Table Of ContentMONOIDAL 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-monoidalcate-
gories,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.
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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
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