arXiv:math/0604414v3 [math.CT] 12 Jul 2006WeakComplicial SetsASimplicial Weak-CategoryTheoryPartI: BasicHomotopyTheoryDominicVerityToRossStreetontheoccasionofhis60thbirthday.Abstract. This paper develops the foundations of a simplicial theory of weak-categories, whichbuilds uponthe insights originallyexpoundedbyRossStreetinhis1987paperonorientedsimplices. Theresultingtheoryof weakcomplicial sets provides acommongeneralisationof the theories of (strict)-categories, Kancomplexes andJoyals quasi-categories. We generalise anumberof resultsdue tothecurrent authorwithregardto complicialsetsandstrict -categories toprovideanarmouryof well behavedtechnical devices,suchasjoinsandGraytensorproducts,whichwillbeusedtostudythesetheweak-categorytheoryof these structuresin a series of companionpapers. Inparticular,we establishtheir basic homotopytheorybyconstructinga Quillenmodelstructureonthecategoryofstratiedsimplicial setswhosebrantob-jects are the weak complicial sets. As a simple corollary of this work we provideanindependentconstructionof Joyalsmodel structureonsimplicial setsforwhichthebrantobjectsarethequasi-categories.Contents1. OverviewandHistory 22. IntroducingWeakComplicialSets 43. JoinsofStratiedSets 154. EquivalencesinWeakComplicialSets 195. GrayTensorProducts 316. QuillenModelStructuresonStratiedSets 437. AppendixA-SomeCategorical HomotopyTheory 60References 642000MathematicsSubjectClassication. Primary18D05,55U10;Secondary18D15,18D20,18D35,18F99,18G30.The author wouldlike tothankFitzwilliamCollege Cambridge whose visitingfellowshipprogrammesupportedthecompletionofthiswork.12 VERITY1. OverviewandHistoryThetheoryof complicial sets datestothemid-1970sandtotheworkof themathematicalphysicistJohnRoberts[16]. Hisoriginalinterestinthistopicgrewfrom hisconvictionthat(strict)-categories constitutedtheappropriate algebraicstructureswithinwhichtovaluenon-abeliancohomologytheories[15]. Thisledhim to dene complicial sets to be simplicial sets equipped witha distinguished setof neutral or thinsimplices,called stratiedsimplicial sets, and satisfying a certainkind of uniquethinhornllercondition. Heconjectured that it shouldbepossibletogeneralisetheclassicalcategoricalnerveconstructiontoprovideafunctorfromthe category of strict -categories to the category of complicial sets and this wouldprovideanequivalencebetweenthesetwocategories.TherststepinrealisinghisvisionwasmadebyRossStreetinhispaperonorientals[19], whichprovidedtherstfullyrigorousdescriptionof suchanerveconstructionandre-formulatedRoberts visionintoaspecicconjecture. Morerecently,theoriginalprogramoutlinedinthesepaperswascompletedbythecur-rent author[24]whoprovidedtherstproofofthefullStreet-Robertsconjecture.That workdemonstratesthatitisindeedthecasethatStreetsnerveconstructionprovidestheequivalencethatRobertsproposed.Thisresultitself providesanewandpowerful approachtostudyingstrict-categoriesthemselves. Forinstance, in[24] weshowhowtoconstructacombina-torially simple tensor product of stratied sets whose reection into the category ofstrict -categories is the laxGraytensorproduct. Calculations involving this latterstructureareknowntobecomplicatedbythefact thatitisusuallypresentedasacolimitof strict-categoriesfreelygeneratedbygeometricproductsof globs. Ontheotherhand, if wearewillingtoworkintheworldof stratiedsimplicial setsthenwemayinsteaddescribethecorrespondingtensordirectlybyequippingtheproductofunderlyingsimplicial setswithasuitablydenedsetofthinsimplices.Incontrast this latter structureis eminentlywell suitedtodirect combinatorialcalculation.However,itwasnotthiskindofapplicationtostrict-categorytheorywhichoriginally encouraged the current authors interest in the Street-Roberts conjecture.InsteaditwaspiquedbyparentheticcommentsinStreetspaperonorientedsim-plices [19] to the eect that we might use it as a foundation upon which to developauseful generalisationofthetheoryofbicategories tohigherdimensions. Atthattimenotrulyworkabletheoryofsuchweak-categorieshadbeenconstructed,al-thoughagrowinggroupofresearcherswerebecomingawareoftherolethatsuchstructuresmightplayinalgebraictopology, theoretical physics, computerscienceandhighercategorytheoryitself. Inbrief, Streetsideawasthatwemightobtainsuchatheorybyagainworkingwithstratiedsimplicial setsbutthistimeweak-eningtheaxiomsthatcharacterisedcomplicialsetsbyonlyinsistingthathornsofthekindidentiedinthattheoryshouldhavesome, notnecessarilyunique, thinller.Thesubsequent20yearshasbeenafertileoneinweak-category theoryandwemightnowidentifyintheliteraturethreeorfourquitedistinctapproachestodeningsuchstructures, eachof whichsplitsintoaplethoraof denitional sub-varieties. Inthis time Streets original side remarkhas remainedlargelyunderinvestigated,indeedforanumberofyearsthecurrentauthorhasavoidedwritingup his own ideas along these lines for fear of simply launching yet another weakenedWEAKCOMPLICIALSETSI 3higher category denition on the world. Spurred on, however, by Streets 2003 workonweak-categories[21], whichreformulatedandrenedhisoriginalinsightandintroducedthetermweakcomplicial set andJoyals work onquasi-categories[10],Iamnowoftheviewthatathoroughexplicationofthistheoryiswelloverdue.Sowhymight webe interestedinstudyingweak-categorytheories basedupon simplicial rather than globular geometries?From a philosophical perspective,Streethimselfsumsthecaseupbestinthefollowingpassagefrom[21]:Simplicial setsarelovelyobjectsabout whichalgebraictopologistsknowalot. Ifsomethingisdescribedasasimplicialset,itisreadytobeabsorbedintotopology. Or,inotherwords, nomatterwhichdenitionofweak-categoryeventuallybecomesdominant,itwillbevaluabletoknowitssimplicial nerve.Inshort, anyweak-categorytheoryworthitssaltshouldcomeequippedwithasimplicialnervefunctordescribingitsplaceinalgebraictopology. Furthermore,itis reasonable to expect that this would, at the very least, map each weak -categorytoaweakcomplicial set. Itfollows, therefore, thatanystudyof weakcomplicialsetsthemselveswill remainvaluableregardlessofwhichparticularformulationoftheweak-category notionmightbecomedominantinthefuture.Morepragmatically,theanswertothisquestionisreallyoneofutility. Asweshall seeherethetheoryof weakcomplicial setsisonewhichimmediatelygener-alises the most widely accepted 0-trivial and 1-trivial weak -categorical structures(Kan sets and quasi-categories respectively) and at the same timeencompasses thetheoryofstrict-categories. Furthermore, weshall alsodemonstrateherethatitsupports aplethora of wellbehaved -categorical constructions,such asjoins(sec-tion3)andGraytensorproducts(section5), andadmitsarichhomotopytheory(section 6). In a companion paper [22], we derive a nerve construction for categoriesenriched, in the classical sense of [12], in weak complicial sets with respect to a Graytensorproduct(calledcomplicial Gray-categories)whichfaithfullyrepresentssuchstructuresasweakcomplicialsets. Inparticular,thisdemonstratesthatthetotal-ityof all weakcomplicial setsandtheirhomomorphisms, strongtransformations,modicationsandsoforthisitselfrepresentableasaveryrichlystructured(large)weakcomplicialset.The actual categorytheoryof these structures will be exploredinanothercompanionpaper[23], whereinwerepresentweakcomplicialsetsascertainkindsofcompliciallyenrichedquasi-categories. Thisprovidesuswithanatural contextinwhichtogeneralisetraditionalcategorytheorytoakindofhomotopycoherentquasi-categorytheorywithintheQuillenmodel categoryof weakcomplicial setsitself. This approachallows us totranslateall of the basicconstructions of n-category theory into theweak complicial context and at thesame timeto establishforithomotopical versionsofthetheoriesofdiscretebrations, Yonedaslemma,adjunctions,limitsandcolimitsandsoforth.Whileall of this speakstotheexpressivenessof weakcomplicial set theory,we must also convince ourselves that it provides a strong enough framework withinwhich to establish certain natural coherence theorems. While work in this directionis still at a relatively early stage, studies to date indicate that it is likely that a directanalogueof thewell-knowncoherencetheoremsforbicategoriesandtricategoriesholdsinthis contextaswell. Tobeprecise, therearestrongreasonstosuspectthateveryweakcomplicial setsatisfyingcertainverymildconditionsonitsthin4 VERITY1-simplices (related to our work here in section 4) is equivalent to the nerve of somecomplicialGray-category.Herein, however, werestrictourselvestothemodesttaskof establishingthefoundationalhomotopytheoryofweakcomplicialsetsuponwhichall ofourlaterworkinthisareawill bebased. Section2introducesthesestructuresandestab-lishes the associated theory of anodyne extensions between stratied simplicial sets.Section 3 studies the join operation on stratied sets, introduces the correspondingdecalage constructionanddemonstratesthattheseareappropriatelywellbehavedwithrespecttoweakcompliciality. Thisworkisthenappliedinsection4toshowthatwecanusefullyreplacetheconditionwhichstipulatesthatweakcomplicialsetsmusthavethinllersforoutercomplicialhornswithonewhichsimplystatesthatallthin1-simplicesareactuallyequivalencesinsomesuitablesense.Section 5, which is combinatorially the most involved of this work, re-introducesthe (lax) Gray tensor products of [24] and studies their properties with regard weakcomplicial sets and anodyne extensions. In particular, this allows us to show that wemay construct weak complicial sets of homomorphisms, (lax) transformations, (lax)modicationsandsoforthbetweenanypairof weakcomplicial setsandtherebyenrich the category of these structures over itself in three distinct but related ways.Subsection 5.3 provides a new characterisation of strict complicial sets as those weakoneswhicharewell-tempered, inthesensethat forthesethinness isasucientpropertyforthedetectionofdegeneratesimplices.Finally section 6 draws together these various threads by constructing a Quillenmodelstructureonthecategoryofstratiedsimplicialsetswhosecobrationsarethe inclusions andwhosebrant objects arepreciselythe weakcomplicial sets.Furthermore,weshowthatthisisamonoidalmodel categorywithrespecttotheGray tensor products studied in section 5. Finally, we round out our presentation bylocalising our model structure andtransporting it to thecategory of simplicialsetsitself, in order to provide an independent construction of a model category structureonthatlattercategorywhosebrantobjectsareJoyalsquasi-categories [10].2. IntroducingWeakComplicialSetsHerewerecall thestandardnotationof thetheoryofsimplicialsets,introducetheirstratiedgeneralisations andestablishthebasicmachinery requiredtodeneandstudyweakcomplicial sets.2.1. StratiedSimplicialSets.Notation1 (simplicial operators). As usual we let +denote the (skele-tal)categoryof niteordinalsandorderpreservingmapsbetweenthemandusethenotationtodenoteitsfull subcategoryofnon-zeroordinals. Followingtra-ditionwe let [n] denote the ordinal n+ 1as anobject of +andrefer toar-rowsof+assimplicial operators. WewillgenerallyuselowercaseGreekletters, , ... :[m]

[n]todenotesimplicialoperators andlet im()denotethesub-seti [n] [ j [m] . (j) = i [n]knownastheimageoftheoperator. Wewillalsousethefollowingstandardnotationandnomenclaturethroughout: The injective maps in + are referred to as face operators. For each j [n]we usethe nj:[n 1]

[n] todenote the elementary faceoperatordistinguishedbythefactthatitsimagedoesnotcontaintheintegerj.WEAKCOMPLICIALSETSI 5 Thesurjectivemapsin+arereferredtoasdegeneracyoperators. Foreach j [n] we use nj:[n + 1]

[n] to denote the elementarydegener-acyoperatordeterminedbythepropertythattwointegersinitsdomainmaptotheintegerjinitscodomain. Foreachi [n] theoperatorni:[0]

[n] givenbyni (0) =iiscalledtheithvertexoperatorof[n]. Wealso usethenotations n:[n]

[0]and n:[1]

[n]todenotetheuniquesuchsimplicialoperators.Unlessdoingsowouldintroduceanambiguity, wewill tendtoreducenotationalclutterbydroppingthesuperscriptsoftheseelementaryoperators.Notation2(simplicial sets). ThecategorySimpof simplicial sets andsim-plicial maps betweenthemis simplythe functor category[op, Set], whereSetdenotes the(large) categoryof all (small) sets andfunctions betweenthem. IfX:op

Set is a simplicial set then we will often simplify our notation by usingXnfor theobject X([n]) / and Xfor thefunction X():X([m])

X([n]).Wealsoadoptthestandardlatinnotationsdni , sniandvnifortheactionsof theelementarysimplicialoperators ni ,niandnirespectively.Inpractice, it is ofteneasier tothinkof asimplicial set as asingleset en-dowed withapartially denedright actionof thesimplicial operators. Tobemoreprecise, thisdescriptionpresentsasimplicial setasatripleconsistingof asingleset X, adimensionfunctiondim:X

N, andapartial right actionxofarr()onxXwhichisdenedwheneverthedimensionof xXequalsthatofthecodomainof. Underthispresentation,simplicialmapsbecomefunc-tions ofunderlying setswhichpreserve bothdimensionandaction. Wesay that Xisasimplicial subset ofasimplicialsetY ,denotedXsY , ifXisasubsetofYwhich is closed in thereunder the action of simplicial operators and thusinherits asimplicialsetstructurefromit. Weadoptthefollowingtraditionaldenotationsofafewfundamentalsimplicialsets: Thestandardn-simplex[n]whichistherepresentablesimplicialseton[n], whoser-simplices areoperators:[r]

[n] acteduponbyrightcomposition. Theboundaryof then-simplex [n] whichis thesimplicial subset of[n] of thosesimplices:[r]

[n] whicharenotsurjective. Noticethattheboundaryof the0-simplex[0] issimplytheemptystratiedset. The(n 1)-dimensional k-hornk[n] whichisthesimplicial subset of[n] consistingofthosesimplices:[r]

[n] forwhichthereissomei [n]which is neither intheimage of nor equal to k(that isfor which[n],=im() k). Inotherwords,thisisthesmallestsimplicialsubsetof[n]containingthesetof(n 1)-faces ni: i [n] k.Wesaythatasimplexxof asimplicial setXisdegenerate ithereissomenon-identitydegeneracyoperatorandasimplexxXsuchthat x=x .Morespecicallywesaythatxisdegenerateat kif x=x kforsomesimplexxX, inwhichcasewewouldhavex=xk=xk+1. TheEilenberg-Zilberlemmatellsusthateverysimplexy Xmayberepresenteduniquelyasy = xwhereisadegeneracyoperatorandxisanon-degeneratesimplex.Finally,recall that Yonedaslemmaforsimplicial setstellsusthat thereexistsa natural bijection between the n-simplices of a simplicial set Xand simplicial maps6 VERITY[n]

X. ThisidentiesxXnwiththesimplicial mapxthatcarriesthesimplex [n], whichissimplyasimplicialoperatorwithcodomain[n], tothesimplexx()def=xinX.Notation3. Weintroducethefollowingnotationstodenotethesimplicesofthestandardsimplex[1]: 0r:[r]

[1]istheoperatorwhichmapseachi [r]to0 [1]. 1r:[r]

[1]istheoperatorwhichmapseachi [r]to1 [1]. ri :[r]

[1](1 i r)istheoperatordenedbyri(j) =

0 ifj< i,1 ifj i.As above, we shall adopt the conventionof omitting the superscripts ontheseoperatorsunlessdoingsowouldintroduceanambiguity. Lateronitwill becomeconvenient toindexthe r-simplices of [1] usingthe doubly pointed set [[r]]def=, +, 1, 2, ..., r, bylettingr=0r, r+=1randdeningrias abovefor anarbitrary integer(non-point)in[[r]].Observation4(nerves of categories). We shall also assume that the reader isfamiliarwiththeclassical nerveconstructionwhichfunctoriallyassociatesasim-plicial setN(C)toeachcategoryC. Thisisformedbyregardingtheorderedsets[n] tobecategoriesintheusual wayandapplyingKansconstruction[11] totheinclusionofasafullsubcategoryintoCat(thecategoryofsmallcategories),toobtainanadjointpair:CatN

SimpF.Inother words, the n-simplices of N(C) arefunctors f :[n]

Cuponwhichsimplicialoperatorsactbypre-composition.Definition5(stratied simplicial sets). A straticationon a simplicial set Xisasubset1tXofitssimplicessatisfyingtheconditionsthat no0-simplexofXisintX,and allofthedegeneratesimplicesofXareintX.Astratiedset isapair(X, tX)consistingofasimplicialsetXandachosenstraticationtXthe elements of whichwe call thin simplices. Inpractice, wewill electtonotionallyconfusestratiedsetswiththeirunderlyingsimplicial setsX, Y, Z, ... anduniformlyadoptthenotationtX, tY, tZ, ... for correspondingsetsof thinsimplices. Then, wheredisambiguationis required, weusethenotationX,Y ,Z, ...todenotetheunderlyingsimplicialsetsofthesestratiedsets.A stratied map f :X

Yis simply a simplicial map of underlying simplicialsetswhichpreservesthinness inthesensethatforallxtXwehavef(x)tY .Identities and composites of stratied maps are clearly stratied maps,from whichitfollowsthatwehaveacategoryStratofstratiedsetsandmaps.1Note thattXis merely a subset ofX,not a simplicial subset,in general it will not be closedinXundertheactionofsimplicialoperators.WEAKCOMPLICIALSETSI 7Definition6(stratiedsubsets,inverseanddirectimages). SupposethatUandXarestratiedsets, thenwesaythatUisastratiedsubset of X, denotedUsX,ifUisasimplicialsubset ofXanditsstratication tUisasubsetoftX.Iff :X

Y isastratiedmapthenthe: direct imageof the stratied subset UsXalong fis the stratied subsetf(U)sY withunderlyingsimplicial setf(x)[ xUandinwhichy f(U)isthinithereissomex tUwithf(x) = y. inverseimage of thestratiedsubset V sY alongf isthestratiedsubsetf1(V )sXwithunderlyingsimplicial setxX[f(x)V andinwhichx f1(V )isthinif(x)isthininV .Observation7 (inclusions of stratiedsets). We call the monomorphismsinStrat stratiedinclusions andthese arecustomarilydenotedbyarrows withhookeddomainsi :X

Y . AstratiedsubsetXof Y clearlygivesrisetoacorresponding stratied inclusion which we denote by X

s

Y . Indeed, wherevernecessary we mayalways replace an arbitrary stratiedinclusion byanisomorphicsubsetinclusion.The forgetful functor from Strat to Set which carries a stratied set to its set ofsimplicespreservescolimitsandreectsmonomorphisms. Itfollowsthattheclassof stratied inclusions is closed in Strat under pushout, transnite composition andretractionsincethisisthecasefortheclassofinjectivefunctionsinSet. Further-moretheclassof all stratiedinclusions is thecellular completionof theset ofboundaryandthinsimplexinclusions:[n]

r

[n] [ n = 0, 1, ... [n]

e

[n]t[ n = 1, 2, ...Definition8(stratied subsets, regularity and entirety). We say that a strat-iedsubsetXofY is: regular,denotedXrY ,iftX=X tY ,and entire,denoted XeY ,ifX=Y .The terms regularsubsetand entire subsetwill always be taken to denote stratiedsubsets which possess the appropriate property. If Wis a subset of simplices of thestratiedsetXthenthestratied(resp. regularorentire)subsetofXgeneratedbyWisdenedtobethesmallestsuchstratiedsubsetofXwhichcontainsW.Extendingthesedenitionstoallstratiedmaps, wesaythatf :X

YStratisregularifitreectsthinsimplices, meaningthatwheneverf(x)isthininY itfollowsthatxisthininX,andentireifitissurjectiveonsimplices.A stratied map f :X

Yadmits two well-behaved canonical factorisations: regular image factorisationXfe

imr(f)

r

Yin which the stratiedmapfeisentireandimr(f),theregularimageoff,istheregular subsetofY whosesetofsimplicesisy Y[ x X . f(x) = y. entire coimage factorisationX

e

coime(f)fr

Yin which the strat-ied map fris regular and coime(f), the entire coimageof f, is the entiresuperset of Xwhose thinsimplices are those x Xfor which f(x) is thininY .Notation9(complicial simplices and horns). The functor from Strat to Simpwhichforgetsstraticationshasbothaleftandarightadjoint, whichassigntoasimplicial setitsminimal andmaximal straticationrespectively. Wewill implic-itlypromoteanysimplicialsetX Simptoastratiedsetusingthe(leftadjoint)8 VERITYminimal stratication, underwhichitssetsofdegenerateandthinsimplicescoin-cide, andtherebyregardSimpasafull subcategoryof Strat. Inparticular, therepresentablesimplicial sets[n] Simpprovideuswithgeometrical modelsforthestandardsimplicesinStrat.Afewotherstratiedsetswill takeonparticularimportanceinourdelibera-tionslateron: The standardthinn-simplex[n]tconstructed from [n] by making thinitsuniquenon-degeneraten-simplexid[n]:[n]

[n] [n]. Thek-complicial n-simplex k[n]constructedfrom [n]bymaking thinall those simplices :[r]

[n] whose image contains the set of integersk1, k, k+1 [n]. Non-degenerate simplices satisfying this latterconditionaresaidtobek-admissible. The (n1)-dimensional k-complicial horn k[n] whichis the regularsubsetof k[n] of thosesimplices:[r]

[n] forwhichtheset[n] (im() k)isnon-empty. Inotherwords,thisistheregularsubsetofk[n] generated by its set i[ i [n] k of all (n1)-faces except k. The stratied set k[n]and itsregular subset k[n]whichare obtainedfromk[n]andk[n](respectively)bymakingall(n 1)-simplicesthin. Theunionk[n] def=k[n] k[n]ek[n]whichmaybeconstructedfromk[n]bymakingthinthe(n 1)-simplicesnk1andnk+1.Whilethestraticationsof thesecomplicial simplicesmayseemalittlelessthanintuitive,theyarehoweverfundamentaltomuchofthetheorythatfollows. Moti-vationforthesechoicesisprovidedbythevariousworksofRoberts[15]and[16],Street[19]and[20]andVerity[24].Observation10(k-admissibilityrecast). Itissometimeuseful torecastthedenitionof k-admissibilityslightly. Tothatend, it iseasilyshownthat anon-degenerater-simplex[n] is k-admissible if andonlyif there exists somel [r]suchthat(i) = k +i lforeachi [r] l 1, l, l + 1.Observation11(Strat as aLFPquasi-topos). Thefull subcategorytofstandardsimplicesandstandardthinsimplicesisdenseinStrat(cf. chapter5ofKelly[12]), therebyprovidingus withareective full embeddingof Strat intothepresheafcategory[top, Set]. Moreexplicitly,tmaybeobtainedfrombyappending extra objects [n]t for n = 1, 2, ... and extra operators nk :[n + 1]t

[n]andn:[n]

[n]tsatisfyingthe relations nkn+1=nk. Apresheaf F [top, Set]isisomorphictosomestratiedsetifandonlyifitmapseachoperatorn:[n]

[n]ttoamonomorphisminSet. ItfollowsthatthecategoryStratislocallynitelypresentable, sinceit isequivalenttothecategoryof modelsforanitelimitsketchont.Theutilityof thisobservation isimmediatelyclear, for instance ittells usthatStrat has limits whichare calculatedpointwise, colimits whichare constructedin [top, Set] and then reected into Strat and that its nitely presented ob-jects are those stratiedsets withonlyanite number of non-degeneratesim-plices. Furthermore,asobservedbyStreetin[21],theleftadjointtotheinclusionStrat

[top, Set] preservespullbacksofpairsofmorphismsinto(imagesof)stratiedsetsfromwhichitfollowsthatStratisaquasi-topos. Inotherwords,foreachstratiedsetXtheslicecategory Strat/XiscartesianclosedandStrathasaclassierforregularsubobjects.WEAKCOMPLICIALSETSI 9Notation12 (skeletaandsuperstructures). Wesaythat astratiedset isn-skeletal if all of itssimplicesof dimensiongreaterthannNaredegenerate.Then-skeletonskn(X) of a stratiedset Xisitsregular subset consisting of thoseof itssimpliceswhosefacesof dimensiongreaterthannareall degenerate. Thisconstructionprovides us withanendo-functor of Strat whose range is the fullsubcategory of n-skeletal stratied sets and which has a right adjoint ckncalled then-coskeletonfunctor.Playingthesamegamewiththinness, wesaythatastratiedsetisn-trivialif all of its simplices of dimensiongreaterthannarethin. Then-trivialisationthn(X) of astratiedset Xisitsentire superset constructedbymaking thinallofits simplices of dimension greater than n. Again thisconstruction provides us withanendo-functorofStratwhoserangeisthefull-subcategoryofn-trivial stratiedsetsandwhichhasarightadjointspncalledthen-superstructurefunctor. Then-superstructure spn(X) may be realised as the regular subset of Xof those simpliceswhosefacesofdimensiongreater thannareallthin.2.2. WeakComplicial Sets. Nowwearereadytoembarkondeningandstudyingweakcomplicialsets:Notation 13 (lifting problems and properties). A commutative square in somecategory (Uu

i

Ep

Vv

Ais called a liftingproblemfrom i to p and it is said to haveasolutionif there existssomediagonalmap(dottedinthediagram)whichmakesbothtrianglescommute.Whensuchasolutionexistswesaythatihastheleftliftingproperty(LLP)withrespecttoporthatphadtherightliftingproperty(RLP)withrespecttoi.We saythat anobject C( has the RLPwithrespect tothe morphismi :U

Vi the unique map:C

1 into the terminal object of (enjoys thatproperty. Insuchacase, aliftingproblemamountstoamorphismu:U

Candasolutiontothisissimplyamorphism u:V

Cforwhich u i = u.Definition14 (elementaryanodyne extensions andweakcomplicial sets).The set of elementaryanodyneextensionsin Strat consists of two families of subsetinclusions: k[n]

r

k[n]forn = 1, 2, ...andk [n],thesearecalledcomplicialhornextensions,and k[n]

e

k[n]for n = 2, 3, ... and k [n], these are called complicialthinnessextensions.Weclassifytheseelementaryanodyneextensions intotwosub-classes, theinneronesforwhichtheindexksatises0