Table Of Content

MONOIDAL CATEGORIES ENRICHED IN BRAIDED MONOIDAL CATEGORIES SCOTTMORRISONANDDAVIDPENNEYS Abstract. WeintroducethenotionofamonoidalcategoryenrichedinabraidedmonoidalcategoryV. Wesetupthebasictheory,andproveaclassificationresultintermsofbraidedoplaxmonoidalfunctorsto theDrinfeldcenterofsomemonoidalcategoryT. 7 1 Eventhebasictheoryisinteresting;itsharesmanycharacteristicswiththetheoryofmonoidalcategories 0 enrichedinasymmetricmonoidalcategory,butlackssomefeatures.Ofparticularnote,thereisnocartesian 2 productofbraided-enrichedcategories,andthenaturaltransformationsdonotforma2-category,butrather n satisfyabraidedinterchangerelation. a Strikingly,ourclassificationisslightlymoregeneralthanwhatonemighthaveanticipatedintermsof J strongmonoidalfunctorsV →Z(T).Wewouldliketounderstandthisfurther;inafuturepaperweshow 3 thatthefunctorisstrongifandonlyiftheenrichedcategoryis‘complete’inacertainsense.Neverthelessit remainstounderstandwhatnon-completeenrichedcategoriesmaylooklike. ] T Oneshouldthinkofourconstructionasageneralizationofde-equivariantization,whichtakesastrong C monoidalfunctorRep(G) → Z(T) forsomefinitegroupG andamonoidalcategoryT,andproducesa newmonoidalcategoryT .Inoursetting,givenanybraidedoplaxmonoidalfunctorV →Z(T),forany . //G h braidedV,weproduceT//V:thisisnotusuallyan‘honest’monoidalcategory,butisinsteadV-enriched. t a IfV hasabraidedlaxmonoidalfunctortoVec,wecanusethistoreducetheenrichmenttoVec,andthis m recoversde-equivariantizationasaspecialcase. [ 1 v 7 1. Introduction 6 5 Whilethesymmetriesofclassicalmathematicalobjectsformgroups,thesymmetriesof‘quantum’ 0 mathematical objects such as subfactors and quantum groups form more general objects which are 0 . best axiomatized as tensor categories. In turn, tensor categories have connections to many branches 1 0 ofmathematics,includingrepresentationtheory,topologicalandconformalfieldtheory,andquantum 7 information. 1 Early in the study of monoidal categories, Eilenberg and Kelly defined the notion of a category : v enrichedinagivenmonoidalcategoryV [EK66](seealso[Kel05]). Anordinarycategoryhasobjectsand i X homsets,whileaV-enrichedcategory C hasobjects,andforeverya,b ∈ C,anassociatedhomobject r C(a →b) ∈ V. TheV-enrichedcategoryalsocomeswithdistinguishedidentityelementsj ∈ V(1 → a c V C(c →c)) foreveryc ∈ C, and acompositionmorphism−◦ − : C(a →b)C(b →c) → C(a →c) for C everya,b,c ∈ C whichmustsatisfycertaincompatibilityandassociativityaxioms. Fromthisperspective, wemaythinkofanordinarycategoryasenrichedinthemonoidalcategorySet,andalinearcategoryas enrichedinthemonoidalcategoryVec. Braided monoidal categories were introduced in [JS93]. They play an essential role as algebraic ingredientsin3-dimensionalquantumtopology. Thisarticleintroducesthenotionofamonoidalcategory enriched in a braided monoidal category. Linear monoidal categories are of course the case when the enrichingcategoryV = Vec. ThespecialcasewhentheenrichingcategoryV = sVechasreceivedsome recentattention[BE16;Ush16;BGHNPR16]. Wewillexpandmoreonrelatedworkin§1.1below. Webelievethenotionofamonoidalcategoryenrichedinasymmetricclosedmonoidalcategoryis well-knowntoexpertsinthefield. However,thefactthattheenrichingcategoryneedonlybebraided, notnecessarilysymmetric,hasnotbeenpursued. Themaindifficultyindefiningthemonoidalstructure 1 ofaV-monoidalcategoryistheexchangerelationfromanordinarymonoidalcategory. If f ∈ C(a →b), 1 f ∈ C(b →c),д ∈ C(d →e),andд ∈ C(e → f),wehave 2 1 2 (f ⊗д ) ◦ (f ⊗д ) = (f ◦ f ) ⊗ (д ◦д ). 1 1 2 2 1 2 1 2 Throughoutthisarticle,wechoosetoreadcompositionlefttorightsothatwedonotneedtochangethe orderofobjects: thatis,compositionisamap C(a →b)C(b →c) → C(a →c). Indeed,weseethetwo morphismsд and f are transposed in the above relation, which tells us that the enriching monoidal 1 2 categoryshouldbebraided. Asenrichedcategoriesdonothavemorphisms,butratherhomobjects,we replacetheordinaryexchangerelationwiththefollowingbraidedinterchangerelation,whichweexpress usingstringdiagramsformorphismsinV: C(ad →cf) C(ad →cf) −◦− −⊗− C(ad →be) C(be →cf) C(a→c) C(d → f) (1.1) = . −⊗− −⊗− −◦− −◦− C(a→b) C(d →e) C(b→c) C(e → f) C(a→b) C(d →e) C(b→c) C(e → f) WereferthereadertoSection2fortheformaldefinitionofa(strict)V-monoidalcategory. In this article, we classify monoidal categories enriched in V in terms of braided oplax monoidal functorsfromV totheDrinfeldcenterZ(T) ofanordinarymonoidalcategory T. (TheDrinfeldcenter wasintroducedin[JS91b].) Recallthatafunctor F isoplaxmonoidal ifthereisafamilyofmorphisms µ : F (uv) → F (u)F (v), which need not be isomorphisms, but must merely satisfy naturality and u,v associativity conditions. We call F braided oplax monoidal if µ also intertwines the braidings (see Proposition5.4). Ourmainresultis: Theorem1.1. LetV beabraidedmonoidalcategory. Thereisabijectivecorrespondence RigidV-monoidalcategories  Pairs(T,FZ)withT arigidmonoidalcategory C,suchthatx (cid:55)→ C(1C →x)  (cid:27) and FZ : V →Z(T) braidedoplaxmonoidal,. admitsaleftadjoint  suchthat F := FZ ◦R admitsarightadjoint  Here,R :Z(T) → T istheforgetfulfunctor,andweusethesuperscripton FZ todistinguishitfrom F : V → T. ThenotionofrigidityforV-monoidalcategoriesisintroducedinSection2.7. Onecoulddressthistheoremupasanequivalenceof2-categories,butwedonotpursuethishere. WealsoworkwithastrictnotionofV-monoidalcategoryforconvenience. Theorem1.1thusgivesusapowerfultooltoconstructV-monoidalcategories. Someexamplesof strongmonoidalfunctors FZ : V →Z(T) asaboveareexploredindetailin[HPT16a,§3.3]. Additional examples of strong monoidal functors include the presence of a full copy of Fib insideZ(Ad(E )) (by 8 [BEK01,Cor.4.9],seealso[Edi17])andafullcopyofAd(SU(3) ) insideZ(Ad(4442)) (usingthemodular 3 data from [GI15], and the classification from [Bru16]). These examples seem very interesting, and we lookforwardtostudyingthemindetail. ForanybraidedV,thereisanuninterestingbraidedoplaxmonoidalfunctorλZ : V →Z(Vec) = Vec, obtained as a left adjoint of the strong monoidal inclusion Vec → V. Under the correspondence, this justinterprets C as‘triviallyenriched’inV: thatis,themorphismobjectsoftheresultingV-monoidal categoryarestilljustvectorspaces,butthoughtofasmultiplesoftheidentityobjectinV. 2 To proceed from left to right in Theorem 1.1, from a V-monoidal category C we first extract an ordinary monoidal category CV (enriched in Vec) by replacing each Hom object C(a → b) with the vectorspaceV(1 → C(a →b)). (SeeSection3formoredetailson CV.)1 Wechosethenotation CV to hintattheideaoftakingfixedpoints,akintoequivariantization. Inparticular,if C isamonoidalcategory enrichedinthesymmetricmonoidalcategoryRep(G),forG afinitegroup,thisjustmeansthatthereisan actionofG onthemorphismsofC,andCRep(G) isthesubcategoryofG-invariantmorphisms. Wecontruct the functor F : V → T in Section 4, and show that it lifts to the centre, giving FZ : V → Z(T), in Section5. To pass from right to left, we use the right adjoint of F together with rigidity to define the hom objects of the V-enriched category T . The category T has the same objects as T, and the hom //F //F objectsaredeterminedbythenaturalisomorphisms V(v → T (a →b)) (cid:27) T (aF (v) →b). //F WedescribethisconstructioninfulldetailinSection6.1. InthecasethatV issemisimple,wegetamore explicitdescriptionby (cid:77) T (a →b) (cid:27) T (aF (v ) →b)v . //F i i v simple i ThenotationT ismeanttoevokethefeelingthattheV-monoidalcategoryissometypeofquotient //F of T by F, akin to de-equivariantization. The usual process of de-equivariantisation begins with a Tannakiansubcategory,thatisacopyofRep(G),forG afinitegroup,insideZ(T),for T somemonoidal category. Thiscanbeviewedasafullyfaithfulbraidedstrongmonoidalfunctor F : Rep(G) → Z(T). Wecanfactorde-equivariantisationintotwosteps: firstapplyingourmaintheoremtoobtaintheRep(G)- monoidalcategory T ,andsecondapplyingthethefibrefunctor(theunderlyingvectorspace)toeach //F Homspace. Inthissenseourconstructionisageneralizationofde-equivariantization,althoughwhenwe ‘quotientout’byV insideZ(T),thereisingeneralnosubsequent‘underlyingvectorspace’fortheHom objectsinV. 1.1. Relatedwork. Asmentionedearlier,wehaveseenrecentinterestinmonoidalcategoriesenriched in V = sVec. Brundan and Ellis defined a super tensor category in [BE16] (see also [Müg13, §6]), and Usherworkedoutmanybasicpropertiesin[Ush16]. Usheralsoindicatedsomeinterestingexamples(his Example6.9)whichwereearlierannouncedbyWalkerinthelanguageofspinplanaralgebras. Recently, [BGHNPR16]definesthenotionofafermionicmodulartensorcategoryasapre-modulartensorcategory whoseMügercenterissVec. Thislatterconditionhasalsobeencalled‘slightlydegenerate’in[DNO13]. Thearticle[BGHNPR16]definesaproceduresimilartode-equivariantizationwhichproducessupertensor categoriesfromfermionicmodulartensorcategories. WewouldliketoacknowledgeexplicitlytheworkofKevinWalkeronenrichmentfor2-categories andhighercategories;althoughmuchofthisisunpublished,weandothersinthefieldhavelearntagreat dealfromhisideas,disseminatedinnotes,conversations,andseminars. WealsopointoutthatrecentworkofHenriques,Penneys,andTener[HPT16a]introducesthenotion of an anchored planar algebra internal to a braided pivotal tensor category, and show that these are equivalent to braided pivotal strong (not merely oplax) monoidal functors FZ : V → Z(T) for some pivotaltensorcategory T suchthat F = FZ ◦R admitsarightadjoint. Thefunctor FZ endows T with 1Thisisaspecialcaseofamoregeneralconstruction:givenabraidedlax monoidalfunctorF :V →W,wecanturn aV-monoidalcategoryintoaW-monoidalcategorybyapplyingF toeachoftheHomobjects.SeeSectionA.1formore detailsonthisconstruction.TheconstructionofCV usesthebraidedlaxmonoidalfunctorV(1V →−). 3 the structure of a module tensor category for V as studied in [HPT16b]. See also [JL16] for the related notionofaparaplanaralgebra. Inasimilarvein,ifweinterpretapivotalbraidedtensorcategoryasadisklike3-category,onecan obtainananalogousclassificationforitsdisklikemodules. Inaconcurrentarticle,MorrisonandWalker studysupertensorcategoriesfromthepointofviewofSpin-disklike2-categories,inthesenseof[MW12]. Thatarticlewillalsoincludemanyexamplesofcategorieswithobjectsofsmalldimension. Also connected to this theorem is the MathOverflow question [MO:51783] which discusses the constructionofV-enrichedmonoidalcategorieswhenV issymmetricandclosedfrombraidedstrong monoidalfunctorstotheDrinfeldcentersofmonoidalcategories. Theactivitytherereinforcesourbelief thatmonoidalcategoriesenrichedinsymmetricclosedmonoidalcategoriesareprobablyknowntoexperts. Interestingly,ourtheoremonlyrequiresthebraidedcentralfunctorbeoplaxmonoidal,andnotstrong monoidal. Weonlyneedanoplaxfunctortopassfromrighttoleft,andallwerecoverwhenpassingfrom lefttorightistheoplaxstructure. Alltheexamplesweknowaboutatthispoint,however,areeitheractuallystrongmonoidalfunctors, or left adjoints of strong monoidal functors (e.g. the ‘trivially enriched’ examples discussed above). It wouldbeveryinterestingtohave‘genuinely’oplaxexamples. 1.2. Future research. In a subsequent paper, we will characterize strong monoidality of F : V → Z(T) in terms of V-completeness of C. A V-monoidal category C is V-complete if there is a V- (cid:68) (cid:68) monoidalfunctorV → C,whereV istheself-enrichment ofV describedin§2.3. Thisistheappropriate generalisationof Π-completenessintroducedin[Ush16]forsupertensorcategories,andistheanalog of C being tensored over V in the sense of [Kel05]. We will explore V-completeness of V-monoidal categories in a followup article. In particular, we will prove that under the bijective correspondence given in Theorem 1.1, V-complete fusion categories correspond to braided strong monoidal functors V →Z(T) forsomerigidmonoidalcategory T. Moreover, we will discuss the V-completion of a V-monoidal category C, which generalises the Π-envelopeintroducedin[BE16]forsuper-tensorcategories. WewillprovethataV-monoidalcategory C isV-completeifandonlyifitisV-equivalenttoitsV-completion. It would be interesting to see if one could weaken the rigidity assumption in Theorem 1.1 to the assumptionthatthemonoidalcategoriesaremerelyclosed. (Forexample,thiscouldhopefullyimprovethe proofofLemma4.6below,whichappearsinAppendixB.)Asweuserigidityforvariousotherpurposes, andas[HPT16b;HPT16a]usespivotalcategories,wearecontenttoremainintherigidworldfornow. In another direction, it seems that we use the fact that the braiding in V has an inverse rather infrequently. Perhapsitispossibletogeneralisethesettingthroughouttomonoidalcategoriesenriched inacategoryV equippedwithalaxbraidinguv →vu asin[DS07;DPS07]. Fornow,however,wehave noapplicationofsuchageneralisation,sowehavenotpursuedit. 2. Basicnotions SupposeV isamonoidalcategory. WesuppressallunitorsandassociatorsinV toeasethenotation. Tensorproductsareindicatedbyjuxtaposition,thatis,omittingall ⊗-symbols,whileallcompositionsare writtenexplicitlywith◦. Wewritecompositionleft-to-rightthroughout. Recallfrom[Kel05]thataV-enrichedcategoryC associatestoeachpaira,b ∈ C ahomobjectC(a → b) ∈ V. For eacha ∈ C, there is a distinguished identity element j ∈ V(1 → C(a → a). For each a V a,b,c ∈ C,thereisadistinguishedcompositionmorphism−◦ − ∈ V(C(a →b)C(b →c) → C(a →c)). C Thesedatamustsatisfythefollowingtwoaxioms. (Wehavetwooptionsfordescribingsuchaxioms, eitherascommutativediagramsorasstringdiagrams[JS91a]inV. Throughoutthisintroductionweuse 4 both,toensureallreadersfindsomethingtheyarecomfortablewith;laterinthepaperweusewhichever ismostconvenient.) • (identity) Foralla,b ∈ C, (j 1 )◦(−◦ −) = 1 and (1 )◦(−◦ −) = 1 : a C(a→b) C C(a→b) C(a→b)j C C(a→b) b C(a →a)C(a →b) C(a →b)C(b →b) ja1C(a→b) 1C(a→b)jb C(a →b) −◦ − and C(a →b) −◦ − C C 1C(a→b) 1C(a→b) C(a →b) C(a →b) Instringdiagramstheaboveaxiomreadsas: C(a→b) C(a→b) C(a→b) −◦ − −◦ − C C = = . j j a b C(a→b) C(a→b) C(a→b) • (associativity) Foralla,b,c,d ∈ C,thefollowingdiagramcommutes: C(a →b)C(b →c)C(c →d) 1(−◦C−) C(a →b)C(b →d) (−◦C−)1 (−◦C−) C(a →c)C(c →d) (−◦C−) C(a →d) whichinstringdiagramsbecomes: C(a→d) C(a→d) −◦ − −◦ − C C = . −◦ − −◦ − C C C(a→b) C(b→c) C(c →d) C(a→b) C(b→c) C(c →d) From thispoint onward, weassumeV is abraided monoidalcategorywherethe braidingin V is denotedby β :uv →vu forallu,v ∈ V. u,v Definition 2.1. A (strict)2 V-monoidal category C3 is a V-enriched category C together with the followingdata: • aunitobject1 ∈ C, C • foreverya,b ∈ C,anobjectab ∈ C,and • foralla,b,c,d ∈ C,atensorproductmorphism−⊗ − ∈ V(C(a →c)C(b →d) → C(ab →cd)) C whichsatisfythefollowingaxioms: 2Itwouldbewonderfulforsomeonetoworkouttheaxiomsforanon-strictV-monoidalcategory! 3OnecanmotivatethisdefinitionbytakingtheusualnotionofaV-enrichedcategory,atfirstnotusingthebraiding, thengivingA×B,forAandBV-enrichedcategories,thestructureofaV-enrichedcategorybydefiningthecomposition usingthebraidinginV intheinevitableway.Afterthis,thedefinitionaboveisjusttheusualdefinitionofa(strict)monoidal category.Perhapssomeonewillproveacoherencetheoremfornot-necessarily-strictmonoidalcategoriesenrichedinabraided monoidalcategory,butfornowwestayinthestrictsetting. 5 • (strictunitorforobjects)Foralla ∈ C,1 a =a1 =a. C C • (strictassociatorforobjects)Foralla,b,c ∈ C, (ab)c =a(bc). • (unitality)Foralla,b ∈ C, (j 1 ) ◦ (−⊗ −) = 1 , (1 j ) ◦ (−⊗ −) = 1 , 1C C(a→b) C C(a→b) C(a→b) 1C C C(a→b) and (j j ) ◦ (−⊗ −) = j . a b C ab • (associativityof−⊗ −)Foralla,b,c,d,e,f ∈ C,thefollowingdiagramcommutes: C C(a →d)C(b →e)C(c → f) 1(−⊗C−) C(a →d)C(bc →ef) (−⊗C−)1 (−⊗C−) C(ab →de)C(c → f) (−⊗C−) C(abc →def). • (braidedinterchange)Foralla,b,c,d,e,f ∈ C,thefollowingdiagramcommutes: C(a →b)C(d →e)C(b →c)C(e → f) (−⊗C−)(−⊗C−) C(ad →be)C(be →cf) −◦ − C 1βC(d→e)C(b→c)1 C(ad →cf). −◦ − C C(a →b)C(b →c)C(d →e)C(e → f) (−◦C−)(−◦C−) C(a →c)C(d → f) Thecorrespondingstringdiagramforthebraidedinterchangerelationwasalreadygivenin(1.1). 2.1. v-gradedmorphisms. AstheobjectsoftheenrichingcategoryV donotnecessarilyhaveunder- lyingsets,wemustbecarefulwhentalkingabout‘morphismsinaV-enrichedcategory’. A1 -graded V morphismfroma tob inaV-monoidalcategory C isamorphism1 → C(a →b) ofV,andav-graded V morphism,forv anobjectofV,isamorphismv → C(a →b). Wecancompose(ortensor)au-graded morphismwithav-gradedmorphismtoobtainauv-gradedmorphism. A1 -gradedmorphism f : 1 → C(a →b) isinvertible ifthereisamorphismд : 1 → C(b →a) V V V calledaninverse suchthatthemaps 1 −−f→д C(a →b)C(b →a) −−−−◦−C→− C(a →a) V 1 −д−→f C(b →a)C(a →b) −−−−◦−C→− C(b →b) V areidentityelements,i.e., (fд) ◦ (−◦ −) = j and (дf) ◦ (−◦ −) = j . Noticethatif f isinvertible,the C a C b usualproofshowsitsinverseisuniqueandcanbedenoted f−1: h = (hj ) ◦ (−◦ −) = (hfд) ◦ (−◦ −◦ −) = (j д) ◦ (−◦ −) =д. a C C C b C Thereareobviousnotionsofmonomorphismsandepimorphismswhichweleavetothereader. 2.2. V-functors. AV-functor F : C → D betweenV-enrichedcategoriesisjustafunctionbetween theobjects,andforeacha,b ∈ C,anelement F ∈ V(C(a →b) → D(F (a) → F (b)),suchthat a→b −◦ − C(a →b)C(b →c) C C(a →c) (2.1) F F F a→b b→c a→c −◦ − D(F (a) → F (b))D(F (b) → F (c)) D D(F (a) → F (c)) 6 commutes,asdoes C(a →a) jC a (2.2) 1V Fa→a . jD F(a) D(F (a) → F (a)) Given V-functors F : C → D and G : D → E, we define the V-functor F ◦ G : C → E by (F ◦G) = F ◦G foralla,b ∈ C. a→b a→b F(a)→F(b) Av-graded natural transformation λ (again, forv ∈ V) between V-functors F,G : C → D is a collectionλ :v → D(F (a) → G(a)) sothatthefollowingdiagramcommutesforallobjectsa,b: a vC(a →b) λaGa→b D(F (a) → G(a))D(G(a) → G(b)) −◦ − D β D(F (a) → G(b)). −◦ − D C(a →b)v Fa→bλb D(F (a) → F (b))D(F (b) → G(b)) (Thisisjust‘thenaturalitysquare,viewedfromoutside’.) Wewriteλ : F ⇒ G. NoticethatweareonlytalkingaboutnaturaltransformationsforV-enrichedcategoriesratherthan V-monoidalcategories,andyetthebraidinginV isessentialtothedefinition! Lemma 2.2. Suppose λ : F ⇒ G is au-graded natural transformation, and µ : G ⇒ H is av-graded naturaltransformation. Thenthereisauv-gradednaturaltransformationλ◦µ : F ⇒ H,calledthevertical composite,definedby (λ◦µ) :uv −λ−a−µ→a D(F (a) → G(a))D(G(a) → H(a))) a −◦ − −−−D−→D(F (a) → H(a)). Verticalcompositionisassociative. Theproofissufficientlystraightforwardthatweleaveitasanexercise. Similarly,horizontalcompositionfollowstheusualformula: Lemma 2.3. Suppose κ : F ⇒ G is a u-graded natural transformation where F,G : C → D, and λ : H ⇒ I is v-graded where H,I : D → E. The following formula defines a uv-graded natural transformationcalledthehorizontalcomposite:κλ : F ◦H ⇒ G ◦I by κaλG(a) (κλ) :uv −−−−−−→D(F (a) → G(a))E(H(G(a)) → I(G(a))) a HF(a)→G(a)1 −−−−−−−−−−→E(H(F (a)) → H(G(a)))E(H(G(a)) → I(G(a))) −◦ − −−−−E→E(H(F (a)) → I(G(a))). Again,horizontalcompositionisassociative. 7 Lemma2.4. Naturaltransformationsthemselvessatisfyabraidedinterchange. GivenV-monoidalcategories,functors,andnaturaltransformations H K µ G ν J C D E κ λ F I whereκ,λ,µ,ν arerespectivelyu,v,w,x-gradednaturaltransformations,wehave ((κλ) ◦ (µν)) = (1 β 1 ) ◦ ((κ ◦µ)(λ◦ν)) :uvwx → E(I(F (a)) → K(H(a))). a u v,w x a WedefertheproofsofLemma2.3andLemma2.4toAppendixA.2. Remark2.5. ItwouldbeinterestingtoshowthatV-categories,V-functors,andV-naturaltransformations form a V-enriched 2-category. In doing so, one would define a Hom-object Nat(F ⇒ G) ∈ V for V- functors F,G : C → D. Wecouldthenexpressverticalcompositionasamorphism−◦ − : Nat(F ⇒ Nat G)Nat(G ⇒ E) → Nat(F ⇒ E) and horizontal composition as a morphism − ⊗ − : Nat(F ⇒ Nat G)Nat(H ⇒ I) → Nat(FI ⇒ GI). One would then prove that these morphims satisfied a braided interchange. 2.3. Self-enriched categories. Given a braided rigid category V, we can construct a V-monoidal category V(cid:68) with the same objects as V, and V(cid:68)(u → v) d=ef u∗v. The composition and tensor product mapsaregivenby −◦(cid:68)− : V(cid:68)(u →v)V(cid:68)(v →w) =u∗vv∗w −u−∗−e−v−v−→w u∗w = V(cid:68)(u →w) V = −◦V(cid:68)− −⊗(cid:68)− : V(cid:68)(u →v)V(cid:68)(w →x) =u∗vw∗x −−β−u−∗−1v−,w−−∗→x w∗u∗vx = V(cid:68)(uw →vx) V = −⊗V(cid:68)− Itisanenjoyableexercisetodiscoverthatthesesatisfythebraidedexchangelaw: = . ThisexampleisrelatedtothecanonicalfunctorV →Z(V) whenV isbraided,viaourmaintheorem. It istheanalogueinthebraidedrigidsettingoftheexamplein§1.6of[Kel05]. The category Tangle of (unoriented, framed) tangles is a braided rigid monoidal category. It has a faithfulfunctorfromBraid,thefreebraidedmonoidalcategoryononeobject. TheobjectsofTangleand ofBraidarejustthenaturalnumbers. Wedenotethestandardgeneratorsofthebraidgroupbyσ . i (cid:70) WecanformTangle,thecategoryoftanglesenrichedinitself. Thisallowsustoprovethefollowing usefulresult. 8 Lemma2.6. Supposethatanequationoftheform C(ad →cf) C(ad →cf) −◦ − −◦ − C C = C(ad →be) C(be →cf) C(ad →be) C(be →cf) −⊗ − −⊗ − −⊗ − −⊗ − C C C C C(a→b) C(d →e) C(b→c) C(e → f) C(a→b) C(d →e) C(b→c) C(e → f) γ γ(cid:48) whereγ andγ(cid:48)are4-strandbraidswiththesameunderlyingpermutation,holdsforallV-enrichedcategories, andallchoicesofobjectsa,b,c,d,e,f. Thenγ =γ(cid:48). Proof. We picka = c =d = f = 1, andb = e = 0 in T(cid:70)angle. Then T(cid:70)angle(1 → 0) = T(cid:70)angle(0 → 1) = 1, and(abbreviatingTangletoT) (cid:68) T(2→2) −◦ − C (cid:68) (cid:68) T(2→0) T(0→2) = −⊗ − −⊗ − C C (cid:68) (cid:68) (cid:68) (cid:68) T(1→0) T(1→0) T(0→1) T(0→1) Thus the equation reduces toγσ−1 =γ(cid:48)σ−1 in Tangle, which is equivalent toγ =γ(cid:48). Since braids map 1 1 faithfullyintotangles,wehavetheconclusion. (cid:3) 2.4. TherotationofaV-monoidalcategory. Givenamonoidalcategory C enrichedinasymmetric monoidalcategory,wecantaketheoppositecompositionortheoppositetensorproduct,obtaininganew enrichedmonoidalcategory. Whentheenrichmentismerelybraided,wefindthatthisisnotthecase. Neverthelessthereissomethingwhichwecalltheπ-rotationof C,whichisformedbysimultaneously modifyingthecompositionandthetensorproduct. Thisisanotherpointofdeparturefromthetheoryfor symmetricenrichments. Lemma2.7. Supposethat C isaV-monoidalcategory. Consideranewcompositionandtensorproducton C givenby −◦ − −⊗ − −◦(cid:48) − = C and −⊗(cid:48) − = C C C βk βl forsomek,l ∈ Z. Thesealwayssatisfyassociativity,butsatisfythebraidedinterchangeaxiomifandonlyif k =l. 9 Proof. Thebraidedinterchangeaxiombecomes(bypullingcompositionortensorproductmorphisms throughthetwists) −◦− −◦− −⊗− −⊗− −⊗− −⊗− = ∆ (β)k ∆ (β)(cid:96) 2 2 β(cid:96) β(cid:96) βk βk where ∆ (β) = =σ σ σ σ denotesthetwostrandcablingof β. ByLemma2.6,wehavethat 2 2 1 3 2 σlσl(σ σ σ σ )k =σ σkσk(σ σ σ σ )lσ−1 1 3 2 1 3 2 2 1 3 2 1 3 2 2 (recallthattheσ arethestandardgeneratorsofthebraidgroup). WethenapplytheBuraurepresentation i tothisidentity. Lookingatthefirstcolumnofthelastrowoftheresulting4-by-4matrices,weobtain (cid:16) (cid:17) (cid:16) (cid:17) (−1)k + (−1)l+1 tk+l+2 + (−1)k + (−1)l+1 tk+l+4 −2(−1)kt2k+3 +2(−1)lt2l+3 = 0. Itisrelativelystraightforwardtoseethatthispolynomialint onlyvanishesidenticallywhenk =l or whenk =l±1. Wenowtakethesecondcolumnofthesecondrow,andaftersettingk =l±1andclearing adenominator,obtain (cid:16) (cid:17) 2(−1)l(t −1)2(t +1) t2 +1 t2l±1 = 0 whichisimpossible. Thuswemusthavek =l,andanisotopyverifiesthatthebraidedinterchangeaxiom indeedholds. (cid:3) Noteintheabovethatiftheenrichingcategoryweresymmetric,theintegersk andl wouldonlyhave appearedmodulo2,andindeedallfourchoiceswouldhavegivennewenrichedmonoidalcategories. Definition2.8. Wedefine Crot tobetheV-monoidalcategoryobtainedtakingk =l = 1intheabove lemma. Again,notethatinthesymmetricallyenrichedcase, (Crot)rot = C. Generally,thisisnotthecase,so weobtainanintegerfamilyofrotationsoftheoriginalcategory. Whenwediscussrigiditybelow,wewill seethatachoiceofdualityfunctorisachoiceofanisomorphism C (cid:27) Crot. 2.5. Products of V-monoidal categories (only?) exist when V is symmetric. We now point out asignificantdifferencebetweenthetheoryofmonoidalcategorieswithabraidedenrichment,andthe theoryofmonoidalcategorieswithasymmetricenrichment. If C and D areV-monoidalcategoriesenrichedinasymmetricmonoidalcategoryV,wecandefine theirCartesianproduct C × D,whichisalsoaV-monoidalcategory. First,weproducethe (V ×V)- V enrichedmonoidalcategory C ×D by (C ×D)((a,c) → (b,d)) = C(a →b)D(c →d),andcomposition 10