tensor cathegories notes

7/27/2019 Tensor Cathegories Notes

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TENSOR CATEGORIESP. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik

Contents1. Monoidalcategories 41.1. Thedefinitionofamonoidalcategory 41.2. Basicpropertiesofunitobjectsinmonoidalcategories 51.3. Firstexamplesofmonoidalcategories 91.4. Monoidalfunctors,equivalenceofmonoidalcategories 141.5. Morphismsofmonoidalfunctors 151.6. Examplesofmonoidalfunctors 161.7. MonoidalfunctorsbetweencategoriesC 17G1.8. MacLanesstrictnesstheorem 191.9. TheMacLanecoherencetheorem 251.10. Rigidmonoidalcategories 261.11. Invertibleobjects 301.12. Tensorandmultitensorcategories 311.13. Exactnessofthetensorproduct 341.14. Quasi-tensorandtensorfunctors 361.15. Semisimplicityoftheunitobject 361.16. Grothendieckrings 381.17. Groupoids 391.18. Finiteabeliancategoriesandexactfaithfulfunctors 411.19. Fiberfunctors 421.20. Coalgebras 431.21. Bialgebras 441.22. Hopfalgebras 461.23. Reconstructiontheoryintheinfinitesetting 501.24. MoreexamplesofHopfalgebras 521.25. TheQuantumGroupUq(sl2) 551.26. ThequantumgroupUq(g) 551.27. Categoricalmeaningofskew-primitiveelements 561.28. PointedtensorcategoriesandpointedHopfalgebras 591.29. Thecoradicalfiltration 601.30. Chevalleystheorem 621.31. Chevalleyproperty 631.32. TheAndruskiewitsch-Schneiderconjecture 651.33. TheCartier-Kostanttheorem 661.34. Quasi-bialgebras 68

1


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21.35. Quasi-bialgebras with an antipode and quasi-Hopf

algebras 711.36. TwistsforbialgebrasandHopfalgebras 741.37. Quantumtraces 761.38. Pivotalcategoriesanddimensions 761.39. Sphericalcategories 771.40. Semisimplemultitensorcategories 781.41. IsomorphismbetweenV andV 781.42. Grothendieckringsofsemisimpletensorcategories 791.43. Semisimplicityofmultifusionrings 831.44. TheFrobenius-Perrontheorem 841.45. Tensor categories with finitely many simple objects.

Frobenius-Perrondimensions 861.46. Delignestensorproductoffiniteabeliancategories 901.47. Finite(multi)tensorcategories 911.48. Integraltensorcategories 931.49. Surjectivequasi-tensorfunctors 941.50. Categoricalfreeness 951.51. Thedistinguishedinvertibleobject 971.52. Integralsinquasi-Hopfalgebras 981.53. Dimensionsofprojectiveobjectsanddegeneracyofthe

Cartanmatrix 1002. Modulecategories 1002.1. Thedefinitionofamodulecategory 1002.2. Modulefunctors 1022.3. Modulecategoriesovermultitensorcategories 1032.4. Directsums 1042.5. Examplesofmodulecategories 1042.6. Exactmodulecategoriesforfinitetensorcategories 1062.7. Firstpropertiesofexactmodulecategories 1082.8. Z+modules 1102.9. Algebrasincategories 1112.10. InternalHom 1152.11. MainTheorem 1192.12. Categoriesofmodulefunctors 1202.13. Modulefunctorsbetweenexactmodulecategories 1212.14. Dualcategories 122References 126


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3Introduction

These are lecture notes for the course 18.769 Tensor categories,taughtbyP.EtingofatMITinthespringof2009.

In these notes we will assume that the reader is familiar with thebasic theory of categories and functors; a detailed discussion of thistheorycanbefoundinthebook[ML]. Wewillalsoassumethebasicsof thetheoryofabelian categories(fora moredetailed treatmentseethebook[F]).

IfC isacategory,thenotationX CwillmeanthatX isanobjectofC, andthe set ofmorphismsbetweenX,Y C will be denotedbyHom(X,Y).

Throughoutthenotes,forsimplicitywewillassumethatthegroundfieldkisalgebraicallyclosedunlessotherwisespecified,eventhoughinmanycasesthisassumptionwillnotbeneeded.


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41. Monoidalcategories

1.1. Thedefinitionofamonoidalcategory. Agoodwayofthinkingaboutcategorytheory(whichwillbeespeciallyusefulthroughoutthese notes) is that category theory is a refinement (or categorification) of ordinary algebra. In other words, there exists a dictionarybetween these two subjects, such that usual algebraic structures arerecovered fromthecorrespondingcategoricalstructuresbypassingtothesetofisomorphismclassesofobjects.

Forexample,thenotionofa(small)categoryisacategorificationofthenotionofaset. Similarly,abeliancategoriesareacategorificationofabeliangroups1 (whichjustifiestheterminology).

Thisdictionarygoessurprisinglyfar,andmanyimportantconstructions below will come from an attempt to enter into it a categoricaltranslationofanalgebraicnotion.

Inparticular,thenotionofamonoidalcategoryisthecategorificationofthenotionofamonoid.

RecallthatamonoidmaybedefinedasasetC withanassociativemultiplication operation (x,y) x y (i.e., a semigroup), with an element1suchthat12 = 1 and the maps 1 1 :C C arebijections., Itiseasytoshowthatinasemigroup,thelastconditionisequivalenttotheusualunitaxiom1 x=x 1 =x.

Asusualincategorytheory,tocategorifythedefinitionofamonoid,weshouldreplacetheequalitiesinthedefinitionofamonoid(namely,the associativity equation (xy)z = x(yz) and the equation 12 = 1)byisomorphismssatisfyingsomeconsistencyproperties,andthewordbijection by the word equivalence (of categories). This leads tothefollowingdefinition.Definition 1.1.1. A monoidal category is a quintuple (C,,a,1, )whereC isacategory, :C C C isabifunctorcalledthe tensorproduct bifunctor,

( )isafunctorialisomorphism:a: ( )(1.1.1) aX,Y,Z : (XY)Z X(Y Z), X, Y , Z Ccalledtheassociativityconstraint(orassociativityisomorphism),1 CisanobjectofC,and:11 1 isan isomorphism,subjecttothefollowingtwoaxioms.

1Tobemoreprecise,thesetofisomorphismclassesofobjectsina(small)abeliancategory C is a commutativemonoid, but one usually extends it to a group byconsideringvirtualobjectsoftheformXY,X,Y C.


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51. The pentagon axiom. Thediagram

(1.1.2)

((

WX)Y)ZaWX,Y,Z aW,X,YIdZ

(WX)(Y Z) (W(XY))ZaW,X,YZ aW,XY,Z

W(X(Y Z)) IdWaX,Y,Z W((XY)Z)iscommutativeforallobjectsW,X,Y,Z inC.2.Theunitaxiom. ThefunctorsL1 andR1 ofleftandrightmultiplicationby1areequivalencesC C.

Thepair

(1, )

is

called

the

unit

object

of

C.

2We see that the set of isomorphism classes of objects in a small

monoidal category indeed has a natural structure of a monoid, withmultiplication and unit 1. Thus, in the categorical-algebraic dictionary,monoidalcategoriesindeedcorrespondtomonoids(whichexplainstheirname).Definition 1.1.2. A monoidal subcategory of a monoidal category(C,,a,1, ) is a quintuple (D,,a,1, ), where D C is a subcategory closed under the tensor product of objects and morphisms andcontaining1and.Definition

1.1.3.

The

opposite

monoidal

category

C

opto

C

is

the

categoryCwithreversedorderoftensorproductandinvertedassociativity

somorphism.Remark 1.1.4. Thenotionoftheoppositemonoidalcategory isnottobeconfusedwiththeusualnotionoftheoppositecategory,whichisthecategoryC obtainedfromCbyreversingarrows(foranycategoryC). NotethatifCismonoidal,soisC (inanaturalway),whichmakesiteveneasiertoconfusethetwonotions.1.2. Basic properties of unit objects in monoidal categories.Let(C,,a,1, )beamonoidalcategory. DefinetheisomorphismlX :1X X bytheformula

lX =L11((Id)a1,11,X),andtheisomorphismrX :X1 X bytheformula

rX =R11((Id)aX,1,1).2Wenote that there isnoconditionon the isomorphism , so itcanbechosen

arbitrarily.


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6Thisgivesriseto functorial isomorphisms l :L1 IdC andr :R1 IdC. These isomorphisms are called the unit constraints or unit isomorphisms. They provide the categorical counterpart of the unit axiom1X=X1 =X ofamonoidinthesamesenseastheassociativityisomorphism provides the categorical counterpart of the associativityequation.Proposition 1.2.1. Thetrianglediagram

aX,1,Y(1.2.1) (X1)Y X(1Y)

rXIdY IdXlYXY

iscommutative

for

all

X,

Y

C.

In

particular,

one

has

r1 =l1 =.

Proof. ThisfollowsbyapplyingthepentagonaxiomforthequadrupleofobjectsX,1,1, Y. Morespecifically,wehavethefollowingdiagram:(1.2.2)

((X1)1)Y aX,1,1Id (X(11))Y

rXIdId (Id)Id(X1)Y

aX1,1,Y aX,1,Y aX,11,Y

X(1Y)rXId

(X1)(1Y)aX,1,1Y

Id(Id)Idl1Y X((11)Y)

Ida1,1,Y

X(1(1Y))Toprovetheproposition, itsufficestoestablishthecommutativity

ofthebottom lefttriangle(asanyobjectofC is isomorphictooneofthe

form

1

Y

).

Since

the

outside

pentagon

is

commutative

(by

the

pentagonaxiom),itsufficestoestablishthecommutativityoftheotherpartsofthepentagon. Now,thetwoquadranglesarecommutativeduetothefunctorialityoftheassociativityisomorphisms,thecommutativityoftheuppertriangle isthedefinitionofr,andthecommutativityofthelowerrighttriangleisthedefinitionofl.

ThelaststatementisobtainedbysettingX=Y =1in(1.2.1).


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7Proposition 1.2.2. Thefollowing diagrams commutefor all objectsX,Y C:

a1,X,Y(1.2.3) (1X)Y 1(XY)

lXIdY lXYXY

aX,Y,1(1.2.4) (XY)1 X(Y 1)

rXY IdXrYXY

Proof. Considerthediagram(1.2.5)((X1)Y)Z aX,1,YId (X(1Y))Z

(rXId)Id

(IdlY)Id(XY)Z

aX1,Y,Z aX,Y,Z aX,1Y,Z

X(Y Z)rXId

(X1)(Y Z)aX,1,YZ

Id(lYId)IdlYZ X((1Y)Z)

Ida1,Y,Z

X(1(Y Z))where X,Y,Z are objects in C. The outside pentagon commutes bythe pentagon axiom (1.1.2). The functoriality of a implies the commutativityofthetwomiddlequadrangles. Thetriangleaxiom(1.2.1)impliesthecommutativityoftheuppertriangleandthelowerlefttriangle. Consequently,thelowerrighttrianglecommutesaswell. SettingX = 1 and applying the functor L

11 to the lower right triangle, weobtaincommutativityofthetriangle(1.2.3). Thecommutativityofthetriangle(1.2.4)isprovedsimilarly. Proposition 1.2.3. For any object X in C one has the equalitiesl1X =IdlX andrX1 =rX Id.


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8Proof. Itfollowsfromthefunctorialityoflthatthefollowingdiagramcommutes(1.2.6) 1(1X)IdlX 1X

lXl1X

1X XlX

SincelX isanisomorphism,thefirstidentityfollows. Thesecondidentityfollowssimilarlyfromthefunctorialityofr. Proposition 1.2.4. Theunitobject inamonoidalcategory isuniqueup toaunique isomorphism.Proof. Let (1, ),(1, ) be two unit objects. Let (r,l), (r, l) be thecorresponding unit constraints. Then we have the isomorphism :=l1 (r1 )1 :11.

Itiseasytoshowusingcommutativityoftheabovetrianglediagramsthatmapsto. Itremainstoshowthat istheonlyisomorphismwiththisproperty. Todoso,itsufficestoshowthatifb:1 1isanisomorphismsuchthatthediagram(1.2.7) 11 bb 11

1 1

b

is commutative, then b = Id. To see this, it suffices to note that foranymorphismc:1 1thediagram(1.2.8) 11 cId 11

1 1c

iscommutative(as=r1),sobb=bIdandhenceb=Id. Exercise 1.2.5. Verifytheassertion intheproofofProposition1.2.4thatmapsto.Hint. UsePropositions1.2.1and1.2.2.

Theresultsofthissubsectionshowthatamonoidalcategorycanbealternativelydefinedasfollows:Definition 1.2.6. A monoidal category is a sextuple (C,,a,1,l,r)satisfyingthepentagonaxiom(1.1.2)andthetriangleaxiom(1.2.1).


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9ThisdefinitionisperhapsmoretraditionalthanDefinition1.1.1,but

Definition1.1.1 issimpler. Besides,Proposition1.2.4 impliesthat fora triple (C,, a) satisfying a pentagon axiom (which should perhapsbe called a semigroup category, as it categorifies the notion of asemigroup),beingamonoidalcategoryisapropertyandnotastructure(similarlytohowitisforsemigroupsandmonoids).

Furthermore,onecanshowthatthecommutativityofthetrianglesimpliesthatinamonoidalcategoryonecansafelyidentify1X andX1withX usingtheunitisomorphisms,andassumethattheunitisomorphismaretheidentities(whichwewillusuallydofromnowon).3

In a sense, all this means that in constructions with monoidal categories, unit objects and isomorphisms always go for the ride, andone need not worry about them especially seriously. For this reason,belowwewilltypicallytake lesscaredealingwiththemthanwehavedoneinthissubsection.Proposition 1.2.7. ([SR, 1.3.3.1]) The monoidEnd(1) of endomorphismsof theunitobjectofamonoidalcategoryiscommutative.Proof. Theunit isomorphism :11 1 induces the isomorphism:End(1 1) End(1). Itiseasytoseethat(a 1)=(1 a) =aforanyaEnd(1). Therefore,(1.2.9) ab=((a1)(1b))=((1b)(a1))=ba,foranya,bEnd(1). 1.3. First examples of monoidal categories. Monoidalcategoriesare ubiquitous. You will see one whichever way you look. Here aresomeexamples.Example 1.3.1. The category Sets of sets is a monoidal category,wherethetensorproductistheCartesianproductandtheunitobjectisaoneelementset;thestructuremorphismsa,,l,rareobvious. Thesame holds for the subcategory of finite sets, which will be denotedby Sets 4. This example can be widelygeneralized: one can take thecategoryofsetswithsomestructure,suchasgroups,topologicalspaces,etc.Example1.3.2. Anyadditivecategoryismonoidal,withbeingthedirectsumfunctor,and1beingthezeroobject.

Theremainingexampleswillbeespeciallyimportantbelow.3WewillreturntothisissuelaterwhenwediscussMacLanescoherencetheorem.4Hereandbelow,theabsenceofafinitenessconditioncondition is indicatedby

theboldfacefont,whileitspresence is indicatedbytheRomanfont.


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10Example 1.3.3. Let k be any field. The category kVec of allkvectorspacesisamonoidalcategory,where=k,1=k,andthemorphisms a,,l,r are the obvious ones. The same is true about thecategoryoffinitedimensionalvectorspacesoverk,denotedbykVec.Wewilloftendropk fromthenotationwhennoconfusionispossible.

Moregenerally, ifR isacommutativeunitalring,thenreplacingkby R we can define monoidal categories Rmod of R-modules andRmodofR-modulesoffinitetype.Example 1.3.4. Let G be a group. The category Repk(G) of allrepresentationsofG overk isa monoidal category, with beingthetensorproductofrepresentations: ifforarepresentationV onedenotesbyV thecorrespondingmapG GL(V),then

VW(g):=V(g)W(g).Theunitobject inthiscategory isthetrivialrepresentation1=k. AsimilarstatementholdsforthecategoryRepk(G)offinitedimensionalrepresentations of G. Again, we will drop the subscript k when noconfusionispossible.Example 1.3.5. LetGbeanaffine(pro)algebraicgroupoverk. Thecategories Rep(G) of all algebraic representations of G over k is amonoidalcategory(similarlytoExample1.3.4).

Similarly,ifgisaLiealgebraoverk,thenthecategoryofitsrepresentationsRep(g)andthecategoryofitsfinitedimensionalrepresentationsRep(g)aremonoidalcategories: thetensorproduct isdefinedby

VW(a) =V(a)IdW +IdV W(a)(whereY :g gl(Y)isthehomomorphismassociatedtoarepresentationY ofg),and1isthe1-dimensionalrepresentationwiththezeroactionofg.Example 1.3.6. Let G be a monoid (which we will usually take tobe a group), and let A be an abelian group (with operation writtenmultiplicatively). Let CG = CG(A) be the category whose objects gare labeled by elementsof G(so there isonly one object in each isomorphism class),Hom(g1, g2) = if g1 = g2, andHom(g, g) = A,with the functor defined by g h = gh, and the tensor tensorproductofmorphisms definedby ab=ab. ThenCG isamonoidalcategorywiththeassociativity isomorphismbeingthe identity,and1beingtheunitelementofG. Thisshowsthat inamonoidalcategory,XY neednotbeisomorphictoY X (indeed, itsufficestotakeanon-commutativemonoidG).


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11This example has a linear version. Namely, let k be a field, and

kVecG denote the category of G-graded vector spaces over k, i.e.vectorspacesV withadecompositionV =gGVg. Morphismsinthiscategory are linear operators which preserve the grading. Define thetensorproductonthiscategorybytheformula

(V W)g =x,yG:xy=gVx Wy,and the unit object 1 by 11 = k and 1g = 0 for g = 1. Then,defininga, inanobviousway,weequipkVecG withthestructureof a monoidal category. Similarly one defines the monoidal categorykVecG offinitedimensionalG-gradedk-vectorspaces.

InthecategorykVecG,wehavepairwisenon-isomorphicobjectsg, g G, defined by the formula (g)x = k if x = g and (g)x =0 otherwise. For these objects, we have g = gh. Thus theh categoryCG(k)isa(non-full)monoidalsubcategoryofkVecG. ThissubcategorycanbeviewedasabasisofVecG (andVecG asthelinearspan of CG), as any object of VecG is isomorphic to a direct sum ofobjectsg withnonnegativeintegermultiplicities.

Whennoconfusionispossible,wewilldenotethecategorieskVecG,kVecG simplybyVecG,VecG.Example1.3.7.ThisisreallyageneralizationofExample1.3.6,whichshows that the associativity isomorphism is not always the obviousone.

Let G be a group, A an abelian group, and be a 3-cocycle of Gwithvalues inA. Thismeansthat :GGG A isa functionsatisfyingtheequation(1.3.1)

(g1g2, g3, g4)(g1, g2, g3g4) =(g1, g2, g3)(g1, g2g3, g4)(g2, g3, g4),forallg1, g2, g3, g4 G.

Let us define the monoidal category C = C(A) as follows. As aG Gcategory,itisthesameasthecategoryCG definedabove. Thebifunctorandtheunitobject(1, ) inthiscategory isalsothesameasthoseinCG. Theonlydifferenceisinthenewassociativityisomorphisma,whichisnottheobviousone(i.e.,theidentity)likeinCG,butratherisdefinedbytheformula(1.3.2) ag,h,m =(g,h,m) : (g h)m g (h m),whereg,h,mG.

ThefactthatC withthesestructuresisindeedamonoidalcategoryGfollows fromthepropertiesof. Namely,thepentagonaxiom(1.1.2)followsfromequation(1.3.1),andtheunitaxiomisobvious.


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12Similarly,forafieldk,onecandefinethecategory(k)Vec,whichG

differsfromVecGjustbytheassociativity isomorphism. This isdonebyextendingtheassociativity isomorphismofC byadditivitytoar-Gbitrary direct sums of objects g. This category contains a monoidalsubcategory Vec of finite dimensional G-graded vector spaces withGassociativitydefinedby.Remark1.3.8. Itisstraightforwardtoverifythattheunitmorphismsl,r inVec aregivenon1-dimensionalspacesbytheformulasG

lg =(1,1, g)1Idg, rg =(g,1,1)Idg,andthetriangleaxiomsaysthat(g,1, h) =(g,1,1)(1,1, h). Thus,wehavelX =rX =Idifandonlyif(1.3.3)

(g,

1,

1)

=

(1,

1, g),

foranygGor,equivalently,(1.3.4) (g,1, h) = 1, g, hG.Acocyclesatisfyingthisconditionissaidtobenormalized.Example 1.3.9. Let C be a category. Then the categoryEnd(C) ofall functors from C to itself is a monoidal category, where is givenbycompositionoffunctors. Theassociativityisomorphisminthiscategory isthe identity. Theunitobject isthe identity functor,andthestructuremorphismsareobvious. IfC isanabeliancategory,thesameis true about the categories of additive, left exact, right exact, andexactendofunctorsofC.Example 1.3.10. Let A be an associative ring with unit. Then thecategoryAbimodofbimodulesoverAisamonoidalcategory,withtensor product = A, over A. The unit object in this category istheringAitself(regardedasanA-bimodule).

IfA iscommutative, thiscategoryhasa fullmonoidalsubcategoryAmod, consisting of A-modules, regarded as bimodules in whichthe left and right actions of A coincide. More generally, if X is ascheme, one can define the monoidal category QCoh(X) of quasicoherent sheaves on X; if X is affine and A = OX, then QCoh(X) =Amod.Similarly, if A is a finite dimensional algebra, we can define themonoidalcategoryAbimodoffinitedimensionalA-bimodules. OthersimilarexampleswhichoftenariseingeometryarethecategoryCoh(X)of coherent sheaves on a scheme X, its subcategory VB(X) of vectorbundles (i.e., locally free coherent sheaves) on X, and the categoryLoc(X)oflocallyconstantsheavesoffinitedimensionalk-vectorspaces


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13(alsocalledlocalsystems)onanytopologicalspaceX. Allofthesearemonoidalcategoriesinanaturalway.Example 1.3.11. The category of tangles.

Let Sm,n be the disjoint union of m circles R/Z and n intervals[0,1]. Atangleisapiecewisesmoothembeddingf :Sm,n R2[0,1]suchthattheboundarymapstotheboundaryandthe interiortotheinterior. Wewillabusetheterminologybyalsousingthetermtanglefortheimageoff.

Let x,y,z be the Cartesian coordinates onR2 [0,1]. Any tanglehasinputs(pointsoftheimageoff withz=0)andoutputs(pointsoftheimageoff withz=1). Foranyintegersp,q0,letTp,q bethesetofalltangleswhichhavepinputsandqoutputs,allhavingavanishingy-coordinate.

Let

Tp,q bethesetof isotopyclassesofelementsofTp,q;thus, during an isotopy, the inputs and outputs are allowed to move

(preservingtheconditiony=0),butcannotmeeteachother. WecandefineacanonicalcompositionmapTp,q Tq,r Tp,r, inducedbytheconcatenationoftangles. Namely,ifsTp,q andtTq,r ,wepickrepresentativessTp,q,tTq,r suchthattheinputsoftcoincidewiththeoutputsofs,concatenatethem,performanappropriatereparametrization,andrescalez z/2. Theobtainedtanglerepresentsthedesiredcompositionts.

We will now define a monoidal category T called the category oftangles (see [K, T, BaKi] for more details). The objects of this category are nonnegative integers, and the morphisms are defined byHomT(p,q) =Tp,q,withcompositionasabove. Theidentitymorphismsare the elements idp Tp,p represented byp vertical intervals and nocircles(inparticular, ifp=0,the identitymorphism idp istheemptytangle).

Now let us define the monoidal structure on the category T. Thetensorproductofobjectsisdefinedbymn=m+n. However,wealsoneedtodefinethetensorproductofmorphisms. Thistensorproductisinducedbyunionoftangles. Namely,ift1 Tp1,q1 andt2 Tp2,q2,wepickrepresentativest

1 T

p1,q1, t

2 T

p2,q2 insuchawaythatanypoint

oft1 istothe leftofanypointoft2 (i.e.,hasasmallerx-coordinate).Thent1 t2 isrepresentedbythetanglet1 t2.Weleaveittothereadertocheckthefollowing:1. Theproductt1 t2 iswelldefined,and itsdefinitionmakesa

bifunctor.2. Thereisanobviousassociativityisomorphismfor,whichturns

T intoamonoidalcategory(withunitobjectbeingtheemptytangle).


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141.4. Monoidal functors, equivalence of monoidal categories.As we have explained, monoidal categories are a categorification ofmonoids. Nowwepasstocategorificationofmorphismsbetweenmonoids,namelymonoidalfunctors.Definition1.4.1.Let(C,,1,a,)and(C,,1, a, )betwomonoidal

categories. A monoidal functor from is a pair (F, J) wheretoC C

F : C C is a functor, and J = {JX,Y : F(X) F(Y) F(XY)|X,Y C} isanatural isomorphism,suchthatF(1) is isomorphicto1. andthediagram(1.4.1)

aF(X),F(Y),F(Z)

(F(X) F(Y)) F(Z) F(X) (F(Y) F(Z))JX,YIdF(Z) IdF(X)JY,Z

F(XY) F(Z)JXY,Z

F(X) F(Y Z)

JX,YZF(a

X,Y,Z)F((XY)Z) F(X(Y Z))

iscommutativeforallX,Y,Z C (themonoidalstructureaxiom).AmonoidalfunctorF issaidtobeanequivalenceofmonoidalcate

goriesifitisanequivalenceofordinarycategories.Remark1.4.2. Itisimportanttostressthat,asseenfromthisdefinition,amonoidal functor isnotjusta functorbetweenmonoidalcategories,butafunctorwithanadditionalstructure(theisomorphismJ)satisfying a certain equation (the monoidal structure axiom). As wewill see later, this equation may have more than one solution, so thesamefunctorcanbeequippedwithdifferentmonoidalstructures.

Itturnsoutthat ifF isamonoidal functor,thenthere isacanonical isomorphism :1 F(1). This isomorphism is defined by thecommutativediagram

l

1

F(1) F(1) F(1)

(1.4.2) IdF(X) F(l1)J1,1

F(11)F(1) F(1)wherel,r,l, r aretheunitisomorphismsforC,C definedinSubsection1.2.


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15Proposition 1.4.3. For any monoidalfunctor (F,J) : C C, thediagrams

l

1 F(X) F(X) F(X)

(1.4.3) IdF(X)F(1) F(X)

F(lX)

J1,X F(1X)

andr

F(X) 1 F(X)

(1.4.4) IdF(X)

F(X)F(r

X)JX,1

F(X1)F(X) F(1)arecommutativeforallX C.Exercise 1.4.4. ProveProposition1.4.3.

Proposition1.4.3impliesthatamonoidalfunctorcanbeequivalentlydefinedasfollows.Definition1.4.5.AmonoidalfunctorC C isatriple(F,J,)whichsatisfiesthemonoidalstructureaxiomandProposition1.4.3.

Thisisamoretraditionaldefinitionofamonoidalfunctor.Remark 1.4.6. Itcanbeseen fromtheabovethatforanymonoidalfunctor (F,J) one can safely identify 1 with F(1) using the isomorphism,andassumethatF(1) =1 and=Id(similarlytohowwehaveidentified1X andX1withX andassumedthatlX =rX =IdX). We will usually do so from now on. Proposition 1.4.3 impliesthatwiththeseconventions,onehas(1.4.5) J1,X =JX,1 =IdX.Remark1.4.7. Itisclearthatthecompositionofmonoidalfunctorsisamonoidalfunctor. Also,theidentityfunctorhasanaturalstructureofamonoidalfunctor.1.5. Morphismsofmonoidalfunctors. Monoidalfunctorsbetweentwomonoidalcategoriesthemselvesformacategory. Namely,onehasthefollowingnotionofamorphism(ornaturaltransformation)betweentwomonoidalfunctors.


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16Definition1.5.1.Let(C,,1,a,)and(C,,1, a, )betwomonoidalcategories, and (F1, J1), (F2, J2) two monoidal functors from C toC. A morphism (or a natural transformation) of monoidal functors : (F1, J1) (F2, J2) isanaturaltransformation :F1 F2 such that1 isanisomorphism,andthediagram

J1X,Y

F1(X) F1(Y) F1(XY)(1.5.1) XY XYJ2

X,YF2(X) F2(Y) F2(XY)

iscommutativeforallX,Y C.Remark 1.5.2. It iseasytoshowthat1 1 =2,so ifonemakestheconventionthati =Id,onehas1 =Id.Remark1.5.3. ItiseasytoshowthatifF :C C isanequivalenceofmonoidalcategories,thenthereexistsamonoidalequivalenceF1 :C C such that the functors F F1 and F1 F are isomorphictothe identity functorasmonoidal functors. Thus, foranymonoidalcategory C, the monoidal auto-equivalences of C up to isomorphismformagroupwithrespecttocomposition.1.6. Examplesofmonoidalfunctors. Letusnowgivesomeexamplesofmonoidalfunctorsandnaturaltransformations.Example1.6.1. Animportantclassofexamplesofmonoidalfunctorsisforgetfulfunctors (e.g. functors of forgetting the structure, fromthe categories of groups, topological spaces, etc., to the category ofsets). Suchfunctorshaveanobviousmonoidalstructure. Anexampleimportant in these notes is the forgetful functor RepG Vec fromtherepresentationcategoryofagrouptothecategoryofvectorspaces.More generally, if H G is a subgroup, then we have a forgetful(or restriction) functor RepG RepH. Still more generally, if f :H G isagrouphomomorphism,thenwehavethepullbackfunctorf :RepG RepH. Allthesefunctorsaremonoidal.Example1.6.2. Letf :H Gbeahomomorphismofgroups. Thenany H-graded vector space is naturally G-graded (by pushforward ofgrading). Thuswehaveanaturalmonoidalfunctorf :VecH VecG.IfG isthetrivialgroup,thenf isjusttheforgetfulfunctorVecH Vec.Example 1.6.3. Let Abeak-algebrawithunit, andC =Amodbe the category of left A-modules. Then we have a functor F : A


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17bimod End(C)givenbyF(M) =MA. This functor isnaturallymonoidal. AsimilarfunctorF :Abimod End(C). canbedefinedifAisafinitedimensionalk-algebra,andC=AmodisthecategoryoffinitedimensionalleftA-modules.Proposition1.6.4.ThefunctorF :Abimod End(C)takesvaluesinthefullmonoidalsubcategoryEndre(C)ofrightexactendofunctorsofC, and defines an equivalencebetweenmonoidalcategoriesAbimodandEndre(C)Proof. The first statement is clear, since the tensor product functoris right exact. To prove the second statement, let us construct thequasi-inverse functor F1. Let G Endre(C). Define F1(G) by theformula F1(G) = G(A); this is clearly an A-bimodule, since it is aleftA-modulewithacommutingactionEndA(A) =Aop (theoppositealgebra). We leave it to the reader to check that the functor F1 isindeedquasi-inversetoF. Remark1.6.5. Asimilarstatementisvalidwithoutthefinitedimensionality assumption, if one adds the condition that the right exactfunctorsmustcommutewithinductivelimits.Example1.6.6.LetSbeamonoid,andC=VecS,andIdC theidentityfunctorofC. Itiseasytoseethatmorphisms:IdC IdC correspondtohomomorphismsofmonoids: :S k (wherek isequippedwiththemultiplicationoperation). Inparticular,(s)maybe0forsomes,sodoesnothavetobeanisomorphism.

1.7. Monoidal functors between categories CG. Let G1, G2 begroups,Aanabeliangroup,andi Z3(Gi, A), i= 1,2be3-cocycles.

iLetCi =CGi, i= 1,2(seeExample1.3.7).Anymonoidal functorF :C1 C2 defines, byrestrictiontosimple

objects,agrouphomomorphismf :G1 G2. Usingtheaxiom(1.4.1)ofamonoidal functorweseethatamonoidalstructureonF isgivenby(1.7.1) Jg,h =(g,h)Idf(gh) :F(g)F(h) F(gh),g, hG1,where:G1 G1 Aisafunctionsuchthat

1(g,h,l)(gh,l)(g,h) =(g,hl)(h,l)2(f(g), f(h), f(l)),forallg,h,lG1. Thatis,(1.7.2) f2 =12(),i.e.,1 andf2 arecohomologousinZ3(G1, A).


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18Conversely, givenagroup homomorphismf :G1 G2, a function

: G1 G1 A satisfying (1.7.2) gives rise to a monoidal functorF : C1 C2 defined by F(g) = f(g) with the monoidal structuregivenby formula(1.7.1). This functor isanequivalence ifandonly iff isanisomorphism.

To summarize, monoidal functors C1 C2 correspond to pairsG1 G2(f,), where f : G1 G2 is a group homomorphism such that 1andf2 arecohomologous,andisafunctionsatisfying(1.7.2)(suchfunctions are in a (non-canonical) bijection with A-valued 2-cocyclesonG1). LetFf, denotethecorrespondingfunctor.

LetusdeterminenaturalmonoidaltransformationsbetweenFf, andFf,. Clearly, such a transformation exists if and only if f = f, isalwaysanisomorphism,andisdeterminedbyacollectionofmorphismsg :f(g) f(g) (i.e.,g A),satisfyingtheequation(1.7.3) (g,h)(g h) =gh(g,h)forallg,hG1,i.e.,(1.7.4) =1().Conversely, every function :G1 A satisfying (1.7.4) gives rise toa morphism of monoidal functors : Ff, Ff, defined as above.Thus, functors Ff, and Ff, are isomorphic as monoidal functors ifandonlyiff =f andiscohomologousto.

Thus,wehaveobtainedthefollowingproposition.Proposition 1.7.1. (i) The monoidal isomorphisms Ff, Ff, of

1 2monoidalfunctors Ff,i : CG1 CG2 form a torsor over the groupH1(G1, k

)=Hom(G1, k)ofcharactersofG1;(ii)Givenf, thesetofparametrizingisomorphismclassesofFf,

isatorsoroverH2(G1, k);(iii)Thestructuresofamonoidalcategoryon(CG,)areparametrized

by H3(G,k)/Out(G), where Out(G) is the group of outer automorphismsofG. 5Remark 1.7.2. The same results, including Proposition 1.7.1, arevalidifwereplacethecategoriesC bytheirlinearspansVec,andG Grequire that the monoidal functors we consider are additive. To seethis, it is enough to note that by definition, for any morphism ofmonoidal functors,1 =0,soequation(1.7.3)(withh=g1) implies

5RecallthatthegroupInn(G)ofinnerautomorphismsofagroupGactstriviallyonH(G,A)(foranycoefficientgroupA),andthustheactionofthegroupAut(G)onH(G,A)factorsthroughOut(G).


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19that all g must be nonzero. Thus, if a morphism : Ff, Ff,exists,thenitisanisomorphism,andwemusthavef =f.Remark 1.7.3. The above discussion implies that in the definitionof the categories C and Vec it may be assumed without loss ofG G,generality that the cocycle is normalized, i.e., (g,1, h) = 1, andthus lg =rg =Id(which isconvenient incomputations). Indeed,weclaim that any 3-cocycle is cohomologous to a normalized one. Toseethis, it isenoughtoalter bydividing itby2,where isany2-cochainsuchthat(g,1)=(g,1,1),and(1, h) =(1,1, h)1.Example1.7.4. LetG=Z/nZwheren >1isaninteger,andk=C.ConsiderthecohomologyofZ/nZ.

SinceHi(Z/nZ,C)=0foralli >0,writingthelongexactsequenceofcohomologyfortheshortexactsequenceofcoefficientgroups

0ZCC =C/Z0,weobtainanaturalisomorphismHi(Z/nZ,C)=Hi+1(Z/nZ,Z).

Itiswellknown[Br]thatthegradedringH(Z/nZ,Z)is(Z/nZ)[x]wherexisageneratorindegree2. Moreover,asamoduleoverAut(Z/nZ) =(Z/nZ),wehaveH2(Z/nZ,Z) There=H1(Z/nZ,C) = (Z/nZ).fore, using the graded ring structure, we find that H2m(Z/nZ,Z) =H2m1(Z/nZ,C)=((Z/nZ))m asanAut(Z/nZ)-module. Inparticular,H3(Z/nZ,C)=((Z/nZ))2.

This consideration shows that if n = 2 then the categorificationproblem

has

2solutions

(the

cases

of

trivial

and

non-trivial

cocycle),

whileifnisaprimegreaterthan2thenthereare3solutions: thetrivialcocycle, and two non-trivial cocycles corresponding (non-canonically)toquadraticresiduesandnon-residues modn.

Letusgiveanexplicitformulaforthe3-cocyclesonZ/nZ. Modulocoboundaries,thesecocyclesaregivenby

si(j+k(j+k))(1.7.5) (i,j,k) = n ,where isaprimitiventhrootofunity,sZ/nZ,andforan integermwedenotebym theremainderofdivisionofmbyn.Exercise1.7.5.ShowthatwhensrunsoverZ/nZthisformuladefinescocyclesrepresentingallcohomologyclassesinH3(Z/nZ,C).1.8. MacLanes strictness theorem. As we have seen above, it ismuch simplertowork withmonoidalcategories in whichthe associativityandunitconstrainsaretheidentitymaps.Definition 1.8.1. A monoidal category C is strict if for all objectsX,Y,Z in C one has equalities (X Y)Z = X (Y Z) and


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20X1=X=1X,andtheassociativityandunitconstraintsaretheidentitymaps.Example1.8.2. ThecategoryEnd(C)endofunctorsofacategoryC isstrict.Example1.8.3. LetSetsbethecategorywhoseobjectsarenonnegative integers,andHom(m,n) isthesetofmaps from{0,...,m1}to{0,...,n1}. Definethetensorproductfunctoronobjectsbym n=mn,andforf1 :m1 n1, f2 :m2 n2,definef1f2 :m1m2 n1n2by(f1f2)(m2x+y) =n2f1(x) +f2(y),0xm11, 0ym21.ThenSets isastrictmonoidalcategory. Moreover,wehaveanaturalinclusionSets Sets,whichisobviouslyamonoidalequivalence.Example1.8.4.Thisisreallyalinearversionofthepreviousexample.LetkVecbethecategorywhoseobjectsarenonnegativeintegers,andHom(m,n)isthesetofmatriceswithmcolumnsandnrowsoversomefieldk(andthecompositionofmorphismsistheproductofmatrices).Definethetensorproductfunctoronobjectsbymn=mn,andforf1 :m1 n1, f2 :m2 n2, define f1 f2 :m1m2 n1n2 tobethe Kronecker product of f1 and f2. Then kVec is a strict monoidalcategory. Moreover, we have anatural inclusion kVec kVec,whichisobviouslyamonoidalequivalence.

Similarly,foranygroupGonecandefineastrictmonoidalcategorykVecG, whose objects areZ+-valued functions on G with finitelymanynonzerovalues,andwhichismonoidallyequivalenttokVecG.Weleavethisdefinitiontothereader.

Ontheotherhand,someofthemostimportantmonoidalcategories,suchasSets,Vec,VecG,Sets,Vec,VecG,shouldberegardedasnon-strict(at least ifonedefines them in the usualway). It isevenmoreindisputablethatthecategoriesVec,Vec forcohomologicallynon-G Gtrivialarenotstrict.

However,thefollowingremarkabletheoremofMacLaneimpliesthatinpractice,onemayalwaysassumethatamonoidalcategoryisstrict.Theorem1.8.5. Anymonoidalcategoryismonoidallyequivalenttoastrictmonoidalcategory.Proof. Theproofpresentedbelowwasgiven in[JS]. WewillestablishanequivalencebetweenCandthemonoidalcategoryofrightC-moduleendofunctorsofC,whichwewilldiscussinmoredetaillater. Thenon-categoricalalgebraiccounterpartofthisresultisofcoursethefactthat


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21everymonoidM is isomorphictothemonoidconsistingofmapsfromM toitselfcommutingwiththerightmultiplication.

ForamonoidalcategoryC, letC bethemonoidalcategorydefinedas follows. The objects of C are pairs (F,c) where F : C C is afunctorand

cX,Y :F(X)Y F(XY)is a functorial isomorphism, such that the following diagram is commutativeforallobjectsX,Y,Z inC:(1.8.1)

(

F(X)Y)Z

cX,YIdZ aF(X),Y,Z

F(X

Y

)

Z F

(X)

(Y

Z)

cXY,Z cX,YZ

F((XY)Z) F(X(Y Z)).F(aX,Y,Z)

Amorphism : (F1, c1)(F2, c2) inC isa naturaltransformation :F1 F2 such that the following square commutes for all objectsX,Y inC:

1c

(1.8.2) F1(X)Y X,Y F1(XY)X

IdY XY

F2(X)Y F2(XY)2cX,Y

Compositionofmorphismsistheverticalcompositionofnaturaltransformations. The tensor product of objects is given by (F1, c1)(F2, c2) = (F1F2, c)wherecisgivenbyacomposition

1 2cF2(X),Y F1(cX,Y)(1.8.3) F1F2(X)Y F1(F2(X)Y)F1F2(XY)

forallX,Y C,andthetensorproductofmorphismsisthehorizontalcomposition of natural transformations. Thus C is a strict monoidalcategory(theunitobjectistheidentityfunctor).ConsidernowthefunctorofleftmultiplicationL:C C givenby

L(X) = (X , aX,, ), L(f) =f .Note that the diagram (1.8.1) for L(X) is nothing but the pentagondiagram(1.1.2).

WeclaimthatthisfunctorLisamonoidalequivalence.


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22Firstofall,Lessentiallysurjective: it iseasytocheckthat forany

(F,c) C, (F,c)isisomorphictoL(F(1)).Let us now show thatLis fully faithful. Let L(X) L(Y) be a:

(1.8.4) X X 1 Y 1 Y. We claim that for allZin one has f Id (so that L(f) andC = =Z ZLis full). Indeed, this follows from the commutativity of the diagram

1 Id Id lr aZ X,1,Z X ZX Z (X 1) Z X Z

morphisminC. Definef :X Y tobethecompositer1

X 1 rY

(1.8.5)X

X(1Z)1IdZfIdZ 1Z Z

Y Z (Y 1)Z aY,1,Z IdYlZ Y Z,r1IdZ Y (1Z) Ywheretherowsaretheidentitymorphismsbythetriangleaxiom(1.2.1),theleftsquarecommutesbythedefinitionoff,therightsquarecommutesbynaturalityof,andthecentralsquarecommutessince isamorphisminC.

Next,ifL(f) =L(g)forsomemorphismsf,ginCthen,inparticularfId1 =gId1 sothatf =g. ThusLisfaithful.

Finally,wedefineamonoidalfunctorstructureJX,Y :L(X) L(Y) L(XY)onLby

JX,Y =a1

:X(Y ),((IdX aY,)aX,Y

,)

X,Y, ,

).((XY) , aXY,,

Thediagram(1.8.2)forthelatternaturalisomorphismisjustthepentagon diagram in C. For the functor L the hexagon diagram (1.4.1)in the definition of a monoidal functor also reduces to the pentagondiagraminC. Thetheoremisproved. Remark 1.8.6. The nontrivial nature of MacLanes strictness theoremisdemonstratedbythefollowinginstructiveexample,whichshowsthateventhoughamonoidalcategoryisalwaysequivalenttoastrictcategory, it need not be isomorphic to one. (By definition, an isomorphismofmonoidalcategoriesisamonoidalequivalencewhichisanisomorphismofcategories).

Namely,letCbethecategoryCG. Ifiscohomologicallynontrivial,this category is clearly not isomorphic to a strict one. However, byMaclanesstrictnesstheorem,itisequivalenttoastrictcategoryC.

In fact, in this example a strict category C monoidally equivalenttoC canbeconstructedquiteexplicitly,as follows. LetG beanother


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23group with a surjective homomorphism f : G

G such that the 3

cocycle f is cohomologically trivial. Such G

always exists, e.g., afreegroup(recallthatthecohomologyofafreegroupindegreeshigherthan 1 is trivial, see [Br]). Let C be the category whose objects gare labeled by elements of G,Hom(g, h) = A if g,h have the sameimage in G, and Hom(g, h) = otherwise. This category has anobvious tensor product, and a monoidal structure defined by the 3cocyclef. WehaveanobviousmonoidalfunctorF :C C definedbythehomomorphismG G,and it isanequivalence,eventhoughnotanisomorphism. However,sincethecocyclefiscohomologicallytrivial, the category C is isomorphic to the same category with thetrivialassociativityisomorphism,whichisstrict.Remark1.8.7. 6 Acategoryiscalledskeletalifithasonlyoneobjectineachisomorphismclass. Theaxiomofchoiceimpliesthatanycategory isequivalenttoaskeletalone. Also,byMacLanestheorem,anymonoidal category is monoidally equivalent to a strict one. However,Remark1.8.6showsthatamonoidalcategoryneednotbemonoidallyequivalenttoacategorywhich isskeletalandstrictatthesametime.Indeed, as we have seen, to make a monoidal category strict, it maybenecessarytoaddnewobjectsto it(whichare isomorphic, butnotequal to already existing ones). In fact, the desire to avoid addingsuch objects is the reason why we sometimes use nontrivial associativity isomorphisms, even though MacLanes strictness theorem tellsus we dont have to. This also makes precise the sense in which thecategories Sets, Vec, VecG, are more strict than the category VecGfor cohomologically nontrivial . Namely, the first three categoriesaremonoidallyequivalenttostrictskeletalcategoriesSets,Vec,VecG,whilethecategoryVec isnotmonoidallyequivalenttoastrictskeletalGcategory.Exercise 1.8.8. Show that any monoidal category C is monoidallyequivalenttoaskeletalmonoidalcategoryC. Moreover,CcanbechoseninsuchawaythatlX, rX =IdX forallobjectsX C.

Hint. Without loss of generality one can assume that 1X =X

1

=

X

and

lX, rX = IdX for all objects X C. Now in everyisomorphismclassiofobjectsofCfixarepresentativeXi,sothatX1 =

1,andforanytwoclassesi,jfixanisomorphismij :XiXj Xi j,sothati1 =1i =IdXi. LetC bethefullsubcategoryofC consistingoftheobjectsXi,withtensorproductdefinedbyXiXj =Xi j,and

6Thisremarkisborrowedfromthepaper [Kup2].


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24withallthestructuretransportedusingtheisomorphismsij. ThenCistherequiredskeletalcategory,monoidallyequivalenttoC.


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251.9. The MacLane coherence theorem. In a monoidal category,onecanformn-foldtensorproductsofanyorderedsequenceofobjectsX1,...,Xn. Namely,suchaproductcanbeattachedtoanyparenthesizingoftheexpressionX1...Xn,andsuchproductsare,ingeneral,distinctobjectsofC.

However,forn=3,theassociativity isomorphismgivesacanonicalidentificationofthetwopossibleparenthesizings,(X1X2)X3 andX1(X2X3). AneasycombinatorialargumentthenshowsthatonecanidentifyanytwoparenthesizedproductsofX1,...,Xn,n3,usingachainofassociativityisomorphisms.

Wewould liketosaythat forthisreasonwecancompletely ignoreparentheses incomputations inanymonoidalcategory, identifyingallpossibleparenthesizedproductswitheachother. Butthisrunsintothefollowingproblem: forn4theremaybetwoormoredifferentchainsofassociativityisomorphismsconnectingtwodifferentparenthesizings,andaprioriitisnotclearthattheyprovidethesameidentification.

Luckily, for n = 4, this is settled by the pentagon axiom, whichstatesexactlythatthetwopossible identificationsarethesame. Butwhataboutn >4?

ThisproblemissolvedbythefollowingtheoremofMacLane,whichisthefirstimportantresultinthetheoryofmonoidalcategories.Theorem1.9.1.(MacLanesCoherenceTheorem)[ML]LetX1, . . . , X n C. Let P1, P2 be any two parenthesized products of X1,...,Xn (in thisorder) with arbitrary insertions of unit objects 1. Let f,g : P1 P2be two isomorphisms, obtained by composing associativity and unitisomorphisms and their inverses possibly tensored with identity morphisms. Thenf =g.Proof. WederivethistheoremasacorollaryoftheMacLanesstrictnessTheorem 1.8.5. LetL :C C beamonoidalequivalencebetweenCandastrictmonoidalcategoryC. ConsideradiagraminCrepresentingf andg andapplyLto it. Overeacharrowoftheresultingdiagramrepresenting an associativity isomorphism, let us build a rectangle asin(1.4.1),anddosimilarlyfortheunitmorphisms. Thiswayweobtainaprism

one

of

whose

faces

consists

of

identity

maps

(associativity

and

unitisomorphismsinC)andwhosesidesarecommutative. Hence,theotherfaceiscommutativeaswell,i.e.,f =g.

Aswementioned,this impliesthatanytwoparenthesizedproductsofX1,...,Xn withinsertionsofunitobjectsareindeedcanonicallyisomorphic,andthusonecansafely identifyallofthemwitheachother


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26andignorebracketingsincalculationsinamonoidalcategory. Wewilldosofromnowon,unlessconfusionispossible.1.10. Rigid monoidal categories. Let (C,,1,a,) be a monoidalcategory, and let X be an object of C. In what follows, we suppresstheunitmorphismsl,r.Definition 1.10.1. Aright dualofanobjectX inC isanobjectXin C equipped with morphisms evX : X X 1 and coevX : 1 XX,calledtheevaluationandcoevaluationmorphisms,suchthatthecompositions(1.10.1)

X coevXIdX aX,X,X X(X X) IdXevX(XX)X X,(1.10.2)

a1X IdXcoevX X,X,X evXIdX XX (XX)(X X)X aretheidentitymorphisms.Definition1.10.2.AleftdualofanobjectX inCisanobjectX inCequippedwithmorphismsevX :XX1andcoevX :1XXsuchthatthecompositions(1.10.3)

IdXcoev a1 XIdXX,X,X evXX

X

(X

X)

(X

X)

X

X,

(1.10.4)

XIdX IdXevcoev aX,X,X XX(XX)XX(XX)Xaretheidentitymorphisms.Remark 1.10.3. It isobviousthat ifX isarightdualofanobjectX thenX isa leftdualofX withevX =evX andcoevX =coevX,andviceversa. Also, inanymonoidalcategory,1 =1=1withthestructure morphisms and 1. Also note that changing the order oftensorproductswitchesrightdualsandleftduals,sotoanystatementabout

right

duals

there

corresponds

asymmetric

statement

about

left

duals.Proposition 1.10.4. If X C has a right (respectively, left) dualobject, then itisuniqueup toaunique isomorphism.Proof. LetX1, X2 betworightdualstoX. Denotebye1, c1, e2, c2 thecorrespondingevaluationandcoevaluationmorphisms. Thenwehave


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27amorphism:X1 X2 definedasthecomposition

IdX a1,X,X e1IdX1c2 X1 2 2

X1X1 (XX2)(X1 X)X2 X2.

Similarlyonedefinesamorphism :X2 X1. Weclaimthat and aretheidentitymorphisms,soisanisomorphism. Indeedconsiderthefollowingdiagram:

X1 Idc1 X1 XX1Id

Idc2 Idc2IdX1 XX2 Idc1 X1 XX2 XX1Ide2Id X1 XX1

e1Id e1Id e1IdX2 Idc1 X2 XX1 e2Id X1.

Herewesuppresstheassociativityconstraints. Itisclearthatthethreesmall squares commute. The triangle in the upper right corner commutes by axiom (1.10.1) applied to X2. Hence, the perimeter of thediagramcommutes. Thecompositionthroughthetoprowistheidentity by (1.10.2) applied to X1. The composition through the bottomrow is andso =Id. Theproofof =Id iscompletely similar.

Moreover, it is easy to check that : X1 X2 is the only isomorphismwhichpreservestheevaluationandcoevaluationmorphisms.Thisprovesthepropositionforrightduals. Theprooffor leftdualsissimilar. Exercise1.10.5. FillinthedetailsintheproofofProposition1.10.4.

IfX,Y areobjectsinCwhichhaverightdualsX, Y andf :XYis a morphism, one defines the right dual f : Y X of f as thecomposition

a1Y IdYcoevX Y,X,X

(1.10.5) Y (XX)(Y X)X(IdYf)IdX evYIdX(Y Y)X X.

Similarly, ifX,Y areobjects inC whichhave leftdualsX,Y andf :X Y isamorphism,onedefinesthe leftdualf :Y X off asthecomposition

XIdYcoev aX,X,Y(1.10.6) Y (XX)Y X(XY)

IdX(fIdY) IdXevYX(Y Y)X.


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28Exercise 1.10.6. LetC,Dbemonoidalcategories. Suppose

(F,J) :C Dis a monoidal functor with the corresponding isomorphism : 1 F(1). Let X be an object in C with a right dual X. Prove thatF(X) is a right dual of F(X) with the evaluation and coevaluationgivenby

JX,X F(evX)evF(X) : F(X)F(X)F(X X)F(1) =1,

J1F(coevX) X,XcoevF(X) : 1=F(1)F(XX)F(X)F(X).

Stateandproveasimilarresultforleftduals.Proposition 1.10.7. LetC beamonoidalcategory.

(i) Let U,V,W be objects in C admitting right (respectively, left)duals, and let f : V W, g : U V be morphisms in C. Then (f g) =g f (respectively,(f g) =g f).

(ii)IfU,V haveright(respectively,left)dualsthentheobjectVU(respectively,V U)hasanaturalstructureofaright(respectively,left)dual toUV.Exercise 1.10.8. ProveProposition1.10.7.Proposition1.10.9. (i)IfanobjectV hasarightdualV thentherearenaturaladjunction isomorphisms(1.10.7) Hom(UV,W) Hom(U,WV),(1.10.8) Hom(V U,W) Hom(U,V W).

Thus,thefunctor V isrightadjointto V andV isrightadjoint toV .

(ii)IfanobjectV hasaleftdualV thentherearenaturaladjunctionisomorphisms

(1.10.9) Hom(UV,W) Hom(U,WV),(1.10.10) Hom(V U,W) Hom(U,V W).

Thus,thefunctor V isrightadjointto V andV isrightadjoint toV ).Proof. Theisomorphismin(1.10.7)isgivenby

f (fIdV)(IdU coevV)andhastheinverse

g (IdW evV) (gIdV).


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29The other isomorphisms are similar, and are left to the reader as anexercise. 7 Remark1.10.10.Proposition1.10.9providesanotherproofofProposition1.10.4. Namely,settingU =1andV =X in(1.10.8),weobtainanatural isomorphismHom(X, W) Hom(1, XW) foranyright=dual X of X. Hence, if Y1, Y2 are two such duals then there is anatural isomorphismHom(Y1, W) Hom(Y2, W), whence there is a=canonicalisomorphismY1 =Y2 byYonedasLemma. Theproofforleftdualsissimilar.Definition 1.10.11. A monoidal category C is called rigid if everyobjectX C hasarightdualobjectandaleftdualobject.Example 1.10.12. The category Vec of finite dimensional k-vectorspaces is rigid: the right and left dual to a finite dimensional vectorspaceV areitsdualspaceV,withtheevaluationmapevV :VV kbeingthecontraction,andthecoevaluationmapcoevV :kV Vbeing the usual embedding. On the other hand, the category Vec ofall k-vector spaces is not rigid, since for infinite dimensional spacesthere is no coevaluation maps (indeed, suppose that c : k V Yisacoevaluationmap,andconsiderthesubspaceV ofV spannedbythe first component of c(1); this subspace finite dimensional, and yetthe composition V V Y V V, which is supposed tobe theidentitymap,landsinV - acontradiction).Example 1.10.13. The category Rep(G) of finite dimensionalk-representations of a group G is rigid: for a finite dimensional representation V, the (left or right) dual representation V is the usualdualspace(withtheevaluationandcoevaluationmapsasinExample1.10.12), and with the G-action given by V(g) = (V(g)1). Similarly,thecategoryRep(g)offinitedimensionalrepresentationsofaLiealgebragisrigid,withV(a) =V(a).Example1.10.14.ThecategoryVecG isrigidifandonlyifthemonoidG is a group; namely, g = g = g1 (with the obvious structuremaps). Moregenerally,foranygroupGand3-cocycleZ3(G,k),the category Vec

G is rigid. Namely, assume for simplicity that the

cocycle is normalized (as we know, we can do so without loss ofgenerality). Then we can define duality as above, and normalize thecoevaluationmorphismsofg tobetheidentities. Theevaluationmorphismswillthenbedefinedbytheformulaevg =(g,g1, g).

7AconvenientwaytodocomputationsinthisandpreviousPropositionsisusingthegraphicalcalculus(see [K,ChapterXIV]).


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30It follows from Proposition 1.10.4 that in a monoidal category C

withright(respectively, left)duals,onecandefinethe(contravariant)right (respectively, left) dualityfunctor C C by X X, f f(respectively,X X,f f)foreveryobjectXandmorphismf inC. By Proposition 1.10.7(ii), these functors are anti-monoidal, in thesensethattheydefinemonoidalfunctorsC Cop;hencethefunctorsX X, X X aremonoidal. Also, it follows from Proposition 1.10.9thatthefunctorsofrightandleftduality,whentheyaredefined,arefullyfaithful(itsufficestouse(i)forU =X, V =Y,W =1).

Moreover,itfollowsfromRemark1.10.3thatinarigidmonoidalcategory,thefunctorsofrightandleftdualityaremutuallyquasi-inversemonoidalequivalencesofcategoriesC Cop (so forrigidcategories,thetwonotionsofoppositecategoryarethesameuptoequivalence).This implies that the functors X X, X X are mutually quasi-inversemonoidalautoequivalences. WewillseelaterinExample1.27.2thattheseautoequivalencesmaybenontrivial; inparticular, itispossiblethatobjectsV andV arenotisomorphic.Exercise 1.10.15. Show that if C,D are rigid monoidal categories,F1, F2 :C Daremonoidalfunctors,and:F1 F2 isamorphismofmonoidalfunctors,then isanisomorphism.8Exercise 1.10.16. LetAbeanalgebra. ShowthatM Abimodhasaleft(respectively,right)dualifandonlyifitisfinitelygeneratedprojective when considered as a left (respectively, right) A-module.Sinilarly,ifAiscommutative,M Amodhasaleftandrightdualifandonlyifitisfinitelygeneratedprojective.1.11. Invertible objects. LetC bearigidmonoidalcategory.Definition1.11.1.AnobjectX inCisinvertibleifevX :X X1andcoevX :1XX areisomorphisms.

Clearly, this notion categorifies the notion of an invertible elementinamonoid.Example 1.11.2. Theobjectsg inVec areinvertible.GProposition 1.11.3. LetX bean invertibleobjectinC. Then

(i)X=X andX is invertible;(ii) ifY isanother invertibleobject thenXY is invertible.Proof. Dualizing coevX and evX we get isomorphisms X X = 1and X X 1. = = In any rigid= Hence X XX X X.category the evaluation and coevaluation morphisms for X can be

8Aswehaveseen inRemark1.6.6,this isfalsefornon-rigidcategories.


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31defined by evX := coevX and coevX := evX, so X is invertible.ThesecondstatementfollowsfromthefactthatevXY canbedefinedasacompositionofevX andevY andsimilarlycoevXY canbedefinedasacompositionofcoevY andcoevX.

Proposition1.11.3impliesthatinvertibleobjectsofCformamonoidalsubcategoryInv(C)ofC.Example1.11.4. Gr-categories. Letusclassifyrigidmonoidalcategories C where all objects are invertible and all morphisms are isomorphisms. We may assume that C is skeletal, i.e. there is only oneobject in each isomorphism class, and objects formagroup G. Also,by Proposition 1.2.7,End(1) is an abelian group; let us denote it byA. Then for any g G we can identify End(g) with A, by sendingf End(g)tofIdg1 End(1) =A. ThenwehaveanactionofGonAby

aEnd(1)g(a):=Idg aEnd(g).Letusnowconsidertheassociativity isomorphism. It isdefinedbyafunction:GGG A. Thepentagonrelationgives(1.11.1)(g1g2, g3, g4)(g1, g2, g3g4) =(g1, g2, g3)(g1, g2g3, g4)g1((g2, g3, g4)),for all g1, g2, g3, g4 G, which means that is a 3-cocycle of G withcoefficientsinthe(generally,nontrivial)G-moduleA. Weseethatanysuch 3-cocycle defines a rigid monoidal category, which we will callC(A). Theanalysisofmonoidalequivalencesbetweensuchcategories

Gis similar to the case when A is a trivial G-module, and yields thatfor a given group G and G-module A, equivalence classes of C areGparametrizedbyH3(G,A)/Out(G).

CategoriesoftheformC(A)arecalledGr-categories,andwerestud-Giedin[Si].1.12. Tensor and multitensor categories. Now wewillstart considering monoidal structures on abelian categories. For the sake ofbrevity,wewillnotrecallthebasictheoryofabeliancategories;letus

just recall the Freyd-Mitchell theorem stating that abelian categoriescanbecharacterizedasfullsubcategoriesofcategoriesofleftmodulesover rings, which are closed undertaking direct sums, as well as kernels,cokernels,andimagesofmorphisms. Thisallowsonetovisualizethe main concepts of the theory of abelian categories in terms of theclassicaltheoryofmodulesoverrings.

Recall that an abelian category C is said tobe k-linear (or definedoverk)ifforanyX,Y inC,Hom(X,Y)isak-vectorspace,andcompositionofmorphismsisbilinear.


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32Definition 1.12.1. A k-linear abelian category is said to be locally

finite if it is essentially small9, and the following two conditions aresatisfied:(i) for any two objects X,Y in C, the space Hom(X,Y) is finitedimensional;

(ii)everyobjectinC hasfinitelength.Almostallabelaincategorieswewillconsiderwillbelocallyfinite.

Proposition1.12.2.InalocallyfiniteabeliancategoryC,Hom(X,Y) =0ifX,Y aresimpleandnon-isomorphic,andHom(X,X) =kforanysimpleobjectX.Proof. RecallSchurs lemma: ifX,Y aresimpleobjectsofanabeliancategory,andf Hom(X,Y),thenf =0orf isanisomorphism. Thisimplies thatHom(X,Y) = 0 if X,Y are simple and non-isomorphic,andHom(X,X) is a division algebra; since k is algebraically closed,condition(i) implies thatHom(X,X) =k for any simpleobjectX C.

Also, the Jordan-Holder and Krull-Schmidt theorems hold in anylocallyfiniteabeliancategoryC.Definition 1.12.3. Let C be a locally finite k-linear abelian rigidmonoidal category. We will call C a multitensor category over k ifthe bifunctor is bilinear on morphisms. If in additionEnd(1) k=thenwewillcallC atensorcategory.

A multifusion category is a semisimple multitensor category withfinitelymanyisomorphismsimpleobjects. Afusioncategoryisasemisimpletensorcategorywithfinitelymanyisomorphismsimpleobjects.Example 1.12.4. The categories Vec of finite dimensional k-vectorspaces, Rep(G) of finite dimensional k-representations of a group G(or algebraic representations of an affine algebraic group G), Rep(g)of finite dimensional representations of a Lie algebra g, and Vec ofGG-gradedfinitedimensionalk-vectorspaceswithassociativitydefinedbya3-cocycle aretensorcategories. IfG isafinitegroup,Rep(G)isafusioncategory. Inparticular,Vecisafusioncategory.Example 1.12.5. Let A be a finite dimensional semisimple algebraoverk. LetAbimodbethecategoryoffinitedimensionalA-bimoduleswithbimoduletensorproductoverA,i.e.,

(M,N)MA N.9Recall that a category is called essentially small if its isomorphism classes of

objectsformaset.


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33ThenC isamultitensorcategorywiththeunitobject1=A,the leftdual defined by M Hom(AM,AA), and the right dual defined byM Hom(MA, AA).10 The category C is tensor if and only if A issimple, in which case it is equivalent to kVec. More generally, ifAhas nmatrix blocks, the category C canbe alternativelydescribedas the category whose objects are n-by-n matrices of vector spaces,V = (Vij),andthetensorproductismatrixmultiplication:

(V W)il =jn=1Vij Wjl .ThiscategorywillbedenotedbyMn(Vec). Itisamultifusioncategory.

Ina similarway, onecan definethemultitensorcategoryMn(C) ofn-by-nmatricesofobjectsofagivenmultitensorcategoryC. IfC isamultifusioncategory,soisMn(C).

10NotethatifAisafinitedimensionalnon-semisimplealgebrathenthecategoryoffinitedimensionalA-bimodulesisnotrigid,sincethedualityfunctorsdefinedasabovedonotsatisfyrigidityaxioms(cf.Exercise1.10.16).


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341.13. Exactness of the tensor product.Proposition 1.13.1. (see [BaKi,2.1.8]) Let C be a multitensor category. Then thebifunctor :C C C is exact in bothfactors(i.e.,biexact).Proof. ThepropositionfollowsfromthefactthatbyProposition1.10.9,the functors V and V have left and right adjoint functors (thefunctors of tensoring with the corresponding duals), and any functorbetweenabeliancategorieswhichhasaleftandarightadjointfunctorisexact. Remark 1.13.2. The proof of Proposition 1.13.1 shows that the biadditivityofthe functorholdsautomatically inanyrigidmonoidalabelian category. However, this is not the case for bilinearity of ,andthusconditionofbilinearityoftensorproduct inthedefinitionofamultitensorcategoryisnotredundant.

This may be illustrated by the following example. Let C be thecategoryoffinitedimensionalC-bimodulesinwhichthe leftandrightactionsofRcoincide. Thiscategory isC-linearabelian; namely, it issemisimple with two simple objectsC+ = 1 andC, both equal toC as a real vector space, with bimodule structures (a,b)z = azb and(a,b)z=azb,respectively. Itisalsoalsorigidmonoidal,withbeingthetensorproductofbimodules. ButthetensorproductfunctorisnotC-bilinearonmorphisms(itisonlyR-bilinear).Definition

1.13.3.

A

multiring

category

over

k

is

a

locally

finite

k-linearabelianmonoidalcategoryC withbiexacttensorproduct. If inadditionEnd(1) =k,wewillcallC aringcategory.

Thus, the difference between this definition and the definition of a(multi)tensorcategory isthatwedontrequiretheexistenceofduals,but instead require the biexactness of the tensor product. Note thatProposition1.13.1impliesthatanymultitensorcategoryisamultiringcategory,andanytensorcategoryisaringcategory.Corollary 1.13.4. For any pair of morphisms f1, f2 in a multiringcategoryC onehasIm(f1 f2)=Im(f1)Im(f2).Proof. LetI1, I2 betheimagesoff1, f2. Thenthemorphismsfi :Xi Yi,i= 1,2,havedecompositionsXi Ii Yi,wherethesequences

Xi Ii 0, 0Ii Yiareexact. TensoringthesequenceX1 I1 0withI2,byProposition1.13.1,wegettheexactsequence

X1 I2 I1 I2 0


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35Tenosring X1 with the sequence X2 I2 0, we get the exact se quence

X1 X2 X1 I2 0.Combiningthese,wegetanexactsequence

X1 X2 I1 I2 0.Arguingsimilarly,weshowthatthesequence

0 I1 I2 Y1 Y2isexact. Thisimpliesthestatement. Proposition1.13.5.IfCisamultiringcategorywithrightduals,thentherightdualizationfunctor isexact. Thesameappliesto leftduals.Proof. Let0 X Y Z 0beanexactsequence. Weneedto showthatthesequence0 Z Y X 0 isexact. LetT be anyobjectofC,andconsiderthesequence

0 Hom(T, Z) Hom(T, Y) Hom(T, X). ByProposition1.10.9,itcanbewrittenas

0 Hom(TZ,1) Hom(TY,1) Hom(TX,1), whichisexact,sincethesequence

TX TY TZ 0 is exact, by the exactness of the functor T. This implies that thesequence0 Z Y X isexact.

Similarly,considerthesequence0 Hom(X, T) Hom(Y, T) Hom(Z, T).

ByProposition1.10.9,itcanbewrittenas0 Hom(1, XT) Hom(1, Y T) Hom(1, ZT),

whichisexactsincethesequence0

X

T Y

T Z

T

is exact, by the exactness of the functor T. This implies that thesequenceZ Y X 0isexact. Proposition 1.13.6. LetP beaprojectiveobjectinamultiringcategoryC. IfX C hasarightdual,thentheobjectPX isprojective.Similarly,ifX C hasa leftdual,thentheobjectXP isprojective.


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36Proof. InthefirstcasebyProposition1.10.9wehaveHom(PX,Y) =Hom(P,YX),whichisanexactfunctorofY,sincethefunctorsXandHom(P, )areexact. SoP X isprojective. Thesecondcase issimilar. Corollary1.13.7. IfCmultiringcategorywithrightduals,then1 Cisaprojectiveobjectifandonly ifC issemisimple.Proof. If1isprojectivethenforanyX C,X=1X isprojective.ThisimpliesthatC issemisimple. Theconverseisobvious. 1.14. Quasi-tensor and tensor functors.Definition 1.14.1. LetC, D bemultiringcategoriesoverk, andF :C Dbeanexactandfaithfulfunctor.

(i) F is said to be a quasi-tensor functor if it is equipped with afunctorialisomorphismJ :F( )F( ) F( ),andF(1) =1. (ii)Aquasi-tensor functor(F,J) issaidtobeatensor functor ifJ

isamonoidalstructure(i.e.,satisfiesthemonoidalstructureaxiom).Example1.14.2.ThefunctorsofExamples1.6.1,1.6.2andSubsection1.7(forthecategoriesVecG)aretensorfunctors. TheidentityfunctorVec1 Vec2 fornon-cohomologous3-cocycles1, 2 isnotatensorG Gfunctor,butitcanbemadequasi-tensorbyanychoiceofJ.1.15. Semisimplicity of the unit object.Theorem 1.15.1. In any multiring category,End(1) is a semisimplealgebra, so it is isomorphic to a direct sum offinitely many copies ofk.Proof. ByProposition1.2.7,End(1) isacommutativealgebra,so it issufficient to show that for any a End(1) such that a2 = 0 we havea=0. LetJ =Im(a). ThenbyCorollary1.13.4JJ=Im(aa) =Im(a2 1)=0.

NowletK=Ker(a). ThenbyCorollary1.13.4,KJ istheimageof1aonK1. ButsinceK1isasubobjectof11,thisisthesameastheimageofa1onK1,whichiszero. SoKJ=0.

Now tensoring the exact sequence 0 K 1 J 0 with J, andapplyingProposition1.13.1,wegetthatJ=0,soa=0.

Let{pi}iI betheprimitiveidempotentsofthealgebraEnd(1). Let1i betheimageofpi. Thenwehave1=iI1i.Corollary 1.15.2. In any multiring category C the unit object 1 isisomorphictoadirectsumofpairwisenon-isomorphicindecomposableobjects: 1=i1i.


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37Exercise 1.15.3. Onehas1i 1j =0fori= j. Therearecanonicalisomorphisms1i 1i =1i.=1i,and1i

LetCij :=1i C 1j.Definition1.15.4.ThesubcategoriesCij willbecalledthecomponentsubcategoriesofC.Proposition 1.15.5. LetC beamultiringcategory.

(1) C = i,jI Cij. Thus every indecomposable object of C belongstosomeCij.

(2) The tensor product maps Cij Ckl to Cil, and it is zero unlessj=k.

(3) ThecategoriesCii areringcategorieswithunitobjects1i (whichare

tensor

categories

if

C

is

rigid).

(3) Thefunctorsofleftandrightduals,iftheyaredefined,mapCij

toCji .Exercise 1.15.6. ProveProposition1.15.5.

Proposition 1.15.5 motivates the terms multiring category andmultitensorcategory,assuchacategorygivesusmultipleringcategories,respectivelytensorcategoriesCii.Remark 1.15.7. Thus, amultiring category may be consideredasa2-categorywithobjectsbeingelementsofI, 1-morphisms fromj to iformingthecategoryCij,and2-morphismsbeing1-morphismsinC.Theorem 1.15.8. (i) In a ring category with right duals, the unitobject1 issimple.

(ii) In a multiring category with right duals, the unit object 1 issemisimple,andisadirectsumofpairwisenon-isomorphicsimpleob

jects1i.Proof. Clearly,(i)implies(ii)(byapplying(i)tothecomponentcategoriesCii). Soitisenoughtoprove(i).

LetXbeasimplesubobjectof1(itexists,since1hasfinitelength).Let(1.15.1) 0X

1Y 0

bethecorrespondingexactsequence. ByProposition1.13.5,therightdualizationfunctorisexact,sowegetanexactsequence(1.15.2) 0Y 1X 0.TensoringthissequencewithX ontheleft,weobtain(1.15.3) 0XY XXX 0,


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38SinceX issimpleandXX =0(becausethecoevaluationmorphismis nonzero) we obtain that X X = X. So we have a surjectivecompositionmorphism1XX X. Fromthisand(1.15.1)wehaveanonzerocompositionmorphism1X 1. SinceEnd(1) =k,thismorphismisanonzeroscalar,whenceX=1. Corollary 1.15.9. Inaringcategorywithrightduals, theevaluationmorphismsaresurjectiveandthecoevaluationmorphismsareinjective.Exercise1.15.10.LetCbeamultiringcategorywithrightduals. andX Cij andY Cjk benonzero.

(a) ShowthatXY =0.(b) Deducethatlength(XY)length(X)length(Y).(c) Show that if C is a ring category with right duals then an invertibleobjectinC issimple.(d) Let X be an object in a multiring category with right duals

suchthatXX ShowthatX isinvertible.=1.Example1.15.11. Anexampleofaringcategorywheretheunitob

jectisnotsimpleisthecategoryCoffinitedimensionalrepresentationsof the quiver of type A2. Such representations are triples (V,W,A),whereV,W arefinitedimensionalvectorspaces,andA:V W isalinear operator. The tensor product on such triples is defined by theformula

(V,W,A)(V, W, A) = (V V, WW, AA),withobviousassociativityisomorphisms,andtheunitobject(k,k,Id).Ofcourse,thiscategoryhasneitherrightnorleftduals.1.16. Grothendieckrings. LetCbealocallyfiniteabeliancategoryoverk. IfXandY areobjectsinCsuchthatY issimplethenwedenoteby[X :Y]themultiplicityofY intheJordan-HoldercompositionseriesofX.

RecallthattheGrothendieckgroupGr(C) isthe freeabeliangroupgenerated by isomorphism classes Xi, i I of simple objects in C,andthattoeveryobjectX inC wecancanonicallyassociate itsclass[X]Gr(C)given bythe formula [X] = [X :Xi]Xi. It isobviousithatif

0XY Z0then [Y] = [X] + [Z]. Whennoconfusion ispossible,wewillwriteXinsteadof[X].


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39NowletCbeamultiringcategory. ThetensorproductonC induces

anaturalmultiplicationonGr(C)definedbytheformulaXiXj :=[Xi Xj] = [Xi Xj :Xk]Xk.

kILemma 1.16.1. TheabovemultiplicationonGr(C) isassociative.Proof. Sincethetensorproductfunctorisexact,

[(Xi Xj)Xp :Xl] = [Xi Xj :Xk][Xk Xp :Xl].k

Ontheotherhand,[Xi (Xj Xp) :Xl] = [Xj Xp :Xk][Xi Xk :Xl].

kThus the associativity of the multiplication follows from the isomorphism(Xi Xj)Xp =Xi (Xj Xp).

ThusGr(C) is an associative ring with the unit 1. It is called theGrothendieckringofC.

Thefollowingpropositionisobvious.Proposition1.16.2.LetCandDbemultiringcategoriesandF :C Dbeaquasi-tensorfunctor. ThenF definesahomomorphismofunitalrings [F] :Gr(C) Gr(D).

Thus,weseethat(multi)ringcategoriescategorifyrings(whichjustifiestheterminology),whilequasi-tensor(inparticular,tensor)functors between them categorify unital ring homomorphisms. Note thatProposition1.15.5mayberegardedasacategoricalanalogofthePeircedecompositioninclassicalalgebra.1.17. Groupoids. Themostbasicexamplesofmultitensorcategoriesarisefromfinitegroupoids. Recallthatagroupoid isasmallcategorywhereallmorphismsareisomorphisms. ThusagroupoidGentailsasetXofobjectsofGandasetGofmorphismsofG,thesourceandtargetmapss,t:G X,thecompositionmap:GX G G(wherethe fibered

product

is

defined

using

sin

the

first

factor

and

using

tin

the

secondfactor),theunitmorphismmapu:X G,andthe inversionmap i : G G satisfying certain natural axioms, see e.g. [Ren] formoredetails.

Herearesomeexamplesofgroupoids.(1) AnygroupGisagroupoidG withasingleobjectwhosesetof

morphismstoitselfisG.


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40(2) LetX beasetandletG=XX. Thentheproductgroupoid

G(X):=(X,G)isagroupoidinwhichsisthefirstprojection,t isthe second projection, u is the diagonal map, and i isthepermutationoffactors. Inthisgroupoidforanyx,yX thereisauniquemorphismfromxtoy.

(3) AmoreinterestingexampleisthetransformationgroupoidT(G,X)arising from the action of a group G on a set X. The setof objects of T(G,X) is X, and arrows correspond to triples(g,x,y) where y = gx with an obvious composition law. Inother words, the set of morphisms is GX and s(g,x) =x,t(g,x) =gx,u(x)=(1, x), i(g,x) = (g1,gx).

LetG= (X,G,,s,t,u,i)beafinitegroupoid(i.e.,Gisfinite)andletC(G)bethecategoryoffinitedimensionalvectorspacesgradedbythesetGofmorphismsofG, i.e.,vectorspacesofthe formV =gGVg.IntroduceatensorproductonC(G)bytheformula(1.17.1) (V W)g = Vg1 Wg2.

(g1,g2):g1g2=gThenC(G)isamultitensorcategory. Theunitobject is1=xX 1x,where1x isa1-dimensionalvectorspacewhichsitsindegreeidx inG.Theleftandrightdualsaredefinedby(V)g = (V)g =Vg1.

We invite the reader to check that the component subcategoriesC(G)xy arethecategoriesofvectorspacesgradedbyMor(y,x).

We see that C(G) is a tensor category if and only if G is a group,which is the case of VecG already considered in Example 1.3.6. Notealso that if X = {1,...,n} then C(G(X)) is naturally equivalent toMn(Vec).Exercise 1.17.1. LetCi be isomorphismclassesofobjects inafinitegroupoid G, ni = |Ci|, xi Ci be representatives of Ci, and Gi =Aut(xi)bethecorrespondingautomorphism groups. ShowthatC(G)is(non-canonically)monoidallyequivalenttoiMni(VecGi).Remark 1.17.2. The finite length condition in Definition 1.12.3 isnotsuperfluous: thereexistsarigidmonoidalk-linearabeliancategorywithbilineartensorproductwhichcontainsobjectsof infinite length.AnexampleofsuchacategoryisthecategoryC ofJacobimatricesoffinite dimensional vector spaces. Namely, the objects of C are semi-infinitematricesV ={Vij}ijZ+ offinitedimensionalvectorspacesVijwithfinitelymanynon-zerodiagonals,andmorphismsarematricesoflinear maps. The tensor product in this category is defined by the


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41formula(1.17.2) (V W)il = Vij Wjl ,

jand the unit object 1 isdefinedby thecondition 1ij =kij. The leftandrightdualityfunctorscoincideandaregivenbytheformula(1.17.3) (V)ij = (Vji ).TheevaluationmapisthedirectsumofthecanonicalmapsVijVij 1jj, and the coevaluation map is a direct sum of the canonical maps1ii Vij Vij.

NotethatthecategoryCisasubcategoryofthecategoryC ofG(Z+)gradedvectorspaceswithfinitedimensionalhomogeneouscomponents.NotealsothatthecategoryC isnotclosedunderthetensorproductdefinedby(1.17.2)butthecategoryC is.Exercise 1.17.3. (1) ShowthatifX isafinitesetthenthegroup

of invertible objects of the category C(G(X)) is isomorphic toAut(X).

(2) LetC bethecategoryofJacobimatricesofvectorspacesfromExample 1.17.2. Show that the statement Exercise 1.15.10(d)fails for C. Thus the finite length condition is important inExercise1.15.10.

1.18. Finite abelian categories and exact faithful functors.Definition

1.18.1.

A

k-linear

abelian

category

C

is

said

to

be

finite

if

itisequivalenttothecategoryAmodoffinitedimensionalmodulesoverafinitedimensionalk-algebraA.

Ofcourse,thealgebraAisnotcanonicallyattachedtothecategoryC; rather, C determines the Morita equivalence class of A. For thisreason, it is often better to use the following intrinsic definition,whichiswellknowntobeequivalenttoDefinition1.18.1:Definition 1.18.2. Ak-linearabeliancategoryC isfiniteif

(i)C hasfinitedimensionalspacesofmorphisms;(ii)everyobjectofC hasfinitelength;(iii) C has enough projectives, i.e., every simple object of C has a

projectivecover;and(iv)therearefinitelymanyisomorphismclassesofsimpleobjects.Note that the first two conditions are the requirement that C be

locallyfinite.Indeed,itisclearthatifAisafinitedimensionalalgebrathenA

modclearlysatisfies(i)-(iv),andconversely,ifC satisfies(i)-(iv),then


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42onecantakeA=End(P)op,whereP isaprojectivegeneratorofC(e.g.,P = in=1Pi, where Pi are projective covers of all the simple objectsXi).AprojectivegeneratorP ofCrepresentsafunctorF =FP :C VecfromC tothecategoryoffinitedimensionalk-vectorspaces,givenbythe formula F(X) =Hom(P,X). The condition that P is projectivetranslates into the exactness property of F, and the condition thatP is a generator (i.e., covers any simple object) translates into thepropertythatF isfaithful(doesnotkillnonzeroobjectsormorphisms).Moreover, the algebra A = End(P)op can be alternatively defined asEnd(F), the algebra of functorial endomorphisms of F. Conversely,it is well known (and easy to show) that any exact faithful functorF :C Vecisrepresentedbyaunique(uptoauniqueisomorphism)projectivegeneratorP.NowletCbeafinitek-linearabeliancategory,andF1, F2 :C Vecbetwoexactfaithfulfunctors. DefinethefunctorF1F2 :C CVecby(F1 F2)(X,Y):=F1(X)F2(Y).Proposition1.18.3.ThereisacanonicalalgebraisomorphismF1,F2 :End(F1)End(F2)=End(F1 F2)givenby

F1,F2(1 2)|F1(X)F2(Y) :=1|F1(X) 2|F2(Y),wherei End(Fi), i= 1,2.Exercise 1.18.4. ProveProposition1.18.3.1.19. Fiberfunctors. LetCbeak-linearabelianmonoidalcategory.Definition 1.19.1. A quasi-fiber functor on is an exact faithfulCfunctor F : C Vec from C to the category of finite dimensionalk-vector spaces, such that F(1) = k, equipped with an isomorphismJ :F( )F( ) F( ). If inadditionJ isamonoidalstructure (i.e. satisfiesthemonoidalstructureaxiom),onesaysthatF isafiber

functor.Example1.19.2. TheforgetfulfunctorsVecG Vec,Rep(G) Vecare naturally fiber functors, while the forgetful functor Vec VecG isquasi-fiber,foranychoiceofthe isomorphismJ (wehaveseenthatif is cohomologically nontrivial, then Vec does not admit a fiberGfunctor). Also, the functor Loc(X) Vec on the category of localsystemsonaconnectedtopologicalspaceX whichattachestoa localsystemEitsfiberEx atapointxX isafiberfunctor,whichjustifiestheterminology. (NotethatifX isHausdorff,thenthisfunctorcanbeidentifiedwiththeabovementionedforgetfulfunctorRep(1(X,x))Vec).


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43Exercise1.19.3. ShowthatifanabelianmonoidalcategoryCadmitsaquasi-fiber functor,then it isaringcategory, inwhichtheobject1issimple. SoifinadditionC isrigid,thenitisatensorcategory.1.20. Coalgebras.Definition1.20.1.Acoalgebra(withcounit)overafieldkisak-vectorspaceCtogetherwithacomultiplicaton(orcoproduct):C CCandcounit:C ksuchthat

(i)iscoassociative,i.e.,(Id) = (Id)

asmapsC C3;(ii)onehas

(Id) = (Id) = Id asmapsC C (thecounitaxiom).Definition 1.20.2. A left comodule over a coalgebra C is a vectorspace M together with a linear map : M C M (called thecoactionmap),suchthatforanymM,onehas

(Id)((m))=(Id)((m)), (Id)((m))=m.Similarly,arightcomoduleoverC isavectorspaceM togetherwithalinearmap:M MC,suchthatforanymM,onehas

(Id)((m))=(Id)((m)), (Id)((m))=m.Forexample,C isa leftandrightcomodulewith =, andso is

k,with=.Exercise 1.20.3. (i) Show that if C is a coalgebra then C is analgebra,andifAisafinitedimensionalalgebrathenA isacoalgebra.

(ii) Show that for any coalgebra C, any (left or right) C-comoduleM isa(respectively,rightorleft)C-module,andtheconverseistrueifC isfinitedimensional.Exercise 1.20.4. (i) Show that any coalgebra C is a sum of finitedimensionalsubcoalgebras.

Hint.

Letc

C,

and

let

(Id)(c)=(Id)(c) = c1i c2i c3i.

iShowthatspan(c2i)isasubcoalgebraofC containingc.

(ii)ShowthatanyC-comoduleisasumoffinitedimensionalsubcomodules.


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(

441.21. Bialgebras. Let C be a finite monoidal category, and (F,J) :C Vecbeafiberfunctor. ConsiderthealgebraH :=End(F). Thisalgebra has two additional structures: the comultiplication :H H H and the counit : H k. Namely, the comultiplication isdefinedbytheformula

(a) =1 (a)),F,Fwhere (a)End(FF)isgivenby

X,YaXYJX,Y,(a)X,Y =J1andthecounitisdefinedbytheformula

(a) =a1 k.Theorem

1.21.1.

(i)

The

algebra

H

is

acoalgebra

with

comultiplicationandcounit.

(ii)Themapsandareunitalalgebrahomomorphisms.Proof. Thecoassociativityoffollowsformaxiom(1.4.1)ofamonoidalfunctor. Thecounitaxiomfollowsfrom(1.4.3)and(1.4.4). Finally,observethatforall,End(F)theimagesunderF,F ofboth()()and()havecomponentsJ1 ()XYJX,Y;hence,isanalgebraX,Yhomomorphism(whichisobviouslyunital). Thefactthatisaunitalalgebrahomomorphismisclear. Definition1.21.2. AnalgebraHequippedwithacomultiplicationand

acounit

satisfying

properties

(i),(ii)

of

Theorem

1.21.1

is

called

abialgebra.

Thus, Theorem 1.21.1 claims that the algebra H = End(F) has anaturalstructureofabialgebra.

Now let H be any bialgebra (not necessarily finite dimensional).Then the category Rep(H) of representations (i.e., left modules) ofH and its subcategory Rep(H) of finite dimensional representationsofH arenaturallymonoidalcategories(andthesameappliestorightmodules). Indeed,onecandefinethetensorproductoftwoH-modulesX,Y tobetheusualtensorproductofvectorspacesXY,withtheactionofH definedbytheformula

XY(a) = (X Y)((a)), aH(where X : H End(X), Y : H End(Y)), the associativity iso morphism to be the obvious one, and the unit object to be the 1dimensionalspacekwiththeactionofHgivenbythecounit,a (a).Moreover,theforgetfulfunctorForget:Rep(H) Vecisafiberfunctor.


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45Thusweseethatonehasthefollowingtheorem.

Theorem 1.21.3. The assignments (C, F) H = End(F), H (Rep(H),Forget)aremutually inversebijectionsbetween

1)finite abelian k-linear monoidal categories C with afiberfunctorF,uptomonoidalequivalenceandisomorphismofmonoidalfunctors;

2)finitedimensionalbialgebrasH overk upto isomorphism.Proof. Straightforwardfromtheabove.

Theorem1.21.3iscalledthereconstructiontheoremforfinitedimensionalbialgebras(asitreconstructsthebialgebraH fromthecategoryofitsmodulesusingafiberfunctor).Exercise 1.21.4. Show that the axioms of a bialgebra are self-dualin

the

following

sense:

if

H

is

a

finite

dimensional

bialgebra

with

multiplication : H H H, unit i : k H, comultiplication :H HH and counit :H k, then H is alsoa bialgebra,withthemultiplication,unit,comultiplication,andcouniti.Exercise 1.21.5. (i) Let G be a finite monoid, and C = VecG. LetF :C Vecbe the forgetful functor. Show that H =End(F) is thebialgebraFun(G,k)ofk-valuedfunctionsonG,withcomultiplication(f)(x,y) = f(xy) (where we identify HH with Fun(GG,k)),andcounit(f) =f(1).

(ii) Show that Fun(G,k) = k[G], the monoid algebra of G (withbasisxGandproductx y=xy),withcoproduct(x) =x x,andcounit(x)=1,xG. Notethatthebialgebrak[G]maybedefinedforanyG(notnecessarilyfinite).Exercise1.21.6. LetHbeak-algebra,C=Hmodbethecategoryof H-modules, and F : C Vec be the forgetful functor (we dontassume finite dimensionality). Assume that C is monoidal, and F isgiven a monoidal structure J. Show that this endows H with thestructureofabialgebra,suchthat(F,J)definesamonoidalequivalenceC Rep(H).

Notethatnotonlymodules,butalsocomodulesoverabialgebraHform a monoidal category. Indeed, for a finite dimensional bialgebra,this is clear, as right (respectively, left) modules over H is the samething as left (respectively, right) comodules over H. In general, ifX,Y are, say, right H-comodules, then the right comodule XY isthe usual tensor product of X,Y with the coaction map defined asfollows: ifxX,yY,(x) = xi ai,(y) = yj bj,then

XY(xy) = xi yj aibj.


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46For a bialgebra H, the monoidal category of right H-comodules willbedenotedbyHcomod,andthesubcategoryoffinitedimensionalcomodulesbyHcomod.1.22. Hopf algebras. Let us now consider the additional structureonthebialgebraH=End(F)fromtheprevioussubsectioninthecasewhen the category C has right duals. In this case, one can define alinearmapS :H H bytheformula

S(a)X =aX,whereweusethenaturalidentificationofF(X) withF(X).Proposition1.22.1. (theantipodeaxiom)Let:HH H andi:k H be themultiplicationand theunitmapsofH. Then

(IdS) =i = (SId) asmapsH H.Proof. ForanybEnd(FF)thelinearmap(IdS)(1 (b))X, XF,FC isgivenby(1.22.1)

coevF(X) bX,X evF(X)F(X)F(X)F(X)F(X)F(X)F(X)F(X)F(X),where we suppress the identity isomorphisms, the associativity constraint, and the isomorphism F(X) = F(X). Indeed, it suffices tocheck(1.22.1)forb=,where,H,whichisstraightforward.

Nowthefirstequalityofthepropositionfollowsfromthecommutativityofthediagram

coevF(X)(1.22.2) F(X) F(X)F(X) F(X)

Id JX,XF(coevX)

F(X) F(XX)F(X)1 XX

F(coevX)F(X) F(XX)F(X)

J1Id X,XevF(X)F(X) F(X)F(X) F(X),

foranyEnd(F).Namely, the commutativity of the upper and the lower square fol

lowsfromthefactthatupon identificationofF(X) withF(X),themorphisms evF(X) and coevF(X) are given by the diagrams of Exercise1.10.6. Themiddlesquarecommutesbythenaturalityof. The


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47compositionof leftverticalarrowsgives()IdF(X), whilethecompositionofthetop,right,andbottomarrowsgives (IdS) ().

Thesecondequalityisprovedsimilarly.

Definition 1.22.2. An antipode on a bialgebra H is a linear mapS :H H whichsatisfiestheequalitiesofProposition1.22.1.Exercise 1.22.3. Show that the antipode axiom is self-dual in thefollowing sense: if H is a finite dimensional bialgebra with antipodeSH,thenthebialgebraH alsoadmitsanantipodeSH =SH.

Thefollowing isalinearalgebraanalogofthefactthattherightdual,whenitexists,isuniqueuptoauniqueisomorphism.Proposition1.22.4. AnantipodeonabialgebraH isuniqueifexists.Proof. The proof essentially repeats the proof of uniqueness of rightdual. Let S,S be two antipodes for H. Then using the antipodepropertiesofS,S,associativityof,andcoassociativityof,weget

S=(S[(IdS)]) =(Id)(SIdS)(Id) =(Id)(SIdS)(Id) =

([(SId)]S) =S.

Proposition1.22.5. IfSisanantipodeonabialgebraH thenSisanantihomomorphismofalgebraswithunitandofcoalgebraswithcounit.Proof. Let

1 2 3(Id) (a)=(Id) (a) = a a ai,i ii

(Id) (b)=(Id) (b) = b 21j bj bj

Thenusingthedefinitionoftheantipode,wehave3

j.

332211321ib)aiS(ai) = ibj)aibjS(bj)S(ai) =S(b)S(a).

i i,jThusSisanantihomomorphismofalgebras(whichisobviouslyunital).The fact that it is an antihomomorphism of coalgebras then followsusingtheself-dualityoftheaxioms(seeExercises1.21.4,1.22.3),orcanbeshownindependentlybyasimilarargument.

S(ab) = S(a S(a


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48Corollary 1.22.6. (i) If H is a bialgebra with an antipode S, thenthe abelian monoidal category C = Rep(H) has right duals. Namely,

foranyobjectX, therightdualX is theusualdualspaceofX,withactionofH givenbyX(a) =X(S(a)),

and the usual evaluation and coevaluation morphisms of the categoryVec.

(ii) If in addition S is invertible, then C also admits left duals, i.e.isrigid(inotherwords,C is tensorcategory). Namely,foranyobjectX,theleftdualX istheusualdualspaceofX,withactionofHgivenby

X(a) =X(S1(a)),and

the

usual

evaluation

and

coevaluation

morphisms

of

the

category

Vec.Proof. Part(i)followsfromtheantipodeaxiomandProposition1.22.5.Part(ii)followsfrompart(i)andthefactthattheoperationoftakingtheleftdualisinversetotheoperationoftakingtherightdual. Remark 1.22.7. Asimilarstatementholdsforfinitedimensionalcomodules. Namely, if X is a finite dimensional right comodule over abialgebraHwithanantipode,thentherightdualistheusualdualXwith

(X(f), x):=((IdS)(X(x)), f),x X,f X, H. If S is invertible, then the left dual X isdefinedbythesameformulawithS replacedbyS1.Remark 1.22.8. ThefactthatS isanantihomomorphismofcoalgebrasisthelinearalgebraversionofthecategoricalfactthatdualizationchangestheorderoftensorproduct(Proposition1.10.7(ii)).Definition 1.22.9. AbialgebraequippedwithaninvertibleantipodeS iscalledaHopfalgebra.Remark 1.22.10. We note that many authors use the term Hopfalgebraforanybialgebrawithanantipode.

Thus,Corollary

1.22.6

states

that

if

H

is

aHopf

algebra

then

Rep(H)

is a tensor category. So, we get the following reconstruction theorem

forfinitedimensionalHopfalgebras.Theorem 1.22.11. The assignments (C, F) H = End(F), H (Rep(H),Forget)aremutually inversebijectionsbetween

1)finite tensor categories C with afiberfunctor F, up to monoidalequivalenceand isomorphismofmonoidalfunctors;


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492)finitedimensionalHopfalgebrasoverk up to isomorphism.

Proof. Straightforwardfromtheabove. Exercise 1.22.12. The algebra of functions Fun(G,k) on a finitemonoidG isa Hopfalgebra if andonly if G isagroup. Inthiscase,theantipodeisgivenbytheformulaS(f)(x) =f(x1),xG.

More generally, if G is an affine algebraic group over k, then thealgebra O(G) of regular functions on G is a Hopf algebra, with thecomultiplication,counit,andantipodedefinedasinthefinitecase.

Similarly, k[G] is a Hopf algebra if and only if G is a group, withS(x) =x1,xG.

Exercises1.21.5and1.22.12motivatethefollowingdefinition:Definition 1.22.13. In any coalgebra C, a nonzero element g Csuchthat(g) =gg iscalledagrouplikeelement.Exercise 1.22.14. Showthatifg isagrouplikeofaHopfalgebraH,theng is invertible, withg1 =S(g). Also,showthattheproductoftwogrouplikeelements isgrouplike. Inparticular, grouplikeelementsofany Hopfalgebra H form a group, denotedG(H). Showthat thisgroup can also be defined as the group of isomorphism classes of 1dimensionalH-comodulesundertensormultiplication.Proposition 1.22.15. If H is afinite dimensional bialgebra with anantipodeS, thenS is invertible,soH isaHopfalgebra.Proof. Let Hn be the image of Sn. Since S is an antihomomorphismof algebras and coalgebras, Hn is a Hopf subalgebra of H. Let m bethe smallest n such that Hn = Hn+1 (it exists because H is finitedimensional). We need to show that m = 0. If not, we can assumethatm=1byreplacingH withHm1.

WehaveamapS :H1 H1 inversetoS. ForaH,letthetriplecoproductofabe

1 2 3ai ai ai.i

Considertheelementb= S(S(ai1))S(ai2)ai3.

iOn the one hand, collapsing the last two factors using the antipodeaxiom,wehaveb=S(S(a)). Ontheotherhand,writingbas

b= S(S(ai1))S(S(S(ai2)))ai3i


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50andcollapsingthefirsttwofactorsusingtheantipodeaxiom,wegetb=a. Thusa=S(S(a))andthusaH1,soH=H1,acontradiction. Exercise1.22.16. Letop andop beobtainedfrom,bypermutationofcomponents.

(i) Show that if (H,,i,,,S) is a Hopf algebra, then Hop :=(H,op,i,,,S1),Hcop :=(H,,i,op,,S1),Hcop :=(H,op,i,op,,S)opareHopfalgebras. ShowthatHisisomorphictoHcop,andHop toHcop.op

(ii) Suppose that a bialgebra H is a commutative (=op) or cocommutative ( = op). Let S be an antipode on H. Show thatS2 =1.

(iii) Assume that bialgebras H and Hcop have antipodes S and S.ShowthatS =S1,soH isaHopfalgebra.Exercise 1.22.17. Show that if A,B are bialgebras, bialgebraswithantipode,orHopfalgebras,thensoisthetensorproductAB.Exercise 1.22.18. A finite dimensional module or comodule over aHopfalgebraisinvertibleifandonlyifitis1-dimensional.1.23. Reconstruction theory inthe infinitesetting. Inthissubsection we would like to generalize the reconstruction theory to thesituationwhenthecategoryC isnotassumedtobefinite.

LetCbeanyessentiallysmallk-linearabeliancategory,andF :C Vec an exact, faithful functor. In this case one can define the spaceCoend(F

)as

follows:

Coend(F):=(XCF(X) F(X))/E

whereE isspannedbyelementsoftheformyF(f)xF(f)yx,xF(X),y F(Y),f Hom(X,Y); inotherwords,Coend(F) =limEnd(F(X)). ThuswehaveEnd(F)=limEnd(F(X))=Coend(F), whichyieldsacoalgebrastructureonCoend(F). SothealgebraEnd(F)(whichmaybeinfinitedimensional)carriestheinverselimittopology,inwhichabasisofneighborhoodsofzeroisformedbythekernelsKXofthemapsEnd(F)End(F(X)),X C,andCoend(F) =End(F),thespaceofcontinuouslinearfunctionalsonEnd(F).

Thefollowingtheoremisstandard(see[Ta2]).Theorem 1.23.1. Let C be a k-linear abelian category with an exactfaithfulfunctor F : Vec. Then F defines an equivalenceC between C and the category offinite dimensional right comodules overC :=Coend(F)(or,equivalently,withthecategoryofcontinuousfinitedimensional leftEnd(F)-modules).


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51Proof. (sketch) Consider the ind-object Q := XCF(X) X. ForX,Y C andf Hom(X,Y),let

jf :F(Y) XF(X) XF(Y) Y Qbethemorphismdefinedbytheformula

jf =IdfF(f) Id.LetI bethequotientofQbytheimageofthedirectsumofalljf. Inotherwords,I=lim(F(X) X).

Thefollowingstatementsareeasytoverify:(i) I represents the functor F( ), i.e. Hom(X,I) is naturally iso

morphictoF(X);inparticular,I isinjective.(ii) F(I) = C, and I is naturally a left C-comodule (the comod

ulestructureis inducedbythecoevaluati

tensor cathegories notes

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