On the iteration of weak wreath products
Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street's weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl^{(n)}(K), of (n…
Authors: Gabriella B"ohm
ON THE ITERA TION OF WEAK WREA TH PR ODUCTS GABRIELLA B ¨ OHM Abstract. Based on a study of the 2-catego r y of weak distributive laws, we de- scrib e a method o f iterating Street’s w eak wreath product cons tr uction in [17]. That is, for any 2-category K and for a n y non-negative in teger n , we in tro duce 2-categorie s Wdl ( n ) ( K ), of ( n + 1 )-tuples of monads in K pairwis e related b y w eak distributiv e laws ob eying the Y ang-B axter equation. The firs t instance Wdl (0) ( K ) c o incides with Mnd ( K ), the usual 2-ca teg ory of mona ds in K , and for other v alues of n , Wdl ( n ) ( K ) contains Mnd n +1 ( K ) as a full 2-sub catego ry . F or the local idemp otent closure K of K , extending the m ultiplication of the 2-monad Mnd , we equip these 2-categ ories with n p ossible ‘weak wreath pro duct’ 2-functors Wdl ( n ) ( K ) → Wdl ( n − 1) ( K ), such that all of their po ssible n -fold comp osites Wdl ( n ) ( K ) → Wdl (0) ( K ) are equal; i.e. such that the weak wreath pro duct is ‘asso ciative’. Whenever idemp o ten t 2 -cells in K split, this lea ds to pseudofunctors Wdl ( n ) ( K ) → Wdl ( n − 1) ( K ) ob eying the asso cia - tivit y proper t y up-to isomor phism. W e presen t a practically importa nt occurr e nce of an iterated weak wre a th pro duct: the alg ebra of o bserv able quantities in an Ising t y pe quan tum spin chain where the spins take their v a lues in a dual pair of finite weak Hopf alg ebras. W e als o construct a fully faithful em b edding o f Wdl ( n ) ( K ) into the 2- category of comm utative n + 1 dimensional cubes in Mnd ( K ) (hence into the 2-catego ry of co mm utative n + 1 dimensional cubes in K whenever K has Eilenberg - Mo ore ob jects and its idemp otent 2-cells split). Finally , we give a sufficient and necessary condition o n a monad in K to be isomor phic to an n -ar y w e a k wreath pro duct. Intr oduction A t the heart of the iteration of wreath pro ducts in the work [11] of Eugenia Che ng, lies the 2-mo na d M nd on the 2-cat ego ry 2 - Cat of 2-categories, 2-functors and 2-natural transformations, first discuss ed in [16]. F or an y 2- category K , the iteration of its asso ciativ e m ultiplication Mnd n ( K ) → Mnd n − 1 ( K ) → · · · → Mnd ( K ) tak es an ( n + 1)-tuple of monads, pairwise related b y distributiv e laws ob eying the Y ang-Baxter equalit y , to a unique monad in K . The resulting monad can b e interpreted as a n iterated wreath pro duct. The aim of this pa p er is to find a similar iteratio n pro c ess for w eak wreath pro ducts in tro duced by Ross Street in [17] and by Stefaan Caenep eel and Erwin De Gro ot in [10]. These w eak w reath pro ducts are defined in 2-categories in whic h idemp oten t 2-cells split, see [17]. They are induced by w eak distributiv e la ws; i.e. certain 2-cells relating t wo mona ds. They ob ey the usual compatibility conditions of distributiv e la ws with the m ultiplications of the monads, but the compatibility conditions with the units are w eake ned [10], [17]. Making w eak distributiv e laws conceptually differen t from their non-w eak counterparts, they ar e not know n to b e monads in an y 2-category . Date : v1:Oct. 201 1 . v2:Jan 201 2 . 1 2 GABRIELLA B ¨ OHM (Ho we ver, a w eak distributiv e la w can b e c haracterized as a pair of monads in 2 - categories extending Mnd ( K ) and its v ariant Mnd ∗ ( K ), resp ectiv ely , see [4]). In Section 2, for any 2-category K , we construct a 2-category Wdl ( n ) ( K ) for ev ery non-negativ e in teger n . Its ob jects are ( n + 1)- tuples of monads in K pairwise related b y we a k distributiv e laws ob eying the Y ang-Baxter equation. The first one, W dl (0) ( K ) is isomorphic to Mnd ( K ), the 2-category of monads in K a s defined in [16]. The next one, Wdl (1) ( K ) is the 2-category of we ak distributiv e la ws, obtained b y dualizing the definition in [7]. F or ev ery n , Wdl ( n ) ( K ) con tains Mnd n +1 ( K ) as a full 2-sub category . But, in con trast t o the class ical (i.e. non-w eak) case, Wdl ( n ) ( K ) is not kno wn to arise b y the ( n + 1)-fold application of some 2-monad. Although in this w ay w e can not in terpret them as m ult iplicatio ns of some 2-monad, for each v alue of n w e describ e n differen t 2-functors Wdl ( n ) ( K ) → Wdl ( n − 1) ( K ) (where K denotes the lo cal idemp oten t closure of K ). They extend the n p ossible multiplications Mnd n +1 ( K ) → Mnd n ( K ). They giv e rise to a unique comp osite Wdl ( n ) ( K ) → Wdl (0) ( K ) whose v alue on an ob ject of Wdl ( n ) ( K ) is regarded as the (asso ciativ ely iterated) w eak wreath pro duct of the n +1 o ccurring monads in K . Whenev er idemp oten t 2- cells in K split; that is, K and K are biequiv alen t, our construction yields pseudofunctors Wdl ( n ) ( K ) → Wdl ( n − 1) ( K ) giving rise to a comp osite Wdl ( n ) ( K ) → Wdl (0) ( K ) which is unique up-to a pseudonatural equiv alence in the c hoice of the biequiv alence K → K . Our motiv ation to study iterat ed w eak wreath pro ducts comes from mathematical ph ysics. The Ising mo del is a quan tum spin c hain in whic h the spins tak e their v alues in the sign group Z (2). In its v arious generalizations, the spins ma y ta k e their v alues in arbitrary finite g r o ups [18], in finite dimensional Hopf algebras [13] or in finite dimensional w eak Hopf algebras [14], [3]. In all of these, except the last quoted family of mo dels, the algebra of the observ able quantities in a n y finite in terv al is giv en b y an iterated wreath pro duct. In quan tum spin c hains based on w eak Hopf algebras, ho we ver, the algebras of observ ables are iterated w eak wreath pro ducts. In Section 3 w e presen t this example in some detail. The definition of W dl ( n ) ( K ) is further motiv a ted in Section 4 b y a fully faithful em b edding of it into the 2-categor y of n + 1 dimensional cubes in the 2-catego r y of monads in K . Whenev er idemp oten t 2- cells in K split, this yields a f ully fait hf ul em b edding of Wdl ( n ) ( K ) in to the 2-category of n + 1 dimensional cubes in the 2- category of monads in K ; and a lso in to the 2-category of n + 1 dimensional cubes in K whenev er in addition K a dmits Eilen b erg-Mo ore o b jects f o r monads. In our final Section 5 we analyze the n -ary factorization problem a sso ciated to w eak distributiv e la ws. That is, w e give a complete c ha r acterization of those monads in the lo cal idemp otent closure of a 2-category which arise a s iterated w eak wreath pro ducts of n monads. Throughout, for a tec hnical simplification, w e w o rk with 2-categories. There is no difficult y , how eve r, to extend our considerations to bicategories. Ac knowledgem ent. I would like to express my gratitude to Ross Street for his helpful commen ts on this w ork. P artial supp ort b y the Hungarian Scien tific Researc h F und OTKA K68195 is gratefully a ckno wledged. ON THE ITERA TIO N OF WEAK WR EA TH PRODUCTS 3 1. Preliminaries on weak distribut ive la ws In this se ction we rev isit some rec ent ‘w eak’ generalizations of the f ormal theory of monads that will b e used in t he sequel. 1.1. Lo cal idempot en t closure. T o an y 2-category K w e asso ciat e another 2- category K b y freely splitting idemp oten t 2-cells. In more detail, the 0-cells o f K are the same as those in K . The 1-cells in K are pairs consisting of a 1-cell v and a 2-cell v : v → v in K suc h that v .v = v ; i.e. v is idemp oten t. The 2-cells ( v , v ) → ( v ′ , v ′ ) in K are 2- cells ω : v → v ′ in K suc h t ha t v ′ .ω = ω = ω .v . Horizontal and v ertical comp ositions in K are induced b y those in K . The iden tit y 2-cell is v : ( v , v ) → ( v , v ). Throughout, w e shall use the no t ation seen a b o ve : If it is not otherwise stated, in a 1 - cell in K , for the idempot en t 2-cell part we use the ov erlined v ersion of the same sym b ol that denotes the 1-cell part. F or an y 2-category K , there is an eviden t inclusion 2-f unctor K → K , acting on the 0-cells as the iden tit y map, taking a 1-cell v to ( v , v ) – i.e. the 1 -cell with iden tit y 2-cell part – and acting on the 2-cells aga in as the iden tit y map. W e sa y that idemp oten t 2 - cells in a 2 - category K split if, for an y idemp oten t 2 -cell θ : v → v there exist a 1- cell w and 2-cells ι : w → v and π : v → w suc h that π .ι = w a nd ι.π = θ . If the splitting exists then it is unique up-to an isomorphism of w . Clearly , in K idempo ten t 2-cells split for a ny 2-category K . Whenev er in K idemp oten t 2-cells split, the inclusion K → K b ecomes a biequiv- alence. (Since it a cts on the 0 -cells as the identit y map, this simply means that it induces an equiv alence of the hom cat ego ries.) Hence there is a pseudofunctor K → K whic h is its inv erse (in the sens e of in v erse biequiv alences). On the 0-cells also this pseudofunctor acts as t he iden tity map. On a 1-cell ( v , v ) its action is constructed via a chosen splitting of the idemp oten t 2-cell v . If ( ι : w → v , π : v → w ) is this c hosen splitting, then the image of ( v , v ) is w . A 2- cell ω : ( v , v ) → ( v ′ , v ′ ) is tak en to w ι / / v ω / / v ′ π ′ / / w ′ . Let us stress t ha t the biequiv alence K → K is not a 2-functor in general a nd it is unique only up-to a pseudonatural equiv alence arising from the c hoice of the splitting of eac h idemp otent 2-cell. 1.2. Demimonads. In simplest terms, a demimonad in a 2-category is a monad ( A, ( t, t )) in the lo cal idemp otent closure, cf. [8]. Explicitly , it is given b y a 1-cell t : A → A and 2-cells µ : t 2 → t and η : 1 A → t suc h that the following diagrams comm ute. t 3 µt / / tµ t 2 µ t 2 µ / / t t ηt / / tη t 2 µ t 2 µ / / t 1 A ηη / / η ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ t 2 µ t t 2 ηt 2 / / µ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ t 3 µ 2 t By the unitality condition, the idemp oten t 2-cell t m ust b e equal to µ .tη = µ.η t (hence it is a redundan t information that w ill b e often omitted in the seq uel). This structure o ccurred in [4] under the name ‘pre-monad’. A dem imonad ( A, ( t, t )) is t he image of a mona d under the inclusion K → K if and only if t is the iden tit y 2-cell t . 4 GABRIELLA B ¨ OHM 1.3. W eak distributiv e la ws. Extending the notion of distributiv e la w due to Jon Bec k (see [2]), w eak distributiv e laws in a 2-category were in tro duced b y Ross Street in [17] as follows. They consist of tw o monads ( A, t ) and ( A, s ) on the same ob ject, and a 2-cell λ : ts → st suc h that the follow ing diag rams comm ute. (1.1) t 2 s tλ / / µs tst λt / / st 2 sµ ts λ / / st s ηs / / sη ts λ st stη / / sts sλ / / s 2 t µt / / st ts 2 λs / / tµ sts sλ / / s 2 t µt ts λ / / st t tη / / ηt ts λ st ηst / / tst λt / / st 2 sµ / / st The same set o f a xioms o ccurred a lso in [10]. By [17, Prop osition 2.2 ], the second and fourth diagrams can b e replaced by a single diagram (1.2) st ηst / / stη tst λt / / st 2 sµ sts sλ / / s 2 t µt / / st . The equal paths around (1.2) give rise to an idemp o ten t 2- cell λ : st → st (whic h o ccurs also in the b ottom ro ws of the second and fourth diagrams in (1.1)). It is an iden tity if a nd only if λ is a distributiv e law in the strict sense. Note that a w eak distributiv e law in K is the same as a w eak distributive law in the horizontal opp osite of K . A weak distributiv e la w in K is then giv en by demimonads ( A, t ) and ( A, s ) and a 2- cell λ : ts → st rendering comm uta tiv e the diagrams in (1.1) and ob eying in addition t he normalization conditions ts tµ.tηs / / λ λ " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ts λ ts µs.tηs / / λ λ # # ● ● ● ● ● ● ● ● ts λ st µt.sηt / / st st sµ.sη t / / st . In the sequel w e shall need some iden tities on w eak distributiv e law s ( in K ). The axioms imply comm utativity of the follow ing diagrams, see [17]. (1.3) ts λ / / λ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ ✾ st λ s 2 t 2 s λt / / µµ s 2 t 2 µµ sts sλ / / λs s 2 t µt tst λt / / tλ st 2 sµ sts sλ tst λt st st λ / / st s 2 t µt / / st st 2 sµ / / st Moreo v er, b y the asso ciativit y of µ , t he left-b ott o m path in the last diagram in (1.1) comm utes with t he m ultiplication by t on the righ t. Hence so do es the top-right path ON THE ITERA TIO N OF WEAK WR EA TH PRODUCTS 5 meaning the comm utativit y o f the first diagram in (1.4) t 2 µ tηt / / tst λt / / st 2 sµ s 2 µ sηs / / sts sλ / / s 2 t µt t tη / / ts λ / / st s ηs / / ts λ / / st. Comm utativity of the second dia g ram follow s symm etrically . 1.4. The 2-category of we ak distributiv e la ws. Dua lizing in the appropriate sense the definition of the 2-category of mixe d w eak distributiv e la ws in [7 ], the fo l- lo wing 2- category Wdl ( K ) of weak distributive la ws in K is obta ined (see [6, P aragra ph 1.9]). The 0-cells a re the w eak distributiv e law s λ : ts → s t . The 1-cells λ → λ ′ are triples consisting of a 1 - cell v : A → A ′ and 2-cells ξ : t ′ v → v t and ζ : s ′ v → v s in K , suc h that ( v , ξ ) : ( A, t ) → ( A ′ , t ′ ) and ( v , ζ ) : ( A, s ) → ( A ′ , s ′ ) are 1- cells in Mnd ( K ) (also called monad morphisms in [16]) and the following diagram commute s. (1.5) t ′ s ′ v t ′ ζ / / λ ′ v t ′ v s ξ s / / v ts vλ s ′ t ′ v s ′ ξ / / s ′ v t ζ t / / v st vλ / / v st The 2- cells ( v , ξ , ζ ) → ( v ′ , ξ ′ , ζ ′ ) are 2- cells ω : v → v ′ in K whic h are 2-cells in Mnd ( K ) (i.e. mo n ad tr ans formations b y the terminolog y of [16]); b oth ( v , ξ ) → ( v ′ , ξ ′ ) and ( v , ζ ) → ( v ′ , ζ ′ ). Horizon tal and v ertical comp ositions are induced by those in K . This definition can b e inte rpreted in terms of (w eak) liftings as in [7]. There is a fully faithful em b edding Mnd 2 ( K ) → Wdl ( K ) as follows . It tak es a 0-cell (( A, t ) , ( s, λ )) to the distributiv e law λ : ts → st , regarded as a w eak distributiv e la w. It takes a 1-cell (( v , ξ ) , ζ ) to ( v , ξ , ζ ) and it acts on the 2-cells as the identit y ma p. 1.5. W eak wreath pro duct. The weak wreath pro duct induced b y a w eak dis- tributiv e law in a 2- catego ry in whic h idemp otent 2-cells split, w as discusse d by Ross Street in [17, Theorem 2.4]. In the particular case of the monoidal category (i.e. one ob ject bicategory) of mo dules o ver a commutativ e ring, it app eared in [10, Theorem 3.2]. F or an arbitrary 2-category K , there is a we ak wr e a th pr o duct 2- functor Wdl ( K ) → Mnd ( K ), whic h sends a we a k distributiv e law λ : ts → st to the monad ( s t, λ ) in K , with mu ltiplication and unit ( st ) 2 sλt / / s 2 t 2 µµ / / st and 1 ηη / / ts λ / / st . It sends a 1-cell (( v , v ) , ξ , ζ ) : λ → λ ′ to the monad morphism with the same 1-cell part ( v , v ) a nd the 2-cell part s ′ t ′ v s ′ ξ / / s ′ v t ζ t / / v st v λ / / v st . On the 2-cells it a cts as the iden tity map. Whenev er idemp otent 2-cells in K split, the biequiv alence K ≃ K induce s a pseudo- functor W dl ( K ) ≃ → Wdl ( K ) → Mnd ( K ) ≃ → Mnd ( K ). (It can b e c hosen, in fact, to b e a 2-functor by c ho osing the biequiv alence K → K adopting the con v en tio n that w e s plit 6 GABRIELLA B ¨ OHM iden tity 2-cells trivially; i.e. via iden tity 2- cells.) Its ob j ect map yields Street’s w eak wreath pro duct in K . 1.6. Binary fa c torization. Let K b e any 2-category . As pro v ed in [6], a demimonad ( A, r ) is isomorphic to a w eak wreath pro duct induc ed b y some w eak distributiv e law ts → st in K if and only if the follo wing hold. (a) There are 1-cells in Mnd ( K ) with trivial 1-cell parts ( A, ( t, t )) ( A, ( r , r )) (( A,A ) ,α ) o o (( A,A ) ,β ) / / ( A, ( s, s )) ; (b) The 2-cell π := ( st, st ) β α / / ( r r , r r ) µ / / ( r , r ) in K p ossesse s section ι (meaning π .ι = r ≡ µ.r η ) whic h is an s - t bimo dule morphism with respect to the t - and s -actions induced on r by α and β , resp ectiv ely . Indeed, for the w eak wreath pro duct induced by a w eak distributiv e law λ : ts → st , w e ha ve 1-cells ( A, ( t, t )) ( A, ( s t, λ )) (( A,A ) ,λ.tη ) o o (( A,A ) ,λ.ηs ) / / ( A, ( s, s )) in Mnd ( K ). Moreov er, the 2-cell π in pa rt (b) comes out as ( st, st ) λλ.ηs tη / / ( stst, λ λ ) µµ.sλt / / ( st, λ ) = ( st, st ) λ / / ( st, λ ) , whic h is split b y the bimo dule morphism λ : ( st, λ ) → ( st, st ). Con v ersely , if prop erties (a ) and (b) hold, then ts αβ / / r r µ / / r ι / / st is a weak distributiv e la w with corresp onding idemp otent equal to ι.π : st → st . The isomorphism b et w een the induced w eak wreath pro duct a nd ( A, r ) is pro vided b y ( st, ι.π ) π / / ( r , r ) ι o o in K . F or the details of the pro of we refer to [6]. 2. 2-ca tegories of we ak distributive la ws and the itera ted weak wrea th product Throughout this section, K is an arbitrary 2- cat ego ry and K stands for its lo cal idemp oten t closure. F or a n y non-negativ e integer n , w e define a 2-catego ry Wdl ( n ) ( K ). Its ob jects are ( n + 1)-tuples of monads pairwise related b y we ak distributiv e law s ob eying the Y ang-Ba xter condition. F or each v alue o f n , we construct n different 2- functors Wdl ( n ) ( K ) → Wdl ( n − 1) ( K ) corresp onding to taking the w eak wreath pro duct of tw o cons ecutive monads o f the n + 1 o ccurring ones. W e show tha t these 2-f unctors giv e rise t o a unique comp osite W dl ( n ) ( K ) → W dl (0) ( K ) = M nd ( K ). W e regard its ob ject map as the n -ary w eak wreath pro duct of the in v olv ed monads. ON THE ITERA TIO N OF WEAK WR EA TH PRODUCTS 7 2.1. The 2-category Wdl ( n ) ( K ). F or any non-negativ e integer n , a 0-cell of Wdl ( n ) ( K ) is give n b y n + 1 monads s 0 , s 1 , . . . , s n together with weak distributiv e laws λ i,j : s j s i → s i s j for all 0 ≤ i < j ≤ n , ob eying for all 0 ≤ i < j < k ≤ n the Y ang- Baxter relation s k s j s i λ j,k s i / / s k λ i,j s j s k s i s j λ i,k / / s j s i s k λ i,j s k s k s i s j λ i,k s j / / s i s k s j s i λ j,k / / s i s j s k . The 1-cells consist of a 1-cell v and 2-cells ξ i : s ′ i v → v s i in K for all 0 ≤ i ≤ n , suc h that ( v , ξ i , ξ j ) is a 1-cell λ i,j → λ ′ i,j in Wdl ( K ) (see P a ragraph 1.4), for all 0 ≤ i < j ≤ n . The 2-cells are those 2-cells ω : v → v ′ in K whic h are 2-cells ( v , ξ i , ξ j ) → ( v ′ , ξ ′ i , ξ ′ j ) in W dl ( K ) (in t he sense of Paragraph 1.4), fo r all 0 ≤ i < j ≤ n . Since Wdl ( K ) is closed under the ho rizon tal a nd v ertical compo sitions in K , so is Wdl ( n ) ( K ). Hence it is a 2 -category with the horizontal and v ertical comp ositions induced b y those in K . Recall from [11] that a 0-cell in Mnd n ( K ) is giv en b y n monads, pairwise related b y distributiv e laws ob eying the Y ang-Baxter condition. The 1- cells consist of n monad morphisms fo r the n in v olved monads w it h a c o mmon underlying 1-cell, ob eying (1.5) (in the simplified f orm when the o ccurring idemp oten ts are iden tities). The 2- cells are those 2- cells in K whic h are mona d tra nsformations fo r all of the n monad mor phisms. With this description in mind, exte nding that in P a ragraph 1.4, there is an eviden t fully faithful em b edding Mnd n +1 ( K ) → Wdl ( n ) ( K ). T aking an y m + 1- elemen t subset of { 0 , 1 , . . . , n } induces an eviden t 2 -functor Wdl ( n ) ( K ) → Wdl ( m ) ( K ). Lemma 2.2. T ake any ob j e ct { λ i,j : s j s i → s i s j } 0 ≤ i 1 , and for any 0-c el l { λ i,j : s j s i → s i s j } 0 ≤ i 1 , the c om p osite Wdl ( n ) ( K ) C 1 / / Wdl ( n − 1) ( K ) C 1 / / . . . C 1 / / Wdl ( K ) takes a 0-c el l { λ i,j : s j s i → s i s j } 0 ≤ i 1 then let λ i,j b e giv en b y the flip map σ . If i is o dd, then let λ i,i +1 b e equal to λ defined as H ⊗ ˆ H σ / / ˆ H ⊗ H ˆ ∆ ⊗ ∆ / / ˆ H ⊗ ˆ H ⊗ H ⊗ H ˆ H ⊗ ev ⊗ H / / ˆ H ⊗ H, and if i is eve n then let λ i,i +1 b e equal to ˆ λ given b y ˆ H ⊗ H σ / / H ⊗ ˆ H ∆ ⊗ ˆ ∆ / / H ⊗ H ⊗ ˆ H ⊗ ˆ H H ⊗ σ ⊗ ˆ H / / H ⊗ ˆ H ⊗ H ⊗ ˆ H H ⊗ ev ⊗ ˆ H / / H ⊗ ˆ H . The symmetry σ : X ⊗ Y → Y ⊗ X , for X , Y ∈ { H , ˆ H } , is a distributiv e la w hence a w eak distributive la w. W e sho w tha t λ : H ⊗ ˆ H → ˆ H ⊗ H is a w eak distributive la w. The morphism ξ := ˆ H ⊗ H ˆ ∆ ⊗ H / / ˆ H ⊗ 2 ⊗ H ˆ H ⊗ ev / / ˆ H is an asso ciativ e (and eviden tly unital) action in the sense of comm utativity of (3.2) ˆ H ⊗ H ⊗ 2 ˆ ∆ ⊗ H ⊗ 2 / / ˆ H ⊗ σ ˆ ∆ ⊗ H ⊗ 2 ) ) ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ˆ H ⊗ 2 ⊗ H ⊗ 2 ˆ H ⊗ ev ⊗ H / / ˆ ∆ ⊗ ˆ H ⊗ H ⊗ 2 ˆ H ⊗ H ˆ ∆ ⊗ H ˆ H ⊗ 2 ⊗ H ⊗ 2 ˆ H ⊗ ˆ ∆ ⊗ H ⊗ 2 / / ˆ H ⊗ 2 ⊗ σ ˆ H ⊗ 3 ⊗ H ⊗ 2 ˆ H ⊗ 2 ⊗ ev ⊗ H / / ˆ H ⊗ 2 ⊗ H ˆ H ⊗ ev ˆ H ⊗ H ⊗ 2 ˆ ∆ ⊗ H ⊗ 2 / / ˆ H ⊗ µ ˆ H ⊗ 2 ⊗ H ⊗ 2 ˆ H ⊗ 2 ⊗ µ ˆ H ⊗ H ˆ ∆ ⊗ H / / ˆ H ⊗ 2 ⊗ H ˆ H ⊗ ev / / ˆ H where the b ottom-right region comm utes b y the first iden tit y in (3.1). In terms of ξ , λ = H ⊗ ˆ H σ / / ˆ H ⊗ H ˆ H ⊗ ∆ / / ˆ H ⊗ H ⊗ 2 ξ ⊗ H / / ˆ H ⊗ H . ON THE ITERA TIO N OF WEAK WR EA TH PRODUCTS 19 Using this form of λ , its compatibility with the m ultiplication of H follo ws by com- m utativity of the diagram b elo w. H ⊗ 2 ⊗ ˆ H H ⊗ σ / / H ⊗ ∆ ⊗ ˆ H + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ σ H ⊗ H , ˆ H ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ µ ⊗ ˆ H H ⊗ ˆ H ⊗ H H ⊗ ˆ H ⊗ ∆ / / H ⊗ ˆ H ⊗ H ⊗ 2 H ⊗ ξ ⊗ H / / σ H, ˆ H ⊗ H ⊗ H H ⊗ ˆ H ⊗ H σ ⊗ H H ⊗ 3 ⊗ ˆ H H ⊗ σ H ⊗ H , ˆ H 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ∆ ⊗ H ⊗ 2 ⊗ ˆ H ˆ H ⊗ H ⊗ 3 ξ ⊗ H ⊗ 2 / / ˆ H ⊗ H ⊗ ∆ ⊗ H ˆ H ⊗ H ⊗ 2 ˆ H ⊗ ∆ ⊗ H H ⊗ 4 ⊗ ˆ H σ H ⊗ 4 , ˆ H ˆ H ⊗ H ⊗ 4 ξ ⊗ H ⊗ 3 / / ˆ H ⊗ σ ⊗ H ⊗ 2 (3.2) ˆ H ⊗ H ⊗ 3 ξ ⊗ H ⊗ 2 ˆ H ⊗ H ⊗ 2 ˆ H ⊗ ∆ ⊗ ∆ / / ˆ H ⊗ µ ˆ H ⊗ H ⊗ 4 ˆ H ⊗ H ⊗ σ ⊗ H / / ˆ H ⊗ σ H ⊗ H ,H ⊗ H 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ˆ H ⊗ H ⊗ 4 ˆ H ⊗ µ ⊗ H ⊗ 2 / / ˆ H ⊗ H ⊗ 3 ξ ⊗ H ⊗ 2 / / ˆ H ⊗ H ⊗ µ ˆ H ⊗ H ⊗ 2 ˆ H ⊗ µ H ⊗ ˆ H σ / / ˆ H ⊗ H ˆ H ⊗ ∆ / / ˆ H ⊗ H ⊗ 2 ξ ⊗ H / / ˆ H ⊗ H The region at the middle of the b ottom ro w comm utes b y the first w eak bialgebra axiom. Symmetrically , in terms of ζ := ( ev ⊗ H ) . ( ˆ H ⊗ ∆), w e can write λ = ( ˆ H ⊗ ζ ) . ( ˆ ∆ ⊗ H ) .σ . With this form of λ at hand, its compatibilit y with the m ultiplication of ˆ H follows symmetrically . It remains to c hec k the weak unitality condition (1.2). F or that consider the (idemp oten t) morphism ¯ ε s := H η ⊗ H / / H ⊗ 2 ∆ ⊗ H / / H ⊗ 3 H ⊗ µ / / H ⊗ 2 H ⊗ ε / / H . Recall from [5] (equations (4) and (8 ), resp ectiv ely) that the following diagrams in- v olving ¯ ε s comm ute. (3.3) H ⊗ 2 ∆ ⊗ H / / H ⊗ ¯ ε s H ⊗ 3 H ⊗ µ / / H ⊗ 2 H ⊗ ε H η ⊗ H / / ∆ H ⊗ 2 ∆ ⊗ H / / H ⊗ 3 H ⊗ µ H ⊗ 2 µ / / H H ⊗ 2 ¯ ε s ⊗ H / / H ⊗ 2 Using the definitions in (3.1), the first identit y in (3.3) is equiv alen t to ξ . ( ˆ H ⊗ ¯ ε s ) = ˆ µ. ( ˆ H ⊗ ξ ) . ( ˆ H ⊗ ˆ η ⊗ H ), implying comm utativity of the b otto m- righ t r egion in ˆ H ⊗ H η ⊗ ˆ H ⊗ H / / ˆ H ⊗ η ⊗ H 2 2 H ⊗ ˆ H ⊗ H σ ⊗ H / / ˆ H ⊗ H ⊗ 2 ˆ H ⊗ ∆ ⊗ H / / (3.3) ˆ H ⊗ H ⊗ 3 ξ ⊗ H ⊗ 2 / / ˆ H ⊗ H ⊗ µ ˆ H ⊗ H ⊗ 2 ˆ H ⊗ µ ˆ H ⊗ H ˆ H ⊗ ∆ / / ˆ H ⊗ H ⊗ ˆ η ˆ H ⊗ ˆ η ⊗ H ' ' P P P P P P P P P P P P ˆ H ⊗ H ⊗ 2 ˆ H ⊗ ¯ ε s ⊗ H / / ˆ H ⊗ ˆ η ⊗ H ⊗ 2 ˆ H ⊗ H ⊗ 2 ξ ⊗ H / / ˆ H ⊗ H ˆ H ⊗ H ⊗ ˆ H ˆ H ⊗ σ / / ˆ H ⊗ 2 ⊗ H ˆ H ⊗ 2 ⊗ ∆ / / ˆ H ⊗ 2 ⊗ H ⊗ 2 ˆ H ⊗ ξ ⊗ H / / ˆ H ⊗ 2 ⊗ H ˆ µ ⊗ H / / ˆ H ⊗ H . An y path in this diagram yields an alt ernat iv e express io n of the idemp oten t λ : ˆ H ⊗ H → ˆ H ⊗ H , prov ing that λ is a weak distributiv e la w. By symmetrical considerations so is ˆ λ . 20 GABRIELLA B ¨ OHM With some routine computations using the w eak bialgebra axioms, one chec ks that λ is equal to the iden tity map – i.e. λ is a distributiv e law in the strict sens e – if and only if ∆ .η = η ⊗ η ; i.e. H is a bialgebra in the strict sense. Our next task is to chec k the Y ang-Baxter conditions. The symmetry op erators among themselv es ob ey the Y ang-Baxter condition, hence for { i, j, k } such that j − i > 1 and k − j > 1 w e are done. F or { i − 1 , i, j } and { i, j, j + 1 } , suc h that j − i > 1, the Y ang-Baxter conditions follo w b y naturalit y of the symmetry . So w e are le ft with the case { i − 1 , i, i + 1 } . Assume first that i is o dd. Then the Y ang-Baxter condition follo ws b y comm utativity of H ⊗ ˆ H ⊗ H σ ⊗ H / / H ⊗ σ ˆ H ⊗ H ⊗ 2 ˆ ∆ ⊗ ∆ ⊗ H / / σ ˆ H ⊗ H ,H ˆ H ⊗ 2 ⊗ H ⊗ 3 ˆ H ⊗ ev ⊗ H ⊗ 2 / / σ ˆ H ⊗ 2 ⊗ H ⊗ 2 ,H ˆ H ⊗ H ⊗ 2 ˆ H ⊗ σ ˆ H ⊗ H ⊗ 2 σ ⊗ H H ⊗ 2 ⊗ ˆ H σ H,H ⊗ ˆ H / / H ⊗ ∆ ⊗ ˆ ∆ H ⊗ ˆ H ⊗ H H ⊗ ˆ ∆ ⊗ ∆ / / ∆ ⊗ ˆ ∆ ⊗ H H ⊗ ˆ H ⊗ 2 ⊗ H ⊗ 2 H ⊗ ˆ H ⊗ ev ⊗ H / / ∆ ⊗ ˆ ∆ ⊗ ˆ H ⊗ H ⊗ 2 H ⊗ ˆ H ⊗ H ∆ ⊗ ˆ ∆ ⊗ H H ⊗ 3 ⊗ ˆ H ⊗ 2 H ⊗ 2 ⊗ ev .σ ⊗ ˆ H H ⊗ 2 ⊗ ˆ H ⊗ 2 ⊗ H H ⊗ 2 ⊗ ˆ H ⊗ ˆ ∆ ⊗ ∆ / / H ⊗ ev .σ ⊗ ˆ H ⊗ H H ⊗ 2 ⊗ ˆ H ⊗ 3 ⊗ H ⊗ 2 H ⊗⊗ ev . σ ⊗ ˆ H ⊗ 2 ⊗ H ⊗ 2 H ⊗ 2 ⊗ ˆ H ⊗ 2 ⊗ H H ⊗ ev .σ ⊗ ˆ H ⊗ H H ⊗ 2 ⊗ ˆ H σ H,H ⊗ ˆ H * * ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ σ ⊗ ˆ H H ⊗ 2 ⊗ ˆ H H ⊗ σ / / H ⊗ ˆ H ⊗ H H ⊗ ˆ ∆ ⊗ ∆ / / H ⊗ ˆ H ⊗ 2 ⊗ H ⊗ 2 H ⊗ ˆ H ⊗ ev ⊗ H / / H ⊗ ˆ H ⊗ H . The case when i is ev en is treated symmetrically . This prov es that the construction in this paragraph yields an ob ject in Wdl ( n ) ( V ec ) (whic h is an ob ject of Mnd n +1 ( V ec ) if a nd only if H is a bialgebra in the strict sense). Hence by Theorem 2.10 there is a corresp onding w eak wreath pro duct monad (i.e. F - algebra) g iv en as the image of the idemp oten t (2.13). Since σ is unital, one obtains the following explicit for ms of this idemp oten t. If n is o dd, then it comes out as ( H ⊗ ˆ H ) ⊗ n +1 2 ˆ λ ⊗ n +1 2 / / H ⊗ ( ˆ H ⊗ H ) ⊗ n − 1 2 ⊗ ˆ H H ⊗ λ ⊗ n − 1 2 ⊗ ˆ H / / ( H ⊗ ˆ H ) ⊗ n +1 2 and if n is eve n, then ( H ⊗ ˆ H ) ⊗ n 2 ⊗ H ˆ λ ⊗ n 2 ⊗ H / / H ⊗ ( ˆ H ⊗ H ) ⊗ n 2 H ⊗ λ ⊗ n 2 / / ( H ⊗ ˆ H ) ⊗ n 2 ⊗ H . If H is a bialgebra in the s trict sense (e.g. it is the linear span of a finite group), then these idemp otents b ecome identit y maps and so the ab o ve w eak wreath pro ducts reduce wreath pro ducts in the strict sense. In the quantum spin c ha ins in [1 4 ] and [3], where the spins take their v a lues in a dual pair of finite dimensional we a k Hopf algebras, this (n+1)-ary weak wreath pro duct is regarded a s the algebra of observ able quan tities lo calized in the interv al [0 , n ] o f t he one dimensional lattice. In particular, in spin c ha ins built on dual pairs of finite dimensional Hopf algebras (e.g. pairs o f a finite group a lg ebra and the a lgebra of linear functions on this group), the observ able algebra is a prop er (n+1)-a r y wreath ON THE ITERA TIO N OF WEAK WR EA TH PRODUCTS 21 pro duct. In the classical Ising mo del – where the spins only hav e ‘up’ and ‘down’ p ositions – these dual Hopf a lgebras are b oth isomorphic to the linear span of the sign group Z (2). 4. A full y f aithful embedding In this section we sho w that, for any 2 -category K , and any non-negativ e in teger n , Wdl ( n ) ( K ) admits a fully faithful em b edding into the p ow er 2-category Mnd ( K ) 2 n +1 . Whenev er idemp otent 2-cells in K split, this gives rise to a fully faithful em b edding Wdl ( n ) ( K ) → Mnd ( K ) 2 n +1 . If in addition K admits Eilen b erg-Mo ore ob jects, this amoun ts to a fully faithful em b edding Wdl ( n ) ( K ) → K 2 n +1 . 4.1. The 2-category K 2 n . The 2-catego ry 2 has tw o 0-cells 0 and 1; an only no n- iden tity 1-cell 1 → 0; and all of its 2-cells are iden tit ies. F or an y 2-catego ry K , there is a 2-category K 2 of 2-functors 2 → K , 2-natural transformations and modifications. Iterativ ely , for n > 1 w e define K 2 n as ( K 2 n − 1 ) 2 . T hat is, K 2 n is isomorphic to the 2-category of 2-functors f r om the n -fold Cartesian pro duct 2 × · · · × 2 to K , 2-natural transformations and mo difications. An explicit description is giv en as follows . The 0-cells are the n dimensional orien ted cubes whose 2-faces are comm utative squares of 1-cells in K . A 1-cell from an n - cub e of e dg es { v p ,q : A p → A q } to { v ′ p,q : A ′ p → A ′ q } consists of 1-cells { u p : A p → A ′ p } in K suc h that the n + 1 - cub e { v p,q : A p → A q , u p : A p → A ′ p , v ′ p,q : A ′ p → A ′ q } is comm utative . That is, for all v alues of p and q , v ′ p ,q .u p = u q .v p,q . Finally , 2- cells consist of 2-cells ω p : u p → ˜ u p in K suc h that v ′ p ,q .ω p = ω q .v p,q . In Cartesian co o r dinates, the v ertices of an n -cub e can b e lab elled b y the ele- men ts p = ( p 1 , . . . , p n ) of the set { 0 , 1 } × n . Sometimes w e represen t p ∈ { 0 , 1 } × n b y listing those v alues o f i for whic h p i = 1. F or example, 12 = (1 , 1 , 0 , . . . , 0), 3 = (0 , 0 , 1 , 0 , . . . , 0) , etc.. The n -cub e has an edge p → q if and only if there is some in teger 1 ≤ i ≤ n such that p j = q j for all j 6 = i , p i = 0 and q i = 1. W e denote this situation b y q = p + i . F or p, q ∈ { 0 , 1 } × n , we say that p < q if, fo r an y 1 ≤ i, j ≤ n , the equalit y p i q j = 1 implies i < j . F or p < q w e define p + q ∈ { 0 , 1 } × n putting ( p + q ) i := p i + q i . W e denote by 0 := (0 , 0 , . . . , 0) and 1 := ( 1 , 1 , . . . , 1) the constan t elemen ts of { 0 , 1 } × n . The construction of the promised 2-functor Wdl ( n ) ( K ) → Mnd ( K ) 2 n +1 relies on a few lemmas b elo w. A routine computat ion prov es the first one: Lemma 4.2. F or any obje ct { λ i,j : s j s i → s i s j } 0 ≤ i
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment