IfwehaveaLsystemuoniIandwehaveamapfIdMJwithinotinJwecande neasystemufoniJForinstanceiftisaLsystemwede netibtobetheconditionsa t ibfor independentofianda t ibifib isinLThisisasy ID: 333653
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Ifwehaveconnectionsandnotdiagonal,wecannowdescribeamorphismI!JasamapI!D(J)whereD(J)isthefreedistributivelatticeonJandtwodistinctelementsofIaresenttoelementofdisjointsupport.Ifwetakeawaythedisjointsupportrestrictionweaddthediagonaloperation.Allthishasstillageometricinterpretationwithmaxandminoperationson[0;1].Finallywecanalsoaddanoperationcorrespondingto1�xon[0;1]byreplacingdistributivelatticebydeMorganalgebra.ToaddconnectionsallowstoreducetheKanllingoperationtotheKancompositionoperation.Thisholdsprovidedwehaveregularity(aliftingofaconstantpathisconstant).2SemanticsTypesareinterpretedby(covariant)presheavesoverthefollowingcategory.Theobjectsarenitesets(of\symbols")andamapI!JisasettheoreticmapI!dM(J)wheredM(J)isthefreedeMorganalgebraonJ.AdeMorganalgebraisadistributivelatticewithanoperation1�xwhichsatises1�0=01�1=01�(x_y)=(1�x)^(1�y)1�(x^y)=(1�x)_(1�y)(Thedi erencewithBooleanalgebrasisthatwedonotrequireneitherx^(1�x)=0norx_(1�x)=1.)Thereisa\nominal"descriptionofacovariantpresheafoverthiscategory.Wehaveasetofelementsuwhichmaydependsonanitesetofsymbolsu(i1;:::;in)andwecandosubstitutions,replacingi1;:::;inbyelementsindM(J)foranyJ=j1;:::;jm.IfwerestrictourselvestomapsI!dM(J)whichsendsdistinctsymbolstoelementswithdisjointsupport,wegetthenotionofcubicalsetswithconnections.IfweconsiderarbitrarymapsI!dM(J)weallowinparticulartohavethemapi=jwhichcorrespondstorestrictiontoadiagonal.Aparticularoperationthatwecandoonanelementu=u(i;j;k)istoreplaceiby0or1.Wewrite(ib)thisoperationforb=0;1sothatu(i0)=u(0;j;k)forinstance.Thiscorrespondstorestrictiontoafaceofu.3RemarksonthebasecategoryThebasecategoryChasforobjectsnitesetsofsymbolsI;J;K;:::andamorphismI!JisamapI!dM(J).Atypewillbeinterpretedasa(covariant)presheafonC,whileatypeA`IdependingonsymbolsinIisinterpretedbyapresheafonInC.ThatissuchatypeisafamilyofsetsAfforf:I!JwithrestrictionmapsAf!Afgu7�!ugforg:J!K.Wesaythatamapf:I!Jisstrictififisneither0nor1foralliinI.Onekeyremarkisthefollowing.Lemma3.1Iff:I!Jisstrictand indM(I)suchthat f=b(wherebis0or1)then =b.ThisdoesnotholdforBooleanalgebra.Forinstancethemap(i=j):fi;jg!fjgisstrictand(i^(1�j))(i=j)=0buti^(1�j)isneither0nor1.Eachfacemap :I!I isepi.Iff:I!Jwewritef6 tomeanthatthereexistsamapf0(uniquelydetermined)suchthatf= f0.Thismeansthatif=i foralliinthedomainof .Corollary3.2Iffg6 andgisstrictthenf6 .WeconsiderthepartialmeetsemilatticeMgeneratedbythefaceoperations(ib).AnelementofMcanbethoughtasanitesequenceoftheform(i0)(j1)(k0):::.Wedenoteby ;;:::anelementofM.InparticularMcontainstheemptysequence1M.ThereisacanonicalpartialorderonMand2 IfwehaveaL-system~uoni;Iandwehaveamapf:I!dM(J)withinotinJwecandeneasystem~ufoni;J.Forinstanceif~tisaL-systemwedene~t(ib)tobetheconditionsa =t (ib)for independentofianda =t (ib)if(ib) isinL.ThisisasystemofA(ib).Foranotherexample,ifweconsiderthesystemai0;ai1andthesubstitutioni=i^jweobtainthesystembi0=ai0;bj0=ai0;b(i1);(j1)=ai1.Hereisanexample(duetoGeorgesGonthier)whichshowstheproblemifweuseaBooleanalgebrainsteadofadeMorganalgebra.Considerthesystemu(i0)=ai0.Ifwetakei=j^kthesystembecomesu(j0)=ai0;u(k0)=ai0.Ifwethentakek=1�jwegetu(j0)=ai0;u(j1)=ai0.Butifwetakedirectlyi=j^(1�j)=0thenwegetthesystemu=ai0instead.Sothereisacoherenceproblem.AnelementainAiscompatiblewithorsatisestheJ-system~uifwehavea =u forall inJ.Wecanthenconsideratobeasolutionoftheconstraintsdenedby~u.Weonlyhaveoneoperation(0;1playsasymmetricrole)compiA;~u(ai0):A(i1)whereLisindependentofiandwhereai0:A(i0)isanelementindependentofiandsatisfyingtheL-system~u(i0)andwhichproducesanelementsatisfyingtheL-system~u(i1).Thesymboliisboundinthisoperation(butimayoccurinAandu ).ThisoperationshouldberegularinthesensethatcompiA;~u(ai0)=ai0wheneverA;~uisindependentofi.Fromthisoperationwecandene~a=lliA;~u(ai0)=compjA(i^j);~u(i^j)(ai0)elementoftypeAwhichsatises~a(i0)=ai0byregularityand~a =u (i^j)(j1)=u .Theuniformityconditionwhichisrequiredisthatwehaveiff:I!dM(J)andIcontainsthefreesymbolsofcompiA;~u(ai0)compiA;~u(ai0)f=compjAg;~ug(ai0f)whereg:I;i!dM(J;j)isanyextensionoffwithg(i)=jnotinJ(whichre ectsthatiisboundinthisoperationandai0isindependentofi).Lemma4.1IfwehaveaL-systemt ofAandai0inA(i0)andbothuandvinAsatisfyu =v =t u(i0)=v(i0)=ai0thenthereisaL-pathbetweenu(i1)andv(i1).Proof.WeintroduceafreshsymboljanddeneaL;(j0);(j1)-system~wbytakingw =t andwj0=uandwj1=v.WecanthenconsidercompiA;~w(ai0)whichisaL-pathbetweenu(i1)andv(i1). Hereisaspecialcaseofthecompositionoperationwhichwillbeconvenient.Lemma4.2GivenatypeAandainAandaL-systemof\lines"t :a !u inA,thereexistsa0inAsuchthata0 =u forall inL.Proof.Letibeafreshsymbol.Wecandenea0=compiA;~v(a)wherev =t i,sothatv (i0)=a andv (i1)=u . Wewritea0=comp(~t;a).Noticethat,byregularity,wehavecomp(~t;a)=aifalllinest areconstant.5ContractibletypesAtypeAiscontractiblei wecansolveinanuniformwayanyL-systeminA.ForinstancethefreeBooleanalgebraonthesetofsymbolsiscontractible.Thisfollowsfromthefactthatthereisexactlyonewaytollacubegivenitscorner.Forinstance,sincewehavea=(1�i)a(i0)+ia(i1)anylineu:u0!iu1hastobeu=(1�i)u0+iu1.4 7RepresentationofcubicalsetsIfwehaveawell-foundedsetX,wedenewhatisaX-set.ThisisinductivelygivenbyafamilyofpredicatesA for inXwhereA isaset-valuedpredicateonthesetofsequencesa; suchthataisinA(a ) .ToagivenX-setA wecanassociatethesetTofsequences(a )suchthata isinA (a) .IfweletT bethesetofsequencesa; suchthataisinA(a ) ,thenA isafamilyofsetsoverT .WealsohaveacanonicalrestrictionmapT!T forall inXandT !Tif .ForinstanceifXistheordinal012thenaX-setconsistsinasetA0withafamilyofsetsA1overA0andafamilyofsetsA2(a0;a1)fora0inA0anda1inA1(a0).ItalsocanbenoticedthatX-setsforminanaturalwayamodeloftypetheorywith;anduniverses.Forinstance,forX=01wegetthemodelwhereatypeisinterpretedbyasetA0togetherwithafamilyofsetsA1(a0).LetXIbethenite(andhencewell-founded)posetoffaceoperationsonI.IfforinstanceI=i;jthenaXIsetconsistsin4setsAib;jc,andset-valuedrelationsAib(xib;j0;xib;j1)andAjc(xi0;jc;xi1;jc)andarelationA1(xi0;j0;xi0;j1;xi1;j0;xi1;j1;xi0;xi1;xj0;xj1).WeshallwritesimplyI-setsinsteadofXI-sets.WerenethesemanticsofaclosedtypeA`I.ItisgivenbyafamilyofsetsAfforf:I!JandrestrictionmapsAf!Afgifg:J!K.WerequireAftobeaJ-set,andAf!Afgisthecanonicalrestrictionmapifgisafacemap:J!J.Itcanbecheckedthatalltype-formingoperationsproduceobjectsofthisform.ForinstanceifFandGareanypresheavesonCthenGF(I)isasetofsequencesfinF(J)!G(J)forf:I!J,satisfyingthecondition(fu)g=fgug,andthishasanaturalstructureofI-set.WedenethefollowingoperationonI-sets.LetAbeaI-setandT beaL-systemofI -sets,withcompatiblemaps :T !A .ThenwecanconsidertheI-set(~T;A),theelementofwhicharesequences(u )whereu isinT if 6LandinA otherwise.Forinstance,ifI=iandAisgivenbyAi0;Ai1;A1andwehavei0:Ti0!Ai0then(~T;A)isthesetofsequences(ti0;ai1;a1)suchthata1isinA1(i0ti0;ai1).Itisnaturaltowritetheelementsontheform(~t;a).Thekeyremarkisthenthatifeach istheidentitymapwehave(~T;A)=Aandwehave(~t;a)=aifwetaket =a .Thisbasicoperationwillbeusedtodeneglueing(whichtransformsequivalencetoequality)andthecompositionoperationintheuniverse.Ineachcase,wewillgetthesameunderlyingtypeifallmapsareidentities.Inthecaseofglueinghowever,theKancompositionoperationsdoesnotneedtostaythesame,whileitwillbethesameforcomposition,whichensuresregularityforcompositionintheuniverse.IfwehaveB=(~T;A)thenthereisacanonicalmapofI-sets:B!A.Forinstance,ifI=iandAisgivenbyAi0;Ai1;A1andwehavei0:Ti0!Ai0thenBisthesetofsequences(ti0;ai1;a1)suchthata1isinA1(i0ti0;ai1)andwedene(ti0;ai1;a1)=(i0ti0;ai1;a1).8GlueingoperationInordertointerpretunivalenceweexplainhowtotransformanequivalencetoanequality.GivenaL-system~TinUandatypeAtogetherwithacompatiblesystemofequivalences :T !A ,wedeneanewtypeB=glue(~T;A;~)AsacubicalsetBis(~T;A).Anelementofthistypeisoftheform(~t;a)witht :T anda:Asuchthata = t .IfL=f1Mg,wehaveonlyoneequivalence1:T1!AandwetakeB=T1.IfLisemptywetakeB=A.Iff:I!JwedeneBf=glue(~Tf;Af;~f)Ifwehaveoneequivalence:T!A,thenintroducingafreshsymboli,wehaveA=A(i0)andB=glue(Ti0;A;i0)withTi0=Tandi0=.ThistypeBwillbesuchthatB(i0)=TandB(i1)=A(i1)=A.Sowehaveanoperationtransforminganequivalencetoanequality.(Weareonly6 Proof.WedeneaL-systeminEe =lljE (t )andei0=lljE(i0)(ti0)sothate (j0)=t e (j1)=a ei0(j0)=ti0ei0(j1)=ai0Ifei1=compiE;~e(ei0)wealsohaveei1(j0)=ti1andei1(j1)=ai1.Wedenee0i1=lljE(ti1)sothate0i1(j0)=ti1ande0i1(j1)=(i1)ti1.Ifkisafreshsymbol,wetakeuk0=e0i1anduk1=ei1and~ai1=compjE(i1);~u(ti1)Wehavethen~ai1(k0)=(i1)ti1and~ai1(k1)=ai1.FurthermoreifEisindependentofjthene0i1=ti1byregularity.Wehavee =t ei0=ti0aswellbyregularity,sothatei1=ti1.Itfollowsthat~ai1=ti1alsobyregularity. Asforglueing,wedeneanelementofBtobeoftheform(~t;a)withainAandt inT suchthat t =a .TheoperationcompiB;~v(vi0)isalmostthesameasforglueing.Werstbuilda andai0andai1=compiA;~a(ai0).Thedi erenceisintherststep.For inL0,wecanconsiderinthetypeT (i1)ti1=compiT ;~v (vi0 )Webuildaline (i1)ti1!ai1 notbyusingLemma4.1butbybuildingdirectlyalinebetweenti1andai1 byusingLemma9.1.ThislinereducesthentoaconstantifE isindependentofj.10IdentitytypeTheKanoperationforidentitytypeissimilartotheonein[1].11FunctionextensionalityGivenf;goftypeAFandpsuchthatpu:fu!guwecomputeext(i;p):f!igforafreshi.Thisamountstodeneext(';p):AFforadeMorganformula'.Sincewedon'thavelinearitycondition,wedeneext(';p)u=pu'.12Propositionaltruncation13OperationalsemanticsWelimitourselvestothedescriptionofthesystemwithoutuniverses.Thepointistoexplainhowwecanjustifyfunctionextensionalitywithoutusingfunctionextensionalityatthemetalevel.Thesyntaxforthetermsist;A;F::=xjttjx:tjIdAttjAFjhiitjcompiA;~t(t)jexttttj'where'representsanelementinthefreedeMorganalgebraonthesymbols.Inthissyntax,hiitrepresentsthepathabstractionoperation,andbindsthesymboli.Similarly,compiA;~t(t)representsKancomposition;itbindsthesymboliand~trepresentsasystemofterms.Forinstance~tmaybeoftheformtj0;tj1oroftheformtj0;tk0;t(j1)(k1).Itmayalsobeempty,inwhichcasewewritesimplycompiA(t).Wehavetheusual-reductionrule(x:t)u=t(x=u)8 WecantheninterprettheusualJeliminationrule.Becauseoftheregularitycondition,thecompu-tationruleforJisinterpretedasajudgementalequality.Wecanaddt;A;F::=Njzerojs(t)jnatrecFttwiththeusualcomputationrulesnatrecFafzero=anatrecFafs(n)=fn(natrecFafn)ThecomputationrulesforcompiN;~u(ui0)arethefollowing.FirstwehavecompiN;~u(ui0)=zeroifui0=zeroand~uistheconstantsystemu =zero.SecondwehavecompiN;~u(ui0)=s(compiN;~v(vi0))ifui0=s(vi0)andu =s(v ).14TypingrulesWehavejudgementoftheforms�`I;�`I`Aand�`It:Arelativizedata\level"(nitesetofsymbols)I.TherulesaretheusualrulesoftypetheoryatalllevelsI,withtherestrictionrule�`It:A �f`Jtf:Afiff:I!dM(J).Thenewrulesarethenthefollowing.�`I�`I;iA�`Iai0:A(i0)� `I ;iu :A �`compiA~u(ai0):A(i1)withI =I�dom( ).Wealsohave�`IA�`Ia0:A�`Ia1:A �`IIdAa0a1�`IA�`Ia0:A�`Ia1:A�`I;it:A�`It(i0)=a0:A�`It(i1)=a1:A �`Ihiit:IdAa0a1Inparticular,wegetthere exivityproofofa:Abydeningreflaastheconstantpathfunctionhiia�`It:IdAa0a1 �`I;iti:A�`It:(x:A)!B�`Iu:(x:A)!B�`Ip:(x:A)!IdB(tx)(ux) �`Iexttup:Id((x:A)!B)tu15GeneralremarksaboutthemodelTherstremarkisthatallpathsinNareconstant,asexpected.Proposition15.1IisthepresheafdenedbyI(J)=dM(J)andNistheconstantpresheafN(J)=N.AnynaturaltransformationI!NisconstantandisdeterminedbytheimageofibythemapI(fig)!N.10