SIMPLICIAL APPROXIMATION

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1、TheoryandApplicationsofCategories,Vol.12,No.2,2004,pp.34–72.SIMPLICIALAPPROXIMATIONJ.F.JARDINEABSTRACT.Thispaperdisplaysanapproachtotheconstructionofthehomotopytheoryofsimplicialsetsandthecorrespondingequivalencewiththehomotopytheoryoftopologicalspaceswhichisbasedo

2、nsimplicialapproximationtechniques.Therequiredsimplicialapproximationresultsforsimplicialsetsandtheirproofsaregiveninfull.Subdivisionbehaveslikeacoveringinthecontextofthetechniquesdisplayedhere.IntroductionThepurposeofthispaperistodisplayadifferentapproachtotheconst

3、ructionofthehomotopytheoryofsimplicialsetsandthecorrespondingequivalencewiththehomo-topytheoryoftopologicalspaces.Thisapproachisanalternativetoexistingpublishedproofs[4],[10],butisofamoreclassicalflavourinthatitdependsheavilyonsimplicialapproximationtechniques.Theve

4、rificationoftheclosedmodelaxiomsforsimplicialsetshasareputationforbeingoneofthemostdifficultproofsinabstracthomotopytheory.Inessence,thatdifficultyisaconsequenceofthetraditionalapproachofderivingthemodelstructureandtheequivalenceofthehomotopytheoriesofsimplicialsetsandt

5、opologicalspacessimultaneously.Themethoddisplayedherestartswithusinganideafromlocalizationtheory(specifically,aboundedcofibrationcondition)toshowthatthecofibrationsandweakequivalencesofsimplicialsets,aswe’vealwaysknownthem,togethergenerateamodelstructureforsimplicials

6、etswhichisquiteeasytoderive(Theorem1.6).Thefibrationsforthetheoryarethosemapswhichhavetherightliftingpropertywithrespecttoallmapswhicharesimultaneouslycofibrationsandweakequivalences.Thisisthecorrectmodelstructure,butitisproducedatthecostofinitiallyforgettingaboutKan

7、fibrations.PuttingtheKanfibrationsbackintothetheoryintheusualway,andderivingtheequivalenceofhomotopycategoriesisthesubjectoftherestofthepaper.Theequivalenceofthecombinatorialandtopologicalapproachestoconstructinghomotopytheoryisreallythecentralissueofinterest,andisth

8、etruesourceoftheobserveddifficulty.RecoveringtheKanfibrationsandtheirbasicpropertiesaspartofthetheoryisdoneinawaywhichavoidstheusualtheoryofminimalfi