A convenient category of locally preordered spaces

arXiv:0709.3646v2[math.AT]25Sep2007ACONVENIENTCATEGORYOFLOCALLYPREORDEREDSPACESSANJEEVIKRISHNANAbstract.Asapracticalfoundationforahomotopytheoryofabstractspace-time,weproposeaconvenientcategoryS′,whichweshowtoextendacategoryofcertaincompactpartiallyorderedspaces.Inparticular,weshowthatS′isCartesianclosedandthattheforgetfulfunctorS′→T′tothecategoryT′ofcompactlygeneratedspacescreatesalllimitsandcolimits.1.IntroductionAhomotopytheorywhichrespectstheflowoftimeonamachinestatespaceXcandetectbehaviorinvisibletotheclassicalhomotopytypeofX,asshownin[5,6,9,12,21].TakeasanexampleX=S1,thestatespaceofacyclicalprocess.Wemightwritex6S1ytoexpressthereachabilityofstateyfromstatex,buttheresultingpreorder6S1hasgraphS1×S1andthereforefailstodistinguishbetweenclockwiseandcounterclockwisetraversalsofS1.Asinglepreorderdoesnotadequatelydescribethe“local”behavioroftime.Theliteratureadoptsseveraldistinctformalismstoencodethetimeinabstractspacetime:an“atlas”onXofpartialordersin[5],adistinguishedchoiceofpathsonXin[10],aquotientmapE→XofspacestogetherwithapreorderonEin[13],andstructuremapsturningXintoasmall,topologicallyenrichedcategoryin[14];theresultingcategoriesofabstractspacetimesharecertaincharacteristicfeaturesidentifiedin[15].Weproposeanalternativeaxiomatizationofabstractspacetimewhichgeneralizesthepartiallyorderedspacesweencounterinnaturewhileformingacategoryconvenientforahomotopytheorist.Asin[17],wedefineastream(X,6)in§3.1tobeaspaceXequippedwithacirculation6,acoherentpreorderingU7→6Uofitsopensubsets.ThecategorySofmapspreservingallstructureinsightiscompleteandcocomplete.Colimits,limits,andsubstreamsarecolimits,limits,andsubspacesofspacesequippedwithuniversalcirculations,whichweconstructaspushforwardsandcosheafaficationsofpullbacksin§3.2.WecanthinkofSasanextensionofthecategoryKoflocallyconvex,compactHausdorffpartiallyorderedspaceswhoseboundedintervalsareclosedandconnected.THEOREM4.7.Thereexistsafullandconcreteembedding(1)K֒→SsendingeachK-objecttoauniquestreamitunderlies.Theimageof(1)containsallcompactHausdorffstreamshavinglocallyconvexunderlyingpreorderedspaceswhoseboundedintervalsareclosed.ThecategorySisnotCartesianclosed.In§5,wereplaceSwithitsfullsubcategoryS′ofcompactlyflowingstreams,analogoustothecategoryT′of12SANJEEVIKRISHNANcompactlygeneratedspaces.AstreamiscompactlyflowingifitislocallycompactHausdorff,forexample.OurnewforgetfulfunctorS′→T′createslimitsandcolimitsasbefore,ouroldembeddingK֒→ScorestrictstoanewembeddingK֒→S′,theformationofproductsandcolimitsinS′remainsintuitive,andS′containsallstreamsofinterest-butnowwehaveanadditionalconvenience.THEOREM5.13.ThecategoryS′isCartesianclosed.Wetranslateourformalismintootherswheretopical.Examples3.7,3.13,4.3,and5.3comparestreamswiththed-spacesof[10],whileExamples3.24,3.19,4.8,and5.12comparestreamswiththelocallypartiallyorderedspacesof[5].Thecomparisonfunctorsintheexamplesfacilitatetheconstructionofabstractspacetimeinallthreesettings.Wesuggestapossiblelineofresearchin§6.2.PreorderedspacesWefixsomeorder-theoreticnotationin§2.1andrecallthebasicdefinitionsofpreorderedspacesin§2.2.2.1.Someorder-theoreticconventions.RecallthatarelationRonasetX,generalizingafunctionX→X,encapsulatesthedataofitsdomaindomain(R)=Xanditsgraphgraph(R),asubsetofS×S.Example2.1.ArelationfonasetXisafunctionf:X→Xifforeachx∈X,thereexistsauniquey∈Ysuchthat(x,y)∈graph(f).ForarelationR,wewritexRyif(x,y)∈graph(R).Wewritex0R1x1R2...xn−1RnxnforasequenceR1,...,Rnofrelationsifxi−1Rxiforeach0i≤n.Restriction,inverses,andapplicationoffunctionsX→Xgeneralizetoarbitraryrelations.Definition2.2.ConsiderarelationRonasetX.Foreachx∈X,wewriteR[x]forthesubset{y|xRy}⊂X.WewriteR−1fortherelationonXhavinggraph{(y,x)|(x,y)∈graph(R)}.ForeachsubsetA⊂S,wewriteR↾AfortherelationonAhavinggraphgraph(R)∩(A×A).Amajorreasonwechoosetodistinguisharelationfromitsgraphissothatwecanunambiguouslydenote“productrelations,”generalizationsofproductsoffunctionsX→X.Definition2.3.ConsiderasetIandarelationRionasetXiforeachi∈I.LetX=Qi∈IXiandforeachi∈I,letπi:X→Xidenoteprojection.WewriteQi∈IRifortherelationonXhavinggraph\i∈I{(x,y)∈X×X|πi(x)Riπi(y)}.IfIconsistsoftwoelements,say0and1,wewriteQi∈IRiasR0×R1.Arelation6Xisapreorderifx6Xxforallxandx6Xzwheneverx6Xy6Xz.Furthermore,apreorder6Xisapartialorderifx=ywheneverx6Xy6Xx.Apreorderedset(poset)(X,6X)isasetXequippedwithapreorder(partialorder)6XonX.ACONVENIENTCATEGORYOFLOCALLYPREORDEREDSPACES3Example2.4.TheidentityfunctionidX:X→XonasetXisthepreorderonXwithsmallestgraph.Example2.5.Thestandardorder≤ontherealnumbersRisapartialorder.WecangeneralizetheclosedintervalsandconvexsubsetsofRtothesettingofarbitrarypreorderedsets.Definition2.6.Considerapreorderedset(X,6X).Theboundedintervalsof(X,6X)areallsubsetsofXoftheform6X[x]∩6−1X[y],x,y∈X.Moregenerally,asubsetC⊂Xisconvexin(X,6X)ify∈Cwheneverx6Xy6Xz,forallx,z∈Candally∈X.Example2.7(Geometricconvexity).ConsideravectorspaceVandletx,y,vdenotepointsinV.Foreachv,definearelationR(v)onVbytherulexR(v)yify−x=λzforsomescalarλ.AsubsetA⊂VisconvexintheusualsenseifandonlyifAisconvexin(V,R(v))foreachv.Wedenotethetra