2DENNISGAITSGORYLemma1.1.1.ForC1;C2asabove,anyF2Functcont(C1;C2)admits - PDF document

2DENNISGAITSGORYLemma1.1.1.ForC1;C2asabove,anyF2Functcont(C1;C2)admitsarightadjointinFunct(C2;C1).Thefullsubcategoryof(Funct(C2;C1))opobtainedasrightadjointsofobjectsfromFunctcont(C1;C2)isFunctcocont(C2;C1)Funct(C2;C1);whichconsistsoffunctorsthatcommutewithlimits.1.2.The2-categoryofDGcategories.We'dliketoviewthetotalityofDGcategoriesasan(1;2)-categoryin2ways,denotedDGCatcontandDGCat,respectively,whereinbothcasestheobjectsareDGcategories,andthe1-morphismsareFunctcont(C1;C2)andFunct(C1;C2);respectively.Forthemostpart,however,we'llbeworkingwithDGCatcont.Inthiscase,we'llalsousethenotationHom(C1;C2):=Functcont(C1;C2):1.2.1.Thetroubleis,however,thatthetheoryof(1;2)-categorieshasn'tbeenadequatelydocumentedatthetimeofwriting.Thepaper[Lu3]developsthenotionof(1;2)-category,butdoesn'tshowthatDGcategoriesanditsvariantsconsideredinthesequelforma(1;2)-category.Apossiblewayoutisasfollows:formostapplications(suchascomputationoflimitsandcolimits),itwouldbesucienttoviewDGCatcontandDGCatasjust1-categories(i.e.,(1;1)-categories),bydiscradingnon-invertible2-morphisms,i.e.,byconsideringas1-morphismsthecorrespondingmaximalsub-groupoidsFunctcont(C1;C2)Functcont(C1;C2)andFunct(C1;C2)Funct(C1;C2):1.2.2.Bydenition,whenconsideringDGCatcontorDGCatasan(1;1)-category,weonlyconsidernaturaltransformationsthatareisomorphisms.However,onecanrecoverallnaturaltransformationsasfollows.Namely,let(0!1)betheDG-categorygeneratedbytwoobjects0and1withauniquearrow0!1.ThenforC1andC2asabove,wecanconsiderFunctcont(C1 (0!1);C2).ThebersofthemapFunctcont(C1 (0!1);C2)!Functcont(C1;C2)Functcont(C1;C2)correspondingtothetwofunctorsVect(0!1),overagivenpairF0;F002Functcont(C1;C2)isthegroupoidofallnaturaltransformationsF0)F00.AsimilarremarkappliestoDGCatcontreplacedbyDGCat.1.3.LimitsandcolimitsinDGCatcontandDGCat.Lemma1.3.1.(1)The1-categoriesDGCatcontandDGCatadmitlimitsandcolimits.(2)TheforgetfulfunctorDGCatcont!DGCatcommuteswithlimits.Thisisdonein[Lu1,Sect.5.5.3].Note,however,thattheforgetfulfunctorDGCatcont!DGCatdoesnotcommutewithcolimits. 4DENNISGAITSGORY1.3.5.AssumethatintheabovesettingthefunctorsGalsobelongtoFunctcont(Cj;Ci).AssumealsothatforeverydiagramA=i1 j!i2,thecategoryIA=iscontractible.(Thishappens,e.g.,whenIisltered.)Lemma1.3.6.Undertheabovecircustamstances,thefunctorCj0insj0�!colimj2I;GCj'limi2Iop;GCievi0�!Ci0iscanonicallyisomorphictocolimk2I;:j0!k;:i0!kGF;wherethecolimitistakeninFunctcont(Cj0;Ci0).Proof.Considerthecategory0C:=lax:lim �i2Iop;GCiofallassignments(1)i7!(ci2Ci);( :i!i0)7!(G (ci0) �!ci2Ci);equippedwiththedataofmakingthemaps coherentlyassociative.However,themaps arenotrequiredtobeisomorphisms.WehaveafullyfaithfulembeddingC:=lim �i2Iop;GCj,!lax:lim �i2Iop;GCi=:0Cwhoseessentialimageconsistsofthoseobjects(1),forwhichthemaps areisomorphisms.Foragivenindexjandcj2Cj,theassignment(2)i7!colimk2I;:j!k;:i!kGFnaturallyupgradestoanobjectof0C.Indeed,for :i!i0,thecorrespondingmapis(3)G colimk2I;:j!k;0:i0!kG0F(cj)'colim�!k2I;:j!k;0:i0!kG G0F(cj)''colim�!k2I;:j!k;0:i0!kG0 F(cj)!colim�!k2I;:j!k;:i!kGF(cj);wheretherstisomorphismresultsfromthecontinuityofthefunctorG ,andthelastarrowcorrespondstothefunctorofindexcategories(4)fk2I;:j!k;0:i0!kg!fk2I;:j!k;:i!kg;givenbypre-compositionwith .Itisclearthattheassignment(2)denesafunctorCj!0C.Denotethisfunctorby0insj.Itfollowsfromtheconstructionthatforcj2Cjandc2C0C,wehaveacanonicalisomorphismMaps0C(0insj(cj);c)'MapsCj(cj;evj(c)):Hence,bythe(insj;evj)-adjunction,itremainstoshowthatwhenIsatisestheassumtionofthelemma,theessentialimageofthefunctor0insjbelongstoC0C. 6DENNISGAITSGORY2.1.Dualitydatum.LetCbeaDG-category.WesaythatCisdualizable,ofthereexistsanotherDG-categoryC_endowedwithmorphismsinDGCatcont:Vect!C_ Cand:C C_!Vect;satisfyingtheusualdualityaxioms,i.e.,thecompositionsCIdC �!C C_ C IdC�!CandC_ IdC�!C_ C C_IdC �!C_areisomorphictotheidentityfunctor.2.1.1.Tautologically,onecansaythatCisdualizableifitissuchasa0-objectofthe(1;1)-categoryDGCatcont,seee.g.Sect.5forareminderwhatitmeanstobeadualizableobjectinamonoidalcategory.2.1.2.Alternatively,CisdualizableifthereexistsaDG-categoryC_endowedwithapairingC C_!VectinDGCatcontwhichinducesanequivalence(5)C_ D!Functcont(C;D)foranyD2DGCatcont.Inparticular,C_canbeidentiedwithFunctcont(C;Vect).2.1.3.Equivalently,thereshouldexistafunctorVect!C_ CinDGCatcontwhichforanyD1;D2inducesanequivalence(6)Functcont(C D1;D2)'Functcont(D1;C_ D2):2.1.4.SincethetensorproductonDGCatcontissymmetric,wehavethatC_isthedualofCifandonlyifCisthedualofC_.2.1.5.Dualizablecategoriesenjoynicepropertiesregardinglimitsandcolimits:Lemma2.1.6.Leti7!CibeafunctorI!DGCatcont.LetDbedualizable.(1)ThenaturalfunctorD limICi!limI(D Ci)isanequivalence.(2)ThenaturalfunctorcolimIFunctcont(D;Ci)!Functcont(D;colimICi)isanequivalence,i.e.,DiscompactasanobjectofDGCatcont(thecolimitinLHSistakenwithinDGCatcont).Proof.Forpoint(1),theLHScanberewrittenasFunctcont(D_;limICi)'limIFunctcont(D_;Ci);whichisequivalenttotheRHS.Point(2)followsfromequation(5).2.2.Dualfunctors.IfC1;C2aredualizable,thereexistsacanonicalequivalenceFunctcont(C1;C2)'C_1 C2'Functcont(C_2;C_1);whichwe'lldenoteF7!F_. 8DENNISGAITSGORYinducedby(8)isanequivalence.Tothisend,onechecksthattheabovefunctorisisomorphictothecompositionoftheequivalences(lim �i2Iop;GCi) D'(colim�!i2I;FCi) D'colim�!i2I;(F Id)(Ci D)'lim �i2Iop;(G Id)(Ci D)''lim �i2IopFunctcont(C_i;D)'Functcont0@(colim�!i2I;G_C_i);D1A;wheretherstandthirdequivalenceareobtainedbyapplyingLemma1.3.3,andthelimitofFunctcont(C_i;D)istakenwithrespecttothefunctorsFunctcont(C_j;D)�!Functcont(C_i;D);F7!FG_;:i!j:Thecharacterizationoftheequivalence(10)givenby(7)followsfromthedenitionofthepairing(8).2.2.3.It'salsoeasytoseethatif,undertheabovecircumstances,eachofthecategoriesCiiscompactlygenerated,thensoisC:=colimI;FCi.Indeed,thefunctorsi:Ci!Csendcompactobjectstocompactones.2.3.Compactlygeneratedcategories.AssumenowthatCiscompactlygenerated.I.e.,wecanwriteC'Ind(Cc),whereCcisasmall(non-cocomplete)DGcategoryconsistingofcompactobjectsofC.ForanyDG-categoryDwehavethatFunctcont(C;D)isequivalenttothecategoryFunct(Cc;D)ofjustk-linearfunctorsCc!D.NotethatwehaveacanonicalpairinginDGCatcont:Ind(Cc) Ind((Cc)op)!Vect;givenbytheYonedapairingCc(Cc)op!Vect.Proposition2.3.1.TheabovepairingmakesInd((Cc)op)intoadualofC.Inparticular,anycompactlygeneratedDG-categoryisdualizable.Proof.We'llcheckthatforanyDtheabovepairingdenesanequivalenceInd((Cc)op) D!Functcont(C;D)'Funct(Cc;D):ItwillbeconvenienttousethefollowingcharacterizationofthethetensorproductoperationonDGCatcont:D1 D2'(Functcont(D1;Dop2))op:Hence,weobtainthatInd((Cc)op) D'(Funct((Cc)op;Dop))op'Funct(Cc;D);asrequired. 10DENNISGAITSGORY3.2.Monadsandtensorproducts.LetC1;C2beDG-categories,andletAi:Ci!Ci,i=1;2bemonadsthatbelongtoFunctcont(Ci;Ci).LetC:=C1 C2,andletA=A1 A2.Proposition3.2.1.ThenaturalfunctorA1-modC1 A2-modC2!A-modCisanequivalence.Proof.LetFi;GidenotethepairofadjointfunctorsFi:Ai-modCiCi:Gi:ConsidertheforgetfulfunctorG1 G2:A1-modC1 A2-modC2!C1 C2'C:ItsleftadjointisF1 F2,andtheresultingmonadonCisA.BySect.3.1.2,itsucestoshowthatG1 G2isconservative.ThelatterisequivalenttotheimageofF1 F2generatingA1-modC1 A2-modC2.However,thisfollowsfromthefactthattheimageofFigeneratesAi-modCi,sinceGiisconservativebyassumption.4.Modulecategories4.1.Theset-up.LetObeamonoidalcategory.We'llalwaysbeassumingthatthemonoidaloperationmultO:O O!ObelongstoFunctcont(O O;O).ByanO-modulewe'llmeanacategoryCendowedwithanassociativeactionactO;C:O C!C,suchthatthisfunctorbelongstoFunctcont(O C;C).FortwoO-modulecategoriesC1andC2,weshalldenotebyHomO-mod(C1;C2)theDGcategoryoffunctorsC1!C2thatarecompatiblewiththeO-action,andbelongtoFunctcont(C1;C2).4.1.1.WemakeO-modulecategoriesintoan(1;2)-category,denotedO-mod,bysetting1-morphismstobeHomO(C1;C2).However,thesamereservationpertainingtothenotionof(1;2)-categoryasinthecaseofDGCatcont(seeSect.1.2.1)applies.AsinthecaseofDGCat,wecanalternativelyviewO-modasan(1;1)-category,bydiscardingthenon-invertible2-morphisms,i.e.,byconsideringthemaximalsub-groupoidHomO-mod(C1;C2)HomO-mod(C1;C2):AsforDGCat,the(1;2)-categorystructurecanbeessentiallyrecoveredfromthe(1;1)-categorybyconsideringthearrowscategory,usingthefactthatO-modistensoredoverDGCat.Whenconsideringafunctor:O1-mod!O2-modas(1;1)-categories,wecanrecoveritasa2-functorbetweenthecorresponding(1;2)-categoriesonceisendowedwithastructureofbeingtensoredoverDGCat. 12DENNISGAITSGORYProof.LetG:A-modO!OandGop:Aop-modOop!Odenotetheforgetfulfunctors,andletFandFopdenotetheiradjoints.WehaveapairofmutuallyadjointfunctorsF Fop:O'O OOA-modO OAop-modOop:G Gop:TheresultingmonadonOcorrespondstothecategoryA-bimodO.Hence,byProposition3.1.2,itsucestocheckthatthefunctorG Gopisconservative.ThelatterisequivalenttothefactthattheimageofF FopgeneratesA-modO OAop-modOop.However,thelatterfollowssincetheimageofF(resp.,Fop)generatesA-modO(resp.,Aop-modOop),sincethefunctorsGandGopareconservative.4.7.LetOandAbeasabove.Proposition4.7.1.ThecategoriesA-modOandAop-modOoparemutuallyO-dual.Proof.Thepairing:Aop-modOop A-modO!OistheusualHochschildhomlogyfunctor.Thefunctor:Vect!A-modO Aop-modOopcorrespondstotheobjectA2A-bimodO'A-modO Aop-modOop:Corollary4.7.2.ForaleftO-modulecategoryC,wehaveanaturalequivalence:HomO-mod(Aop-modOop;C)'A-modO OC:4.8.LetCbealeftO-modulecategory.ForAasabove,wecanconsiderthemonadAConCgivenbytensorproductwithA.LetA-modCdenotethecorrespondingcategoryofmodules.Proposition4.8.1.ThenaturalfunctorA-modO OC!A-modCisanequivalence.Proof.TheprooffollowsagainfromProposition3.1.2:itsucestoobservethattheforgetfulfunctorA-modO OC!O OC'Cisconservative.4.9.Compactgenerationoftensorproducts.LetObeamonoidalDGcategory,andletC1andC2beleftandrightO-modulecategories,respectively.AssumenowthatOandthatthemonoidaloperationO O!Oadmitsacontinuousrightadjoint,andthatsodotheactionfunctorsO C1!C1andC2 O!C2.Proposition4.9.1.Undertheabovecircumstances,therightadjointtothetautologicalfunctorC2 C1!C2 OC1iscontinuous. 14DENNISGAITSGORY5.2.LetusnowconsiderOasamoduleoveritself.RecallthatanobjectX2Oissaidtobeleft-dualizableifthereexistsanobjectX_2Oendowedwiththe1O!X X_andX_ X!1O;satisfyingtheusualaxioms.Recallalsothefollowing:Lemma5.2.1.(1)IfXisdualizable,wehaveHom(Y;X Z)'Hom(X_ Y;Z)andHom(Z X;Y)'Hom(Z;Y X_):(2)IfeitherXorYisleft-dualizable,thenHom O(X;Y)'Y Hom (X;1O):(3)AnobjectXisleft-dualizableifandonlyifthereexistsanobjectX_endowedwithafunctorialisomorphismHom O(X;Y)'Y X_.5.2.2.Evidently,ifanobjectXisleft-dualizable,thenit'srelativelycompact.Proposition5.2.3.SupposethatOisgeneratedbyleft-dualizableobjects.Theneveryrelativelycompactobjectisleft-dualizable.Proof.LetXberelativelyO-leftcompact.WeneedtoestablishtheisomorphismY Hom O(X;1O)'Hom O(X;Y):Byassumption,bothsidescommutewithcolimitsinY.Hence,it'senoughtoestablishitforageneratingsetofY's.However,theisomorphismdoesholdwheneverYisleft-dualizable.6.Rigidmonoidalcategories6.1.LetObeamonoidalcategory.LetmultOdenotethetensorproductfunctorO O!O,andunitO:Vect!Otheunit.WeshallsaythatOisrigidifthefollowingconditionshold:TherightadjointofmultO,denotedmultO,belongstoFunctcont(O;O O).ThefunctormultO:O!O OiscompatiblewiththeleftandrightactionsofO.1TherightadjointofunitO,denotedunitO,belongstoFunctcont(O;Vect)(equivalently,theobjectunitO(k)2Oiscompact).Ifthishappens,it'seasytoseethatthedataof:O OmultO�!OunitO�!Vectand:VectunitO�!OmultO�!O OdeneanisomorphismO!O_;let'sdenotethisisomorphism1O.6.1.1.NotethatwhenOiscompactlygenerated,theconditionthatOberigidisequivalenttoOcbeingarigidmonoidalcategoryintheusualsense(i.e.,everyobjectadmitsaleftandarightdual). 1Apriori,it'sonlylaxcompatible 16DENNISGAITSGORYwrittenabove,andtheotheris(multO).WeconsiderbothsidesasendowedwithanactionofOontheright.TheassumptiononOsaysthatbothfunctorsarecompatiblewiththisaction.Hence,itisenoughtoidentifythetwocompositionsVectunitO!OO Ocoincide.Butthisiseasytoseethatbothidentifywith.Asacorollary,weobtain:Corollary6.2.4.LetOberigid,andletF:C1!C2beafunctorbetweenO-modulecat-egories.SupposethatF,whenviewedasafunctorbetweenjustDGcategoriesadmitsaleft(resp.,continuousright)adjointG.ThenthediagramO C1IdO F �����O C2actO;C1??y??yactO;C2C1 ����C2;thataprioricommutesuptoanaturaltransformation,actuallycommutes.Inparticular,GhasanaturalstructureoffunctorbetweenO-modulecategories.6.3.Hohschildhomologyandcohomology.LetObeamonoidalcategoryandKabi-modulecategory.Recallthatinthiscasewecanformthe"Hochschildhomology"categoryHHO(K),denedasthegeometricrealizationofthesimplicialcategoryKO K:::Inparticular,ifK=C1 C2withC1beingarightmoduleandC2aleftmodule,wehaveHHO(C1 C2)=:C1 OC2,thetensorproductofC1andC2overO.LetOandKbeasbefore.Wecanalsodenethe"Hochschildcohomology"categoryCHO(K),denedasthetotalizationoftheco-simplicialcategoryKO_ K:::Inparticular,fortwoleftmodulecategoriesC1;C2,bysettingK=Hom(C1;C2),wehaveCHO(Hom(C1;C2))'HomO-mod(C1;C2).ForC1aleftO-moduleandC2arightO-module,wewillusethenotationC1O C2:=CHO(C1 C2):6.3.1.AssumenowthatOisrigid.FromLemma1.3.3andProposition6.2.3,weobtain:Proposition6.3.2.HHO(K)'HCO(K0),whereK0isthesameasKasarightO-module,andtheleftO-modulestructureistwistedby'.Corollary6.3.3.LetObearigidmonoidalcategory.(1)LetC1andC2betwoleftO-moduleswithC1dualizableasacategory.ThenHomO-mod(C1;C2)'C_1 OC02;whereC02isobtainedfromC2bytwistingtheactionby'.(2)LetC1andC2berightandleftO-modules,bothdualizableascategories.ThenC1 OC2isdualizableanditsdualidentieswithC02_ OC_1.