ThegregariousmodelstructurefordoublecategoriesAlexanderCampbellCentreo - PDF document

ThegregariousmodelstructurefordoublecategoriesAlexanderCampbellCentreofAustralianCategoryTheoryMacquarieUniversityMasarykUniversityAlgebraSeminar18June2020 AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20201/27 Plan 1 Motivation:Whatisadouble1-category? 2 Thegregariousmodelstructurefordoublecategories 3 Doublequasi-categories AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20202/27 Plan 1 Motivation:Whatisadouble1-category? 2 Thegregariousmodelstructurefordoublecategories 3 Doublequasi-categories AlexanderCampbell(CoACT) Thegregariousmodelstructu

refordoublecategories MUNIAlgebraSeminar18June20203/27 Doublecategories Adoublecategoryconsistsof: objectsa, horizontalmorphismsf:a// b, verticalmorphismsu:a // c, doublecellsaf//  u  b v cg// d horizontalcompositionoperationsforhorizontalmorphismsanddoublecells, verticalcompositionoperationsforverticalmorphismsanddoublecells, identityhorizontalandverticalmorphismsandidentitydoublecells,allsubjecttoassociativity,unit,andinterchangeaxioms.DoublecategoriesanddoublefunctorsformthecategoryDblCat. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20204/

27 Doublecategoriesasbisimplicialsets LetssSet=[
op
op;Set]denotethecategoryofbisimplicialsets.LetXbeabisimplicialset.Foreachn0,wecallthesimplicialsetsXn;andX;nthenthcolumnandthenthrowofXrespectively.ThereisafullyfaithfulnervefunctorN:DblCat�!ssSet,inducedbythefunctor

�!ssSetthatsends([m];[n])totheexteriorproduct[m][n](thefree-livingm-by-ngridofdoublecells).So,foreachdoublecategoryA,(NA)0;0,(NA)1;0,(NA)0;1,and(NA)1;1arethesetsofobjects,verticalmorphisms,horizontalmorphisms,anddoublecellsofArespectively.AbisimplicialsetXisisomorphictothenerveofadoublecategoryifandonlyifeachcolumnXm;a

ndeachrowX;nofXisisomorphictothenerveofacategory,i.e.ifandonlyifthehorizontalandverticalSegalmapsXm;n�!Xm;1Xm;0


Xm;0Xm;1Xm;n�!X1;nX0;n


X0;nX1;narebijectionsforallm;n0. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20205/27 DoubleSegalspacesI Question Whatisthecorrect1-categoricalgeneralisationofthenotionofdoublecategory? LetSdenotethe1-categoryof1-groupoids. Candidate1 AdoubleSegalspaceisafunctorX:
op
op�!SsuchthateachcolumnXm;andeachrowX;nisaSegalspace. Drawback:Thisdenitionistoogeneral.Itlacksaco

mpletenesscondition,àlaRezk'scompleteSegalspaces,totiethehomotopicalstructureofadoubleSegalspacetoitscategoricalstructure.Butwhatisthemissingcompletenesscondition? AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20206/27 DoubleSegalspacesII Question WhatshouldbeacompletedoubleSegalspace? Candidate2 Adouble1-categoryisaSegalobjectinthe1-categoryof1-categories,i.e.adoubleSegalspaceXsuchthateachcolumnXm;isacompleteSegalspace. Drawback:Thisdenitionisnotsymmetricinthehorizontalandverticaldirections.Sothetransposeofadouble1-categoryisnotalwaysadouble1-category. Cand

idate3 AdoublecompleteSegalspaceisadoubleSegalspaceXsuchthateachcolumnXm;andeachrowX;nisacompleteSegalspace. Drawback:Thisdenitionistoorestrictive.Forexample,twoobjectsinsuchanXareequivalentinthecompleteSegalspaceX0;ifandonlyiftheyareequivalentinthecompleteSegalspaceX;0.Thisexcludesimportantexamples,suchasthepseudodoublecategoryofrings,ringhomomorphisms,andbimodules.(Ringisomorphism6=Moritaequivalence.) AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20207/27 Plan 1 Motivation:Whatisadouble1-category? 2 Thegregariousmodelstructurefordoublecategories 3 Doublequasi-categor

ies AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20208/27 Modelstructures AmodelstructureonacategoryEenablesonetodohomotopytheoryinE. Modelstructures AmodelstructureonacategoryEconsistsofthreeclassesofmorphisms(C;W;F)inEcalledcobrations,weakequivalences,andbrationssuchthat: Wsatisesthe2-out-of-3property,and (C;W\F)and(C\W;F)areweakfactorisationsystemsinE. AnobjectAiscobrantif0�!Aisacobration.AnobjectXisbrantifX�!1isabration.ThemorphismsintheclassesC\WandW\Farecalledtrivialcobrationsandtrivialbrationsrespectively. Lemma Amo

delstructureonacategoryisdeterminedbyitsbrantobjectsandtrivialbrations. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June20209/27 Thegregariousmodelstructurefordoublecategories Lemma Amodelstructureonacategoryisdeterminedbyitsbrantobjectsandtrivialbrations. Lack'smodelstructurefor2-categories Thereexistsauniquemodelstructureon2-Catinwhich: every2-categoryisbrant,and a2-functorisatrivialbrationiitissurjectiveonobjects,fullon1-morphisms,andfullyfaithfulon2-cells. Theorem(Thegregariousmodelstructurefordoublecategories) ThereexistsauniquemodelstructureonDblCatinwhic

h: everydoublecategoryisbrant,and adoublefunctorisatrivialbrationiitissurjectiveonobjects,fullonhorizontalmorphisms,fullonverticalmorphisms,andfullyfaithfulondoublecells. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202010/27 Somefunctorsfrom2-CattoDblCat LetAbea2-category.ThedoublecategorySq(A)calledthedoublecategoryofsquaresofAhasthesameobjectsasA,itshorizontalandverticalmorphismsarethemorphismsofA,anditsdoublecellsarethesquaresinA:af// u  bv cg// dLetH(A)denotethelocallyfullsubdoublecategoryofSq(A)whosehorizontalmorphismsarethemor

phismsofA,andwhoseverticalmorphismsaretheidentitymorphismsofA.Dually,onehasthedoublecategoryV(A)whoseverticalmorphismsarethemorphismsofA,butwhosehorizontalmorphismsaretheidentitymorphismsofA.ThesethreeconstructionsdenefunctorsSq;H;V:2-Cat�!DblCat. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202011/27 SomefunctorsfromDblCatto2-Cat LetAbeadoublecategory.LetH(A)thehorizontal2-categoryofAdenotethe2-categorywiththesameobjectsasA,whosemorphismsarethehorizontalmorphismsofA,andwhose2-cellsarethedoublesquaresofAoftheformdisplayedontheleftbelow.af//  b ag// ba u &

#15;  av  b bDually,letV(A)thevertical2-categorydenotethe2-categorywiththesameobjectsasA,whosemorphismsaretheverticalmorphismsofA,andwhose2-cellsarethedoublecellsofAoftheformdisplayedontherightabove.ThesetwoconstructionsdenefunctorsH;V:DblCat�!2-Cat.Weshallsometimesrefertotheinvertible2-cellsinthe2-categoriesH(A)andV(A)asglobularisomorphismsinA. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202012/27 Companionpairs LetAbeadoublecategory. Companionpairs AcompanionpairofmorphismsinAconsistsof: ahorizontalmorphismf:a// binA, averticalmorphismu:

a // binA,and doublecellsinAa  au  af// baf// u  " b b bsuchthat  "=1fand"=1u. AcompanionpairinAamountspreciselytoadoublefunctorSq(2)�!A.Wesaythatahorizontalmorphismf:a�!binAhasaverticalcompanionifitextendstoacompanionpairinA.Duallyforverticalmorphisms. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202013/27 GregariousequivalencesI LetAbeadoublecategory. Denition(gregariousequivalence) Acompanionpairofmorphisms(f;u)inAissaidtobeagregariousequivalenceinAiffisanequivalenceinthehorizontal2-categoryH(A)andui

sanequivalenceinthevertical2-categoryV(A). LetEdenotethefree-livingadjointequivalence2-category. Lemma Acompanionpairofmorphisms(f;u)inAisagregariousequivalenceithereexistsanextensionSq(2)(f;u)//  ASq(E) 9== AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202014/27 GregariousequivalencesII Lemma Letf:a�!bbeahorizontalmorphisminA.Thefollowingareequivalent. 1 fextendstoagregariousequivalenceinA. 2 fhasaverticalcompanionuinA,fisanequivalenceinH(A),anduisanequivalenceinV(A). 3 fhasapseudo-inverseginH(A),andbothfandghaveverticalcompanionsinA. Byanabuseoflanguage,weshallsay

thatahorizontalmorphisminAisagregariousequivalenceifitsatisestheequivalentpropertiesofthelemma.Duallyforverticalmorphisms. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202015/27 Weakequivalencesinthegregariousmodelstructure Proposition AdoublefunctorF:A�!Bisaweakequivalenceinthegregariousmodelstructurefordoublecategoriesiitis: 1 surjectiveonobjectsuptogregariousequivalence(i.e.foreveryobjectb2B,thereexistsanobjecta2AandagregariousequivalenceFa!binB), 2 fullonhorizontalmorphismsuptoglobularisomorphism, 3 fullonverticalmorphismsuptoglobularisomorphism,and 4 fullyfaithfulondoublecel

ls. By(2)wemeanthatforeachpairofobjectsa;binA,andeachhorizontalmorphismg:Fa�!FbinB,thereexistsahorizontalmorphismf:a�!binAandaverticallyinvertibledoublecellinBasbelow.Fa Ff// = Fb Fag// FbDuallyfor(3). AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202016/27 Fibrationsinthegregariousmodelstructure Proposition AdoublefunctorF:A�!Bisabrationinthegregariousmodelstructurefordoublecategoriesi: 1 foreveryobjecta2A,andeverygregariousequivalenceg:Fa!binB,thereexistsagregariousequivalencef:a!a0inAsuchthatF(f)=g, 2 the2-functorH(F)isanisobrationonhom-categories,and

3 the2-functorV(F)isanisobrationonhom-categories. LetI2denotethefree-livinginvertible2-cell2-category. Proposition Adoublefunctorisabrationinthegregariousmodelstructurefordoublecategoriesiithastherightliftingpropertywithrespecttothefollowingthreedoublefunctors: 1 f0g�!Sq(E), 2 H(2)�!H(I2),and 3 V(2)�!V(I2). AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202017/27 Theproofofexistenceofthegregariousmodelstructure Theproofoftheexistenceofthegregariousmodelstructurefollowsbystandardargumentsfromthefollowingpropositions.ThisfollowstheargumentofLack'sproofoftheexiste

nceofhismodelstructureforbicategories. Proposition TheclassofgregariousweakequivalencesinDblCatsatsesthe2-out-of-6property. Proposition AgregariousbrationinDblCatisagregariousweakequivalenceiitisatrivialbration. Proposition ThebrationsandtrivialbrationsaretherightclassesofcobrantlygeneratedweakfactorisationsystemsinDblCat. Proposition EverydoublecategoryAadmitsapathobject,i.e.afactorisationofthediagonalA�!AAintoaweakequivalencefollowedbyabration. TheobjectsofourchoiceofpathobjectofAarethegregariousequivalencesinA. AlexanderCampbell(CoACT) Thegregariousmodel

structurefordoublecategories MUNIAlgebraSeminar18June202018/27 Cobrationsinthegregariousmodelstructure Recallthata2-functorisacobrationinLack'smodelstructureon2-CatiitsunderlyingfunctorhastheLLPinCatwrtallsurjective-on-objects-and-fullfunctors. Proposition AdoublefunctorF:A�!Bisacobrationinthegregariousmodelstructureithe2-functorsH(F)andV(F)arecobrationsinLack'smodelstructureon2-Cat. Corollary AdoublecategoryAiscobrantinthegregariousmodelstructureiitsunderlyingcategoriesH(A)0andV(A)0ofhorizontalmorphismsandverticalmorphismsarefree. Adoublepseudofunctorisakindofmorphismofdoublecategoriesthatisweak(i.e

.pseudofunctorial)inbothdirections. Proposition LetAbeadoublecategory.ThenthedoublepseudofunctorclassierQAofAisacobrantdoublecategory,andthecounitdoublefunctorQA�!Aisabijective-on-objectsweakequivalence.Thatis,QAisacobrantreplacementofA. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202019/27 Thedoublecategoryofsquaresfunctor ThedoublecategoryofsquaresfunctorSq:2-Cat�!DblCathasbothaleftadjoint(calledStringbyEhresmann&Ehresmann),andarightadjointC:DblCat�!2-Cat.Notethat,foreachdoublecategoryA,themorphisms(resp.equivalences)inC(A)arethecompanionpairs(resp.gregari

ousequivalences)inA. Proposition ThefunctorSq:2-Cat�!DblCat: isbothleftandrightQuillenwrtLack'smodelstructureon2-CatandthegregariousmodelstructureonDblCat, createsLack'smodelstructureon2-CatfromthegregariousmodelstructureonDblCat, ishomotopicallyfullyfaithful. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202020/27 Workinprogress HereareacoupleofpropertiesofthegregariousmodelstructurewhichIexpecttobetrue,butwhichIhaven'tyetproved. ThegregariousmodelstructureonDblCatisproper. ThegregariousmodelstructureonDblCatismonoidalwithrespecttoBöhm'sGraytensorproductfordoublecategories. Theorem(90%com

plete) ThereexistsaleftBouseldlocalisationofthegregariousmodelstructureonDblCatinwhich: 1 adoublefunctorisalocalweakequivalenceiitissurjectiveonobjectsuptohorizontalequivalence,fullonhorizontalmorphismsuptoglobularisomorphism,surjectiveonverticalmorphismsuptohorizontalequivalence,andfullyfaithfulondoublecells(i.e.aweakequivalenceintheMoserSarazolaVerdugomodelstructure,seearXiv:2004.14233); 2 adoublefunctorF:A�!BisalocalbrationiH(F)isanequibration,V(F)isanisobrationonhoms,andanyhorizontalequivalenceinAsentbyFtoagregariousequivalenceinBisagregariousequivalenceinA.Inparticular,adoubl

ecategoryAisbrantinthismodelstructureifandonlyifeveryhorizontalequivalenceinAhasaverticalcompanion. [Afterthistalk,itwaspointedouttomethatthisdescriptionofthelocalisedmodelstructurecannotbecorrect,becausethepurportedtrivialcobrationsarenotstableunderpushout!] AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202021/27 Plan 1 Motivation:Whatisadouble1-category? 2 Thegregariousmodelstructurefordoublecategories 3 Doublequasi-categories AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202022/27 Cisinskimodelstructures InspiredbyAra'sdenitio

nof2-quasi-categories,weshalldenedoublequasi-categoriesasthebrantobjectsofaCisinskimodelstructure. Localisers LetAbeasmallcategory.Amorphismin[Aop;Set]isatrivialbrationifithastherightliftingpropertywithrespecttoallmonomorphismsin[Aop;Set].AnA-localiserisaclassWofmorphismsin[Aop;Set]suchthat: 1 Wsatisesthetwo-out-of-threeproperty, 2 everytrivialbrationbelongstoW, 3 theclassofmonomorphismsbelongingtoWisstableunderpushoutandtransnitecomposition. Cisinskimodelstructures LetSbeasmallsetofmorphismsin[Aop;Set].Thereexistsauniquemodelstructureon[Aop;Set]whosecobrationsarethemonomorphisms,andwhoseclassofweakequivalen

cesisthesmallestA-localisercontainingS.WewillcallthismodelstructuretheCisinskimodelstructuregeneratedbyS. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202023/27 Doublequasi-categories Let:sSetsSet�!ssSetdenotetheexteriorproductfunctor,denedby(XY)m;n=XmYn.Foreachn2,letI[n]�!
[n]denotethespineinclusionofthefree-livingn-simplex. Themodelstructurefordoublequasi-categories Wedenethemodelstructurefordoublequasi-categoriesonssSettobetheCisinskimodelstructuregeneratedbythemorphisms: I[m]
[n]�!
[m]
[n],forallm2,n0,
[m]I[n]&#

0;!
[m]
[n],forallm0,n2, NH(I2�!2), NV(I2�!2). Denition Adoublequasi-categoryisabisimplicialsetthatisbrantinthemodelstructurefordoublequasi-categories. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202024/27 Ahomotopycoherentnervefordoublecategories Watson'snervefunctor DenethecoherentnervefunctorNW:DblCat�!ssSettobethenerve(orsingular)functorinducedbythefunctor

�!DblCatthatsends([m];[n])tothenormaldoublepseudofunctorclassierof[m][n]. So,foreachdoublecategoryA,theelementsoftheset(NWA)m;narethenormaldoublepseudofunctors[m][n

]�!A. Theorem ThecoherentnervefunctorNW:DblCat�!ssSet: isarightQuillenfunctorfromthegregariousmodelstructurefordoublecategoriestothemodelstructurefordoublequasi-categories; right-inducesthegregariousmodelstructurefordoublecategoriesfromthemodelstructurefordoublequasi-categories; ishomotopicallyfullyfaithful. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202025/27 CompletedoubleSegalspaces SaythatadoubleSegalspaceX:
op
op�!Sislocalwithrespecttoamorphismofbisimplicialsetsf:A�!BwhentheinducedmapbetweenweightedlimitsftX:BtX�!AtXisanequivalenceof1-groupoids. Denition(c

ompletedoubleSegalspace) AdoubleSegalspaceXiscompleteifitislocalwithrespecttothefollowingmorphismsofbisimplicialsets: 1 NH(I2�!2), 2 NV(I2�!2), 3 theprojectionNSq(I)
[m;n]�!
[m;n],forallm;n0. Iexpect(buthavenotproved)thatissucestotakem=n=0in(3). Theorem The1-categoryofcompletedoubleSegalspacesisthe1-categorypresentedbythemodelcategoryofbisimplicialsetsequippedwiththemodelstructurefordoublequasi-categories. AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18June202026/27 Thankyou! AlexanderCampbell(CoACT) Thegregariousmodelstructurefordoublecategories MUNIAlgebraSeminar18Jun