Torsors, herds and flocks

T orsors, herds and flo c ks Thomas Bo ok er and Ross Street June 11, 2018 2000 Mathematics Subje ct Classific ation. 18D35; 1 8D10; 20J06 Key wor ds and phr ases. torsor; herd; flo c k; monoidal category; descent morphism. ——————————————————– Abstract This pap er presen ts non-commutative and structural notions of torsor. The tw o are related by the machinery of T annak a-Krein duality . 1 In tro duction Let us descr ibe br iefly how the concepts a nd results dealt with in this pa per ar e v a rian ts o r gener a lizations o f those existing in the literature. While a ffine functions b et ween vector spa c es take lines to lines , they are not linear functions. In par ticular, tra nslations do not preserve the origin. This phenomenon appears at the ba sic level with g roups: left m ultiplication a − : G → G b y an elemen t a of a group G is not a group morphism. Ho wever, left multiplication do es preserve the ternar y o peration q : G 3 → G defined b y q ( x, y , z ) = xy − 1 z . Clearly , the op eration q , to gether with a c ho ice of a ny element as unit, determine the g roup structure. A herd is a set A with a ter na ry op eration q sa tisfying three simple axioms motiv a ted by the group example. References for this term go back a long wa y , originating with the German form “ Schar”: s e e [1 3], [1], [8]. Because Hopf alge- bras generalize groups, it is natural to consider the corresp onding generaliza tio n of herds. Such a gener alization, in volving algebr a s with a ternary “co-o peration”, is indeed considered in [11]; for later developmen ts see [2] and re fer ences there. Moreov er , using the term “quantum hea ps” o f his 2002 thesis, Škoda [14] proved an equiv a lence b et ween the ca t eg ory o f cop oint ed quantum heaps and the cat- egory of Hopf alge br as. In this paper, we use the term herd for a coa lgebra with an appropriate ternary op eration. As far as p ossible w e work with como noids in a braided monoidal categor y V (admitting reflexiv e co equalizers preser v ed by X ⊗ − ) rather than the s p ecial case of categories o f mo dules ov er a commutativ e r ing in which comonoids interpret as coalge br as ov er the ring. The algebra version is included by taking V to be a dual catego ry o f mo dules (plus so me fla tn es s assumptions). When V is the ca tegory of sets, we obtain the class ical notion of herd. The theory of torsor s for a group in a top os E app ears in [6]. T orso rs g iv e an interpretation of the first co homology gr oup of the top os with co efficien ts in 1 the group. In Section 2 we review how, when working in a top os E , tor sors are herds with chosen elements existing lo cally . Lo cally here means after applying a functor − × R : E → E /R where R → 1 is an epimorphism in E . Such a functor is conserv ative and, since it has b oth adjoints, is mo nadic. In Section 3 we define herds in a bra ided mo noidal ca tegory and make ex- plicit the co descen t condition causing them to be tors ors for a Hopf mo noid. This gener alizes the top os case and s ligh tly extends asp ects of rec en t work of Grunspan [7] and Schauenb ur g [15], [1 6 ]. Finite dimensiona l r epresen tations o f a Hopf algebra form a n autonomous (= compact = r igid) mo no idal categor y . In fact, as explained in [4], Hopf al- gebras and autonomous monoidal categ ories a re structures of the same kind: autonomous pseudomo no ids in different a utonomous mono ida l bica tegories. In Section 4, we int r o duce a general structure in an autonomous monoida l bicat- egory we call a “flo ck”. It generalizes herd in the same way that autonomous pseudomonoid generalizes Hopf algebra . How ever, by lo oking at the autono mous monoidal bicatego ry V - M od , in Section 4 this g iv es us the notion of enriched flo c k, ro ughly descr ibed as a n autonomous monoidal V -category without a cho- sen unit for the tensor pro duct. W e b eliev e our use of the term flo ck is new. How ever, our us e is clos e to the concept o f “heapy catego ry” in [14]. The como dules admitting a dual ov er a herd in V form a V -flo c k. In Section 5 we adapt T annak a duality (as pr esen ted, for example, in [9]) to r elate V -flo c ks and herds in V . In Section 3 we need to know that the existence of a n antipo de in a bimonoid is equiv alent to the inv er tib ility o f the so -called fusion op erator. F or complete- ness, we include in an App endix a direct pro of shown to us by Mi cah McCurdy at the genera lit y required. It is a s tandard re s ult for Hopf alg ebras over a field. 2 Recollections on torsors f or groups Let G b e a monoid in a ca rtesian closed, finitely complete and finitely co complete category E . A G -torso r [6] is an ob ject A with a G -actio n µ : G × A − → A such that: (i) the unique ! : A − → 1 is a regula r epimorphism; a nd, (ii) the morphism ( µ, pr 2 ) : G × A − → A × A is inv ertible. The inv erse to ( µ, pr 2 ) must hav e the form (  , pr 2 ) : A × A − → G × A wher e  : A × A − → G has the prop ert y that the following t wo comp osites are equal to the first pro- jections. G × A 1 × δ − → G × A × A µ × 1 − → A × A  − → G A × A 1 × δ − → A × A × A  × 1 − → G × A µ − → A 2 If the epimorphism A − → 1 is a retra ction with right inv er se a : 1 − → A then G 1 × a − → G × A µ − → A is an inv ertible morphism of G -actions where G acts on itself by its own multi- plication. It follows that the existence of such a torsor forces G to b e a g roup. If A is a G -tor sor in E , we obtain a ternary op eration on A as the comp osite q : A × A × A  × 1 − → G × A µ − → A. The following prop erties hold: A × A × A × A × A 1 × 1 × q q × 1 × 1 A × A × A q A × A × A q = A (2.1) A × A × A q A × A 1 × δ pr 1 = A (2.2) A × A × A q A × A δ × 1 pr 2 = A (2.3) That is, A b ecomes a herd (“Schar” in Germa n) in E ; refer ences for this term are [13], [1], [8]. Conv ersely , g iv en a her d A in E for which ! : A − → 1 is a regula r epimor - phism, a group G is obtained as the co equalizer  : A × A − → G of the tw o morphisms A × A × A 1 × 1 × δ A × A × A × A q × 1 A × A and A × A × A 1 × 1 × ! A × A with mult iplicatio n induced by A × A × A × A q × 1 − → A × A. The unit for G is c onstructed using the co mposite A δ − → A × A  − → G and the co equalizer A × A − → − → A − → 1 . Moreov er , there is an a ction of G o n A induced by q whic h causes A to be a G -torsor. Our purp ose is to gener alize this to the ca se o f comonoids in a monoida l category V in pla ce of the ca r tesian monoidal E and to examine a higher dimen- sional version. W e rela te the tw o concepts using the T annakian adjunction; see Chapter 16 of [17]. 3 3 Non-comm utativ e torsors Let V be a braided mono idal ca teg ory with tensor pro duct ⊗ having unit I , and with reflexive co equalizers pr e s erv ed by each X ⊗ − . F or a comonoid A = ( A, δ : A → A ⊗ A, ε : A → I ) in V , we write A ◦ for the opp osite co monoid ( A, A δ − → A ⊗ A c A,A − → A ⊗ A, ε : A → I ) . F or como no ids A and B , there is a comonoid A ⊗ B = ( A ⊗ B , A ⊗ B δ ⊗ δ A ⊗ A ⊗ B ⊗ B 1 ⊗ c A,B ⊗ 1 A ⊗ B ⊗ A ⊗ B , A ⊗ B ǫ ⊗ ǫ I ) . Definition 1 A comono id A in V is called a her d when it is e quipped with a comonoid morphism q : A ⊗ A ◦ ⊗ A − → A (3.1) for which the following co ndit io ns hold: A ⊗ A ⊗ A ⊗ A ⊗ A 1 ⊗ 1 ⊗ q q ⊗ 1 ⊗ 1 A ⊗ A ⊗ A q A ⊗ A ⊗ A q = A (3.2) A ⊗ A ⊗ A q A ⊗ A 1 ⊗ δ 1 ⊗ ǫ = A (3.3) A ⊗ A ⊗ A q A ⊗ A δ ⊗ 1 ǫ ⊗ 1 = A (3.4) Suc h structure s , stated dually , o ccur in [11] and [2], for ex a mple. F or any comonoid A in V we hav e the ca tegory Cm l A of left A -como dules δ : M − → A ⊗ M . There is a mona d T A on Cm l A defined by T A ( M δ − → A ⊗ M ) =  A ⊗ M δ ⊗ 1 − → A ⊗ A ⊗ M  , (3.5) with mult iplicatio n and unit for the monad having co mp onents A ⊗ A ⊗ M 1 ⊗ ε ⊗ 1 A ⊗ M and M δ − → A ⊗ M . (3.6) There is a co mparison functor K A : V − → (Cm l A ) T A in to th e category of Eilenberg - Moore alg ebras taking X to the A -comodule δ ⊗ 1 : A ⊗ X − → A ⊗ A ⊗ X with T A -action 1 ⊗ ε ⊗ 1 : A ⊗ A ⊗ X → A ⊗ X . W e say that ε : A → I is a c o desc ent morph ism when K A is fully faithful. If K A is a n equiv a lence of categ ories (that is, the rig h t a dj o int to the underly ing functor Cm l A − → V is monadic) then we say ε : A → I is a n effe ctive c o desc ent morphism . 4 Definition 2 A torsor in V is a herd A for which the co unit ε : A → I is a co descen t morphism. Let A b e a her d. W e asymmetrically introduce morphisms σ and τ : A ⊗ A ⊗ A → A ⊗ A defined by σ =  A ⊗ A ⊗ A 1 ⊗ 1 ⊗ δ A ⊗ A ⊗ A ⊗ A q ⊗ 1 A ⊗ A  and τ =  A ⊗ A ⊗ A 1 ⊗ 1 ⊗ ε A ⊗ A  . These form a r eflexiv e pa ir using the common rig h t inv er se A ⊗ A 1 ⊗ δ − → A ⊗ A ⊗ A. Let  : A ⊗ A − → H b e the co equalizer of σ a nd τ . It is easily seen that there is a uni que comonoid s tructure o n H suc h that  : A ⊗ A ◦ − → H becomes a comonoid morphism which mea ns the following diagrams commutes (where c 1342 is the p ositive braid whose underlying p erm utation is 1 342 ). H δ A ⊗ 2  = ǫ ⊗ ǫ A ⊗ 2 δ ⊗ δ  = H ⊗ 2 I A ⊗ 4 c 1342 A ⊗ 4  ⊗  H ǫ (3.7) Since H is a reflexive co equalizer, we hav e the co equalizer A ⊗ 6 σ ⊗ σ − → − → τ ⊗ τ A ⊗ 4  ⊗  − → H ⊗ 2 . (3.8) It is readily chec ked that A ⊗ 6 σ ⊗ σ − → − → τ ⊗ τ A ⊗ 4 q ⊗ 1 − → A ⊗ 2  − → H (3.9) commut es . So there exists a unique µ : H ⊗ 2 − → H such that A ⊗ 4  ⊗  − → H ⊗ 2 µ − → H = A ⊗ 4 q ⊗ 1 − → A ⊗ 2  − → H . (3.10) Prop osition 1 This µ : H ⊗ 2 → H is asso ciative and a c omonoid morphism. Pro of Asso ciativity follows easily from equation (3.2) and (3.10). T o show that µ is a comono id morphism we need to pr ove the equations H ⊗ 2 δ ⊗ δ − → H ⊗ 4 1 ⊗ c ⊗ 1 H ⊗ 4 µ ⊗ µ H ⊗ 2 = H ⊗ 2 µ − → H δ − → H ⊗ 2 (3.11) H ⊗ 2 ǫ ⊗ ǫ − → I = H ⊗ 2 µ − → H ǫ − → I . (3.12) 5 The following diag r am proves (3.11) while equatio n (3.12) follows ea sily from the second diagr am o f (3.7) and the fa c t that  and q preser ve counits. H ⊗ 4 1 ⊗ c ⊗ 1 H ⊗ 2 δ ⊗ δ H ⊗ 4 µ ⊗ µ A ⊗ 8  ⊗  ⊗  ⊗  c 12563478 A ⊗ 4  ⊗  q ⊗ 1  ⊗  δ ⊗ δ ⊗ δ ⊗ δ A ⊗ 8 c 13425786 c 14523678 c 15623784 A ⊗ 8  ⊗  ⊗  ⊗  q ⊗ 1 ⊗ q ⊗ 1 H ⊗ 2 A ⊗ 8 c 12356748 q ⊗ q ⊗ 1 ⊗ 1 A ⊗ 4  ⊗  A ⊗ 2 δ ⊗ δ  A ⊗ 4 c 1324 H ⊗ 2 µ H δ Prop osition 2 If A is a torsor then H is a Hopf monoid and A is a left H - torsor. Pro of In or der to construct a unit for the multiplication on H , we us e the co descen t condition on ε : A → I . W e define η : I − → H b y providing an Eilen b erg-Mo ore T A -algebra morphism fro m K A I =  A, A ⊗ A 1 ⊗ ε A  to K A H =  A ⊗ H , A ⊗ A ⊗ H 1 ⊗ ε ⊗ 1 A ⊗ H  and using the a ssumption that K A is fully faithful. The morphism is A δ − → A ⊗ A c A,A − → A ⊗ A 1 ⊗ δ − → A ⊗ A ⊗ A 1 ⊗  − → A ⊗ H . (3.13) It is an ( A ⊗ − ) -algebr a morphism since (1 ⊗ ε ⊗ 1) (1 ⊗ 1 ⊗  ) (1 ⊗ 1 ⊗ δ ) (1 ⊗ c ) (1 ⊗ δ ) = (1 ⊗  ) (1 ⊗ δ ) ( 1 ⊗ ε ⊗ 1) (1 ⊗ c ) (1 ⊗ δ ) = (1 ⊗  ) (1 ⊗ δ ) = (1 ⊗  ) (1 ⊗ 1 ⊗ 1 ⊗ ε ) c 2341 ( δ ⊗ δ ) = (1 ⊗  ) (1 ⊗ q ⊗ 1) (1 ⊗ δ ⊗ δ ) c 231 (1 ⊗ δ ) = (1 ⊗  ) (1 ⊗ δ ) c 21 δ (1 ⊗ ε ) . 6 So indeed we can define η by 1 A ⊗ η = (1 ⊗  ) (1 ⊗ δ ) c A,A δ . (3.14) Here is the pr o of that η is a right unit: (1 ⊗ µ ) (1 ⊗ 1 ⊗ η ) (1 ⊗  ) = (1 ⊗ µ )  c − 1 ⊗ 1  (1 ⊗ 1 ⊗ η ) c (1 ⊗  ) = (1 ⊗ µ )  c − 1 ⊗ 1  (1 ⊗ 1 ⊗  ) (1 ⊗ 1 ⊗ δ ) (1 ⊗ cδ ) c (1 ⊗  ) = (1 ⊗ µ ) ( 1 ⊗  ⊗  δ ) ( c 231 ) − 1 (1 ⊗ 1 ⊗ cδ ) c 231 = (1 ⊗  ) (1 ⊗ q ⊗ 1) (1 ⊗ 1 ⊗ δ ) ( c 231 ) − 1 (1 ⊗ 1 ⊗ cδ ) c 231 = (1 ⊗  ) (1 ⊗ 1 ⊗ 1 ⊗ ε ) ( c 231 ) − 1 (1 ⊗ 1 ⊗ cδ ) c 231 = 1 ⊗  . Here is the pr o of that η is a left unit: (1 ⊗ µ ) (1 ⊗ η ⊗ 1) ( 1 ⊗  ) = (1 ⊗ µ ) (1 ⊗  ⊗ 1) ( 1 ⊗ δ ⊗ 1) ( cδ ⊗ 1) (1 ⊗  ) = (1 ⊗ µ ) (1 ⊗  ⊗  ) (1 ⊗ δ ⊗ 1 ⊗ 1) ( cδ ⊗ 1 ⊗ 1) = (1 ⊗  ) (1 ⊗ q ⊗ 1) (1 ⊗ δ ⊗ 1 ⊗ 1) ( cδ ⊗ 1 ⊗ 1) = (1 ⊗  ) (1 ⊗ ε ⊗ 1 ⊗ 1) ( cδ ⊗ 1 ⊗ 1) = 1 ⊗  . T o complete the pro of that H is a bimonoid, one easily chec ks the rema inin g prop erties: I η − → H δ − → H ⊗ H = I η ⊗ η − → H ⊗ H (3.15) I η − → H ε − → I = I 1 − → I . (3.16) Using the co equalizer A ⊗ 4 σ ⊗ 1 − → − → τ ⊗ 1 A ⊗ 3  ⊗ 1 − → H ⊗ A , (3.17) we can define a mor phism µ : H ⊗ A − → A (3.18) b y the condition A ⊗ A ⊗ A  ⊗ 1 − → H ⊗ A µ − → A = A ⊗ A ⊗ A q − → A . (3.19) It is easy to see that this is a comono id mor phism and satisfies the tw o a xioms for a left action of the bimonoid H o n A . Moreov er , the fusion morphism v =  H ⊗ A 1 ⊗ δ − → H ⊗ A ⊗ A µ ⊗ 1 − → A ⊗ A  (3.20) has inv erse A ⊗ A 1 ⊗ δ − → A ⊗ A ⊗ A  ⊗ 1 − → H ⊗ A. (3.21) 7 In fa c t, H is a Hopf mo noid. T o see this, co nsider the fusion mor phism v =  H ⊗ H 1 ⊗ δ − → H ⊗ H ⊗ H µ ⊗ 1 − → H ⊗ H  . (3.22) It suffices to prove this is inv ertible (see the Section 8 App endix for a pr oof that this implies the existence of an antipo de). Again we app eal to the co descen t prop ert y of ε : A → I ; we only require that V − → Cm l A is conserv ative (reflects in vertibilit y). F or, we hav e the fusion equation H ⊗ H ⊗ A 1 ⊗ v H ⊗ A ⊗ A v ⊗ 1 H ⊗ H ⊗ A v ⊗ 1 1 ⊗ v = A ⊗ A ⊗ A H ⊗ A ⊗ A c ⊗ 1 A ⊗ H ⊗ A 1 ⊗ v (3.23) which holds in Cm l A . All mo r phisms her e are known to b e inv ertible with the exception of H ⊗ H ⊗ A v ⊗ 1 − → H ⊗ H ⊗ A . So that p ossible exception is also in vertible. Alternatively , for the herd A , we can introduce mor phi sms σ ′ and τ ′ : A ⊗ A ⊗ A → A ⊗ A defined by σ ′ =  A ⊗ A ⊗ A δ ⊗ 1 ⊗ 1 A ⊗ A ⊗ A ⊗ A 1 ⊗ q A ⊗ A  and τ ′ =  A ⊗ A ⊗ A ε ⊗ 1 ⊗ 1 A ⊗ A  . These form a r eflexiv e pa ir using the common rig h t inv er se A ⊗ A δ ⊗ 1 − → A ⊗ A ⊗ A. Let  ′ : A ⊗ A − → H ′ be the co equalizer of σ ′ and τ ′ . Then there is a unique comonoid structure on H ′ such that  ′ : A ◦ ⊗ A − → H ′ bec o mes a co monoid morphism. Symmetrically to H , we see that A beco mes a r igh t H ′ -torsor for the Hopf mo noid H ′ . Indeed, A is a torso r from H to H ′ in the sense that the actions make A a left H -, right H ′ -bimo dule. 8 4 Flo c ks Flo c ks ar e a higher -dimensional version of her ds. O ur use of the term may b e in conflict with other uses in the literature (such as [3 ]). Definition 3 Let M denote a rig ht autono mous monoidal bicategor y [5]. So each o b ject X has a bidual X ◦ with unit n : I − → X ◦ ⊗ X and counit e : X ⊗ X ◦ − → I . A left flo ck in M is an ob ject A equipp ed with a morphism q : A ⊗ A ◦ ⊗ A − → A (4.1) and 2-cells A ⊗ A ◦ ⊗ A ⊗ A ◦ ⊗ A 1 ⊗ 1 ⊗ q q ⊗ 1 ⊗ 1 A ⊗ A ◦ ⊗ A q A ⊗ A ◦ ⊗ A q φ ∼ = A (4.2) A ⊗ A ◦ ⊗ A q A 1 ⊗ n 1 α A (4.3) A ⊗ A ◦ ⊗ A e ⊗ 1 q β A (4.4) satisfying the following three conditions (where 1 n = n z }| { 1 ⊗ ... ⊗ 1 ): q (1 2 ⊗ q )( q ⊗ 1 4 ) ∼ = q ( q ⊗ 1 2 )(1 4 ⊗ q ) φ (1 4 ⊗ q ) q ( q ⊗ 1 2 )( q ⊗ 1 4 ) φ ( q ⊗ 1 4 ) q ( φ ⊗ 1 2 ) q (1 2 ⊗ q )(1 4 ⊗ q )) q ( q ⊗ 1 2 )(1 2 ⊗ q ⊗ 1 2 ) φ (1 2 ⊗ q ⊗ 1 2 ) q (1 2 ⊗ q )(1 2 ⊗ q ⊗ 1 2 ) q (1 2 ⊗ φ ) = (4.5) 1 1 ∼ = ( e ⊗ 1 )( 1 ⊗ n ) β (1 ⊗ n ) q (1 ⊗ n ) α 1 = (4.6) q (1 2 ⊗ q )(1 1 ⊗ n ⊗ 1 2 ) φ − 1 (1 1 ⊗ n ⊗ 1 2 ) q ( q ⊗ 1 2 )(1 1 ⊗ n ⊗ 1 2 ) q ( α ⊗ 1 2 ) q (1 2 ⊗ e ⊗ 1 1 )(1 1 ⊗ n ⊗ 1 2 ) ∼ = q 1 3 ∼ = q q (1 2 ⊗ β )(1 1 ⊗ n ⊗ 1 2 ) 1 q ∼ = q 1 3 = (4.7) 9 Example 1 Suppos e A is a left auto no mous ps eudomonoid in M in the sense of [4] and [12]. A ps eudomonoid co nsists o f a n ob ject A , morphisms p : A ⊗ A → A and j : I → A , 2 -cells φ : p ( p ⊗ 1 ) ⇒ p (1 ⊗ p ) , λ : p ( j ⊗ 1) ⇒ 1 and ρ : p (1 ⊗ j ) ⇒ 1 , satisfying co her ence ax ioms. It is left autonomous when it is eq uipped with a left dualization mor phism d : A ◦ → A having 2 -cells α : p ( d ⊗ 1) n ⇒ j and β : j e ⇒ p (1 ⊗ d ) , satisfying tw o a xioms. Put q =  A ⊗ A ◦ ⊗ A 1 ⊗ d ⊗ 1 A ⊗ A ⊗ A p ⊗ 1 A ⊗ A p A  φ : q ( q ⊗ 1 2 ) = p ( p ⊗ 1) (1 ⊗ d ⊗ 1) ( p ⊗ 1 2 ) ( p ⊗ 1 3 ) ( 1 ⊗ d ⊗ 1 3 ) ∼ = p ( p ⊗ 1) ( p ⊗ 1 2 ) ( p ⊗ 1 3 ) (1 ⊗ d ⊗ 1 ⊗ d ⊗ 1) ∼ = p 5 (1 ⊗ d ⊗ 1 ⊗ d ⊗ 1 ) ∼ = p ( p ⊗ 1) (1 2 ⊗ p ) (1 2 ⊗ p ⊗ 1) (1 ⊗ d ⊗ 1 ⊗ d ⊗ 1) ∼ = q (1 2 ⊗ q ) α : q (1 ⊗ n ) ∼ = p (1 ⊗ ( p ( d ⊗ 1) n )) p (1 ⊗ α ) p (1 ⊗ j ) ∼ = 1 β : e ⊗ 1 ∼ = p ( j ⊗ 1 ) ( e ⊗ 1) ∼ = p (( j e ) ⊗ 1) p ( β ⊗ 1) p (( p (1 ⊗ d )) ⊗ 1) ∼ = q . The ax io ms for the flo c k ( A, q , φ, α, β ) follow from thos e on ( A, p, φ, λ, ρ, d, α, β ) in [4]. Remark As noted in [5], Baez-Do la n coined the ter m “micro cosm principal” for the phenomenon whereby a concept finds its appr o priate lev el of g e ner alit y in a higher dimensio nal version of the concept. In pa rticular, monoids in the ca tegory of s ets generalize to mono ids in monoida l categ ories. Similarly , monoida l cate- gories generalize to pseudo monoids in monoidal bicatego ries. F or autonomous pseudomonoids the context is an a utonomous monoida l bicate g ory . Exa mple 1 shows that autonomous monoida l categorie s b ecome flo c ks. Although we shall not explicitly define a biflo c k, an a uto no mous monoidal bicategor y w ould be an example. The gener al context for flo c k would b e biflo c k. Given a left flo c k A , cons ider the mate ˆ q : A ◦ ⊗ A − → A ◦ ⊗ A (4.8) of q under the biduality A ⊣ b A ◦ ; that is, ˆ q is the comp osite A ◦ ⊗ A n ⊗ 1 2 − → A ◦ ⊗ A ⊗ A ◦ ⊗ A 1 ⊗ q − → A ◦ ⊗ A. (4.9) Define µ : ˆ q ˆ q = ⇒ ˆ q to b e the co mposite (1 ⊗ q ) ( n ⊗ 1 2 ) (1 ⊗ q ) ( n ⊗ 1 2 ) ∼ = (1 ⊗ q ) (1 ⊗ q ⊗ 1 ) (1 ⊗ n ⊗ 1) ( n ⊗ 1 2 ) (1 ⊗ q ) (1 ⊗ α ⊗ 1)( n ⊗ 1 2 ) (1 ⊗ q ) ( n ⊗ 1 2 ) (4.10) 10 and η : 1 2 = ⇒ ˆ q to b e the comp osite 1 2 ∼ = (1 ⊗ e ⊗ 1) ( n ⊗ 1 2 ) (1 ⊗ β )( n ⊗ 1 2 ) (1 ⊗ q ) ( n ⊗ 1 2 ) . (4.11) Prop osition 3 ( ˆ q , µ, η ) is a monad on A ◦ ⊗ A . Pro of Conditions (4.5 ), (4.6) and (4.7 ) translate to the mo na d axioms. Assume a Kleisli ob ject K e xists for the mona d ˆ q . So we hav e a morphism h : A ◦ ⊗ A − → K (4.12) and a 2- cell χ : h ˆ q = ⇒ h (4.13) forming the universal Eilen b erg-Mo ore algebra for the monad defined by pr e- comp osition with ˆ q . It fo llows that h is a map in the bicategor y M ; that is, h has a right adjoint h ∗ : K − → A ◦ ⊗ A . Prop osition 4 The morphism 1 ⊗ q : A ◦ ⊗ A ⊗ A ◦ ⊗ A − → A ◦ ⊗ A induc es a pseudo-asso ciative multiplic ation p : K ⊗ K − → K . Pro of W e obtain a mona d o pmorphism (1 ⊗ q , ψ ) : ( A ◦ ⊗ A ⊗ A ◦ ⊗ A, ˆ q ⊗ ˆ q ) − → ( A ◦ ⊗ A, ˆ q ) wher e ψ is the comp osite 2- cell (1 ⊗ q ) ( ˆ q ⊗ ˆ q ) = (1 ⊗ q ) (1 3 ⊗ q ) (1 ⊗ q ⊗ 1 4 ) (1 4 ⊗ n ⊗ 1 2 ) ( n ⊗ 1 4 ) ∼ = (1 ⊗ q ) ( 1 3 ⊗ q ) (1 3 ⊗ q ⊗ 1 2 ) (1 4 ⊗ n ⊗ 1 2 ) ( n ⊗ 1 4 ) = ⇒ (1 ⊗ q ) (1 3 ⊗ q ) ( n ⊗ 1 4 ) ∼ = (1 ⊗ q ) ( n ⊗ 1 2 ) (1 ⊗ q ) ∼ = ˆ q (1 ⊗ q ) . (4.14) Since the Kleisli ob ject for the monad ˆ q ⊗ ˆ q is K ⊗ K , the mo r phism p : K ⊗ K − → K is induced by (1 ⊗ q , ψ ) . The invertible 2-cell φ of (4.2) induces an inv er ti ble φ : p ( p ⊗ 1 ) = ⇒ p (1 ⊗ p ) satisfying the a ppropriate “p en tago n” condition following from condition (4.5) . Prop osition 5 The morphism q : A ⊗ A ◦ ⊗ A − → A induc es a pseudo-action q : A ⊗ K − → A. If q is a map then so is q . Pro of The Kle is li o b ject for the monad 1 ⊗ ˆ q on A ⊗ A ◦ ⊗ A is A ⊗ K . So there exists a unique q : A ⊗ K − → A such that  q (1 ⊗ ˆ q ) ∼ = q (1 ⊗ h ) (1 ⊗ ˆ q ) q (1 ⊗ χ ) q (1 ⊗ h ) ∼ = q  =  q (1 ⊗ ˆ q ) = q (1 2 ⊗ q ) (1 ⊗ n ⊗ 1 2 ) ∼ = q ( q ⊗ 1 2 ) (1 ⊗ n ⊗ 1 2 ) q ( α ⊗ 1 2 ) q  . Prop osition 6 The morphism h : A ◦ ⊗ A − → K is a p ar ametric left adjoint for q : A ⊗ K − → A . That is, the mate K n ⊗ 1 − → A ◦ ⊗ A ⊗ K 1 ⊗ q − → A ◦ ⊗ A (4.15) of q is right adjoint to h : A ◦ ⊗ A − → K . 11 5 Enric he d flo c ks Suppos e V is a base monoidal catego ry of the kind considered in [10] and we will use the enriched ca tegory theor y develop ed there. In particular reca ll that the end R A T ( A, A ) of a V -functor T : A op ⊗ A → V is constructed as the equalizer R A T ( A, A ) Q A T ( A, A ) Q A,B V ( A ( A, B ) , T ( A, B )) F or a V -functor T : A op ⊗ A → X , the end R A T ( A, A ) and co end R A T ( A, A ) are defined by V -natural isomorphisms X ( X, Z A T ( A, A )) ∼ = Z A X ( X, T ( A, A )) and X ( Z A T ( A, A ) , X ) ∼ = Z A X ( T ( A, A ) , X ) . Definition 4 A left V - fl o ck is a flo ck A in V - M od for which the structure mo dule Q : A ⊗ A op ⊗ A − → A (5.1) of (4.1) is a V -functor. More ex plicit ly , we hav e a V -categor y A and a V -functor Q : A ⊗ A op ⊗ A − → A equipp ed with V -na tural transformations φ : Q ( Q ( A, B , C ) , D , E ) ∼ = − → Q ( A, B , Q ( C, D , E )) (5.2) α B A : Q ( A, B , B ) − → A (5.3) β A B : B − → Q ( A , A, B ) , (5.4) with φ inv ertible, such that the fo llowing three conditions hold: Q ( Q ( A, B , C ) , D , Q ( E , F , G )) φ Q ( A, B , Q ( C , D , Q ( E , F , G ))) Q ( Q ( Q ( A, B , C ) , D , E ) , F , G ) φ Q ( φ, 1 , 1) Q ( Q ( A, B , Q ( C , D , E )) , F , G ) φ = Q ( A, B , Q ( Q ( C , D , E ) , F , G ) Q (1 , 1 ,φ ) (5.5) Q ( A, B , Q ( B , B , C )) φ − 1 Q ( Q ( A, B , B ) , B , C ) Q ( α, 1 , 1) Q ( A, B , C ) Q (1 , 1 ,β ) 1 Q ( A, B , C ) = (5.6) Q ( A, A, A ) α A β 1 = A (5.7) 12 The following observ ation characterize s the s pecial case of Example 3 (b elo w) where H = A . Prop osition 7 Su pp ose A is a V -fl o ck which has an obje ct J for which α K A : Q ( A, J, J ) → A and β K B : B − → Q ( J, J, B ) ar e invertible for al l A and B . Then A b e c omes a left aut onomo us monoidal V -c ate gory by defin ing A ⊗ B = Q ( A, J, B ) and A ∗ = Q ( J, A, J ) . (5.8) The asso ciativity and u nit c onstr aints ar e define d by instanc es of φ , α and β , while the c ounit and un i t for A ∗ ⊣ A ar e α A K and β A K . Now, given any V -flo ck, we hav e the Kleisli V - c a tegory K a nd a V -functor H : A op ⊗ A − → K (5.9) constructed as in (4.12). The ob jects o f K are pair s ( A, B ) as fo r A op ⊗ A and the hom-ob jects are defined by K (( A, B ) , ( C, D )) = A ( B , Q ( A, C , D )) . (5.10) Comp osition Z C,D K (( C , D ) , ( E , F )) ⊗ K (( A, B ) , ( C , D )) − → K (( A, B ) , ( E , F )) (5.11) for K is defined to b e the morphism Z C,D A ( D, Q ( C , E , F )) ⊗ A ( B , Q ( A, C, D )) − → A ( B , Q ( A, E , F )) equal to the comp osite o f the canonical Y oneda isomorphism with the comp osite Z C A ( B , Q ( A, C, Q ( C , E , F ))) R C A (1 ,φ − 1 ) Z C A ( B , Q ( Q ( A , C, C ) , E , F )) R C A (1 ,Q ( α, 1 , 1)) A ( B , Q ( A, E , F )) . The V -functor H of (5.9) is the identit y on o b jects a nd its effect o n hom-o b jects A ( C, A ) ⊗ A ( B , D ) H − → A ( B , Q ( A, C , D )) is the V -natural family corr esponding under the Y oneda Lemma to the comp os- ite I j B A ( B , B ) A (1 ,β ) A ( B , Q ( A, A, B )) . Example 2 Let X and Y be V -ca tegories for a suitable V . There is a V -categor y Adj( X , Y ) of adjunctions b et ween X and Y : the ob jects F = ( F , F ∗ , ε , η ) consist of V - fu nctor s F : X → Y , F ∗ : Y → X , and V -na tural tr ansformations ε : F F ∗ ⇒ 1 Y and η : 1 X ⇒ F ∗ F whic h are the co unit and unit for an adjunction F ⊣ F ∗ ; the hom ob jects are defined by Adj( X , Y ) ( F , G ) = [ X , Y ] ( F, G ) = Z X X ( F X, GX ) ( ∼ = [ Y , X ] ( G ∗ , F ∗ )) 13 where [ X , Y ] is the V -functor V -categ o ry [10]. W e obtain a V -flo c k A = Adj( X , Y ) b y defining Q ( F, G, H ) = ( H G ∗ F, F ∗ GH ∗ , ε , η ) where ε and η come b y co mposition fro m those for F ⊣ F ∗ , G ⊣ G ∗ and H ⊣ H ∗ . In this case φ is a n equality while α and β are induced by the appropria te counit ε and unit η . Notice that K (( F , G ) , ( H , K )) = [ Y , Y ] ( GF ∗ , K H ∗ ) . Example 3 Let H b e a left autono mo us monoida l V -categ ory for s uit a ble V . F or each X ∈ H , we hav e X ∗ ∈ H a nd ε : X ∗ ⊗ X − → I and η : I − → X ⊗ X ∗ inducing isomor phisms H ( X ∗ ⊗ Y , Z ) ∼ = H ( Y , X ⊗ Z ) and (5.12) H ( Y ⊗ X , Z ) ∼ = H ( Y , Z ⊗ X ∗ ) . (5 .13) So X Z = Z ⊗ X ∗ acts as a left internal ho m for H and h X, Y i = X ∗ ⊗ Y (5.14) acts as a left internal co hom. Notice that X ( Z ⊗ Y ) ∼ = Z ⊗ Y ⊗ X ∗ ∼ = Z ⊗ X Y and (5.15) h X , Y ⊗ Z i ∼ = X ∗ ⊗ Y ⊗ Z ∼ = h X , Y i ⊗ Z . (5.16) Suppos e A is a right H - actegory; that is, A is a V -ca teg ory equipped with a V -functor ∗ : A ⊗ H − → A (5.17) and V -natura l isomorphisms ( A ∗ X ) ∗ Y ∼ = A ∗ ( X ⊗ Y ) and (5.18) A ∗ I ∼ = A (5.19) satisfying the obvious cohere nce conditions. Suppo se further that each V -functor A ∗ − : H − → A (5 .20) has a left adjoint h A, −i : A − → H . (5.21) It follows that we hav e a V -functor h− , −i : A op ⊗ A − → H (5.22) and a V -natural isomorphism H ( h A, B i , Z ) ∼ = A ( B , A ∗ Z ) . (5.23) T o encapsulate: A op is a tensor ed H op -categor y . Obs erv e that we ha ve a canon- ical isomorphism h A, B ∗ Z i ∼ = h A, B i ⊗ Z. (5.24) 14 F or, we hav e the natura l iso morphisms H ( h A, B ∗ Z i , X ) ∼ = A ( B ∗ Z , A ∗ X ) ∼ = A ( B , ( A ∗ X ) ∗ Z ∗ ) ∼ = A ( B , A ∗ ( X ⊗ Z ∗ )) ∼ = H ( h A, B i , X ⊗ Z ∗ ) ∼ = H ( h A, B i ⊗ Z, X ) . There are also cano nical mo rphisms e : h A, B i − → I and ( 5.2 5) d : B − → A ∗ h A, B i (5.2 6) corresp onding re spectively under isomorphism (5.23) to A ∼ = A ∗ I and 1 : h A, B i − → h A, B i . Finally , we come to o ur exa mple of a left V -flo c k. The V -categ o ry is A a nd the V -functor Q : A ⊗ A op ⊗ A − → A of (5.1) is Q ( A, B , C ) = A ∗ h B , C i . (5.27) The isomor phism φ o f (5.2) is derived fr om (5.24) as Q ( Q ( A, B , C ) , D , E ) = A ∗h B , C i∗h D , E i ∼ = A ∗h B , C ∗ h D , E ii ∼ = Q ( A, B , Q ( C , D , E )) . (5.28) The natural transforma tions α and β of (5 .3 ) a nd (5.4) ar e Q ( A, B , B ) = A ∗ h B , B i 1 ∗ e − → A ∗ I ∼ = A and (5.29) B d − → A ∗ h A, B i = Q ( A , A, B ) . (5.30) In this case the K le is li V -catego ry K of (5.9) is given by the V -functor h− , −i of (5.22). Notice tha t K is c lo sed under binary tenso ring in H since, by (5.24), we hav e h A, B i ⊗ h C, D i ∼ = h A, B ∗ h C, D ii . How ever, K may not contain the I of H . Definition 5 Suppo se F : A − → X is a V -functor betw een V -flo cks A and X . W e call F flo ckular when it is equipp ed with a V -natura l family consis t ing of maps ρ A,B ,C : Q ( F A, F B , F C ) → F Q ( A, B , C ) such that Q ( F Q ( A, B , C ) , F D, F E ) ρ F Q ( Q ( A, B , C ) , D , E ) F φ Q ( Q ( F A, F B , F C ) , F D , F E ) Q ( ρ, 1 , 1) φ F Q ( A, B , Q ( C, D , E )) Q ( F A, F B , Q ( F C, F D , F E )) Q (1 , 1 ,ρ ) = Q ( F A, F B , F Q ( C, D , E )) ρ 15 F Q ( A, B , B ) F α Q ( F A, F B , F B ) ρ α = F A (5.31) Q ( F A, F A, F B ) ρ F B β F β = F Q ( A, A, B ) (5.32) (where w e hav e suppressed the subscripts of ρ ) co mm ute. W e call F str ong flo ckular when each ρ A,B ,C is inv ertible. 6 Herd como dules Let A b e a herd in V in the sense of Section 3. In the first instance, A is a comono id. W rite Cm r f A for the V -c a tegory of right A -co modules whose underlying ob jects in V hav e duals. W rite V f for the full sub category of V consisting of the ob jects w ith duals. W e sha ll show that Cm r f A is a V -flo c k in the s e ns e of Section 5 and that the forgetful V -functor U : Cm r f A − → V f is strong flo c kular. The flo c k structure on V f is Q ( L, M , N ) = L ⊗ M ∗ ⊗ N whic h is a sp ecial case of Example 3 with A = H = V f . W e wish to lift this flo ck s tr uctu r e on V f to Cm r f A . So, assuming L , M and N are r igh t A -como dules with duals in V , we need to provide a right A -como dule str uctu r e on L ⊗ M ∗ ⊗ N . This is defined as the co mposite L ⊗ M ∗ ⊗ N 1 ⊗ 1 ⊗ η ⊗ 1 L ⊗ M ∗ ⊗ M ⊗ M ∗ ⊗ N δ ⊗ 1 ⊗ δ ⊗ 1 ⊗ δ L ⊗ A ⊗ M ∗ ⊗ M ⊗ A ⊗ M ∗ ⊗ N ⊗ A 1 ⊗ 1 ⊗ ε ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 L ⊗ A ⊗ A ⊗ M ∗ ⊗ N ⊗ A c 145236 L ⊗ M ∗ ⊗ N ⊗ A ⊗ A ⊗ A 1 ⊗ 1 ⊗ 1 ⊗ q L ⊗ M ∗ ⊗ N ⊗ A. (6.1) In terms of str ings we can write this as (6.2) 16 Theorem 8 F or al l right A -c omo dules L , M and N with duals in V , the c om- p osite (6.1) re nders Q ( L, M , N ) = L ⊗ M ∗ ⊗ N a right A -c omo dule such t hat the c anonic al morphisms φ : Q ( Q ( L , M , N ) , R, S ) → Q ( L, M , Q ( N , R , S )) , α : Q ( L , M , M ) → L and β : M → Q ( L, L, M ) in V ar e right A -c omo dule morphisms. F urther, Cm r f A is a flo ck such that the for getful functor U : Cm r f A → V f is str ong flo ckular. Pr o of. The main coaction axiom for the comp osite (6.1) follo ws by using a dualit y (“snake”) identi ty for M ∗ ⊣ M , the coa ction axio ms fo r L , M and N , and that q is a comono id mor phism. = = = = The fact that φ , α a nd β a re right como dule morphisms follow from dualit y iden tities, comonoid axioms, and equations (3.2) , (3.3) and (3.4). The calculation for φ is straight forward = 17 The one for α is = = = = = and the one for β is = = = = Since the for g etfu l functor Cm r f A → V f is faithful, the flo c k axioms hold in Cm r f A beca use they do in V f , it follows that Cm r f A is a flo c k. Clear ly also the V -functor U : Cm r f A − → V f is strong flo c kular . 18 7 T annak a dualit y f or flo c ks and herds Given a strong flo ckular V -functor F : A − → V f , we show tha t , when the co end E = End ∨ F = Z A ( F A ) ∗ ⊗ F A, (7.1) exists in V , it is a her d in V . T o simplify notation, let us put e X = X ∗ ⊗ X for X ∈ V so that, for A ∈ A , we have a copro jection copr A : eF A − → E . It is well known that E is a como noid (see [17 ] for example). The comult i- plication δ : E → E ⊗ E is defined by commutativit y of eF A 1 ⊗ η ⊗ 1 copr A eF A ⊗ eF A copr A ⊗ copr A E δ E ⊗ E (7.2) The counit ε : E → I r e stricts along the copro jection copr A : eF A → E to yield the counit for the duality F A ∗ ⊣ F A . W e have the following isomor phi sms E ⊗ 3 = Z A eF A ⊗ Z B eF B ⊗ Z C eF C ∼ = Z A,B ,C eF A ⊗ eF B ⊗ eF C ∼ = Z A,B ,C (( F A ) ⊗ ( F B ) ∗ ⊗ F C ) ∗ ⊗ F A ⊗ ( F B ) ∗ ⊗ F C ∼ = Z A,B ,C Q ( F A, F B , F C ) ∗ ⊗ Q ( F A, F B , F C ) ∼ = Z A,B ,C ( F Q ( A, B , C )) ∗ ⊗ F Q ( A, B , C ) = Z A,B ,C eF Q ( A, B , C ) compatible with the copro jections. The morphism q : E ⊗ E ⊗ E → E is defined b y co mm utativit y of eF Q ( A, B , C ) copr A,B,C copr Q ( A,B,C ) E ⊗ 3 q E (7.3) Theorem 9 F or any str ong flo ckular V -functor F : A → V f , t he c o en d (7.1), and diagr ams (7.2) and (7.3), define a her d E i n V . 19 Pro of Once again we pro ceed by string s. As usual, morphisms in s tr ing dia- grams are depicted as no des shown as circles with the mo rphism’s name written inside. Howev er, copro jections copr A : eA → E ar e lab eled a s A . W e define the comonoid multiplication by = and the counit by = . The q for the herd structure is = . There are also the α and β where, for example β = 20 β ∗ = for which we have the following iden tifications : = = = . F rom now on we will drop the lab els on the strings and take them to b e under- sto od. 21 By Definiti o n 1 w e require q to b e a comonoid mor phi sm. W e pro ceed as follows: = = = = = 22 = . The map q defined ab o ve is also a ssociative since: = = = 23 = = = = . The calculation for comultiplying on the left is: = 24 = = = = = = . 25 The one for comultiplying on the right is: = = = = = 26 = . 8 App endix: In v ertible F usion Impl ies An tip o de. F o r completeness we pr ovide a proo f shown to us by Micah Blake McCurdy . This pro of applies in any braided monoidal categor y . The result for Hopf a lgebras is classical. Prop osition 10 A bimo noid H is a Hopf monoid if and only if the fusion morphism (3.22) is invertible. Pro of If H has a n antipo de ν : H → H denoted b y a blac k no de with o ne input and one output, define v by v = so that v v = = = = = 1 A ⊗ 2 v v = = = = = 1 A ⊗ 2 ; ν is inv ertible. 27 Conv ersely , (M.B. McCurdy) suppose v has an inv erse v denoted by a blac k no de with tw o inputs and t wo outputs. Define ν by ν = . W e shall use ( i ) A ⊗ 3 1 ⊗ v µ ⊗ 1 A ⊗ 3 µ ⊗ 1 ( ii ) A δ η ⊗ 1 A ⊗ 2 v A ⊗ 2 v A ⊗ 2 A ⊗ 2 ( iii ) A ⊗ 2 1 ⊗ δ v A ⊗ 3 v ⊗ 1 ( iv ) A ⊗ 2 1 ⊗ ǫ v A ⊗ 2 µ A ⊗ 2 1 ⊗ δ A ⊗ 3 A which follow from the mor e o b vious ( i ) ′ ( µ ⊗ 1)(1 ⊗ v ) = v ( µ ⊗ 1) ( ii ) ′ δ = v ( η ⊗ 1) ( iii ) ′ (1 ⊗ δ ) v = ( v ⊗ 1)(1 ⊗ δ ) ( iv ) ′ µ = (1 ⊗ ǫ ) v which as strings ar e ( i ) ′ = ( ii ) ′ = ( iii ) ′ = ( iv ) ′ = 28 Now = = (i) = = (iv) = = = (iii) = = (ii) = so that ν is an antipo de. ——————————————————– References [1] R. Baer , Zur Einführung des Scharb e griffs , Journa l für die reine und a nge- wandte Mathematik 160 (1 929) 199– 207. [2] T. Brzezinski and J. 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