, η, δ, ǫ ) in V is an ob ject A ∈ V equipp e d with the structure of a monoid ( A, µ, η ) and a comonoid ( A, δ, ǫ ) satisfying the fo llowing equatio ns. (b)    ? ? ? ? ? ?    = J J J J J t t t t (v)   9 9 9 9     =     w w w G G G =     (w)       9 9 9 9 =     G G G w w w =     Suppo se A and B ar e w eak bimonoids in V . A morphism of we ak bimonoids f : A / / B is a morphism f : A / / B in V whic h is b oth a mono id morphis m a nd a co monoid mor phism. Let A be a weak bimonoid and define the sourc e a nd tar get morphisms s, t : A / / A o f A as follows: s =     t =     . Notice that s : A / / A is inv ariant under r otation by π , while t : A / / A is inv ar ian t under hor izon tal re flection and the in verse braiding. Impor tan tly , under either o f these trans formations • (m) and (c) are interchanged 2 , • (b) is inv aria n t, a nd • (v) a nd (w) a re interc hanged. Note that these are not the “usual” so ur ce and tar g et morphisms. They w ere chosen, as mentioned in the int ro duction, pr e cisely b ecause of the need for them to satisfy the following three prop erties: (1) the “g lobular” identities ts = s and st = t hold; (2) the source sub comonoid and tar get s ubcomonoid co incide (up to isomor - phism), and is denoted by C ; 2 The (m) and (c) here refer to the monoid and comonoid i de n tities found in A pp endix A. 6 CRAIG P ASTR O AND ROSS STREET Under (b) and (w) Under (b) and (v) (1)   s    7 7 7 =     s J J J     =     s X X     s 7 7 7    =     s J J J     =     s X X   (2)     s   =      : : :   s   =       s   =   : : :      s   =   (3)   s   s   =   s    7 7 7   s   s   =   s 7 7 7    (4)   s      7 7 7 =   s ? ? ? ?   s ? ? 7 7 7    =   s   s ? ? 7 7 7    =   s   s      7 7 7 =   s ? ? ? ? (5)   s   s   s ? ?   =   s   s ? ?     s   s   s   ? ? =   s   s   ? ? Under (b) (6)   s o o o O O O o o o o o O O O O G G w w w = Under (b) and (w) or (v) (7)   s   s =   s Figure 1. P roperties of s (3) s : A / / C ◦ and t : A / / C are c omonoid morphisms. These pr operties will b e prov ed in this section. Note tha t we will run in to the usual source morphism (which we call r ) in the definition of weak Hopf monoids (Definition 3 .1 ). A ta ble of prop erties of the sourc e morphism s is giv en in Fig ure 1 a nd table of prop erties of the target mor phism t in Figure 2 . Prop erties inv olving the int eraction of s and t are given in Fig ure 3. P roo fs of these pro perties may b e found in Appendix B. In the sequel A = ( A, µ, η , δ, ǫ ) will alw ays denote a weak bimo noid and s , t : A / / A the source and target morphis ms. W e see from prop ert y (7) in Figures 1 and 2 respectively that b oth s and t are idempo ten ts. In the following we will w ork in the Ca uc h y c o mpletion (= completio n under idemp otent s = Kar oubi envelope) Q V of V . W e do this rather than assume that idemp oten ts split in V . 2.2. Cauc hy com pletion. Given a categor y V , its Cauchy c ompletion Q V is the category whose ob jects are pairs ( X, e ) with X ∈ V and e : X / / X ∈ V an WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 7 Under (b) and (w) Under (b) and (v) (1)   t    7 7 7 =     t J J J     =     t X X     t 7 7 7    =     t t t t + + + + =     t , , / / (2)     t   =      : : :   t   =       t ? ? =   : : :      t   =   (3)   t   t   =   t    7 7 7   t   t ? ? =   t 7 7 7    (4)   t      7 7 7 =   t ? ? ? ?   t ? ? 7 7 7    =   t   t ? ? 7 7 7    =   t       t      7 7 7 =   t (5)   t   t   t : :   =   t   t : :     t   t   t   : : =   t   t   : : Under (b) (6)   t 7 7      7 7 7 = Under (b) and (w) or (v) (7)   t   t =   t Figure 2. P roperties of t idempo ten t. A morphism ( X , e ) / / ( X ′ , e ′ ) in Q V is a morphis m f : X / / X ′ ∈ V such that e ′ f e = f . Note that the identit y morphism of ( X , e ) is e itself. The p oin t of working in the Cauch y completion is that every idempotent f : ( X, e ) / / ( X, e ) in Q V has a splitting, viz., ( X, e ) ( X, e ) ( X, f ) . f / / f   < < < < < < < < f @ @         If V is a monoidal category , then Q V is a monoidal catego r y via ( X, e ) ⊗ ( X ′ , e ′ ) = ( X ⊗ X ′ , e ⊗ e ′ ) . The catego ry V may b e fully embedded in Q V by sending X ∈ V to ( X , 1 ) ∈ Q V and f : X / / Y ∈ V to f : ( X , 1) / / ( Y , 1), which is obviously a morphism in Q V . When working in Q V we will often identify an ob ject X ∈ V with ( X, 1) ∈ Q V . 8 CRAIG P ASTR O AND ROSS STREET Under (w) (8)   s   t =   s   t   s =   t (9)   t   s   =   t    7 7 7   s   t   =   s    7 7 7 Under (v) (10)   t   s   ? ? =   s   t z z D D A A A A A | | ~ ~ Under (w) and (v) (11)   t ? ? ? ? =   s Figure 3. Interactions of s a nd t 2.3. Prope rties of the source and target morphism s . Let A = ( A, 1) b e a weak bimonoid in Q V . F ro m the definition of the Cauch y co mpletio n the r e s ult of splitting the source morphism s is ( A, s ), and similarly , the result of splitting the target morphism t is ( A, t ). The following pro position shows that these t wo ob jects are isomo rphic. Prop osition 2 . 2. The idemp otent t : ( A, 1 ) / / ( A, 1) has the fol lowing two split- tings. ( A, 1) ( A, 1) ( A, t ) t / / t   < < < < < < < < t @ @         ( A, 1) ( A, 1) ( A, s ) t / / t   < < < < < < < < s @ @         In this c ase s : ( A, s ) / / ( A, t ) and t : ( A, t ) / / ( A, s ) ar e inverse morph isms, and henc e ( A, t ) ∼ = ( A, s ) . Pr o of. This result follows fro m the identities ts = s and st = t (pro perty (8) in Figure 3 ).  W e will de no te this ob ject b y C = ( A, t ) and call it the obje ct-of-obje cts of A . In the next pro positions we will sho w that C is a comonoid, and furthermore, that it is a se parable F rob enius monoid, similar to what was done in [19] (there called F rob enius-separa ble). Prop osition 2.3. The obje ct C = ( A, t ) e quipp e d with δ =  C C ⊗ C δ / / C ⊗ C t ⊗ t / /  ǫ = C I ǫ / / WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 9 is a c omonoid in Q V , and if furthermor e e quipp e d with µ =  C ⊗ C C ⊗ C t ⊗ t / / C µ / /  η = I C η / / then C is a sep ar able F r ob enius monoid in Q V (se e Defin it ion A.5). Pr o of. W e first observe that ( t ⊗ t ) δ : C / / C ⊗ C and ǫ : C / / I are in Q V which follows from (5) and (2) resp ectively . The comonoid identities a re given as   t   t   t   t   ? ?   ? ? (5) =   t   t   t    ? ?   7 7 (c) =   t   t   t ? ? ?   7 7   (5) =   t   t   t   t ? ?   ? ?   and   t   t     ? ? (2) =     t   ? ? (c) =   t (c) =     t ? ?   (2) =   t   t   ? ?   . T o see that C is a separ able F rob enius mono id we firs t observe that µ a nd η are morphisms in Q V from (5) and (2), and the monoid iden tities a re dual to the comonoid identities. The following ca lc ula tion prov es that the F ro benius condition holds.   t   t   t   t ? ? ? ? ? ? (7) =   t   t   t G G G G (5) =   t   t   t   t   t G G G G (3) =   t   t   t   t J J J J J (4) =   t   t   t   t ? ?     ? ? (4) =   t   t   t   t t t t t t (3) =   t   t   t   t   t w w w w (5) =   t   t   t w w w w (7) =   t   t   t   t       Finally , that this is a separable F r obenius monoid follows from µδ =   t   t   t   t   ? ? ? ?   (7) =   t   t   7 7 7 7   (5) =   t   t   t   7 7 7 7   (3) =   t   t    7 7 7 7 7   (6) =   t = 1 C .  Corollary 2.4. Every morphism of we ak bimonoids induc es an isomorphi sm on the “obje cts-of-obje cts”. That is, if ( A, 1) and ( B , 1) ar e we ak bimonoids, and f : ( A, 1) / / ( B , 1 ) is a m orphism of we ak bimonoids, then the induc e d morphism tf t : ( A, t ) / / ( B , t ) is an isomorphism. Pr o of. Note that if f : A / / B is a morphism of weak bimonoids then f t = tf and f s = st . The cor ollary now follows fro m Pr opositio n 2.3 and Prop osition A.3.  10 CRAIG P ASTR O AND ROSS STREET Prop osition 2.5. If we write C ◦ for the c omonoid C with t he “opp osite” c omul- tiplic ation define d via C C ⊗ C δ / / C ⊗ C t ⊗ t / / C ⊗ C c / / =   t   t z z D D A A A A A | | ~ ~ then s : A / / C ◦ and t : A / / C ar e c omonoid morphisms. That is, the diagr ams A C C ⊗ C A ⊗ A s / / c ( t ⊗ t ) δ   δ   s ⊗ s / / A C C ⊗ C A ⊗ A t / / ( t ⊗ t ) δ   δ   t ⊗ t / / c ommu te. Pr o of. The second diagram express e s   t   t   t   ? ? =   t   t   ? ? which is exactly (5 ), and the following calcula tion   s   s   ? ? (5) =   s   s   s   ? ? (8) =   s   s   s   t   ? ? (3) =   s   t   s   ? ? (10) =   s   s   t z z D D A A A A A | | ~ ~ (9) =   s   t   t   s z z D D F F F F w w y y (8) =   s   t   t z z D D A A A A A | | ~ ~ shows that the firs t diagram commutes.  3. Weak Hopf monoids In this section we intro duce weak Hopf monoids. Usually in the literature, a weak Hopf monoid is a weak bimono id H equipp ed with an antipo de ν : H / / H satisfying the three axioms ν ∗ 1 = t, 1 ∗ ν = s, a nd ν ∗ 1 ∗ ν = ν, where f ∗ g = µ ( f ⊗ g ) δ is the conv olution pr oduct. O ur definition is slig htly different as, instead of ch o osing our sour ce mor phis m in the seco nd a xiom, we replace it with 1 ∗ ν = r, where r is defined b elo w. This turns out to be the usual definition of w eak Hopf monoids a s found in the liter a ture; in the symmetric case s ee [4], and in the braided case see [1, 2]. 3.1. The endomorphi sm r and weak Hopf monoids. Define a n endomo rphism r : A / / A by ro tating the ta r get mor phism t : A / / A by π , i.e ., r =     . Since r is just t ro ta ted by π , all th e iden tities for t in Figure 2 r otated by π hold for r . W e list s ome additional ide ntities of r in teracting with s and t . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 11 (12)   r   s =   s   s   r =   r (13)   t   r   ? ? =   r   t z z D D A A A A A | | ~ ~   t   r ? ?   =   r   t ? ? ? ? ? J J t t     (14)   s   r   =   s    7 7 7   r   s   =   r    7 7 7 The pro ofs o f these pr operties ma y also b e f ound in Appe ndix B. Definition 3.1. A weak bimonoid H is called a we ak Hopf monoid if it is equipp ed with a n endo morphism ν : H / / H , called the antip o de , satisfying   ν    ? ? ? =   t ,   ν ? ? ?    =   r ,   ν   ν ? ? ?       ? ? ? =   ν . The axioms o f a weak Ho pf monoid immediately imply the following identities   ν =   t   ν ? ? ?       ? ? ? =   ν   r ? ? ?       ? ? ? . The antipo de is unique since if ν ′ is ano ther ν ′ = ν ′ ∗ 1 ∗ ν ′ = t ∗ ν ′ = ν ∗ 1 ∗ ν ′ = ν ∗ r = ν ∗ 1 ∗ ν = ν. If H and K ar e weak Hopf monoids in V , then a morphism of we ak Hopf monoids f : H / / K is a mo rphism f : H / / K in V which is a monoid and como no id morphism tha t also pre serv es the an tipo de, i.e., f ν = ν f . W e list some proper ties of the an tipo de ν : H / / H . Prop osition 3.2. (15)   s   ν =   r (16)   ν   t =   r   ν =   r   t   ν   r =   t   ν =   t   r   ν   =     ν    7 7 7 =   ν   ν z z D D A A A A A | | ~ ~     ν =     ν 7 7 7    =   ν   ν z z D D A A A A A | | ~ ~ (17) 12 CRAIG P ASTR O AND ROSS STREET The last identit y (17 ) sta tes that ν : A / / A is b oth an an ti-comonoid morphism and an an ti-monoid mor phism. Pr o of. The calculatio n   s   ν ( ν ) =   s   t   ν    ? ? ? ? ? ?    (9) =   s   ν ? ? ?    ( ν ) =   s   r (12) =   r verifies the iden tit y (15 ), a nd the following verifies the first identit y o f (16):   ν   t ( ν ) =   t   ν   t    ? ? ? ? ? ?    (3) =   ν   t ? ? ?    ( ν ) =   r   t ( ν ) =   r   ν    ? ? ? (3) =   r   ν   r    ? ? ? ? ? ?    ( ν ) =   r   ν . The seco nd identit y o f (16) follows from a similar calculation. T o prov e (17) w e will only prov e tha t ν is an an ti-comonoid morphism. That ν is a n anti-monoid morphism follows by rotating all the dia grams use d t o pr o v e this statement by π . The proof of the counit property is easy enough:   ν   ( ν ) =     t   ν ? ?     ? ? (2) =     ν   ? ? ( ν ) =   r   (2) =   . The following ca lc ulation prov es that the antipo de is an ti-comultiplicativ e.   ν    ? ? ? ( ν ) =   ν   r    ? ? ? ? ? ?       ? ? ? (b) =   ν   r    ? ? ? O O O O O O o o o o (3) =   ν   r   r    ? ? ? ? ? ? ? ? ? ?       ( ν ) =   ν   ν   r    ? ? ? ?    ? ? ? ? ? ? ? ? ? ?       (4) =   ν   ν   r    ? ? ? ?     / / R R R R R R R     ( ν ) =   ν   ν   ν    ? ? ? ?     / / G G w w L L L L L L L      (c) =   ν   ν   ν        ? ? ? ? 4 4 4 4     O O O O O O 0 0 0 0 0 0 0 0 0 0 0 o o o o v v v v v v v v v v (b) =   ν   ν   ν        ? ? ? ? ? ? ?     / / / / /        ? ? ? ? / / / / / / / / / / / t t t t t t t t t t (c, ν ) =   t   ν   ν     D D D D J J J J J 1 1 1 1 1 1 1 1 1 w w w w w w w WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 13 (3) =   t   t   ν   ν     D D D D J J J J J 1 1 1 1 1 1 1 1 1 w w w w w w w ( ν ) =   ν   ν   t   ν          ? ? ? J J J J J ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) l l l l l l l (4) =   ν   ν   ν   t        ? ? ? ? w w w G G G ( ( ( ( ( ( ( ( ( ( ( ( ( m m m m m m m m (c, ν ) =   ν   t   r   ν     ? ? ? ? ? ? ? ? ? ? ? ? ?          ? ? ? O O O O O O O O O O o o o o o o o o (13) =   ν   r   t   ν     ? ? ? ? ? ? ? ? ? ? ? ? ?          ? ? ? O O O O O o o o o o G G G G G G w w w w w w (c, ν ) =   ν   ν    ? ? ? ? ? ? ? ? ?       4. The mono id al ca tegor y of A -comodules Suppo se A = ( A, 1) is a weak bimonoid in Q V and let C = ( A, t ). In this section we describ e a monoidal structure on the categor ie s Bicom o d ( C ) of C -bicomo dules in Q V , and Como d ( A ) of right A -como dules in Q V such that the underlying functor U : Como d ( A ) / / Bicomo d ( C ) is strong monoidal. If A is a weak Hopf monoid then we show that Como d f ( A ), the s ubcategor y consis ting of the dualizable ob jects, is left autono mous. This sectio n is fair ly standard in the V = V ect case (see [6], [15], or [16] for example) a nd carr ies o ver rather stra igh tforw ardly to the general braided V case (cf. [1 0 ]). 4.1. The monoidal structure on C -bicomo dule s . Supp ose, for this s e ction, that C ∈ V is just a comono id, a nd that M ∈ V is a C - bic o module with coac tion γ : M / / C ⊗ M ⊗ C. A left C -coaction and a right C -c oaction a re obtained fro m γ b y in volving the counit ǫ : γ l =  M C ⊗ M ⊗ C γ / / C ⊗ M 1 ⊗ 1 ⊗ ǫ / /  γ r =  M C ⊗ M ⊗ C γ / / M ⊗ C ǫ ⊗ 1 ⊗ 1 / /  . Suppo se now that N is another C -bicomo dule. The tenso r pro duct of M and N ov er C is defined to be the equa lizer M ⊗ C N M ⊗ N M ⊗ C ⊗ N ι / / γ r ⊗ 1 / / 1 ⊗ γ l / / . Obviously the morphism M ⊗ C N M ⊗ N ι / / C ⊗ M ⊗ N ⊗ C γ l ⊗ γ r / / 14 CRAIG P ASTR O AND ROSS STREET equalizes the tw o morphisms C ⊗ M ⊗ N ⊗ C / / C ⊗ M ⊗ C ⊗ N ⊗ C and so induces a morphism γ : M ⊗ C N / / C ⊗ M ⊗ C N ⊗ C , which is the co action on M ⊗ C N . That this defines a mono idal str ucture on the ca tegory Bicomo d ( C ) with tensor pro duct ⊗ C and unit C is standard. 4.2. The tensor pro duct of A -como dules. Let A = ( A, 1) b e a weak bimo noid in Q V and let C = ( A, t ). The monoida l structur e on the catego ry of right A - como dules will b e ⊗ C , t he tensor pro duct o v er C , with unit C . Suppo se that M is a r igh t A -co module. W e know that s : A / / C ◦ and t : A / / C are comonoid morphisms and that prop ert y (10) holds, where recall that prop ert y (10) expr esses the co mm utativit y of the following diagra m. A A ⊗ A δ 8 8 r r r r r r C ⊗ C s ⊗ t / / C ⊗ C c   A ⊗ A δ & & L L L L L L t ⊗ s / / Therefore, M may b e ma de into a C -bico module via γ =  M M ⊗ A γ / / M ⊗ A ⊗ A 1 ⊗ δ / / A ⊗ M ⊗ A c − 1 ⊗ 1 / / C ⊗ M ⊗ C s ⊗ 1 ⊗ t / /  , which is γ = M A C M C   s   t ? ? ? o o o o o / / in strings . The left and right C -c o actions are γ l =   s J J J k k k k and γ r =   t G G . The tensor product of t w o A -como dules M and N ov er C then ma y b e defined as in § 4.1. W e der iv e an explicit description of M ⊗ C N . Before doing so we will need the follo wing definition. Definition 4.1. Let f , g : X / / Y b e a pa rallel pair in V . This pair is called c osplit when there is an arr o w d : Y / / X such that d f = 1 X and f dg = g dg . It is not har d to se e tha t, in this case , dg : X / / X is an idemp oten t and a splitting of dg , i.e., X X Q dg / / x   ? ? ? ? ? ? ? y ? ?        Q Q X 1 / / y   ? ? ? ? ? ? ? x ? ?        provides a n a bsolute equalizer ( Q, y ) for f and g . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 15 Now supp ose M a nd N ar e A -como dules. Two morphisms M ⊗ N / / M ⊗ C ⊗ N are given as γ r ⊗ 1 = M N A M C N   t G G and 1 ⊗ γ l = M N A M C N   s J J J q q q q . Prop osition 4.2. The p ai r γ r ⊗ 1 and 1 ⊗ γ l ar e c osplit by d = M C N   t   M N G G G . Pr o of. That d is a mor phism in Q V follows immediately as t is idempotent. The calculation d ( γ r ⊗ 1) =   t   t   G G G G G (7) =   t   G G G G G (c) =   t   O O O 4 4      4 4 4 (6) =   ? ? (c) = = 1 M ⊗ N shows that d ( γ r ⊗ 1) = 1 and the identit y ( γ r ⊗ 1) d (1 ⊗ γ l ) = (1 ⊗ γ l ) d (1 ⊗ γ l ) follows from: ( γ r ⊗ 1) d (1 ⊗ γ l ) =   s   t     t J J J m m m m G G G G G (8) =   s     t J J J m m m m G G G G G (2) =     t J J J m m m m m G G G G G (c) =     t O O O   ? ? J J J l l l l (12) =   s   4 4 4 J J J r r r r t t j j j j (2) =   s   s   4 4 4 J J J m m m m m t t / / k k k k (c) =   s     s J J J m m m m G G G J J J m m m m (8) =   s   t     s J J J m m m m G G G J J J m m m m = (1 ⊗ γ l ) d (1 ⊗ γ l ) .  16 CRAIG P ASTR O AND ROSS STREET The idemp oten t d (1 ⊗ γ l ) will be denoted by m . The following calculatio n gives a simpler representation o f m :   s   t   J J J m m m m G G G (8) =   s   J J J m m m m G G G (2) =   J J J m m m m m G G G = m. A s plitting of m , i.e., ( M ⊗ N , 1) ( M ⊗ N , 1) ( M ⊗ N , m ) m / / m A A A A A A A A A m > > } } } } } } } } } ( M ⊗ N , m ) ( M ⊗ N , m ) ( M ⊗ N , 1) m / / m A A A A A A A A A m > > } } } } } } } } } provides an abso lute e q ualizer ( M ⊗ N , m ) of ( γ r ⊗ 1) and (1 ⊗ γ l ). Thu s, the tensor pro duct of M a nd N ov er C is M ⊗ C N = ( M ⊗ N , m ) . 4.3. The coaction on the tensor pro duct. If Como d ( A ) is to be a mono idal category with underlying functor U : Co mo d ( A ) / / Bicomo d ( C ) stro ng monoidal, then the tenso r pro duct of t w o A -como dules m ust a lso b e an A -como dule. In this section we show that the obvious coactio n on M ⊗ C N , na mely , γ = Q Q Q Q Q Q 7 7 7 : M ⊗ C N / / M ⊗ C N ⊗ A do es the jo b. Lemma 4.3. The c o action γ : M ⊗ C N / / M ⊗ C N ⊗ A , as define d ab ove , is a morphism in Q V . That is, the fol lowing e quation hold s.   ? ? r r r r ? ? ? Q Q Q Q Q Q 7 7 7 = Q Q Q Q Q Q 7 7 7 =   Q Q Q Q Q Q 7 7 7 ? ? r r r r ? ? ? Pr o of. The first equalit y is giv en by   ? ? r r r r ? ? ? Q Q Q Q Q Q 7 7 7 (c) =   O O O O O O O O O O O O J J J J J t t t t (b) =   Q Q Q Q Q Q 7 7 7 ? ? (c) = Q Q Q Q Q Q 7 7 7 , WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 17 and the second by a similar calculation:   Q Q Q Q Q Q 7 7 7 ? ? r r r r ? ? ? (c) =   O O O O O O O O O O O O J J J J J t t t t (b) =   R R R R R R R ? ? ? ?   (c) = Q Q Q Q Q Q 7 7 7 .  Prop osition 4.4. ( M ⊗ C N , γ ) is an A -c omo dule. Pr o of. Coasso ciativity is prov ed as usua l, R R R R R R R ? ? ? ?    / / / (b) = O O O O O O O O O O O O J J J J J t t t t (c) = T T T T T T T T J J J J J Q Q Q Q Q Q 7 7 7 and the counit co ndition a s   S S S S S 4 4 4 (L. 4.3) =     ? ? r r r r ? ? ? Q Q Q Q Q Q 7 7 7 (c) =      ? ? ? ? o o o 9 9 9 9     (b) =     ? ? o o o o   ? ? G G G (c) =   ? ? r r r r ? ? ? = 1 M ⊗ C N .  4.4. Como d ( A ) is a monoidal category. W e no w set out to prov e the cla im a t the beginning of this section, tha t ( Como d ( A ) , ⊗ C , C ) is a monoidal category . It will turn out that asso ciativit y is a strict equality (if it is so in V ) and the unit conditions are only up to isomorphism. W e state this a s a theorem and devote the r emainder o f this s ection to its pro of. Theorem 4. 5. Como d ( A ) = ( Como d ( A ) , ⊗ C , C ) is a monoid al c ate gory. First off no te that C itself is an A -como dule with coaction C C A   t    7 7 7 . Before proving this theorem it will b e useful to have the follo wing lemma. Lemma 4. 6. The fol lowing identities hold.   t Q Q Q Q Q Q 7 7 7 =   t   t J J J J J J J   t Q Q Q Q Q Q 7 7 7 =   t   s O O O O O ? ? r r r t t t t 18 CRAIG P ASTR O AND ROSS STREET Pr o of. The first iden tit y is pro ved by   t Q Q Q Q Q Q 7 7 7 (9) =   t   s Q Q Q Q Q Q 7 7 7 (11) =   t   t J J J J J (c) =   t   t J J J J J J J and the second by   t Q Q Q Q Q Q 7 7 7 (3) =   t   t Q Q Q Q Q Q 7 7 7 (11) =   t   s J J J J J J   t t t t t (c) =   t   s O O O O O ? ? r r r t t t t .  Pr o of of The or em 4.5. Consider ( M ⊗ C N ) ⊗ C P and M ⊗ C ( N ⊗ C P ) in Q V . T he former is ( M ⊗ N ⊗ P , u ) and the latter ( M ⊗ N ⊗ P , v ) where u =     J J J ? ? l l l l W W W W W / / / / / ? ? l l l l and v =     J J J ? ? l l l l J J J ? ? l l l l o o o o ? ? ? . Since, by Lemma 4.3, γ is a morphism in Q V , both u and v may b e r ewritten as   V V V V V 7 7 7 ? ? l l l l proving the (strict) equality ( M ⊗ C N ) ⊗ C P = M ⊗ C ( N ⊗ C P ) in Q V (since w e are writing as if V w ere strict). It r emains to pr o v e M ⊗ C C ∼ = M ∼ = C ⊗ C M . By definition M ⊗ C C = ( M ⊗ C,   t   ? ? l l l l J J J ) and C ⊗ C M = ( C ⊗ M ,   t   ? ? l l l l J J J ) . W e will show that th e morphisms   t   G G G : M ⊗ C C / / M and   t G G : M / / M ⊗ C C will establis h the is omorphism M ⊗ C C ∼ = M , and   t   ? ? o o o o : C ⊗ C M / / M and   s ? ? o o o : M / / C ⊗ C M WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 19 the isomorphism M ∼ = C ⊗ C M . These morphisms ar e easily seen to b e in Q V , and the fact that they a r e mutually inv erse pair s is given in one direction by Lemma 4.6, and in the other b y an easy string calculation making use of the identit y (6). It now remains to show that these four mor phisms ar e A -como dules mo rphisms, i.e., that they are in Como d ( A ). Note that M ⊗ C C and C ⊗ C M are A -como dules via the coactions   t Q Q Q Q Q Q 7 7 7 and   t Q Q Q Q Q Q 7 7 7 resp ectiv ely . W e then have: •   t   G G G : M ⊗ C C / / M is an A -como dule morphism as:   t   t   Q Q Q Q Q Q / / /   ? ?     (L. 4.6) =   t   t   t   O O O O J J J J J (7) =   t   t   O O O O J J J J J (c,6) =   t   L L L L L L (c) =   t G G G (c) =   t   L L L L     ? ? (4) =   t   L L L L (c) =   t   G G G ? ? ? ? . •   t G G : M / / M ⊗ C C is an A -como dule morphism as:   t   t O O Q Q Q Q Q 7 7 7 (7) =   t O O Q Q Q Q Q 7 7 7 (L. 4.6) =   t   t O O O O O O O O O (c) =   t   t O O O O O O O (6) =   t J J . •   t   ? ? o o o o : C ⊗ C M / / M is an A -como dule mo rphism as:   t   t   / / X X X X X X 7 7 7 ? ? l l l l     (L. 4.6) =   t   s   t   T T T T T 7 7 7 ? ? l l l ? ? l l l l     (8) =   t   s   T T T T T 7 7 7 ? ? l l l ? ? l l l l     (c,6) =   t   S S S S 7 7 7 ? ?     (c) =   t M M M M 2 2 2 2 (c) =   t   M M M M 2 2 2 2 ? ? (4) =   t   R R R R R R J J J 7 7 7 (c) =   t   ? ? J J J j j j j    / / / . 20 CRAIG P ASTR O AND ROSS STREET •   s ? ? o o o : M / / C ⊗ C M is an A -como dule mo rphism as:   s   t ? ? o o o V V V V V 7 7 7 (L. 4.6) =   s   t   s ? ? o o o V V V V 7 7 7 ? ? o o o (8) =   s   s ? ? o o o M M M M 7 7 7 ? ? o o o (c,6) =   s ? ? ? ? ? o o o . Thu s, M ⊗ C C ∼ = M ∼ = C ⊗ C M in Q V .  Thu s, Como d ( A ) = ( Como d ( A ) , ⊗ C , C ) is a mo noidal categor y . 4.5. The forgetful functor from A -com odul es to C -bicom odul es. Ther e is a fo r getful functor U : Como d ( A ) / / Bicomo d ( C ) which as signs to each A - como dule M a C -bicomo dule U M which is M itself with coaction M A C M C   s   t ? ? ? o o o o o / / . A morphis m of A -como dules f : M / / N is automatica lly a morphism of the underlying C -bicomo dules f : U M / / U N . Prop osition 4. 7 . The for getful functor U : Como d ( A ) / / Bicomo d ( C ) is str ong monoidal. Pr o of. W e m ust es tablish the C -bicomo dule isomo rphisms C ∼ = U C and U M ⊗ C U N ∼ = U ( M ⊗ C N ) . The first is obvious. T o establish the second isomorphism we obser v e that the ob ject U M ⊗ C U N is ( M ⊗ C N , m ) with coaction     s   t ? ? ? ? ? r r r r ? ? j j j G G and U ( M ⊗ C N ) is also ( M ⊗ C N , m ) but with coaction   s   t T T T T T T 2 2 2 2 i i i i i i i ' ' . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 21 The follo wing calculation sho ws that these t w o coa ctions ar e the sa me, a nd hence the is omorphism U ( M ⊗ C N ) ∼ = U M ⊗ C U N .     s   t ? ? ? ? ? r r r r ? ? j j j G G (2) =   s     s   t 7 7 7 ? ? o o o o o ? ? j j j G G (c,10) =   s     s   t 7 7 7 ? ? o o o o o ? ? j j j * * * (2) =     s   t J J J ? ? l l l l ? ? j j j * * * (4) =     t   s Y Y Y Y Y Y ? ? ? ?   4 4 ? ? l l l (c) =   s   t T T T T T T O O O O O O L L L L j j j j j j j (4) =   s   t X X X X X X 7 7 7 j j j j j j j / /  This may seem to be a strict equality , but as tensor pro ducts are r eally only defined up to isomorphism we pr e fer “strong”. 4.6. Como d f ( H ) is left a utonomous. Let V f denote the sub category of V con- sisting of the ob jects with a left dua l (since V is bra ided left duals are right duals). There is a for g etful functor U l : Como d ( H ) / / V defined as the comp osite of the tw o forg etful functor s Com od ( H ) / / Bicomo d ( C ) a nd Bicom o d ( C ) / / V . This comp osite U l : Como d ( H ) / / V is sometimes called the lo ng for getful func- tor , as oppos e d to the short for getful functor U : Com od ( H ) / / Bicomo d ( C ). Let us say a n ob ject M ∈ Como d ( H ) is d ualizable if U l M has a left dua l in V , i.e., U l M ∈ V f . Denote by Com od f ( H ) the sub category of Como d ( H ) consisting of the dualizable ob jects. The goal o f this section is to prov e the following pro position. Prop osition 4 . 8. If H is a we ak Hopf monoid then the c ate gory Como d f ( H ) is left autonomous (= left rigid = left c omp act). Suppo se M ∈ Como d f ( H ) has a left dual M ∗ in V . Using the an tipo de of H a co action on M ∗ is defined as M ∗ M ∗ A   ν J J J J . By (17) it is eas y to see that this defines a como dule structure on M ∗ . W e claim that M ∗ is the left dual of M in Com o d f ( H ). Define mor phisms e : M ∗ ⊗ C M / / C and n : C / / M ⊗ C M ∗ via e =   t ? ? and n =   r   t   O O O O O   . Prop osition 4.9. Supp ose M ∈ Como d f ( H ) with underlyi ng left dual M ∗ . Then M ∗ with evaluation and c o evaluation morphisms e and n r esp e ctively is the left dual of M in Co mo d f ( H ) . That is, Como d f ( H ) is left autonomous. 22 CRAIG P ASTR O AND ROSS STREET Pr o of. Let M , M ∗ , e , and n be as ab ov e. W e will first show that e and n a re como dule morphisms, and s econdly that they satisfy the triangle iden tities. The f ollowing ca lculation shows that e is a como dule morphism.   ν   t X X X X X J J J J (tri) =   t   ν J J J J S S S S (c) =   ν   t H H H H H J J K K (4) =   ν   t J J J J   ( ν ) =   t   t ? ?   (3,7) =   t   t ? ?    ? ? ? T o sho w that n is a como dule mor phis m we must establish the equality   r   t     ν O O O O O   O O O T T T T T T T T =   r   t   O O O O O    which is prov ed by the follo wing calcula tion.   r   t     ν O O O O O   O O O T T T T T T T T (tri) =   ν   r   t   O O O O O O O Q Q Q Q   U U U U U U ( ν ,c) =   ν   ν   t   O O O O / / T T T S S S S S S S S S S / / /    5 5 5 5 5 5 (17) =   ν   t   O O O J J : : : : :    R R R R j j j j j G G G (b) =   ν   t   O O O 2 2 2 Q Q Q Q Q Q ? ?    ( ν ) =   r   t   O O O O O O   (4) =   r   t O O O O O   (4) =   r   t   O O O O O    It remains to show that e and n sa tisfy the triangle identities, i.e., that the following comp osites a re the iden tit y : (i) M ∼ = C ⊗ C M M ⊗ C M ∗ ⊗ C M n ⊗ 1 / / M ⊗ C C ∼ = M 1 ⊗ e / / (ii) M ∗ ∼ = M ∗ ⊗ C C M ∗ ⊗ C M ⊗ C M ∗ 1 ⊗ n / / C ⊗ C M ∗ ∼ = M ∗ e ⊗ 1 / / . Recall that M ∼ = M ⊗ C C and M ∼ = C ⊗ C M via   t G G ,   t   G G G and   s ? ? o o o ,   t   ? ? o o o o resp ectiv ely . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 23 The f ollowing ca lculation prov es (i):   s   r   t     t   T T T ? ?   M M M M M M q q q q q - - - - ? ? y y y (tri) =   s   t   r     J J J J J J M M M M O O O O O (c,13) =   s   r   t     J J J J J J J J J J J (2,c) =   r     t   O O O O O O O O   ? ? ? ? ?       ? ? ? ? ?   (6) = = 1 M , and (ii) is given by:   ν   t   t   r     ν   t   t   O O O J J O O O O O O O O O O O O O O Q Q Q Q (tri) =   ν   r   t   ν     t   T T T T T T T T T T T T S S C C C C C C O O O O   ? ? ?       (13) =   ν   t     r   ν   t   T T T T T T S S T T T E E E T T T J J J   ? ? ?     (2, ν ) =   r     r   ν   t   T T T S S T T T E E E T T T J J J   ? ? ?     (2,c) =   ν   r   t   U U U U U U   G G G ? ? ? ? G G   w w w w w (13) =   ν   t   r   T T T T T T   ? ? ? ? ?   (2) =   ν   T T T T T T   ? ? ? ? ?    ( ν ) =   t   T T T (2,c) = (tri) = = 1 M ∗ . This completes the pro of that M ∗ is the left dual of M in Com o d f ( H ), and hence that Como d f ( H ) is left autonomous.  5. Frobenius monoid example Let R b e a se pa rable F rob enius mono id in V . In this section we prove that R ⊗ R is an example of a weak Hopf mo noid with an in v ertible antipo de. In the case V = V ect , this exa mple is essent ially the same as in [4, Appendix]. Let R b e a F rob enius monoid in V . Then R ⊗ R b ecomes a comonoid via δ = and ǫ = . (where, for simplicit y , in this section we will adopt the simpler notation      ? ? ? = and   ? ? ?    = ) , and a monoid via µ = A A A A A A = = = = = =      F F y y t t t t t t and η =     . 24 CRAIG P ASTR O AND ROSS STREET The comonoid structure is via the como nad gener a ted by the adjunction R ⊣ R . The monoid s tr ucture is the usual mo noid structure (viewing R as a mo noid) on the tens o r pro duct R ◦ ⊗ R , where R ◦ is the opp osite monoid of R . Prop osition 5.1. If R is sep ar able, me aning µδ = 1 R , then R ⊗ R is a we ak bimonoid. An invertible antip o de ν on R ⊗ R is given by ν = . which makes R ⊗ R into a we ak Hopf monoid. The following three sets of calculatio ns establish r e spectively the axioms (b), (v), and (w), and hence the first claim. The axiom (b) is given by: ( µ ⊗ µ )(1 ⊗ c ⊗ 1 )( δ ⊗ δ ) = 9 9 9 9 9 9 9 9 ? ?   ? ?   9 9 9 9 9 9 9 9 9 9 9 ' ' ' ' ' ' ' '                       (nat) = H H H H H D D D D D      O O O   o o o o o o (sep) = H H H H H D D D D D      O O O   o o o o o o = δ µ. Axiom (v) is se en from t he diagrams:   9 9 9 9     : 6 6 6 6 6      O O O O O O        : : : : : ? ?   P P P P P P P P P E E    t t t t t t              w w w G G G :      / / / / /      ( ( ( ( ( : : : : : 6 6 6 6 6 : : : : : :      ? ?   J J J o o r r r r r r t t t t t t (c,tri) = 4 4 4 4 4 4 4       : : : : : ? ?   P P P P P P P P P E E    t t t t t t         (c) = 6 6 6 6 6      O O O O O O        : : : : : ? ?   P P P P P P P P P E E    t t t t t t              :     2 2 2 2 . . . . . . . , , , , , , , ,                  ? ?   J J J   v v v v v v v v v v < < < < < < < < @ @ @ @ @ @ @ @        (c) = ? ?       / / / / 2 2 2 2 9 9 9 9 + + + + + + + +                   ? ?   x x x x x x x x x x n n n n n n > > > > > > > >       (tri) = : : : : : ? ?   6 6 6 6 6      Q Q Q Q Q    O O O O O O O t t t t t t (c) = 6 6 6 6 6      O O O O O O        : : : : : ? ?   P P P P P P P P P E E    t t t t t t          F or (w), by the naturality of the braiding and the counit prop erty of R ea c h equation in (w), i.e.,       9 9 9 9 ,     G G G w w w ,     WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 25 is eas ily s een to b e equal to the following diagra m      . Thu s, R ⊗ R is a w eak bimonoid. W e next prov e that tha t R ⊗ R is a weak Hopf monoid with in vertible a n tipo de ν = . An inverse to ν is easily see n to b e g iv en b y ν − 1 = : : , and so the antipo de is invertible. W e note that (in simplified form) r =   7 7   and t =   ? ? ?    . The following ca lc ulations then prov e the an tipo de axioms. µ ( ν ⊗ 1) δ =     9 9 9 9 2 2 2 2 2 2 2 = = = = = = 7 7 7 7 7 7       F F y y y y y y y y z z z z (tri) = ? ? ?    (sep) =   ? ? ?    = t µ (1 ⊗ ν ) δ =     9 9 9 9 % % % % % % / / / / / / / = = = = = = 7 7 7 7 7 7       F F y y y y y y y y z z z z (nat) = 7 7   (sep) =   7 7   = r µ 3 ( ν ⊗ 1 ⊗ ν ) δ 3 =        ? ? ? ? ? ? ? % % % % % % / / / / / / / = = = = = = 7 7 7 7 7 7       F F y y / / / / / / /      N N N N N N N N D D D D D D t t t J J J o o o o o o y y y y y y z z z z z z z z (sep) =     5 5 5 5 5 5 5     z z z z (sep) =   6 6 6 6 6 6 6 6   z z z z (c) = = ν 26 CRAIG P ASTR O AND ROSS STREET Thu s, R ⊗ R is a w eak Hopf monoid with in v ertible antipo de. 6. Quantum gr oupoids In this section we recall the quantum categor ies and qua n tum gr oupoids of D ay and Street [9]. There is a succinct definit ion given in [9, p. 216 ] in terms o f “ba s ic data” and “Hopf ba sic data”. Here w e give the unpack ed definition of quantum category and quan tum groupoid which is ess en tially found in [9, p. 221]; how ev er, we do mak e a correction. Our setting is a braided monoidal c a tegory V = ( V , ⊗ , I , c ) in which the functors A ⊗ − : V / / V with A ∈ V , preserve coreflexive equalizers, i.e., equalizers of pairs o f morphisms with a common left in v erse. 6.1. Quan tum categories. Suppose A and C a re comono ids in V and s : A / / C ◦ and t : A / / C are c o monoid morphisms suc h that the diagra m A A ⊗ A δ 8 8 r r r r r r C ⊗ C s ⊗ t / / C ⊗ C c   A ⊗ A δ & & L L L L L L t ⊗ s / / commutes. Then A may b e viewed as a C -bicomo dule with left and right coactions defined respectively v ia γ l =  A A ⊗ A δ / / A ⊗ C 1 ⊗ s / / C ⊗ A c − 1 / /  γ r =  A A ⊗ A δ / / A ⊗ C 1 ⊗ t / /  . Recall that the tensor pro duct P = A ⊗ C A o f A with itself ov er C is defined as the e q ualizer P A ⊗ A ι / / A ⊗ C ⊗ A γ r ⊗ 1 / / 1 ⊗ γ l / / . The f ollowing dia grams may b e seen to commute P A ⊗ A ι / / C ⊗ A ⊗ A γ l ⊗ 1 / / C ⊗ A ⊗ C ⊗ A 1 ⊗ γ r ⊗ 1 / / 1 ⊗ 1 ⊗ γ l / / P A ⊗ A ι / / A ⊗ A ⊗ C 1 ⊗ γ r / / A ⊗ C ⊗ A ⊗ C γ r ⊗ 1 ⊗ 1 / / 1 ⊗ γ l ⊗ 1 / / and therefore induce resp ectiv ely a left C - and right C -coactio n on P . These coactions ma k e P int o a C - bicomodule. The comm utativit y of the diagra m P A ⊗ A ι / / A ⊗ 4 δ ⊗ δ / / A ⊗ 4 1 ⊗ c ⊗ 1 / / A ⊗ A ⊗ A ⊗ C ⊗ A 1 ⊗ 1 ⊗ γ r ⊗ 1 / / 1 ⊗ 1 ⊗ 1 ⊗ γ l / / WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 27 may b e seen from   t   ι 3 3                 E E E E E E * * * * * * * * * * y y y E E E =   t   ι      ; ; ; ; ; ; ; 2 2 2 2 2 2 y y y y w w 3 3 3 3 3 3 2 2 2      y y y + + + + + + =   s   ι     G G G G G G       ; ; ; ; < < $ $ $ $ $ R R R         7 7 =   s   ι      5 5 6 6    ; ; ;          * * * * * * * * *          : : :     ? ? ? ? , and as 1 ⊗ 1 ⊗ ι is the equa lizer of 1 ⊗ 1 ⊗ γ r ⊗ 1 and 1 ⊗ 1 ⊗ 1 ⊗ γ l , there is a unique morphism δ l : P / / A ⊗ A ⊗ P making the diagram P A ⊗ A ι / / A ⊗ A ⊗ A ⊗ A δ ⊗ δ / / A ⊗ A ⊗ A ⊗ A 1 ⊗ c ⊗ 1   A ⊗ A ⊗ P δ l   1 ⊗ 1 ⊗ ι / / commute. In strings,   ι     : : : : :     / / / /    2 2 2 =   ι '&%$ !"# δ l          * * * 9 9 . It is eas y to see (pos tco mpose with the monomorphism 1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ ι ) that the mo rphism δ l is the left c oaction o f the comonoid A ⊗ A on P making P in to a (left) A ⊗ A -co module. This means that the diagr ams P A ⊗ A ⊗ P δ l   A ⊗ A ⊗ A ⊗ A ⊗ P δ ⊗ δ ⊗ 1   A ⊗ A ⊗ A ⊗ A ⊗ P 1 ⊗ c ⊗ 1 ⊗ 1 / / A ⊗ A ⊗ P δ l / / 1 ⊗ 1 ⊗ δ l   P A ⊗ A ⊗ P δ l / / P ǫ ⊗ ǫ ⊗ 1   1 " " E E E E E E E E E E E E commute. W e are now r eady to s ta te the definition. A quantum c a te go ry in V consists o f the data A = ( A, C, s, t, µ, η ) where A , C , s , t ar e as a bov e, and µ : P = A ⊗ C A / / A and η : C / / A are morphisms in V , called the c omp osition morphism a nd un it morphism resp ectively . This data m ust s atisfy axioms (B1 ) through (B6 ) b elow. (B1) ( A, µ, η ) is a monoid in Bicomo d ( C ). (B2) The following dia gram commutes. P A ⊗ A ⊗ P δ l / / C ⊗ P t ⊗ ǫ ⊗ 1 / / ǫ ⊗ s ⊗ 1 / / C ⊗ A 1 ⊗ µ / / 28 CRAIG P ASTR O AND ROSS STREET Before stating (B3), we use (B2) to sho w that th e diagram P A ⊗ A ⊗ P δ l / / A ⊗ A ⊗ A 1 ⊗ 1 ⊗ µ / / A ⊗ C ⊗ A ⊗ A γ r ⊗ 1 ⊗ 1 / / 1 ⊗ γ l ⊗ 1 / / commutes, a s s een by the calculation   t   µ '&%$ !"# δ l ! ! ! ! ! ! ! !    3 3 3 3 =   t     µ '&%$ !"# δ l , , ,    % % 5 5 5 5 5 5 5 5 =     t   µ '&%$ !"# δ l '&%$ !"# δ l        + + + : : : :   =     s   µ '&%$ !"# δ l '&%$ !"# δ l         # # # 3 3 3 3 9 9 =   µ   s '&%$ !"# δ l . As ι ⊗ 1 is the equalizer of γ r ⊗ 1 ⊗ 1 and 1 ⊗ γ l ⊗ 1 there is a unique morphism δ r : P / / P ⊗ A making the square P A ⊗ A ⊗ P δ l / / A ⊗ A ⊗ A 1 ⊗ 1 ⊗ µ   P ⊗ A δ r   ι ⊗ 1 / / commute. W e can no w sta te (B3). (B3) The following dia gram commutes. P A µ / / A ⊗ A δ   P ⊗ A δ r   µ ⊗ 1 / / (B4) The following dia gram commutes. P A µ / / I ǫ   A ⊗ A ι   ǫ ⊗ ǫ / / (B5) The following dia gram commutes. C A η   I ǫ 7 7 o o o o o o o ǫ ' ' O O O O O O O WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 29 (B6) The following dia gram commutes. C A η = = | | | | | | | A ⊗ A δ / / C ⊗ A s ⊗ 1 / / A ⊗ A η ⊗ 1 ! ! B B B B B B A η ! ! B B B B B B B A ⊗ A δ / / C ⊗ A t ⊗ 1 / / η ⊗ 1 = = | | | | | | A η / / δ / / A consequence of these axioms is that P b ecomes a le ft A ⊗ A -, right A - bicomo dule. The axiom (B6) ma kes C in to a righ t A -como dule via C A η / / A ⊗ A δ / / C ⊗ A s ⊗ 1 / / . W e refer to A as the obje ct-of-arr ows and C a s the obje ct-of-obje cts . 6.2. Quan tum group oids. Suppo se we hav e co monoid isomorphisms υ : C ◦◦ ∼ = / / C and ν : A ◦ ∼ = / / A . Denote by P l the le ft A ⊗ 3 -como dule P with coaction defined b y P A ⊗ A ⊗ P ⊗ A δ / / A ⊗ A ⊗ P ⊗ A 1 ⊗ 1 ⊗ 1 ⊗ ν / / A ⊗ A ⊗ A ⊗ P 1 ⊗ 1 ⊗ c P,A / / , and by P r the le ft A ⊗ 3 -como dule P with coaction defined b y P A ⊗ A ⊗ P ⊗ A δ / / A ⊗ A ⊗ P ⊗ A 1 ⊗ 1 ⊗ 1 ⊗ ν − 1 / / A ⊗ A ⊗ A ⊗ P c − 1 A ⊗ A ⊗ P,A / / . F urthermo r e, s uppose that θ : P l / / P r is a left A ⊗ 3 -como dule isomorphism. W e define a quant um gr oup oid in V to be a quan tum ca tegory A in V equipp ed with an υ , ν , and θ sa tisfying (G1) through (G3) b elow. (G1) sν = t , (G2) tν = υ s , and (G3) the diag r am 3 P C ⊗ C ⊗ C ς / / C ⊗ C ⊗ C c C,C ⊗ C / / C ⊗ C ⊗ C 1 ⊗ 1 ⊗ υ   P θ   ς / / commutes, where the mor phis m ς : P / / C ⊗ 3 is defined by taking either of the equal routes P A ⊗ A ι / / A ⊗ C ⊗ A γ r ⊗ 1 / / 1 ⊗ γ l / / C ⊗ 3 s ⊗ 1 ⊗ t / / . 3 This corrects [9, § 12, p. 223]. 30 CRAIG P ASTR O AND ROSS STREET 7. Weak Hopf monoids are quantum groupoids The goal o f this section is to prov e the following theor e m. Theorem 7.1. A we ak bimonoid in Q V is a quantu m c ate gory in Q V whose obje ct- of-obje cts is a sep ar ab le F r ob enius monoid. If the we ak bimonoid is e quipp e d with an invertible antip o de, making it a we ak Hopf monoid, t he n the quantu m c ate gory b e c omes a quantum gr o up oid. 7.1. W eak bimonoids are quan tum categories. Let A = ( A, 1) b e a w eak bimonoid in Q V with source mo rphism s and target morphism t and set C = ( A, t ). This data along with µ = 7 7 7    : P / / A η = t : C / / A forms a quant um ca tegory in Q V . The morphisms s and t are obviously in Q V , hence so is η = t , and   J J J ? ? l l l l 7 7 7    (nat) =   D D D D D z z z z (b) = 7 7 7    shows that µ is as well. Recall tha t P = ( A ⊗ A, m ) where m =   J J J ? ? l l l l . The mor phis ms δ l : P / / A ⊗ A ⊗ P and δ r : P / / P ⊗ A a re given by δ l =   J J J ? ? l l l l J J J ? ? l l l l     : : : : : * * * *     δ r =   J J J ? ? l l l l Q Q Q Q Q Q . The tw o calculatio ns   J J J ? ? l l l l J J J ? ? l l l l      4 4 4 4 4 4 4 4 4 4   J J J ? ? l l l l     (c) =   ? ? l l l l J J J   J J J ? ? l l l l J J J ? ? l l l l      4 4 4 4 4 4 4 4 4 4     =   J J J ? ? l l l l J J J ? ? l l l l      4 4 4 4 4 4 4 4 4 4     and   Q Q Q i i i i ? ?   J J J ? ? l l l l Q Q Q Q Q Q (c) =     T T T T T T D D D D D J J J l l l l ? ? z z z z (b) =   J J J ? ? l l l l Q Q Q Q Q Q WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 31 show that these ar e morphis ms in Q V . T o see that ( A, µ, η ) is a comonoid in Bicomo d ( C ) no tice that a ssocia tivit y follows f rom the a ssociativ ity of the µ viewed as a weak bimonoid and the counit prop ert y may b e seen fr o m prop erty (6), i.e., µ (1 ⊗ t ) δ = 1 A and µ ( s ⊗ 1) c − 1 δ = 1 A , and so (B1) holds. (B2) follows fr om o ne application o f (12), (B3) f rom (b), (B4) from (c), and (B5) from (2), while the calculation   t   s   t   (3) =   t   t   s   t   (8) =   t   t   t   verifies (B6 ). Thus, A = ( A, C, s, t, µ, η ) is a quantum category in Q V . 7.2. W eak Hopf mo noids are quan tum group oids. Now suppo se that A = ( A, 1) is a weak Hopf monoid in Q V with an in v ertible an tipo de ν : A / / A , and that A = ( A, C , s, t, µ, η ) is as ab o ve. The data for a quantu m g roupoid ( υ , ν, θ ) is υ = tν ν t : C ◦◦ / / C ν = ν : A ◦ / / A θ =   ν : P / / P. In the remainder o f this section w e will v erify this claim. The morphisms υ and ν are o bviously mo rphisms in Q V , and the tw o calcula tions     ν J J J ? ? l l l l (c) =     ν R R R R R R R : : : : : : 4 4 4 4 4 T T T T T T (b) =   ν and   ν   J J J l l l l ? ? (17) =   ν     ν t t J J J (2) =   ν   r     ν t t J J J (16) =   t   ν     ν t t J J J (4) =   t   ν     ν J J J 32 CRAIG P ASTR O AND ROSS STREET (16) =   ν   r     ν J J J (2) =   ν     ν J J J (c) =   ν     ν    4 4 4 4 4   ( ν ) =   r     ν (2,c) =   ν show that θ is a s well. Lemma 7. 2. An inve rse for θ is given by θ − 1 = 7654 0123 ν - 1   O O O O      J J J ? ? l l l l : : : : : , Pr o of. Since   ? ? l l l l J J J O O O O (c) =   J J J E E E E y y y y (b) = J J J J J it is clear that θ − 1 is a morphism in Q V . That θ − 1 is an in verse for θ ma y b e se e n in one directio n fro m θ − 1 θ =   ν 7654 0123 ν -1   O O O O      J J J ? ? l l l l : : : : : (17) =   ν 7654 0123 ν -1 7654 0123 ν -1       4 4 4         J J J ? ? l l l l ? ? ? ? ? ? ? O O O O O = 7654 0123 ν -1         / / / /         J J J ? ? l l l l 9 9 9 9 9 9 9 J J J J (c) = 7654 0123 ν -1      ? ? ? v v v v v ? ? ? ? y y y f f f f f f Q Q Q ? ? i i i i K K K K ( † ) =   s   ? ? j j j k k O O O ? ? j j j j (4) =   s     ? ? o o o o ? ? O O O ? ? j j j j (2) =     ? ? l l l l J J J J J J ? ? l l l l =   ? ? l l l l J J J = 1 P where ( † ) is given by 7654 0123 ν -1    ? ? ? v v v v v y y y ? ? ? ? K K K K =   ν 7654 0123 ν -1 7654 0123 ν -1    ? ? w w w w y y y E E E G G G G (17) =   ν 7654 0123 ν -1 ? ?   ( ν ) =   r 7654 0123 ν -1 (15) =   s , WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 33 and in the other direction by: θθ − 1 = 7654 0123 ν -1     ν O O O O      J J J ? ? l l l l : : : : : ( ‡ ) = 7654 0123 ν -1   ν O O O O     4 4 4 4 4 4 (c) = 7654 0123 ν -1   ν T T T T z z z z ? ? ?    D D D D (17) = 7654 0123 ν -1   ν   ν T T T T 7 7   =   ν T T T T ? ?    (c, ν ) =   t ? ? ? ? (4) =   t   J J ? ? r r r r (2) =   J J J ? ? l l l l = 1 P for which the first step ( ‡ ) holds since θ is a morphism in Q V .  That the antipo de ν : A ◦ / / A is a comonoid is omorphism is our assumption. That υ : C ◦◦ / / C is as well may b e see n from the following calcula tion: ( t ⊗ t ) δ υ = ( t ⊗ t ) δ tν ν t = ( t ⊗ t ) δ ν ν t (3) = ( t ⊗ t ) c ( ν ⊗ ν ) δ ν t (17) = ( t ⊗ t ) c ( ν ⊗ ν ) c ( ν ⊗ ν ) δ t (17) = ( t ⊗ t )( ν ⊗ ν )( ν ⊗ ν ) ccδ t (nat) = ( t ⊗ t )( t ⊗ t )( ν ⊗ ν )( ν ⊗ ν ) ccδ t (7) = ( t ⊗ t )( ν ⊗ ν )( r ⊗ r )( ν ⊗ ν ) ccδ t (16) = ( t ⊗ t )( ν ⊗ ν )( ν ⊗ ν )( t ⊗ t ) ccδ t (16) = ( t ⊗ t )( ν ⊗ ν )( ν ⊗ ν )( t ⊗ t ) ccδ (5) = ( υ ⊗ υ ) ccδ. (5) An inverse for υ is g iv en by the morphism υ − 1 = tν − 1 ν − 1 t, as may b e seen in one direction by the calculation υ − 1 υ = tν − 1 ν − 1 ttν ν t = ttν − 1 ν − 1 ν ν tt (16) = tν − 1 ν − 1 ν ν t (7) = tt = t = 1 C . (7 ) The other direction is similar. 34 CRAIG P ASTR O AND ROSS STREET Recall that the left A ⊗ A -, r igh t A -coac tion δ on P is defined by tak ing the diagonal o f the comm utative sq ua re: P A ⊗ A ⊗ P A ⊗ A ⊗ P ⊗ A. P ⊗ A δ l / / 1 ⊗ 1 ⊗ δ r   δ r   δ l ⊗ 1 / / W e note that δ may b e wr itten a s     U U U U U U J J J J J J     : : : : : ? ? l l l l ? ? l l l l * * * *     =   U U U U U U J J J     : : : : : ? ? l l l l * * * *     (c) =       : : : : : M M M M M E E E E / / / / y y y y     (b) =     : : : : : P P P P P P P * * * *     = δ. W e must s how that θ is a left A ⊗ 3 -como dule iso morphism P l / / P r . That is, we must prove the comm utativit y of the square P l A ⊗ 3 ⊗ P l A ⊗ 3 ⊗ P r P r γ / / 1 ⊗ θ   θ   γ / / where the left A ⊗ 3 -coactions on P l and P r were defined using δ (see § 6 .2 ). The clockwise dir ection ar ound the square is   ν   ν P P P P P P P         < < < < < < < < < < 6 6 6 6 6                (17) =   ν   ν   ν B B B B B B B B B   E E E P P P P P P P 4 4 4 4         7 7 7 7 7 7                 (c) =   ν   ν   ν K K K K K K K K K K K K K    ? ?   ( ( ( ( ( C C C C C C C      C C C C C C C v v v v v v (?) =   ν O O O O O O O     * * * *         : : : : : where the last step (?) is giv en by the fo llo wing ca lculation   ν ? ? ? ?         ? ? ? ? (17) =   ν   ν K K K K K G G w w     ? ? ? ? s s s s (b) =   ν   ν K K K K K K K K K K K s s s s s s s s (17) =   ν   ν   ν   ν O O O O O O O    9 9 9    9 9 9 M M M M M M M M M M - - - - - o o o o o o      q q q q q q q q q (17) =   ν   ν   ν O O O O O O O    , , , ,    , , , , M M M M M M M M M M - - - - -    4 4 4 o o o o o o q q q q q q q q q q (nat) =   ν   ν   ν < < < < < < < <    7 7   ? ? ? ? ? ? ? { { { { { . WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 35 The coun ter-clo c kwise direction is   ν 7654 0123 ν -1 M M M M M ' ' ' , ,      * * * * $ $ $ $     (17) =   ν   ν 7654 0123 ν -1 7654 0123 ν -1   ? ?      C C C { { {               4 4 4 4 4 $ $ $ $ H H H H H H H H      (b) =   ν 7654 0123 ν -1 J J J J J J J       C C C C { { {               4 4 4 4 4 $ $ $ $ + + + + + D D D D D D D t t t t t t      (c) = 7654 0123 ν -1   ν T T T T T T T T T T T L L L L t t t J J J z z z z z z J J J J t t r r r r r r       # # # # # G G G G =   s   ν R R R R R R R R    ? ? s s s s s     % % % 9 9 9 9 (c) =   s   ν L L L L L L L L z z z z z     % % % (11) =   t   ν R R R R R R R R w w w w w     % % %    (c,6) =   ν O O O O O O O     * * * *         : : : : : . Thu s, θ is a left A ⊗ 3 -como dule morphism P l / / P r . The inv erse of θ then is a left A ⊗ 3 -como dule morphism P r / / P l . W e now prove the pro perties (G1) through (G3) required of a quantum gro upoid. The calcula tion   ν   s (12) =   ν   r   s (16) =   t   r   s (12) =   t   s (8) =   t verifies (G1), and the follo wing esta blis hes (G2 ).   ν   t (7) =   ν   t   t (16) =   r   ν   t (15) =   s   ν   ν   t (8) =   s   t   ν   ν   t (def ) =   s   υ It r emains to prov e (G3), i.e., we must show that θ mak es the following square P C ⊗ 3 C ⊗ 3 C ⊗ 3 P ς / / c C,C ⊗ C / / 1 ⊗ 1 ⊗ υ   θ   ς / / commute. 36 CRAIG P ASTR O AND ROSS STREET The clockwise dir ection ar ound the square is     s   s   t   t   ν   ν   t     J J J l l l l ? ? J J J t t t q q q q M M M M           (10) =     s   s   t   t   ν   ν   t   J J J l l l l ? ?   7 7           (4) =   s   s   t   t   ν   ν   t   k k k k k ? ?   7 7           =   ν   s   t   t   ? ? for which the last step holds since tν ν ts = tν ν s (8) = tν r (15) = ttν (16) = tν . (7) The coun ter-clo c kwise direction is   ν     s   t   t Q Q Q i i i i ? ?   7 7 (17) =   ν   ν     t   s   t   J J J   7 7 ? ? (2) =   ν   r   ν     t   s   t   J J J   7 7 ? ? (16) =   t   ν   ν     t   s   t   J J J   7 7 ? ? (4) =   t   ν   ν     t   s   t J J J   7 7 ? ? (16) =   ν   r   ν     t   s   t J J J   7 7 ? ? (2) =   ν   ν     t   s   t J J J   7 7 ? ? ( ν ) =   s   ν     t   s   t   7 7 ? ? (2,c) =   ν   s   t   t   ? ? therefore esta blishing the c o mm utativit y of the square. Corollary 7.3. Any F r ob enius monoid in Q V yields a quantum gr oup oi d. By P ropos ition 5.1 every F rob enius monoid R in Q V leads to a w eak Hopf monoid with in v ertible ant ipo de R ⊗ R . Apply Prop osition 7 .1 to this w eak Hopf monoid with in vertible a n tipo de to get a quantu m g roupoid. Appendix A. String diagrams and basic definitions In this a ppendix we g iv e a quick introductio n to str ing dia grams in a braided monoidal category V = ( V , ⊗ , I , c ) [13] and use these to define monoid, mo dule, WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 37 comonoid, como dule, and s eparable F rob enius monoid in V . The string calculus was shown to be rigorous in [12]. A.1. String diagrams. Suppo s e that V = ( V , ⊗ , I , c ) is a braide d (strict) monoidal category . In a string diag r am, ob jects lab el edg es and morphisms lab el no des. F o r example, if f : A ⊗ B / / C ⊗ D ⊗ E is a morphism in V it is re presen ted a s f = A B C D E '&%$ !"# f / / / /          3 3 3 3 3 where this diagram is here meant to b e r ead top-to -bottom. The identit y morphism on an ob ject will b e r epresen ted as the ob ject itself as in A = A . A s p ecial c a se is the ob ject I ∈ V which is represented as th e empt y edge. If, in V , there are morphisms f : A ⊗ B / / C ⊗ D ⊗ E and g : D ⊗ E ⊗ F / / G ⊗ H then they may b e comp osed a s A ⊗ B ⊗ F C ⊗ D ⊗ E ⊗ F f ⊗ 1 / / C ⊗ G ⊗ H 1 ⊗ g / / which may b e repr esen ted as v ertical co ncatenation (1 ⊗ g )( f ⊗ 1) = '&%$ !"# f '&%$ !"# g , ,     , , (where we hav e left off the o b jects). The tensor pro duct of morphisms, say '&%$ !"# f / / / /         / / / / and '&%$ !"# g / / / /         / / / / , is re pr esen ted as horizontal juxtap osition f ⊗ g = '&%$ !"# f / / / /         / / / / '&%$ !"# g / / / /         / / / / (again leaving o ff the ob jects). The braiding c A,B : A ⊗ B / / B ⊗ A is represented as a left-over-right crossing. The inv erse braiding is then represented as a right-ov er-left cross ing. c A,B = A B B A 2 2 2 2 2 2 2       c − 1 A,B = B A A B        2 2 2 2 2 2 Suppo se A ∈ V has a left dual A ∗ , which we deno te by A ∗ ⊣ A a nd say tha t A ∗ is the left a djoin t of A (it is an a djunction if we were to vie w V as a one o b ject 38 CRAIG P ASTR O AND ROSS STREET bicategory ). The ev aluation and co ev a luation morphisms e A : A ∗ ⊗ A / / I and n A : I / / A ⊗ A ∗ are repr esen ted as e A = A ∗ A and n A = A A ∗ . The triang le e qualities b ecome A A A ∗ = A and A ∗ A ∗ A = A ∗ In what follows, in or der to simplify the str ing diag rams, the no des will be omit- ted from certain morphisms (e.g ., m ultiplication a nd co m ultiplication mo rphisms) or simplified (e.g., unit and c o unit mor phisms). A.2. Monoids and mo dules. A monoid A = ( A, µ, η ) in V is an ob ject A ∈ V equipp e d with morphisms µ = 2 2 2 2     : A ⊗ A / / A and η =   : I / / A, called the multiplic ation and unit of the monoid r espectively , s atisfying (m) 9 9 9 9     = 9 9 9 9       = 9 9 9 9     9 9 and   4 4     = =   9 9 9 9   . If A, B a r e monoids , a monoid morphism f : A / / B is a morphism in V satisfying A A B '&%$ !"# f 7 7 7    = A A B '&%$ !"# f '&%$ !"# f 2 2 2    and A B   '&%$ !"# f = B   . Monoids make se nse in any mo noidal ca tegory , how ev er, in order that the tenso r pro duct A ⊗ B of monoids A, B ∈ V is aga in a monoid ther e m ust be a “switch” morphism c A,B : A ⊗ B / / B ⊗ A in V given by , say , a braiding. In this case A ⊗ B bec omes a mo no id in V via µ = / / / / / / / /         and η =     . Suppo se that A is a monoid in V . A right A -mo dule in V is an ob ject M ∈ V equipp e d with a mo rphism µ = M A M 2 2 2 2     : M ⊗ A / / M WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 39 called the action of A on M satisfying (m) M A A M 9 9 9 9       = M A A M 9 9 9 9     9 9 and M A M   , , , ,   = M . Notice that we use the same la bel “(m)” as the monoid axioms (and “(c)” b elow for the co module axioms). This should no t cause any confusion as the labelling of strings disa m biguates a multiplication and an a ction; how ever, the lab elling will usually b e left off. If M , N are modules, a mo dule morphism f : M / / N is a morphism in V satisfying M A N '&%$ !"# f 7 7 7    = M A N '&%$ !"# f 2 2 2       . A.3. Comonoids and como dules. Co monoids and como dules ar e dual to monoids and mo dules. Explicitly , a c omonoid C = ( C , δ, ǫ ) in V is a n ob ject C ∈ V equipp ed with mo rphisms δ =     2 2 2 2 : A / / A ⊗ A a nd ǫ =   : A / / I , called the c omultiplic atio n and c ounit of the comonoid resp ectively , satisfying (c)     9 9 9 9 =     9 9 9 9 9 9 =     9 9 9 9   and     9 9 9 9 = =       4 4 . If C, D are comono ids, a c o monoid morphism f : C / / D is a morphism in V satisfying B B A '&%$ !"# f    7 7 7 = B B A '&%$ !"# f '&%$ !"# f    2 2 2 and A B '&%$ !"# f   = A   . Similarly here, V must con tain a switch morphism c C,D : C ⊗ D / / D ⊗ C in order that the tensor pro duct C ⊗ D of comonoids C, D ∈ V is again a comonoid. In this case the com ultiplication and counit ar e g iv en by δ =     / / / / / / / /     and ǫ =     . Suppo se that C is a co monoid in V . A right C -c omo dule in V is an ob ject M ∈ V equipp ed with a morphism γ = M M C     2 2 2 2 : M / / M ⊗ C called the c o action of A on M satisfying (c) M C C M 9 9 9 9       = M C C M     9 9 9 9 9 9 and M C M       4 4 = M . 40 CRAIG P ASTR O AND ROSS STREET If M , N ar e C -c o modules, a c omo dule morphism f : M / / N is a morphism in V satisfying N C M '&%$ !"# f    7 7 7 = N C M '&%$ !"# f    * * * * * * . In this pap er we a lso ma ke use of C -bicomo dules. Supp ose that M is b oth a left C - comodule and a right C -co module with coactio ns γ l : M / / C ⊗ M γ r : M / / M ⊗ C. If the square M C ⊗ M M ⊗ C C ⊗ M ⊗ C γ l / / γ r   1 ⊗ γ r   γ l ⊗ 1 / / commutes, mea ning M C M C 9 9 9 9       = M C M C     9 9 9 9 9 9 in string diagrams , then M is called a C -bic omo dule . The diago nal of the square will b e denoted b y γ : M / / C ⊗ M ⊗ C. A.4. F rob enius monoi ds. A F r ob enius monoid R in V is bo th a monoid and a comonoid in V which a dditionally satisfies the “F ro benius co ndition”: R ⊗ R R ⊗ R ⊗ R δ ⊗ 1 / / R ⊗ R. 1 ⊗ µ   R ⊗ R ⊗ R 1 ⊗ δ   µ ⊗ 1 / / In string s the F rob enius condition is display ed as ? ? ? ? ? =      . W e will now review some basic facts ab out F r obenius monoids. Lemma A.1 . (1 ⊗ µ )( δ ⊗ 1) = δ µ = ( µ ⊗ 1)(1 ⊗ δ ) : R ⊗ R / / R ⊗ R . Pr o of. The left-hand iden tit y is pro v ed by the follo wing s tring calculation. ? ? ? ? ? =   ? ? ? ? ? ? ? =   J J J J J =   J J J J J ? ? ? =   o o o o ? ? ? =    ? ? ? ? ? ?    The rig h t-hand ident it y follows from a similar calculation.  WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 41 Define mo rphisms ρ and σ by ρ =  I R η / / R ⊗ R δ / /  =      7 7 7 σ =  R ⊗ R R µ / / I ǫ / /  =   7 7 7    . Prop osition A. 2. The morphisms ρ a nd σ form the unit and c ounit of an adjunc- tion R ⊣ R . Pr o of. One of the tria ng le identities is given as     t t t t t (f) =     J J J J J (m,c) = and the other should no w b e clear .  A morphism of F r o b enius monoids f : R / / S is a morphism in V which is b oth a mono id a nd co mo noid morphism. Prop osition A. 3. Any m orp hism o f F r ob enius monoi ds f : R / / S is an isomor- phism. Pr o of. Giv en f : R / / S define f − 1 : S / / R b y f − 1 = S R ⊗ R ⊗ S ρ ⊗ 1 / / R ⊗ S ⊗ S 1 ⊗ f ⊗ 1 / / R 1 ⊗ σ / / =   '&%$ !"# f   G G G G . It is then an easy calcula tio n to sho w that f − 1 is the inv erse o f f , namely , '&%$ !"# f ?>=< 89:; f − 1 =   '&%$ !"# f '&%$ !"# f   G G ? ?   =   '&%$ !"# f   J J J J J =     J J J J J (f ) =     t t t t t (m,c) = and the same viewed upside do wn.  A s imila r calculation sho ws that S S ⊗ R ⊗ R 1 ⊗ ρ / / S ⊗ S ⊗ R 1 ⊗ f ⊗ 1 / / R σ ⊗ 1 / / =   '&%$ !"# f   w w w w . is als o an inv erse of f . Therefore, Corollary A.4. F or any morphism of F r ob enius monoids f : R / / S we have   '&%$ !"# f   G G G G =   '&%$ !"# f   w w w w . 42 CRAIG P ASTR O AND ROSS STREET Definition A.5. A F rob enius monoid R is said to b e sep ar able if a nd only if µδ = 1 , i.e.,    ? ? ? ? ? ?    = . Appendix B. P r oofs of the proper ties of s , t , and r As we hav e noted in § 2, s : A / / A is inv ar ian t under rota tion by π , t : A / / A is inv ariant under hor izon tal reflection, and r is t r otated b y π . This r educes the nu m ber of pro ofs we present as the others are deriv able. (1)   s    7 7 7 ( s ) =        ? ? ? (c) =         ? ? (w) =       t t t J J J ( s ) =     s J J J       s    7 7 7 ( s ) =        ? ? ? (c) =         ? ? (w) =       E E y y n n n n n P P P P ( s ) =     s X X     t    7 7 7 ( t ) =        ? ? ? (c) =     9 9 9 9      (w) =       O O O j j j j ( t ) =     t J J J       t    7 7 7 ( t ) =        ? ? ? (c) =     9 9 9 9      (w) =       9 9 w w J J J J k k k k k ( t ) =     t X X   (2)     s   ( s ) =          (nat) =       J J J t t t (w) =         9 9 9 9 (c) =      : : :   s   ( s ) =         (m) =     ? ? (c) =       t   (t) =       (nat) =       E E y y n n n n n P P P P (w) =         9 9 9 9 (c) =      : : :   t   (t) =       (m) =     ? ? (c) =   (3)   s   s   (1) =     s   s T T     (2) =     s J J J     (1) =   s    7 7 7   t   t   (1) =     t   t T T     (2) =     t J J J     (1) =   t    7 7 7 WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 43 (4)   s      7 7 7 (b) =   s J J J J J t t t t (1) =     s D D D D D { { { z z z z (m) =     s D D D D D { { { z z z z (b) =   s ? ? ? ?   s ? ? 7 7 7    (b) =   s J J J J J t t t t (1) =     s ? ? ? ? ?   j j j j j j j X X X X X (m) =     s n n n n n n n V V V V V V V (b) =   s   t      7 7 7 (b) =   t J J J J J t t t t (1) =     t D D D D D { { { z z z z (m) =     t D D D D D { { { z z z z (b) =   t ? ? ? ?   t ? ? 7 7 7    (b) =   t J J J J J t t t t (1) =     t ? ? ? ? ?   j j j j j j j X X X X X (m) =     t n n n n n n n V V V V V V V (b) =   t (5)   s   s   s ? ?   (4) =   s   s   s   G G G (3) =   s   s   G G G (4) =   s   s ? ?     t   t   t ? ?   (4) =   t   t   t   G G G (3) =   t   t   G G G (4) =   t   t ? ?   (6)   s t t t J J J t t t t J J J J ? ?    (s) =         + + + + + + (nat) =     J J J J J t t t t (b) =     ? ?       ? ? (m,c) =   t (t) =     J J J J J t t t t (b) =     7 7 7      ? ? (m,c) = (7)   s   s ( s ) =       s (2) =     ( s ) =   s   t   t ( t ) =       t (2) =     ( t ) =   t 44 CRAIG P ASTR O AND ROSS STREET (8)   s   t ( s ) =       t (2) =     ( s ) =   s   t   s ( t ) =       s (2) =     ( t ) =   t (9)   t   s   (3) =   t   t   s   (8) =   t   t   (3) =   t    7 7 7   s   t   (3) =   s   s   t   (8) =   s   s   (3) =   s    7 7 7 (10)   t   s   ? ? ( s, t ) =         (nat) =            ? ? ? x x x x x x x F F F F F F (v) =             ? ? ? ? ( s, t ) =   s   t w w G G K K K K K s s s s (11)   t ? ? ? ? (m) =     t E E E E   (4) =     t 4 4 4 (m) =     t 4 4 4 (2) =   4 4 4 (2) =     s 4 4 4 (4) =   s (12)   r   s (r,s) =         (nat) =       (v) =        9 9 9 9 (c) =     (s) =   s (13)   t   r   ? ? ( t, r ) =         (nat) =            ? ? ? F F F F F F x x x x x x (v) =         ? ? ? ? ?      ( t, r ) =   r   t z z D D A A A A A | | ~ ~ (14)   r   s   (3) =   r   s   s   (12) =   s   s   (3) =   s 7 7 7      s   r   (3) =   s   r   r   (12) =   r   r   (3) =   r 7 7 7    WEAK HOPF MONOIDS IN BRAIDED MONOIDAL CA TEGORIES 45 References [1] J. N. Al onso ´ Alv arez, J. M. F ern´ andez Vilab oa, R . Gonz´ alez Rodr ´ ıguez. W eak br aided Hopf algebras, pr epri n t (2007). [2] J. N. Alonso ´ Alv arez, J. M. F ern´ andez Vilab oa, R. Gonz´ alez Ro dr ´ ıguez. W eak Hopf algebras and weak Y ang-Baxter operators, preprint (2007). [3] Micha el Barr. Nonsymmetric ∗ -autonomous categories, Theoret. Comput. Sci. 139 no. 1- 2 (1995): 115–130. [4] Gabriella B¨ ohm, Florian Nill , and Kor n´ el Szlac h´ an yi. W eak Hopf algebras I. 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