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PAGE 1 ANATURALISOMORPHISMFROMTHEORDEREDHOMOLOGYTOTHEORIENTEDHOMOLOGYOFANINJECTIVESETBYNATHANIELCHANDLERAThesisSubmittedtotheDivisionofNaturalSciencesNewCollegeofFloridainpartialfulllmentoftherequirementsforthedegreeBachelorofArtsUnderthesponsorshipofPatrickMcDonaldSarasota,FloridaApril,2009 PAGE 2 DedicationTomyadvisorPatMcDonald.TomyfriendAdeleFournet.i PAGE 3 ANATURALISOMORPHISMFROMTHEORDEREDHOMOLOGYTOTHEORIENTEDHOMOLOGYOFANINJECTIVESETNathanielChandlerNewCollegeofFlorida,2009ABSTRACTInthebasiccombinatorialtopologyofsimplicialcomplexes,ithasbeenproventhattheorderedhomologyfunctorisnaturallyisomorphictotheorientedhomologyfunctor.Inthisthesiswegeneralizethisresultsubstantially.Weprovethatthereisanaturalisomorphismfromtheorderedhomologytotheorientedhomologyofthesingularsetfunctorassociatedtoawell-behavedsimplicial-typeobjectinacategorysubjecttosomemildrestrictions.Whiletheauthor'sliteraturereviewdidnotturnupthisresult,hemakesnoclaimstooriginality.PatrickMcDonaldDivisionofNaturalSciencesii PAGE 4 Contents1.Introduction12.TopologicalPreliminaries32.1.IntroductoryRemarks32.2.SimplicialComplexes32.3.SimplicialHomology82.4.TheNaturalIsomorphism202.5.ConcludingRemarks203.CategoricalPreliminaries223.1.IntroductoryandMotivationalRemarks223.2.Categories223.3.Functors263.4.NaturalTransformations293.5.ConcludingRemarks324.SomeHomologicalAlgebra334.1.IntroductionandMotivation334.2.ChainComplexes344.3.HomologyofChainComplexes374.4.ChainHomotopy454.5.TheAcyclicCarrierTheorem535.TheNaturalIsomorphism615.1.IntroductoryRemarks615.2.SimplicialandCosimplicialTypeObjects625.3.FreeandOrientedChainComplexes765.4.InjectiveandSingularSets855.5.OrientedandOrderedChainComplexes925.6.NaturalandInfranaturalTransformationsonChainComplexes935.7.NaturalandInfranaturalIsomorphismsonHomology101References103iii PAGE 5 1.IntroductionInbasiccombinatorialtopology,specicallyinthetheoryofsimplicialcomplexes,thereisaresultwhichprovesthattheorderedhomologyfunctorisnaturallyisomor-phictotheorientedhomologyfunctor.Inthisthesis,weestablishageneralizationofthisresultwhichappliesinaratherremarkablywiderangeofsituations.Letusquicklysummarizethesituationinthecaseofsimplicialcomplexes:Re-callthatasimplicialcomplexisacollectionofsimplicesofvaryingdimensionequippedwithgluingruleswhichsatisfythateithertwosimplicesaregluedisomet-ricallyalongasubsimplexortheydonotintersectandthatasimplicialmapfromonesimplicialcomplextoanothersendsverticestoverticesandextendsthatmappinglinearlyoneverysimplex.Therearetwotypesofchaincomplexeswhichweassociatetosuchanobject:First,weassociatetoittheorientedchaincomplexwhichisgeneratedbyorientationsofthesimplicesinthecomplexsubjecttotherelationthatanorientationisequaltotheinverseoftheoppositeorientation.Moreover,wecanassociatetoeachsimplicialmapachainmapwhichsendsanorientationtotheorientationoftheimageofarepresentativeunderthemapping.Second,weasso-ciatetoittheorderedchaincomplexwhichisfreelygeneratedbytuplesofverticeswhichareallcontainedinsomesimplex.Wecanalsoassociatetoeachsimplicialmapachainmapwhichsendsatupleofverticestothetupleofevaluationsofthefunctionateachofthevertices.We,furthermore,candenethehomologyofbothofthesesortsofchaincomplexesandinducemapsonthelevelofhomology.TheclassicresultTheorem2.4.1isthatthereisachainhomotopybetweentheorderedandtheorientedchaincomplexeswhichliftstoanaturalisomorphismatthelevelofhomology.Ourapproachisabitmoreabstract.Webeginwithacategorysatisfyingmildniceness"constraintsandacosimplicial-typesetinthecategorywhichissubjecttosomesomewhatstrongconditions.Wedenethefunctorwhichassociatetoeachobjectinthecategorythesimplicial-typesetofmappingsintotheobjectfromthe1 PAGE 6 componentsofthecosimplicial-typeset.Wethendenetheorientedcomplextobegeneratedbythenaturalequivalenceclassesundertheactionofthesymmetricgroupofinjective"mappingsfromthecomponentsofthecosimplicialobjecttothespeciedobjectinthecategorysubjecttotheconditonthatoppositeequivalenceclassesareinverses.Thereisanicewaytofunctoriallyextendthistomorphismsinthecategoryinducedbycomposition.Wedenetheorderedcomplextobegeneratedbyallincludingdegeneratemorphismsintotheobject.Thisalsoextendstoafunctorviacomposition.ThemainresultofthisthesisTheorem5.7.1isthatthereisaninfranaturalchainhomotopythesetermsmakingupthisexpressionwillbeexplainedinsections3.4and4.4betweenthesetwofunctorswhichliftstoanaturalisomorphismatthelevelofhomology.Inchapter2werecallthebasictheoryofsimplicialcomplexesexpresstheclassicresult.Afterthis,webeginbuildingtowardsthemoregeneralcontextandresult.Inchapter3,wegiveanoverviewoftheelementarycomponentsofcategorytheorywhichwemakeuseofinthisthesis.Theninchapter4,werecallthebasicsofhomologicalalgebra{discussingchaincomplexes,insection4.2,andtheirhomologies,insection4.3{andthen,insection4.5provetheacycliccarriertheoremTheorem4.5.4afterdevelopingthebasictheoryofchainhomotopy,insection4.4.Afterofallthispreliminaryexposition,inchapter5wedosomeoriginalwork:Afterrecalling,insection5.2,thedenitionandbasictheoryofthesimplicialcategory"andgeneralizingthisdenitionandtheseresultsslightly,weconstructtwofunctorsfromthiscategorytothecategoryofchaincomplexesinsection5.Theninsections5.4and5.5wedevelopthecontextinwhichwewillproveourmainresult.Finally,insections5.6and5.7,weproveourtheorem.2 PAGE 7 2.TopologicalPreliminaries2.1.IntroductoryRemarks.Inthischapter,wereviewthehomologytheoryofsimplicialcomplexesandstatetheresultTheorem2.4.1whichwegeneralizeinthisthesis.Insection2.2weintroducethebasicobjectsofstudy.Theninsection2.3weintroduceboththeorientedinsubsection2.3.2andtheorderedinsubsection2.3.3chainandhomologytheories.Insection2.4westatethetheoremTheorem2.4.1whichsaysthattheorderedhomologytheoryisnaturallyisomorphictotheorientedhomologytheory.Finally,insection2.5wegesturetowardsthedirectioninwhichwewillgeneralizeinthisthesis.2.2.SimplicialComplexes.Webeginthissectionbystatingthedenitionofthestandardn-simplexdenition2.2.1{atopologicalspaceanalgoustoatrianglebutwhichhasdimensionn{andofamorphismofstandardsimplicesdenition2.2.3{thelinearextensionofamapdenedonvertices.Inparticular,weintroducethreespecictypesofstandardsimplexmorphisms{knownascofacesdenition2.2.4,codegen-eracies.2.5,andcotranspositionsdenition2.2.6{intoasequenceofwhichanymorphismofstandardsimplicesfactors.Atlastwegivethedenitionofasimplicialcomplexdenition2.2.7{basicallyacollectionofstandardsimplicesofvaryingdi-mensiongluedtogetherlinearlyalongsubsimplices{andofasimplicialmapdenition2.2.6betweensimplicialcomplexes.Thebasicbuildingblockofasimplicialcomplexisaspecicsortoftopologicalspace,astandardsimplex{theanalogueofatriangleintheappropriatedimension:Denition2.2.1.Thestandardn-simplex,denotedbyn,isatopologicalspacedenedforn>1;0;1;:::.Forn)]TJ/F39 11.955 Tf 10.985 0 Td[(1,thestandard1-simplexistheemptyspace.FornC0,thestandardn-simplexisthesetx0;:::;xnSxiC0;nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(0xi)]TJ/F15 11.955 Tf 9.279 0 Td[(1`Rn1equippedwiththesubspacetopology.3 PAGE 8 Inordertostatethegluingrulesbetweencomponentsimplicesinsimplicialcom-plexesviatopologicalquotientingand,wemustdiscussmapsbetweenstandardsim-plices.Whilewecandothisdirectly,itiseasiesttoproceedbyrstintroducingthenotionofastandardsimplex'svertex:Denition2.2.2.Theithvertexofthestandardn-simplexisthepointvni)]TJ/F24 11.955 Tf 9.279 12.036 Td[(0;:::;0i;1;0;:::;0ni1inn.WedenotebyVnthesetvn0;:::;vnnofverticesofn.Inordertoproceedtodenemorphismsofstandardsimplicesusingtheterminologyofvertices,werstusetheterminologytodescribethestandardsimplicesthemselves.Itallowsustodierentlystatethedenitionofthestandardn-simplex,fornC0:thestandardn-simplexisthesetnQi)]TJ/F16 7.97 Tf 4.632 0 Td[(0xivniSxiC0;nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(0xi)]TJ/F15 11.955 Tf 9.278 0 Td[(1oflinearcombinations{wheretheweightsarenon-negativeandsumto1{ofverticesagainequippedwiththetopologypulledbackalongtheinclusionintoRn1.Thisformalismsimpliesthedescriptionofmappingsbetweensimplices:Denition2.2.3.Amorphismofsimplicesn)]TJ/F15 11.955 Tf 14.678 0 Td[(mfromntomisthelinearextensionofasetmapVn)]TJ/F94 11.955 Tf 12.642 0 Td[(Vmonthecorrespondingvertexsetsdenoted,thankstoabuseofnotation,byVn)]TJ/F94 11.955 Tf 12.642 0 Td[(Vm.Indetail,itisgivenbynQi)]TJ/F16 7.97 Tf 4.631 0 Td[(0xivninQi)]TJ/F16 7.97 Tf 4.631 0 Td[(0xivni)]TJ/F19 7.97 Tf 11.908 12.276 Td[(mQj)]TJ/F16 7.97 Tf 4.631 0 Td[(0vmjQvni>1vmjxiHerethedoublesummationisobtainedbyrewritingintermsoftheverticesofnthesinglesummationgivenbylinearextension.Wesaythatthemapn)]TJ/F15 11.955 Tf 13.261 0 Td[(mfromnonsimplicesisinducedbythemapVn)]TJ/F94 11.955 Tf 13.627 0 Td[(Vmonvertexsets.This4 PAGE 9 allowsustodescribethemapn)]TJ/F15 11.955 Tf 12.995 0 Td[(mfromnonsimplicesbyonlyspecifyingtheactiononvertexsets.Itturnsout1thateverymorphismofsimplicesn)]TJ/F15 11.955 Tf 12.642 0 Td[(mfactorsasasequencedimkmdi1k1sinkksi1n1ti`nti1nofcotranspositionmaps"tinfollowedbyasequenceofcodegeneracymaps"sij,followedbyasequenceofcofacemaps"dij.Thisaloneisreasonenoughtomakementionofthesemaps.Butmoreover,thesemapsallowustoexplainexactlythewayinwhichthesimplicesmakingupasimplicialcomplexaregluedtogether.Andfurthermore,thesemapssatisfynicecommutativityrelations2whichallowustodenetheorientedandorderedhomologies.Weintroducerstthecofacemaps:Denition2.2.4.Theithcofaceofthestandardn-simplexistheinclusionmapdinn1)]TJ/F15 11.955 Tf 12.642 0 Td[(ninducedbythemapVn1)]TJ/F94 11.955 Tf 12.643 0 Td[(Vnonvertexsetsgivenbyvn1`vn`if`@ivn`1if`CiIntuitively,thismapincludesn1intothecopyofn1insidenwhichisoppositetheithvertexofn".Whereasthecofacemapsarethecanonicaldimension-increasingmapsbetweensim-plices,thecodegeneracymapsarethecanonicaldimension-decreasingmapsbetweensimplices: 1Wewillnotprovethisdirectly.However,theinterestedreadercandeducethisfromexample5.2.11togetherwithlemma5.2.7.2Theseidentitiesarelistedat5.2.6.5 PAGE 10 Denition2.2.5.Theithcodegeneracyofthestandardn-simplexisthemapsinn1)]TJ/F15 11.955 Tf 12.643 0 Td[(ninducedbythemapVn1)]TJ/F94 11.955 Tf 12.642 0 Td[(Vnonvertexsetsgivenbyvn1`vn`if`Bivn`1if`AiIntuitively,thismapcollapsesonedimensionofthestandardn1-simplexbypinchingtogethertheithandthei1thverticesandextendingthepinchlinearly.Finally,thecodegeneracymapsarethecanonicalmapsbetweensimpliceswhicharedimension-constant:Denition2.2.6.Theithcotranspositionofthestandardn-simplexisthemaptinn)]TJ/F15 11.955 Tf 12.642 0 Td[(ninducedbythemapVn)]TJ/F94 11.955 Tf 12.642 0 Td[(Vnonvertexsetsgivenbyvn`vn`if`@ivn`1if`)]TJ/F77 11.955 Tf 9.279 0 Td[(ivn`1if`)]TJ/F77 11.955 Tf 9.279 0 Td[(i1vn`if`Ai1Intuitively,thismaprotatesthestandardn-simplexthroughRn1sothattheithandthei1thverticesareswapped.Therstofthesethreemapsinparticularallowsustodenetheglue"inasimplicialcomplex.Thisallowsustogivethedenitionatlast:Denition2.2.7.AsimplicialcomplexListhetopologicalspaceequippedwithextradatawhichdescribesitsconstruction.Oneisbuiltfroman^N-graded3collectionComp)]TJ/F77 11.955 Tf 4.551 0 Td[(L)]TJ/F24 11.955 Tf 9.279 0.154 Td[(Compn)]TJ/F77 11.955 Tf 4.552 0 Td[(L)]TJ/F24 11.955 Tf 9.279 0 Td[(Ln)]TJ/F15 11.955 Tf 9.279 0 Td[(n>Jnn>^N 3By^Nwedenotetheaugmentedcollectionofnaturalnumbers,theset1;0;1;:::.6 PAGE 11 ofcopiesofthestandardsimplices,4knownasL-componentstandardn-simplices,whicharegluedtogether.Inparticular,twonotnecessarilydistinctsimplicesLn,Lmeitherarenotgluedtogetherortheyshareasubsimplexinthesensethatinclndinkndi1k1)]TJ/F15 11.955 Tf 9.279 0 Td[(inclmdimkndi1k1ti`kti1kIntuitively,thismeansthatanytwocomponentsimpliceswhicharegluedtogetheralongasubsimplexviaanisometry.Itisimportanttonotethatwhenwespeakofasimplicialcomplexwereferbothtothetopologicalspaceconstructedasdescribedaboveandofthedatawhichdescribesitsconstruction:itscomponentsimplices,inclusionmaps,anditsgluingdata.Example2.2.8.Therstexampleofasimplicialcomplexisthestandardn-simplexn.The^N-gradedsetofcomponentsimplicesisgivenbyComp)]TJ/F15 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 9.279 65.927 Td[(Comp1)]TJ/F15 11.955 Tf 4.552 0 Td[(n)]TJ/F39 11.955 Tf 16.352 0 Td[(gCompn1)]TJ/F15 11.955 Tf 4.552 0 Td[(n)]TJ/F39 11.955 Tf 16.352 0 Td[(gCompn)]TJ/F15 11.955 Tf 4.552 0 Td[(n)]TJ/F24 11.955 Tf 16.352 0 Td[(nCompn1)]TJ/F15 11.955 Tf 4.552 0 Td[(n)]TJ/F39 11.955 Tf 16.352 0 Td[(gItistrivialisalldimensionsbutnwhereitcontainsonecopyofthestandardn-simplex.Havingnowintroducedtheobjectsofinterest,weintroducethemapsbetweenthem:Denition2.2.9.AsimplicialmapfL)]TJ/F77 11.955 Tf 12.84 0 Td[(Kisacontinuousmapfromthetopo-logicalspaceLtothetopologicalspaceKwhichsendseachL-componentstandard 4Whichautomaticallyyieldsan^N-gradedcollectioninclnn)]TJ/F10 6.974 Tf 8.301 -1.494 Td[(Ln`L>Jnn>^Nofinclusionsn)]TJ/F10 6.974 Tf 8.3 -1.494 Td[(Lngivenbyx0;:::;xnx0;:::;xn;.7 PAGE 12 simplexLnviaamorphismofsimplicesf;n;mtoaK-componentstandardsimplexKm.Example2.2.10.Therstexampleofasimplicialmapis"amorphismofstandardsimplices:Letn)]TJ/F15 11.955 Tf 13.96 0 Td[(mmeamorphismofstandardsimplices.Asinexample2.2.8,nisasimplicialcomplexwith^N-gradedsetofcomponentsimplicesgivencomponentwisebyCompi)]TJ/F15 11.955 Tf 4.551 0 Td[(n)]TJ/F39 11.955 Tf 10.447 0 Td[(gforixnandbyCompn)]TJ/F15 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 10.447 0 Td[(nfori)]TJ/F77 11.955 Tf 10.447 0 Td[(n.Similarly,misasimplicialcomplexwith^N-gradedsetofcomponentsimplicesgivencomponentwisebyCompi)]TJ/F15 11.955 Tf 4.551 0 Td[(m)]TJ/F39 11.955 Tf 9.316 0 Td[(gforixmandbyCompm)]TJ/F15 11.955 Tf 4.551 0 Td[(m)]TJ/F24 11.955 Tf 9.316 0 Td[(mfori)]TJ/F77 11.955 Tf 9.316 0 Td[(m.Onesimplicialmapn)]TJ/F15 11.955 Tf 12.748 0 Td[(misthusgivenbysendingthen-componentn-simplexntothem-componentm-simplexmaccordingtothemorphismofstandardsimplices.Itisclearthatthesetwodescriptionsarebothofthesamecontinuousmap.Furthermore,everysimplicialmapbetweenstandardsimplicesis"amorphismofstandardsimplices.Thuseverysimplicialmapbetweenstandardsimplicesdecom-posesaswediscussedabove.Inthissectionweintroducedaclassofobjectsofinterest,simplicialcomplexes,andmapsbetweenthemaswewillseeinsection3.2,thoseconstitutetheobjectsandmorphismsofacategory.Weobserved,inparticular,thatthesimplicialmappingsamongthestandardsimplicesfactorasasequenceofcotranspositionsfollowedbyasequenceofcodegeneraciesfollowedbyasequenceofcofaces.Inthefollowingsection,wewillmakeuseofthisfacttodenetheorientedandorderedhomologyassociations.2.3.SimplicialHomology.2.3.1.IntroductoryRemarks.Inthissection,afterdiscussinglooselythebasicideaofasimplicialhomologytheory,wediscusstwoparticulartheories:orientedhomology,insubsection2.3.2,andorderedhomology,insubsection2.3.3.Beforewegetintothedetailsofeitherhomology,let'sdiscussthegeneralnotionofsimplicialhomology.Thetwopurposesofatopologicalhomologytheoryareto8 PAGE 13 distinguishtwotopologicalspacesonthebasisofthenumber,dimension,andtypeofholes"thatappearinthespacesand2todistinguishtwodierentmapsbetweenthesametopologicalspacesonthebasisofthewaytheytreatthestructuresonthetwospacesdetectedbythehomologytheory.Byahole,"wemeanaloop"whichdoesnotboundaregion".An-dimensionalloopinatopologicalspaceXisacontinuousmapSn)]TJ/F77 11.955 Tf 13.788 0 Td[(Xintothespacefromthen-sphere;wesaythataloopboundsaregionjustincasethemapextendstoamapDn1)]TJ/F77 11.955 Tf 12.643 0 Td[(Xfromthen1-ballwhoseboundaryisthen-spheretothespace.Thesestructurescanbedetectedviamappingsinofsimplicesaswellbecauseann-spherecanbetriangulatedbyn-simplicesasann1-ballcanbetriangulatedbyn1simplices.Thisisbasicallytheapproachthatbothhomologiesdiscussedinthissectiontake,althoughtheclassofmappingsthattheyallowaredierentbothfromeachotherandfromthesingular"casedescribedhere.2.3.2.OrientedHomology.Inthecaseoforientedhomology,weconsideronlyin-jectivemappingsofsimplicesintoasimplicialcomplex.Wedonotlookatthesemappingsdirectlybutratherconsiderequivalenceclassesundertheactionofthesymmetricgroup.Sointhissubsection,werstdeneorientedn-simplicesde-nition2.3.4tobeequivalenceclassesofinjectiven-simplicesdenition2.3.1,andthenwedenetheorientedchaincomplextobethesequenceofabeliangroupsgen-erated,atdimensionn,bytheorientedn-simplicessubjecttotheobviousrelationequippedwiththeboundarymapwhichisthelinearextensionofwhatamountstothealternatingsumofthecofaces,thoughthisnotionitselfisnotwell-dened,oftheorientedsimplices.Afterprovingthattheboundarymapsatisesthenecessarytech-nicalconditionlemma2.3.6,wedenetheorientedhomologydenition2.3.4ofasimplicialcomplex.Finally,weinduceamaponorientedchaincomplexesdenition2.3.8and,afterdemonstratingthatitispossiblecorollary2.3.10,weinduceamaponorientedhomologydenition2.3.11.9 PAGE 14 Tobeginwith,wedeneinjectivemappingsofthesimplicesintoasimplicialcom-plex:Denition2.3.1.Aninjectiven-simplex0ninasimplicialcomplexLisasimplicialmap0nn)]TJ/F77 11.955 Tf 12.643 0 Td[(Lwhichisinjectiveontheunderlyingtopologicalspaces.Aswementioned,wewon'tbelookingattheinjectivemappingsdirectlybutinsteadwillbeconsideringequivalenceclassesofthesemappingsundertheactionofthesymmetricgroup.Below,atdiscussion2.3.3,wetalkaboutjustwhatactionofthesymmetricgroupwemean.Butbeforewedothat,westatethefollowingfactaboutinjectivesimpliceswhichisnecessaryinorderforustounderstandtheseequivalenceclasses:Observation2.3.2.Everyinjectiven-simplex0ninasimplicialcomplexLfactorsas0n)]TJ/F90 11.955 Tf 9.279 0 Td[(inclnkdiknkdi1n1ti`nti1nasequenceofcotranspositionsfollowedbyasequenceofcofacemapsfollowedbyacomponentinclusionmapofL.Proof.Omitted.5Wearenowpreparedtounderstandthenaturalactionofthesymmetricgroupontheinjectivesimplicesandinparticulartounderstandtheorbitofaninjectivesimplexundertheaction:Discussion2.3.3.ThereisastraightforwardactionofthesymmetricgroupSn{thatisthesymmetricgroupontheletters0;:::;n{onthesetMapsinjn;Lofinjectiven-simplicesinasimplicialcomplexLinducedbythecotranspositionmaps.Observerstthattinn)]TJ/F15 11.955 Tf 12.642 0 Td[(ninducestheprecompose-by-tinmaptni)]TJ/F39 11.955 Tf 9.279 0 Td[(XtinMapsinjn;L)]TJ/F75 11.955 Tf 12.642 0 Td[(Mapsinjn;L 5Theinterestedreadercanobtainthisasaresultofproposition5.2.7.10 PAGE 15 whichwetermtheithtranspositionmap.Withalittlework,onecancheckthatthetranspositionmapssatisfythefollowingequations:1tnitni)]TJ/F15 11.955 Tf 9.279 0 Td[(12tnj1tnjtnj1)]TJ/F15 11.955 Tf 9.279 0 Td[(tnjtnj1tnj3tnitnj)]TJ/F15 11.955 Tf 9.279 0 Td[(tnjtniifi@j1ButtheseareexactlytherelationsimposedinthetranspositionpresentationofSn!Inotherwords,`t0;:::;tn1Stiti)]TJ/F15 11.955 Tf 9.279 0 Td[(1;tj1tjtj1)]TJ/F104 11.955 Tf 9.279 0 Td[(tjtj1tj;titj)]TJ/F104 11.955 Tf 9.279 0 Td[(tjtiifi@j1eisapresentationofSn.6ThissuggeststhatweletSnactonMapsinjn;Lbyreplacingtiwithtni".Indetail,denetheactionYSnMapsinjn;L)]TJ/F75 11.955 Tf 12.643 0 Td[(Mapsinjn;LongeneratorsbytiY0nz)]TJ/F15 11.955 Tf 20.604 0 Td[(tni0n)]TJ/F34 7.97 Tf 9.507 7.809 Td[(0ntinWecandescribetheorbitsofthisgroupactionusingobservation2.3.2:Let0n>Mapsinjn;L.Then0n)]TJ/F15 11.955 Tf 9.279 0 Td[(inclnkdiknkdi1n1ti`nti1nbyobservation2.3.2.ThustheorbitU0nUof0nisthesetU0nU)]TJ/F24 11.955 Tf 9.279 0.154 Td[(inclnkdiknkdi1n1tjnt1nSj>N;0B1;:::;jBn1ofallsequencesofcotranspositionsi.e.allpermutations"followedbyinclnkdiknkdi1n1.Moreover,thisorbitisinbijectionwithSn. 6Hereti0;:::;n)]TJ/F51 9.963 Tf 10.759 0 Td[(0;:::;n,givenbykkifk@ik1ifk)]TJ/F11 9.963 Tf 7.887 0 Td[(ik1ifk)]TJ/F11 9.963 Tf 7.887 0 Td[(i1kifkAi1isjusttheithstandardtranspositionontheletters0;:::;n.11 PAGE 16 Abasicfactaboutthesymmetricgroupisthatanytwofactoringsti`ti1andtj`ti1ofapermutationp>Snintoaproductoftranspositionshavethesamelengthmod2i.e.``mod2.ThisallowsustodeneanequivalencerelationonSnbyppjustincaseanyfactoringofp1phaslength0mod2.ClearlythisequivalencerelationpartitionsSnintotwoequivalenceclasses.ThisinducesapartitionoftheorbitU0nUofeachinjectivesimplex0nintotwoequivalenceclasses0naand0nb.Wewilldenoteby0ntheequivalenceclasswhichactuallycontainstheinjectivesimplex0nandby0ntheequivalenceclasswhichdoesnotcontainit.Forthesakeofbrevityandconceptualclarity,weintroducefurtherterminologytodescribetheseequivalenceclasses:Denition2.3.4.Anorientedn-simplex0nistheequivalenceclassofaninjectivesimplex0nwhichwejustconstructedindiscussion2.3.3.Wesaythattheorientedn-simplex0nisoppositetotheorientedn-simplex0n)]TJ/F24 11.955 Tf 9.279 0 Td[(0nforsome.Weassociateanintermediatesimplicialcomplexthealgebraicstructure,knownastheorientedchaincomplex,fromwhichwewilldescribethealgebraicstructureofmaininterest,theorientedhomology,whichtracksthenumber,dimension,andtypeofholes"thatappearinaspace:Denition2.3.5.TheorientedchaincomplexassociatedtoasimplicialcomplexListheZ-gradedsequenceZoriL)]TJ/F39 11.955 Tf 9.278 0 Td[(@n1ZoriL)]TJ/F30 11.955 Tf 18.178 0 Td[(ZorinL@nZoriL)]TJ/F30 11.955 Tf 18.177 0 Td[(Zorin1L@n1ZoriL)]TJ/F39 11.955 Tf 18.178 0 Td[(@0ZoriL)]TJ/F30 11.955 Tf 18.178 0 Td[(Zori1L@1ZoriL)]TJ/F75 11.955 Tf 18.178 0 Td[(0@2ZoriL)]TJ/F75 11.955 Tf 18.178 0 Td[(0@3ZoriL)]TJ/F39 11.955 Tf 18.178 0 Td[(ofabeliangroupsZorinLandhomomorphisms@nZoriLZorinL)]TJ/F30 11.955 Tf 12.642 0 Td[(Zorin1Lknownasboundarymaps.TheabeliangroupsZorinLarepresentedbyZorinL)]TJ/F24 11.955 Tf 9.279 0 Td[(`0n0n>Mapsinjn;LS0n1)]TJ/F39 11.955 Tf 9.279 0 Td[(0ne12 PAGE 17 Theyaregeneratedbytheorientedn-simplicesinLsubjecttotherelationthatoppositesimplicesareinverses.Theboundarymap@nZoriLZorinL)]TJ/F30 11.955 Tf 12.642 0 Td[(Zorin1Lisgivenongeneratorstobe0nz)]TJ/F19 7.97 Tf 24.267 12.276 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(00ndinanalternatingsumoftheorientedn1-simpliceswhicharerepresentedbythecofacesofsomerepresentative0nof0n.Infact,theboundarymapwell-denedascanbededucedfromthediscussioninconstruction5.3.4.AnelementCninlevelnoftheorientedchaincomplexZorinLisknownasanorientedn-chaininL.Thefollowingfactabouttheorientedchaincomplexassociatedtoasimplicialcomplexallowsustodenetheorientedhomology:Lemma2.3.6.LetLbeasimplicialcomplex.TheboundarymapoftheorientedchaincomplexassociatedtoLsatisesthat@n1ZoriL@nZoriL)]TJ/F15 11.955 Tf 9.279 0 Td[(0foralln>Z.Proof.Omitted.Theinterestedreadercandeducethisfromthediscussionincon-struction5.3.4.Equippedwiththisfact,wecannowdenetheorientedhomologyofasimplicialcomplex:Denition2.3.7.TheorientedhomologyofasimplicialcomplexListheZ-gradedsequenceHoriL)]TJ/F24 11.955 Tf 9.279 0.155 Td[(HorinLn>ZwhichisdenedateachlevelbyHorinL)]TJ/F15 11.955 Tf 10.475 11.301 Td[(ker@nZoriL im@n1ZoriLThisquotientmakessensebecausebylemma2.3.6,im@n1ZoriL`ker@nZoriL.Intheintroductoryremarkssubsection2.3.1,wesaidthatsimplicialhomologydetectsholes:"theirnumber,dimension,andtype.Wesaidthatitdetectsloopswhicharenotboundingregions.Infact,orientedhomologydoesthis.Buthow?13 PAGE 18 Loopsofdimensionninasimplicialcomplexcorrespondtogeneratorsinsomerepresentationofthekernelofthenthboundaryoperator.Ageneratorforthekernelofthenthboundaryoperatorisanorientedn-chainwhichtheboundaryoperatortakesto0{thatis,ageneratorforthekernelofthenthboundaryoperatorisasumoforientedn-chainswhoseboundaries7cancel.Inparticular,loopsofdimensionnwhichboundregionsofdimensionn1correspondtogeneratorsinsomerepresentationoftheimageofthen1thboundaryoperator.Ageneratorfortheimageofthen1thboundaryoperatoristhesumoforientedn-simpliceswhichwraparound"theorientedn1-chainunderconsideration.Thusbequotientingthenthkernelbythen1thimage,weobtainagadgetwhichkeepstrackoftheloopsofdimensionnwhichdonotboundregionsofdimensinn1.Butaswestatedintheintroductoryremarks,ahomologydoesnotjustdistinguishtwospacesbutalsodistinguishestwomapsbetweenthesamespaces.Inordertodothis,wewillinducemapsonorientedhomologyfrominjectivemapsonsimplicialcomplexes.Wedothisbyrstinducingmapsonorientedchaincomplexes:Denition2.3.8.TheinducedmorphismoforientedchaincomplexesZorifZoriL)]TJ/F30 11.955 Tf -422.247 -23.083 Td[(ZoriKinducedbyaninjectivesimplicialmapfL)]TJ/F77 11.955 Tf 14.146 0 Td[(KistheZ-gradedsequenceZorif)]TJ/F24 11.955 Tf 9.568 0 Td[(ZorinfZorinL)]TJ/F30 11.955 Tf 12.643 0 Td[(ZorinKn>Zofhomomorphisms{reallythechainmap8{givenateachlevelbyZorif0nz)]TJ/F24 11.955 Tf 20.603 0 Td[(f0nThismapisinfactwell-denedastheinterestedreadercandeducefromobservation5.3.5. 7Unfortunately,thereisnotaverycleargeometricnotionoftheboundaryofanorientedsimplexbecausetheorientedn1-simplexcorrespondingtotheithcofaceofsomerepresentativeoftheorientedn-simplexisalwaysdierentfromtheithcofaceofanotherrepresentativeoftheorientedn-simplex.8Wewillexplainwhatthistermmeanshere.Weprovidethoroughformaldiscussionofchaincomplexesandchainmapsbetweentheminsection4.2.14 PAGE 19 Weneedthattheinducedmaponchaincomplexescooperateswiththeboundaryoperatorofthechaincomplexesinordertodenetheinducedmaponhomology.Inparticularweneedthefollowing:Lemma2.3.9.LetfL)]TJ/F77 11.955 Tf 14.641 0 Td[(Kbeasimplicialmap.ThentheinducedmorphismoforientedchaincomplexesZorifZoriL)]TJ/F30 11.955 Tf 15.363 0 Td[(ZoriKcommuteswiththeboundaryoperator"inthesensethat@nZoriLZorinf)]TJ/F30 11.955 Tf 9.279 0 Td[(Zorin1f@nZoriK.Proof.Omitted.Theinterestedreadercandeducethisfromobservation5.3.5.Beausetheinducedmaponchaincomplexescooperateswiththeboundaryopera-torsonthechaincomplexes,weobtainthatkernelsandimagesoftheboundarymapinthedomainorientedcomplexareincludedintothekernelsandimages,respectively,ofthecodomainorientedcomplex:Corollary2.3.10.LetfL)]TJ/F77 11.955 Tf 13.05 0 Td[(Kbeasimplicialmap.ThentheinducedmorphismoforientedchaincomplexesZorifZoriL)]TJ/F30 11.955 Tf 12.642 0 Td[(ZoriKsatisesthefollowing:Zorifim@n1ZoriL`im@n1ZoriKZorifker@nZoriL`ker@nZoriKProof.Immediate.Theseinclusionsallowustodenetheinducedmorphismonhomologyintheobviousway:"Denition2.3.11.TheinducedmorphismonorientedhomologyHorifHoriL)]TJ/F75 11.955 Tf -422.247 -23.083 Td[(HoriKinducedbyasimplicialmapfL)]TJ/F77 11.955 Tf 12.642 0 Td[(KisthesequenceHorif)]TJ/F24 11.955 Tf 9.279 0.154 Td[(HorinfHorinL)]TJ/F75 11.955 Tf 12.642 0 Td[(HorinKn>Zgivenateachlevelby)]TJ/F94 11.955 Tf 4.551 0 Td[(Cnz)]TJ/F24 11.955 Tf 20.603 0.155 Td[(ZorinfCnThisdenitionmakessensebycorollary2.3.10.Sowehavenowbothassociatedalgebraicstructures,theorientedhomologygroups,andalsotheintermediatealgebraicstructures,theorientedchaincomplexes15 PAGE 20 tosimplicialcomplexesand2associatedmapsofalgebraicstructures,theinducedmapsonhomology,andalsotheinducedmapsoforientedchaincomplexestoinjec-tivesimplicialcomplexes.Theseassociationskeeptrackoftheholes"insimplicialcomplexesandthewaythatmapsbetweensimplicialcomplexestreattheseholesinthewaysketchedintheremarksatsubsection2.3.1.2.3.3.OrderedHomology.Inthissubsection,weproceedinamanneranalogoustothatoftheprevioussubsection.Howeverthemannerinwhichweproceedinthissectionismoreinlinewiththeintroductoryremarksofsubsection2.3.2:weconsiderall,notonlyinjective,simplicialmappingsintoasimplicialcomplexfromthen-simplices{knownasorderedsimplicesinthesimplicialcomplexseedenition2.3.12.Wewilldeneseedenition2.3.13theorderedchaincomplexofasimplicialcomplextobefreelygeneratedbytheorderedsimplicesinthesimplicialcomplexanddenetheboundaryoperatoronthechaincomplextosendanorderedsimplextothealternatingsumofitscofaces.Aftercheckingthatitispossibletodososeelemma2.3.14,wedene,atdenition2.3.15,theorderedhomologyofasimplicialcomplextobethesequenceofquotientsofkernelsbyimagesoftheboundarymapoftheorderedchaincomplexaswedidforthedenitionsee2.3.7oforientedhomology.Subsequently,wewillproceed,asintheprevioussubsection,toinducemapsonorderedhomologyseedenition2.3.19byrstinducingamaponorderedchaincomplexes.Webeginbydeningorderedsimplices:Denition2.3.12.Anorderedn-simplex)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(ninasimplicialcomplexLisasimplicialmap)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(nn)]TJ/F77 11.955 Tf 12.643 0 Td[(L.Thentheorderedchaincomplexassociatedtoasimplicialcomplexisjustthecomplexfreelygeneratedateachlevelbytheorderedsimplicesatthatlevelwhoseboundarymapisgivenbythealternatingsumofcofaces.Indetail:16 PAGE 21 Denition2.3.13.TheorderedchaincomplexassociatedtoasimplicialcomplexListheZ-gradedsequenceZordL)]TJ/F39 11.955 Tf 9.279 0 Td[(@n1ZordL)]TJ/F30 11.955 Tf 19.146 0 Td[(ZordnL@nZordL)]TJ/F30 11.955 Tf 19.147 0 Td[(Zordn1L@n1ZordL)]TJ/F39 11.955 Tf 19.146 0 Td[(@0ZordL)]TJ/F30 11.955 Tf 19.146 0 Td[(Zord1L@1ZordL)]TJ/F75 11.955 Tf 19.147 0 Td[(0@2ZordL)]TJ/F75 11.955 Tf 19.146 0 Td[(0@3ZordL)]TJ/F39 11.955 Tf 19.146 0 Td[(ofabeliangroupsZordnLandhomomorphisms@nZordLZordnL)]TJ/F30 11.955 Tf 12.643 0 Td[(Zordn1Lknownasbound-arymaps.TheabeliangroupsZordnLarepresentedbyZordnL)]TJ/F24 11.955 Tf 9.279 0 Td[(`)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(n)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(n>Mapsn;LeTheyarefreelygeneratedbytheorderedn-simplicesinL.Theboundarymap@nZordLZordnL)]TJ/F30 11.955 Tf 12.643 0 Td[(Zordn1Lisgivenonbasiselementstobe)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(nz)]TJ/F19 7.97 Tf 24.267 12.276 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(0)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(ndinanalternatingsumoftheorderedn1-simpliceswhicharethecofacesof)]TJ/F77 11.955 Tf -0.228 -7.338 Td[(n.AnelementCninlevelnoftheorderedchaincomplexZordnLisknownasanorderedn-chaininL.Inordertodenethehomology,weneedtocheckthatdoubleapplicationoftheboundaryoperatortoachainyieldszerosothatwewillhavethattheimageofthen1thboundaryoperatorisasubgroupofthekernelofthenthboundaryoperator:Lemma2.3.14.LetLbeasimplicialcomplex.TheboundarymapoftheorderedchaincomplexassociatedtoLsatisesthat@n1ZordL@nZordL)]TJ/F15 11.955 Tf 9.279 0 Td[(0foralln>Z.Proof.Omitted.Theinterestedreadercandeducethisfromproposition5.3.2.Thisfactallowsustodenetheorderedhomology"associatedtoasimplicialcomplex:17 PAGE 22 Denition2.3.15.TheorderedhomologyofasimplicialcomplexListheZ-gradedsequenceHordL)]TJ/F24 11.955 Tf 9.279 0.154 Td[(HordnLn>ZwhichisdenedateachlevelbyHordnL)]TJ/F15 11.955 Tf 10.474 11.415 Td[(ker@nZordL im@n1ZordLThisquotientmakessensebecausebylemma2.3.14,im@n1ZordL`ker@nZordL.Thereasonwhythisalgebraicobjectdetectsholes"inasimplicialcomplexisexactlythesameasthereasondiscussedbelowdenition2.3.7.Theloopsinasimplicialcomplexshowupasgeneratorsinsomerepresentationofthekerneloftheboundaryoperatorandthoseloopswhichboundregionsshowupasgeneratorsinsomerepresentationoftheimageoftheboundaryoperator.Thusquotientingthekernelbytheimageyieldsagadgetwhichcountsandgivesadditionalinformationabouttheloopsinthesimplicialcomplexwhichdonotboundregions.Again,wewanttodenehomologynotonlyforsimplicialcomplexesbutalsoformapsofsimplicialcomplexes.Asbefore,werstinduceamaponthelevelofchaincomplexes:Denition2.3.16.TheinducedmorphismoforderedchaincomplexesZordfZordL)]TJ/F30 11.955 Tf -424.697 -23.083 Td[(ZordKinducedbyasimplicialmap9fL)]TJ/F77 11.955 Tf 14.518 0 Td[(KistheZ-gradedsequenceZordf)]TJ/F24 11.955 Tf -425.611 -23.084 Td[(ZordnfZordnL)]TJ/F30 11.955 Tf 12.643 0 Td[(ZordnKn>Zofhomomorphisms{reallythechainmap{givenateachlevelbyZordf)]TJ/F77 11.955 Tf -0.228 -7.339 Td[(nz)]TJ/F77 11.955 Tf 20.603 0 Td[(f)]TJ/F77 11.955 Tf -0.229 -7.339 Td[(nInordertoinduceamaponorderedhomologyviathisinducedmaponorderedchaincomplexes,weneedthatthemaponchaincomplexescooperateswiththeboundaryoperatorineachcomplex:Lemma2.3.17.LetfL)]TJ/F77 11.955 Tf 14.078 0 Td[(Kbeasimplicialmap.ThentheinducedmorphismoforderedchaincomplexesZordfZordL)]TJ/F30 11.955 Tf 15.408 0 Td[(ZordKcommuteswiththeboundaryoperator"inthesensethat@nZordLZordnf)]TJ/F30 11.955 Tf 9.279 0 Td[(Zordn1f@nZordK. 9Notethatthissimplicialmapneednotbeinjective{adierencefromtheorientedcase.18 PAGE 23 Proof.Omitted.Theinterestedreadercandeducethisfromobservation5.3.3.Withthisresult,wecandeducecertaininclusions:Corollary2.3.18.LetfL)]TJ/F77 11.955 Tf 13.05 0 Td[(Kbeasimplicialmap.ThentheinducedmorphismoforderedchaincomplexesZordfZordL)]TJ/F30 11.955 Tf 12.642 0 Td[(ZordKsatisesthefollowing:Zordfim@n1ZordL`im@n1ZordKZordfker@nZordL`ker@nZordKProof.Immediate.Theseinclusionsallowustodeneaninducedmorphismonorderedhomology:Denition2.3.19.TheinducedmorphismonorderedhomologyHordfHordL)]TJ/F75 11.955 Tf -422.247 -23.084 Td[(HordKinducedbyasimplicialmapfL)]TJ/F77 11.955 Tf 12.642 0 Td[(KisthesequenceHordf)]TJ/F24 11.955 Tf 9.279 0.155 Td[(HordnfHordnL)]TJ/F75 11.955 Tf 12.642 0 Td[(HordnLn>Zgivenateachlevelby)]TJ/F94 11.955 Tf 4.552 0 Td[(Cnz)]TJ/F24 11.955 Tf 20.603 0.155 Td[(ZordnfCnThisdenitionmakessensebycorollary2.3.18.Aswedidinthepreceedingsubsection,inthissectionwedevelopedasimplicialhomologyforsimplicialcomplexes.Wehaveconstructedtwolayersofalgebraicstruc-turesoversimplicialcomplexesandsimplicialmaps:intermediatelyorderedchaincomplexesandinducedmapsonorderedchaincomplexesandcentrallyorderedho-mologyandinducedmorphismsonorderedhomology.Asdiscussedintheintroduc-toryremarkssubsection2.3.1theorderedhomologykeepstrackofthenumber,type,anddimensionofholesthatappearinasimplicialcomplex.2.3.4.ConcludingRemarks.Inthissection,wepresentedtwosimilarbutdierentapproachestosimplicialhomologyofsimplicialcomplexes.Aswewillseeinchapter3,whatweactuallyintroducedweretwofunctorsfromthecategorySimpCompofsimplicialcomplexesandsimplicialmapstothecategoryAbofabeliangroupsandhomomorphisms.Itturnsoutthatthesetwofunctorsarerelated.Infact,theorientedandtheorderedhomologyarenaturallyisomorphic.19 PAGE 24 2.4.TheNaturalIsomorphism.Aswesuggestedintheremarksconcludingsec-tion2.3,thereisaconnectionbetweenthetwosimplicialhomologiesweintroduced.Infact,thereisanaturalisomorphism"fromtheorderedtotheorientedhomology.Thismeansthattheorientedhomologygroupsassociatedtoasimplicialcomplexarecanonicallyisomorphictotheorderedhomologygroupsassociatedtothatcomplexandthatfurthermorethemapsinducedonthehomologygroupsbymapsofsimplicialcomplexesarethesame"withrespecttotheseisomorphisms.Thefollowingtheoremstatesthisindetail:Theorem2.4.1.Orderedsimplicialhomologyisnaturallyisomorphic"toorientedsimplicialhomology.Thatis,foreachsimplicialcomplexJthereisanisomorphismTJHordJ)]TJ/F109 11.955 Tf 12.643 0 Td[(HoriJsuchthatforanyinjectivesimplicialmapfL)]TJ/F77 11.955 Tf 12.643 0 Td[(KwehavethatTKHordf)]TJ/F109 11.955 Tf 9.279 0 Td[(HorifTL.Proof.Omitted.Thisfactisaninstanceofthemainresultofthethesis,theorem5.7.1.Theproofisbywayofsomeelementaryresultsonchainhomotopy,discussedinsection4.4,togetherwiththeacycliccarriertheoremtheorem4.5.4,whichweproveinsection4.5.Themostinterestingpartofthistheoremisthenaturality"oftheisomorphismfromtheorderedtotheorientedhomology.Unfortunately,thisaspectofthetheoremismuedbythelanguageusedtostateit.Thetheoremcouldbestatedmuchmorecleanlyastheorderedhomologyfunctorisnaturallyisomorphictotheorientedhomologyfunctor."Thisstatementismorepowerful"becauseitdrawsattentiontotheimportantobjects,thefunctors"ratherthantoonlycertainaspectsoftheseobjects,theirbehavioronspacesalone.Forthisreason,wewillintroduceinthecomingchapterthelanguageofcategories,functors,andnaturaltransformations.2.5.ConcludingRemarks.Inthischapter,weprovidedthebasiccasewhichmoti-vatestheworkdoneintherestofthethesis.Thereaderisencouragedtoworkthroughexamplescomputingtheorientedandorderedhomologiesofasimplicialcomplexand20 PAGE 25 thehomologiesofmapsfromonesimplicialcomplextoanother.Itwillbeveryhelpfultoreferbacktothisgeometriccasewhentheabstractionbecomesoverbearing.Wewillnotbediscussingtheabstractionoftheorem2.4.1untilchapter5becausewerstneedtodevelopthelanguageinwhichtostatetheresult.Tobeginwith,weintroducethelanguageofcategorytheoryinchapter3andthenweintroducetheabstractalgebraicnotionofhomologyinchapter4.21 PAGE 26 3.CategoricalPreliminaries3.1.IntroductoryandMotivationalRemarks.Thelanguageofcategorytheoryisaninvaluableorganizingtoolusedthroughoutmodernmathematics.Butmorethanjustanorganizingtool,itallowsustostateresultsoftheformwheneverasituationisorganizedinthiswayitisfurthermoreorganizedinthisotherway"{itallowsustoradicallyabstractfromconcreteset-basedstatementsaboutanobjecttostatementsbasedonhowtheobjectinteractswithotherobjectsofthesamekind."Weintroducerst,insection3.2,thedenitionofacategory.Then,insec-tion3,weintroducefunctors,mapsbetweencategories,andpseudofunctors,not-necessarilywell-behavedmapsbetweencategories.Finally,insection3.4,wediscussnaturaltransformations,mapsbetweenfunctors,andinfranaturaltransformations,not-necessarilywell-behavedmapsbetweenpseudofunctors.3.2.Categories.Firstofall,weintroducethebasicobjectsofstudy,categories.Denition3.2.1.AcategoryCisacollection10ObC,whoseelements{denotedbyC,C,C,:::{areknownastheobjectsofC,togetherwith,foreachorderedpairingC;CofobjectsinC,acollectionMapsCC;C)]TJ/F75 11.955 Tf 11.562 0 Td[(MapsC;Cwhoseelementsf,f,f,:::areknownasmorphismsormapsfromCtoCinCequippedwithacompositionoperationMapsCC;CMapsCC;C)]TJ/F75 11.955 Tf 14.298 0 Td[(MapsCC;Cwhichsatisesthefollowing:ThereexistidentitymorphismsidCC)]TJ/F77 11.955 Tf 13.167 0 Td[(CsuchthatforanymorphismfC)]TJ/F77 11.955 Tf 13.612 0 Td[(CfromChereCisanyobjectinCtheequalityfidC)]TJ/F77 11.955 Tf 10.248 0 Td[(fholdsandthatforanymorphismfC)]TJ/F77 11.955 Tf 12.647 0 Td[(CtoChereCisanyobjectinCtheequalityidCf)]TJ/F77 11.955 Tf 9.279 0 Td[(fholds.Compositionisassociative.Thatis,forfC)]TJ/F77 11.955 Tf 15.384 0 Td[(C,fC)]TJ/F77 11.955 Tf 15.384 0 Td[(C,andfC)]TJ/F77 11.955 Tf 12.642 0 Td[(Ctheequalityfff)]TJ/F24 11.955 Tf 9.279 0 Td[(fffholds.Thusuniquecomposition 10Wewillnottroubleourselveswithsettheoreticalissueshere.Theinterestedreaderisinvitedtoconsiderthetreatmentoftheseissuesin[2].22 PAGE 27 ofmorphismsisdenotedbyfffordisplayeddiagrammatically:Cf)]TJ/F77 11.955 Tf 35.741 0 Td[(Cf)]TJ/F77 11.955 Tf 35.741 0 Td[(Cf)]TJ/F77 11.955 Tf 35.741 0 Td[(CExample3.2.2.Anexcellentrstexampleofacategoryisthecategoryofsetsandfunctions,denotedbySet.ItsobjectsaresetsS;T;U;:::anditsmorphismsarejustfunctionsbetweensets.Example3.2.3.AnexcellentsecondexamplecategoryisthecategoryofleftR-modulesandmodulemaps{whereRissomering{denotedbyRMod.ItsobjectsaremodulesoverRanditsmorphismsareR-modulemaps.OfcentralimportancefortheexpositioninthisthesisisthecasewhereR)]TJ/F30 11.955 Tf 10.796 0 Td[(Z.WedenotethecategoryZModofmodulesoverZandZ-modulemapsbyAbandrefertoitasthecategoryofabeliangroupsandhomomorphisms.Notallcategorieshaveasobjectssetswithstructureandasmorphismsstructure-preservingmapsbetweenthem,asthefollowingexampledemonstrates.Example3.2.4.LetSbeanyset.ThediscretecategoryonShasasobjectstheelementsofSandhasnomorphismssaveforidentitymorphisms.Itistrivialtocheckthatthisisacategory.Example3.2.5.Anotherexampleofacategoryisthecategoryoftopologicalspacesandcontinuousfunctions,denotedbyTop.ItsobjectsaretopologicalspacesX,Y,Z,:::anditsmapsarecontinuousfunctionsbetweentopologicalspaces.Example3.2.6.AnexamplewithperhapsmorepertinencetothisthesisisthecategoryofsimplicialcomplexesandsimplicialmapsSimpComp.Again,itsobjectsaresimplicialcomplexesanditsmorphismsaresimplicialmapsbetweenthemwhichweintroducedinsection2.2.23 PAGE 28 Example3.2.7.AcategorywhichhasacentralroletoplayinthisthesisisthecategoryChainR,whereRisaxedring,ofchaincomplexesofR-modulesandchainmaps.Seesection4.2.Denition3.2.8.AsubcategoryDofagivencategoryCisacategorywhichhasasitsobjectcollectionObDasubsetoftheobjectcollectionObCofCandhasasitscollectionMapsDD;DofmorphismsfromDtoDasubcollectionofthecollectionMapsCD;DofmorphismsinCfromDtoD.Example3.2.9.Forthepurposesofthisthesis,agoodexampleofasubcategoryisthecategoryInjSetwhoseobjectsarejustsetsandwhosemorphismsareinjectivefunctionsbetweensets.Obviously,thisisasubcategoryofSet,thecategorydiscussedinexample3.2.2.Denition3.2.10.WesaythattwoobjectC,CareisomorphicjustincasetherearemorphismsfCCfsuchthatff)]TJ/F15 11.955 Tf 9.279 0 Td[(idCandthatff)]TJ/F15 11.955 Tf 9.278 0 Td[(idC.Typically,one'spointofviewaboutcategoryisdierentfromone'spointofviewaboutaparticularobjectofstudy,suchasagroup.Tobeginwith,thecategorieswithwhichoneisoftenworkingarelarge,verylarge.Instead,whatoneoftenthinksaboutaresmallbitsofacategory,knownasdiagrams.Adiagraminacategoryisjust11awayofdisplayingsomeobjectsofacategoryandsomemorphismsbetweenthem.Oftenthroughoutthethesisandthroughouttheliteraturethereistalkofsomediagramcommuting.Theintuitiveideaofthisdenitionisthis:inacommutativediagram,anytwopaths"ofmorphismsfromoneobjectCtoanotherCareequal.Forexample,that{insomecategoryC{thefollowingdiagramCf)]TJ/F77 11.955 Tf 38.327 0 Td[(CffCf)]TJ/F77 11.955 Tf 35.741 0 Td[(C 11Infactadiagramistheimageofafunctorfromsomeshapecategory"whoseobjectandmorphismstructureisexactlytheoneillustratedonthepagetogetherwithidentitymorphismsandcomposition,etc..24 PAGE 29 commutesjustmeansthatff)]TJ/F77 11.955 Tf 9.279 0 Td[(ff.Toeverycategoryiscanonicallyassociatedanoppositecategorywhichcontainsexactlythesameinformation".Itisusefultodiscussthesecategoriesinordertodenecontravariantfunctors"seesection3.3withoutthemappearingtobeanythingmorethancovariantfunctors"fromtheoppositecategoryofthedomain.Denition3.2.11.LetCbeacategory.TheoppositecategoryofC,denotedbyCopisthecategorywhichhasthesameobjectsasCandthesamemorphismsasC,butintheoppositedirection"{thatistosaythemorphismsetsMapsCopC;Chavetheirargumentobjectsipped",andthecompositionisbackwards".Indetail,themorphismsetisdenedbyMapsCopC;C)]TJ/F75 11.955 Tf 10.673 0 Td[(MapsCC;CandthecompositionMapsCopC;CMapsCopC;C)]TJ/F75 11.955 Tf 15.092 0 Td[(MapsCopC;CinCopfactorsinthewaydescribedbythefollowingdiagram:MapsCopC;CMapsCopC;Crenamerename)]TJ/F75 11.955 Tf 70.821 0 Td[(MapsCC;CMapsCC;CbraidcomposeCopMapsCC;CMapsCC;CcomposeCMapsCopC;CrenameMapsCC;CwherethelastarrowiscompositioninC.Therearemanydierentsortsofwell-studiedpropertiesandstructureswhichtherearegoodreasonstodemandthatacategoryofdiscussionhave.Forinstance,insection4,wewillremarkuponthefactthataparticularcategoryofinterest,namelyChainZ,hasthestructureofa2-category.Forthisthesis,weonlyeverneedtodemand,insection5,thatacategorysatisfythefollowingtwoproperties:thatthereexistaninitialobject"andthatthereexistaterminalobject".Denition3.2.12.AninitialobjectinacategoryCisanobject0Cwiththepropertythatthereexistsauniquemorphism0C)]TJ/F77 11.955 Tf 12.642 0 Td[(CtoeachobjectCinC.25 PAGE 30 Denition3.2.13.AterminalobjectinacategoryCisanobject1CwiththepropertythatthereexistsauniquemorphismC)]TJ/F75 11.955 Tf 12.643 0 Td[(1CfromeveryobjectCinC.3.3.Functors.Intheprevioussection,weintroducedatypeofobject,namelyacategory.Afterintroducingaclassofstructuredobject,theusualnextstepistointroducestructure-preservingmapsbetweenthem.Inthecaseofcategories,thereisoneauthenticsortofstructurepreservingmap{knownasafunctor{butalsoamoregeneralsortofmapwhich,althoughitdoesnotpreserveasmuchofthestructure,isstillusefultospeakof{knownasapseudofunctor.Denition3.3.1.ApseudofunctorFC)]TJ/F112 11.955 Tf 12.642 0 Td[(DisafunctionObC)]TJ/F75 11.955 Tf 12.642 0 Td[(ObDwhichassignstoeachobjectCinCanobjectFCinDtogetherwithforeachorderedpairingC,CafunctionMapsCC;C)]TJ/F75 11.955 Tf 13.121 0 Td[(MapsDFC;FCwhichassignstoeachmorphismfC)]TJ/F77 11.955 Tf 12.642 0 Td[(CamorphismFfFC)]TJ/F77 11.955 Tf 12.643 0 Td[(FC.Denition3.3.2.AcontravariantpseudofunctorFC)]TJ/F112 11.955 Tf 15.456 0 Td[(DisapseudofunctorFCop)]TJ/F112 11.955 Tf 12.642 0 Td[(D.NoticethatacontravariantpseudofunctorsendsdiagramsoftheformCf)]TJ/F77 11.955 Tf 35.741 0 Td[(CtodiagramsFCFfFCintheopposite"direction.Becauseofthisterminology,weoftenrefertonon-contravariantpseudofunctorsC)]TJ/F112 11.955 Tf 12.643 0 Td[(Dascovarianttohighlightthefactthattheyarenotcontravariant.Denition3.3.3.AfunctorFC)]TJ/F112 11.955 Tf 15.323 0 Td[(Disapseudofunctorwhichpreservesthestructureofcompositioninthefollowingsense:26 PAGE 31 Identitymorphismsarepreserved.Thatis,forCanobjectinC,thefunctorFtakestheidentitymorphismidConCtotheidentitymorphismidFConFC.Insymbols,FidC)]TJ/F15 11.955 Tf 9.279 0 Td[(idFC.Compositionsarepreserved.Thatis,wherewehavethefollowingdiagramCf)]TJ/F77 11.955 Tf 35.741 0 Td[(Cf)]TJ/F77 11.955 Tf 35.741 0 Td[(CinC,Fff)]TJ/F77 11.955 Tf 9.279 0 Td[(FfFf.Again,acontravariantfunctorFC)]TJ/F112 11.955 Tf 12.643 0 Td[(DisjustafunctorCop)]TJ/F112 11.955 Tf 12.643 0 Td[(D.AnaccurateintuitivedescriptionofafunctorFC)]TJ/F112 11.955 Tf 14.741 0 Td[(Disobtainedbythink-ingaboutndingacopyofthedomaincategoryCinsidethecodomaincategoryDwhereweallowmorphismstobestretchedapart"{i.e.senttocompositions{ortobecollapsed"{i.e.senttoidentitymorphisms.Example3.3.4.AverysimplekindoffunctorisaforgetfulfunctorfromacategoryCwhoseobjectsarestructured-setstoSetwhichisjustgivenbyforgetting"thestructure.Forexample,thecategoryTopoftopologicalspacesandcontinuousmapsisacategoryofstructuredsets{atopologicalspaceisasetequippedwithatopology;theforgetfulfunctorsendsaspacetoitsunderlyingsetandacontinuousmaptotheunderlyingsetmap.Example3.3.5.LetDbeasubcategoryofC.AsecondsimplesortoffunctoristheinclusionfunctorinclD)]TJ/F112 11.955 Tf 13.583 0 Td[(CwhichtakeseachobjectDinDtothatsameobjectD)]TJ/F15 11.955 Tf 9.279 0 Td[(inclDinCandwhichtakeseachmorphismgD)]TJ/F77 11.955 Tf 12.642 0 Td[(D.Itistrivialtocheckthatthissatisesdenition3.3.3.Example3.3.6.AmuchmoreinterestingfunctoristhesingularhomologyfunctorHTop)]TJ/F77 11.955 Tf 13.315 0 Td[(RModwhichassignstoeachtopologicalspaceitshomologymoduleswithcoecientsinR.Ofcourse,fortheclassicalcase,takeR)]TJ/F30 11.955 Tf 9.279 0 Td[(Z.27 PAGE 32 Example3.3.7.Insection2.3,weassociatedtoeverysimplicialcomplexachaincomplex,theorientedchaincomplex,andtoeachinjectivesimplicialmapachainmapfromtheorientedchaincomplexassociatedtothedomaintothatassociatedtothecodomain.Itiseasytocheckthatthisassociationpreservesbothidentitymorphismsandcomposition.Therefore,weinfactdenedafunctorZoriInjSimpComp)]TJ/F75 11.955 Tf -422.247 -23.084 Td[(ChainZknownastheorientedcomplexfunctorfromthesubcategoryofSimpCompwhichincludesonlyinjectivesimplicialmaps.Similarly,wedened,alsoinsection2.3,theorderedcomplexfunctorZoriSimpComp)]TJ/F75 11.955 Tf 12.642 0 Td[(ChainZ.Example3.3.8.Insection2.3wedenedtwoadditionalfunctors:theorientedhomologyfunctorHoriSimpComp)]TJ/F75 11.955 Tf 12.682 0 Td[(AbwhichassignstoeachsimplicialcomplexitsorientedhomologyabeliangroupsandtheorderedhomologyfunctorHordSimpComp)]TJ/F75 11.955 Tf 13.997 0 Td[(Abwhichassignstoeachsimplicialcomplexitsorderedhomologyabeliangroups.12Allthefunctorsdiscussedinexample3.3.6infactfactorasacompositionoffunctorsoftheformC)]TJ/F75 11.955 Tf 14.417 0 Td[(ChainRH)]TJ/F77 11.955 Tf 11.486 0 Td[(RModwhereCisthedomainappropriateforeachfunctorandRistheringappropriateforeachfunctor.Denition3.3.9.LetFC)]TJ/F112 11.955 Tf 12.643 0 Td[(DandGD)]TJ/F112 11.955 Tf 12.642 0 Td[(Ebepseudofunctors.ThecompositionpseudofunctorGFC)]TJ/F112 11.955 Tf 12.642 0 Td[(EisthefunctorgivenonobjectsbysendingCtoGFCandbysendingmorphismsfC)]TJ/F77 11.955 Tf 12.967 0 Td[(CtoGFfGFC)]TJ/F77 11.955 Tf 12.967 0 Td[(GFC.Obviouslythisdenesapseudofunctor.InthecasethatFandGarefunctors,thiscompositiondenesafunctor,thecompositionfunctorC)]TJ/F112 11.955 Tf 12.642 0 Td[(E.Wenowhaveaclassofobjects{namelycategories{andadenitionforstructurepreservingmapsbetweenthem{namelyfunctors{whichwecancompose.Thisallowsustodenethefollowingcategory: 12Forboththeorderedandorientedhomologyfunctors,asisthecasewiththesingularhomologyfunctor,wecanwithequaleasedenetheorientedorderedhomologyfunctorHSimpComp)]TJ/F11 9.963 Tf -423.7 -11.955 Td[(RMod.28 PAGE 33 Denition3.3.10.Thecategoryofcategoriesandfunctors,13denotedbyCat,hasasobjectsallsmallcategoriesandasmorphismsallfunctors.Denition3.3.11.WesaythatafunctorFC)]TJ/F112 11.955 Tf 12.642 0 Td[(DisfaithfulifitsmapsMapsCC;C)]TJ/F75 11.955 Tf -446.942 -23.083 Td[(MapsDFC;FConmorphismsetsareinjectiveforallorderedpairingsC;CofobjectsinC.WesaythatafunctorFC)]TJ/F112 11.955 Tf 14.685 0 Td[(DisinjectiveonobjectsifitsmapObC)]TJ/F75 11.955 Tf 15.021 0 Td[(ObDisinjective.AfunctorwhichisbothfaithfulandinjectiveonobjectsisknownasanembeddingofCinD.Denition3.3.12.WesaythatafunctorFC)]TJ/F112 11.955 Tf 12.642 0 Td[(DisfullifitsmapsMapsCC;C)]TJ/F75 11.955 Tf -427.515 -23.083 Td[(MapsDFC;FConmorphismsetsaresurjectiveforallorderedpairsC;Cofob-jectsinC.WesaythatafunctorFC)]TJ/F112 11.955 Tf 15.153 0 Td[(DissurjectiveonobjectsifitsmapObC)]TJ/F75 11.955 Tf 12.642 0 Td[(ObDissurjective.AfunctorwhichisbothfullandsurjectiveonobjectsisknownasancoveringofDbyC.Analogously,wesaythatasubcategoryisfulljustincasetheinclusionfunctorisfull.Thefollowingprovidesausefulexampleofthesenotionsofinjectivityandsurjec-tivityforfunctors:Example3.3.13.ThefunctorSet)]TJ/F75 11.955 Tf 12.642 0 Td[(CatwhichtakesasetStothediscretecategoryDSonthatsetseethedenitionatexample3.2.4isafullembedding.IfwerestrictthecodomaincategorytothecategoryDCatofdiscretecategories,thenthefunctorisalsoacoveringofDCatbySet.3.4.NaturalTransformations.Wehavenowdiscussedaclassofobjects{categories{andstructure-preservingmapsbetweenthem{functors.Classically,wewouldbedoneatthisstage.However,wearenotdone.Wenowdenenaturaltransformationsbe-tweenfunctors.Thisiswherethecategoricalpointofviewrstshowspromiseofallowingustothinkofandspeakofconnectionswecouldnotbefore:mapsbetween 13Inordertobecareful,weshouldsaythelargecategoryofsmallcategoriesandfunctorsbetweenthem.29 PAGE 34 mapsthemselvesonobjectsandmaps.Infact,oratleastasthefolkloretellsit,thisisthereasonthatEilenbergandMacLanedevelopedcategorytheory.Aswasthecasewithfunctors,thereisoneauthentictypeofstructure-preservingmapbetweenfunctors{namelyanaturaltransformation{butalsoamoregeneraltypeofmapofwhichitisusefultospeak{namelyaninfranaturaltransformation.Denition3.4.1.LetF;GC)]TJ/F112 11.955 Tf 12.642 0 Td[(DbetwopseudofunctorsfromCtoD.Aninfranatu-raltransformationTFGassignstoeachobjectCaD-morphismTCFC)]TJ/F77 11.955 Tf 12.642 0 Td[(GD.WerefertothemapTCFC)]TJ/F77 11.955 Tf 12.643 0 Td[(GCastheC-componentofT.Denition3.4.2.LetF;GC)]TJ/F112 11.955 Tf 12.752 0 Td[(Dbetwofunctors14fromCtoD.Anaturaltrans-formationTFGisaninfranaturaltransformationsothatforeverymorphismfC)]TJ/F77 11.955 Tf 12.642 0 Td[(CthefollowingdiagramFCTC)]TJ/F77 11.955 Tf 37.283 0 Td[(GCFfGfFCTC)]TJ/F77 11.955 Tf 35.741 0 Td[(GCcommutes.Denition3.4.3.LetF;GC)]TJ/F112 11.955 Tf 15.964 0 Td[(DbetwofunctorsfromCtoD.AnaturalisomorphismTFGisannaturaltransformationsothateverycomponentTCFC)]TJ/F77 11.955 Tf 12.642 0 Td[(GCisanisomorphism.Example3.4.4.Theexampleofcentralimportancetothisthesisisthenaturaltrans-formationTHoriHordfromtheorientedhomologyfunctorHoriSimpComp)]TJ/F75 11.955 Tf -422.247 -23.084 Td[(AbtotheorderedhomologyfunctorHordSimpComp)]TJ/F75 11.955 Tf 13.301 0 Td[(Abdiscussedinsection2.4andgeneralizedinthesequel.Thenaturaltransformationinthepreceedingexampleisactuallyobtainedasthepush-forwardalongH,thehomologyfunctor,ofanaturaltransformationTZord 14WhereasforinfranaturaltransformationsitwassucientFandGbepseudofunctors,itisneces-saryforthedenitionofnaturaltransformationsthatthesemapsbefunctors.30 PAGE 35 ZorifromtheorderedcomplexfunctorZordSimpComp)]TJ/F75 11.955 Tf 13.291 0 Td[(ChainZ,formalizedatexample3.3.7,totheorientedcomplexfunctorZoriSimpComp)]TJ/F75 11.955 Tf 12.642 0 Td[(Ab,alsoformal-izedat3.3.7.Infactweobtainthatthenaturaltransformationofexample3.4.4isanaturalisomorphismviathepush-forwardalongHofaninfranaturaltransformationTZoriZordfromtheorientedcomplexfunctortotheorderedcomplexfunctor.Denition3.4.5.LetF;GC)]TJ/F112 11.955 Tf 12.643 0 Td[(DbepseudofunctorsfromCtoD.LetTFGbeaninfranaturaltransformation.LetHD)]TJ/F112 11.955 Tf 12.642 0 Td[(Ebeapseudofunctor.Thepush-forwardofTalongHistheinfranaturaltransformationwhicharisesastheimageHFCHTC)]TJ/F77 11.955 Tf 35.741 0 Td[(HFCofthediagramFCTC)]TJ/F77 11.955 Tf 35.741 0 Td[(GCunderthepseudofunctorH.Denition3.4.6.LetF;GC)]TJ/F112 11.955 Tf 13.581 0 Td[(DbefunctorsfromCtoD.LetTFGbeanaturaltransformation.LetHD)]TJ/F112 11.955 Tf 12.642 0 Td[(Ebeafunctor.Thepush-forwardofTalongHisthenaturaltransformationwhicharisesastheimageHFCHTC)]TJ/F77 11.955 Tf 37.283 0 Td[(HGCHFfHGfHFCHTC)]TJ/F77 11.955 Tf 35.741 0 Td[(HGCofthediagramFCTC)]TJ/F77 11.955 Tf 37.283 0 Td[(GCFfGfFCTC)]TJ/F77 11.955 Tf 35.741 0 Td[(GCunderthefunctorH.Example3.4.7.LetPbeacategorywhichweknoweverything"about;wecallitaprobecategory.LetCbesomecategorywhichwewanttogetinformationabout.A31 PAGE 36 P-objectinCisjustacontravariantfunctorP)]TJ/F112 11.955 Tf 12.89 0 Td[(C.ThenamapfromoneP-objectFtoanotherGisjustanaturaltransformationTFG.WewouldliketobeabletocomposemapsbetweenP-objectsinC.Todoso,wemustdeneverticalcompositionofnaturaltransformations:Denition3.4.8.LetF;G;HC)]TJ/F112 11.955 Tf 12.642 0 Td[(Dbefunctors.15LetTFGandTGHbenaturaltransformations.WedenethecompositionnaturaltransformationT)]TJ/F77 11.955 Tf -425.611 -23.084 Td[(TTFHoncoordinatesbyTC)]TJ/F77 11.955 Tf 10.264 0 Td[(TCTC.Wecheckthatthisdenesanaturaltransformationbycomposingcommutativediagramstoobtainanewcommutativediagram".Thefollowingsortofcategory,whichwecandeneonlynowthatwehaveintro-ducedallofthislanguage,hasanimportantparttoplayinthisthesis.Denition3.4.9.LetC;Dbecategories.ThefunctorcategoryDCisthecategorywhoseobjectsarefunctorsandwhosemorphismsarenaturaltransformations.3.5.ConcludingRemarks.Overthecourseofsection3,wehaveintroducedthemostfundamentaldenitionsofcategorytheory.Althoughtherecertainlyarethingsinthisthesiswherecategoricallanguagewasusedalthoughitwasnotnecessary,themaintheoremofthethesis{thenaturalisomorphismbetweenorientedandorderedhomology"{aswellasmanyotherresults,couldnotbestatedinanyconciseandcomprehensiblewaywithoutthislanguage.Itservesasaninvaluablewaynotonlytoorganizeinformationbutalsotoabstractawayfromtraditionalnotionsofinformationtoprovedeeptheorems. 15Wecoulddeneaswelltheverticalcompositionofinfranaturaltransformationsbetweenpseud-ofunctors,butthatnotiondoesnotplayaroleinthisthesis.32 PAGE 37 4.SomeHomologicalAlgebra4.1.IntroductionandMotivation.Insection2.3,weassociatedtoeachsimplicialcomplextwochaincomplexes:1theorientedchaincomplexseedenition2.3.5andtheorderedchaincomplexseedenition2.3.13.Wealsoassociatedtoeachsimplicialmapbetweensimplicialcomplexesawell-behavedorientedchainmapbetweentheassociatedorientedchaincomplexesseedenition2.3.8andawell-behavedorderedchainmapbetweentheassociatedorderedchaincomplexesseedenition2.3.16.Aswediscussedinchapter3,wedenedtwofunctorsfromthecategoryofsimplicialcomplexesandmapstothecategoryofchaincomplexesofZ-modules:theorientedchainfunctorandtheorderedchainfunctor.Whatwereallywanttounderstand,however,isthehomologyofchaincomplexes.Rememberthatbymeansoftheorientedandorderedchaincomplexes,weassociatedtoeachsimplicialcomplextwoabeliangroups:theorientedhomologydenition2.3.7and2theorderedhomologydenition2.3.15.Furthermore,bymeansofthesemapsonchaincomplexes,weassociatedtoeachsimplicialmapahomomorphismbetweentheassociatedorientedhomologiesseedenition2.3.11andahomomor-phismbetweentheassociatedorderedhomologiesseedenition2.3.19.Infact,wehadinthiswaydenedtwofunctorsfromthecategoryofsimplicialcomplexesandsimplicialmapstothecategoryofabeliangroups:theorientedandorderedhomologyfunctors.Inthissection,wedevelopthemachinerywhichwillallowustounderstandtherela-tionshipbetweenthesefunctors.WebeginbyintroducingabstractchaincomplexesofZ-modules.WethenproceedtodenehomologyZ-modulesfortheseobjects.Infact,wedenethehomologyfunctorHChainZ)]TJ/F75 11.955 Tf 12.642 0 Td[(ChainZ.Next,weintroducethenotion,centralforourpurposeofunderstandingtheconnectionbetweentheorientedandorderedhomologyfunctors,ofchainhomotopyandweinvestigatethewaychainhomotopyinteractswithhomology.Finally,weprovetheacycliccarriertheorem,atheoreminwhichchainhomotopyplaysacentralrole.33 PAGE 38 4.2.ChainComplexes.Webeginbyintroducing,inanabstractsettingforthersttime,chaincomplexesofZ-modulesandchainmaps{thecategoryofchaincomplexesofZ-modules.Wethenproceedtointroducetwospecictypesofchaincomplexeswhichwebegantodiscussinsection2.3andwillcontinuetodiscussthroughoutthethesis:chaincomplexeswithnon-trivialgradingoverN,thenaturalnumbers,andchaincomplexeswithnon-trivialgradingover^N,theaugmentednaturalnumbers,16aswellassometechnicalpropertiesoftherelationshipbetweenthetwotypes.Denition4.2.1.AchaincomplexofrightZ-modules17isasequenceC)]TJ/F24 11.955 Tf -425.611 -23.084 Td[(Cii>ZofabeliangroupsCiequippedwithhomomorphisms,knownasboundarymaps@i)]TJ/F77 11.955 Tf 9.733 0 Td[(@iCCi)]TJ/F75 11.955 Tf 13.097 0 Td[(Ci1foralli>Zwhichsatisfythat@i1@i)]TJ/F15 11.955 Tf 9.733 0 Td[(0.WewillrefertosuchaCexclusivelyassimplyachaincomplex.Denition4.2.2.Achaincomplexmap,orsimplyachainmap,C)]TJ/F75 11.955 Tf 15.1 0 Td[(DisasequenceofhomomorphismsiCi)]TJ/F75 11.955 Tf 12.643 0 Td[(Dii>ZwhichcommutewiththeboundaryoperatorsinthesensethatthediagramCi@iC)]TJ/F75 11.955 Tf 36.038 0 Td[(Ci1iiDi@iC)]TJ/F75 11.955 Tf 35.741 0 Td[(Di1iscommutativei.e.@iDi)]TJ/F77 11.955 Tf 9.279 0 Td[(i1@iC.Inpractice,wewillbediscussingonlytwospecialtypesofchaincomplexes:thosechaincomplexeswhereallthegroupsindimensionstrictlylessthan0vanish,andthosewhereallthegroupsindimensionlessthan1vanish.AlthoughbothsortsofcomplexesareinfactstillgradedoverZ,thenon-vanishinggroupsi.e.thoseof 16Bytheaugmentednaturalnumbers,wemeantheset,denotedby^N,denedby^N)]TJ/F51 9.963 Tf 7.887 0 Td[(1;0;1;2;:::.17EverythinginthissectionworksifweworkoveranarbitraryringRratherthanZ.Theproofsaredoneonlywiththisassumption,and,althoughthesymbolisusedrepeatedly,themorespecicstructureofZisnotused.34 PAGE 39 interestaregradedoverasubsetZ.ThuswewillspeakofthoseoftherstsortasbeingN-gradedandthoseofthesecondsortasbeing^N-graded.Denition4.2.3.Anaugmentedchaincomplexisan^N-gradedchaincomplexCwhichsatisesthelowest-gradednot-necessarily-trivialgroupC1isjustZtheboundarymap@0CC0)]TJ/F75 11.955 Tf 12.643 0 Td[(C1issurjective.Thisterminologymayseemstrangeatrst.Onemightaskinwhatsensedoesachaincomplexwhichvanishesineverydimensionbelow1augmentanarbitrarily-gradedchaincomplex?"Theansweristhatitdoesnotaugmentanarbitrarily-gradedchaincomplex,butthatitdoesaugmentanarbitrarychaincomplexwhereweareabusingthewordchaincomplex"totheclassofobjectsN-gradedchaincomplex".WewillbeinterestedespeciallyinaugmentedchaincomplexeswhicharisefromN-gradedchaincomplexes,sincethesewillallowustodenereducedhomology.Denition4.2.4.AnaugmentationofchaincomplexCisanaugmentedchaincom-plexACwhichsatises:theithgroupsAiCoftheaugmentedcomplexarethesameasthegroupsCiforix1theboundarymaps@i)]TJ/F77 11.955 Tf 10.101 0 Td[(@iACAiC)]TJ/F45 11.955 Tf 13.465 0 Td[(Ai1Coftheaugmentedcomplexarethesameastheboundarymaps@iCCi)]TJ/F75 11.955 Tf 13.852 0 Td[(Ci1oftheoriginalcomplexforix0;1WewillalsorefertoACasaC-augmentation.Ingeneraltherearemanyaugmentationsforagivenchaincomplex.Soinordertonotationallyhighlightthenon-functorialityofobtaininganaugmentedcomplex,"wewilldenotetheaugmentationsoftwochaincomplexesCandDbyACandADrespectively{ratherthanbyACandAD,whichmightseemtosuggest,evenforamoment,thatAisafunctor.Wenowintroducetheclassofmapsbetweenaugmentationsofchaincomplexesinwhichwewillbeinterested.35 PAGE 40 Denition4.2.5.TheaugmentionofachainmapC)]TJ/F75 11.955 Tf 12.642 0 Td[(DonN-graded18C)]TJ/F75 11.955 Tf 12.642 0 Td[(Disthesequence)]TJ/F24 11.955 Tf 9.279 0.154 Td[(ii>Zwherei)]TJ/F24 11.955 Tf 9.279 18.118 Td[(iCi)]TJ/F75 11.955 Tf 12.643 0 Td[(Diifix1idZZ)]TJ/F30 11.955 Tf 12.642 0 Td[(Zifi)]TJ/F39 11.955 Tf 9.279 0 Td[(1Wewillalsorefertoasa-augmentation.Denition4.2.6.AnaugmentationofachainmapC)]TJ/F75 11.955 Tf 13.251 0 Td[(Disaugmentation-preserving{withrespecttotheaugmentationsACandADofCandDrespectively{ifthesequence)]TJ/F24 11.955 Tf 9.8 0.155 Td[(ii>ZisachainmapfromACtoAD{seedenition4.2.2.Ifitisreallynecessary,wemightsaythatisAC;AD-augmentation-preserving.NotethateveryAC;AD-augmentation-preserving-augmentationAC)]TJ/F45 11.955 Tf -422.247 -23.083 Td[(ADisachainmapAC)]TJ/F45 11.955 Tf 14.529 0 Td[(AD.Theconverseisnottrue:thetimes2"mapfromtheuniqueaugmentationA0ofthezerochaincomplex0toitselfisachainmap,obviously,butcertainlydoesnotariseasa0-augmentationsinceitisnotgivenbytheidentityindimension1.Inotherwords,thesubcategoryofchaincomplexaugmentationsandchainmapaugmentationsisnotfullinthecategoryofchaincomplexes,denedbelow:Denition4.2.7.TheZ-modulechaincomplexcategory,denotedbyChainZ,hasasobjectschaincomplexesofZ-modulesandasmorphismschainmaps.TheZ-moduleN-gradedchaincomplexcategory,denotedbyNChainZ,hasasobjectsN-gradedchaincomplexesofZ-modulesandasmorphismschainmaps.ThecategoryofaugmentationsofN-gradedchaincomplexes,denotedbyANChainZ,hasasobjectsaugmentationsofN-gradedchaincomplexesandaugmentation-preservingaugmentedchainmaps.Thelasttwocategoriesdenedindenition4.2.7arebothsubcategoriesofChainZ.ObservethatNChainZisafullsubcategoryofChainZsinceeverymapinChainZ 18Thisclauseisparentheticalizedbecause{aswementioneddirectlyafterdenition4.2.3{wewillrefertoN-gradedchaincomplexessimplyaschaincomplexes".Thiswillnotbepointedoutagain.36 PAGE 41 betweentwoN-gradedchaincomplexesisachainmap,asiseverymapNChainZ.Asmentionedinthecommentsabovedenition4.2.7,ANChainZisnotafullsubcat-egoryofChainZsincewerequirethatmorphismsinANChainZnotonlybechainmapsbutthattheybe,inaddition,augmentation-preservingandwegaveexplicitlyan{albeittrivial{exampleofachainmapwhichisnotaugentation-preserving.Fur-thermore,thereisastraightforwardfunctorA1ANChainZ)]TJ/F30 11.955 Tf 14.274 0 Td[(NChainZgivenonobjectsbyACCandonmorphismsby;thisfunctorjustforgets"theaugmentation.Infact,thisfunctorisfullyfaithful.Inthecomingsection4.3,wewilldiscussreducedhomologybymeansofthisfunctor.4.3.HomologyofChainComplexes.Inthissection,wewilldenethehomologyfunctorwhichassociatestoachaincomplexCanabeliangroupknownasitshomologyandtomapsbetweenchaincomplexesmapsbetweentheassociatedhomologygroups.Further,thisallowsthedenition,viaanarbitraryaugmentationAC,ofasecondabeliangroupknownasitsreducedhomology.Theseobjectsandmorphismsareniceforavarietyofreasons.Forpurelyalgebraicreasons,theyaremuchsimplerthanthechaincomplexesandchainmapstowhichtheyareassociated.Moreover,inpractice,achaincomplexisfunctoriallyassociatedtoasomeobjectC>ObCandachainmaptosomemapf>MapsC;itturnsoutthatpassingfromassociatedchaincomplexestoassociatedhomologyoftenamountstoquotientingoutbyincidentalinformation.Forinstance,twodierenttriangulationsofatopologicalspaceyieldtwodistinctchaincomplexesbothofwhichhavethesamehomologygroup;inthiscase,theincidentalinformationwascomprisedofthedetailsofthetriangulations.Denition4.3.1.ThehomologyofachaincomplexCisthesequenceHC)]TJ/F24 11.955 Tf -425.611 -23.084 Td[(HiC)]TJ/F15 11.955 Tf 9.279 0 Td[(ker@i~im@i1i>Zofquotientsofthekernelsofthenthboundarymapsbytheimagesofthen1stboundarymaps.Thisquotientmakessensebecauseofthe37 PAGE 42 conditionontheboundarythat@i@i1)]TJ/F15 11.955 Tf 9.336 0 Td[(0.Equivalently,19werefer,bythehomologyofC,totheZ-gradedorderedsumHC)]TJ/F21 11.955 Tf 11.026 -0.941 Td[(>n>ZHnC.Denition4.3.2.Thereducedhomology,ifitexists,20ofachaincomplexC,denotedbyHC,isjustthestandardhomologyofanaugmentationACoftheoriginalchaincomplex.InsymbolsHC)]TJ/F75 11.955 Tf 10.04 0 Td[(HAC.Asbefore,werefer,byreducedhomologyofC,totheZ-gradedorderedsumHC)]TJ/F21 11.955 Tf 11.026 -0.941 Td[(>n>ZHC)]TJ/F21 11.955 Tf 11.026 -0.941 Td[(>n>ZHnAC.Apriori,reducedhomologyisnotawell-denedobjectforaxedchaincomplexsinceitiscontingentonachoiceofaugmentation.Weobtainthatitis,however,welldenedasanimmediateconsequenceofthefollowing:Observation4.3.3.LetCbeachaincomplex.SupposeACisanaugmentationofC.ThenHiC)]TJ/F109 11.955 Tf 9.279 0 Td[(HiACforix0andH0CH0AC`ZProof.FirstnotethattheidentityHiC)]TJ/F75 11.955 Tf 10.169 0 Td[(HiACforix0;1isanimmediatecon-sequenceoftheequalities@iC)]TJ/F77 11.955 Tf 10.33 0 Td[(@iACandCi)]TJ/F45 11.955 Tf 10.33 0 Td[(AiCforix0;1fromthedenitionofanaugmentation.ToseethatH1C)]TJ/F75 11.955 Tf 10.368 0 Td[(H1C,simplyobservethat,since@0ACissurjective,H1ACistrivial,asisH1C.Fori)]TJ/F15 11.955 Tf 9.279 0 Td[(0,wewillconstructashortexactsequence0)]TJ/F75 11.955 Tf 12.642 0 Td[(H0ACf)]TJ/F75 11.955 Tf 11.486 0 Td[(H0Cg)]TJ/F30 11.955 Tf 11.487 0 Td[(Z)]TJ/F75 11.955 Tf 12.643 0 Td[(0ofZ-moduleswhichwillsplitbecauseZisobviouslyaprojectiveZ-module. 19Hereisthenatureofthisequivalence:Firstnotethatitispossibletoconstructthesequencefromthesumandthesumfromthesequence.AmaphN)]TJ/F11 9.963 Tf 11.318 0 Td[(M{fromoneZ-gradedsequenceN)]TJ/F51 9.963 Tf 7.887 0 Td[(Nnn>ZofZ-modulestoanotherM)]TJ/F51 9.963 Tf 7.887 0 Td[(Mnn>Z{isaZ-gradedsequencehnNn)]TJ/F11 9.963 Tf 10.76 0 Td[(Mnn>ZofmapsfromthenthlevelmoduleNnoftherstsequencetothenthlevelmoduleMnofthesecondsequence.WeidentifytherstsequenceNwiththeorderedsum>n>ZNnandthesecondsequenceMwiththeorderedsum>n>ZMn,andweallowexactlythosemapsontheleveloforderedsumswhichoccuronthelevelofsequences.Thusthemapoforderedsumscorrespondingtohistheorderedsumh)]TJ/F48 9.963 Tf 9.908 -0.775 Td[(>n>Zhndescribedbyh>n>Znn)]TJ/F48 9.963 Tf 9.908 -0.775 Td[(>n>Zhnnn.20Itisclearthatreducedhomologyisnotdenedforthezerochaincomplexsinceitwouldinvolveasurjection0)]TJ/F30 9.963 Tf 10.759 0 Td[(Z.38 PAGE 43 Toconstructthissequence,wechasearrowsinthefollowingdiagram00)]TJ/F15 11.955 Tf 35.741 0 Td[(im@1ACincl)]TJ/F15 11.955 Tf 35.741 0 Td[(ker@0ACq)]TJ/F75 11.955 Tf 35.74 0 Td[(H0AC)]TJ/F75 11.955 Tf 35.741 0 Td[(0[incl0)]TJ/F15 11.955 Tf 39.188 0 Td[(im@1Cincl)]TJ/F15 11.955 Tf 39.188 0 Td[(ker@0Cq)]TJ/F75 11.955 Tf 40.787 0 Td[(H0C)]TJ/F75 11.955 Tf 35.741 0 Td[(0@0ACZ0wherethecolumnandbothrowsareshortexactsequences.Thetworowsareexactjustbythedenitionofhomology,andthecolumnisexactbecauseA0C)]TJ/F15 11.955 Tf 9.279 0 Td[(ker@0Cand@0ACissurjectivebythedenitionofanaugmentation.Weconstructourshortexactsequenceinthethirdcolumntotheright.WerstdenegH0C)]TJ/F30 11.955 Tf 14.883 0 Td[(Z.Let)]TJ/F94 11.955 Tf 4.551 0 Td[(C>H0C.ThenthereissomeC>ker@0CsuchthatqC)]TJ/F24 11.955 Tf 11.25 0 Td[()]TJ/F94 11.955 Tf 4.552 0 Td[(Csinceqsurjective{moreover,qCB)]TJ/F24 11.955 Tf 11.25 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(CforanyB>im@1C.Infactq1)]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F24 11.955 Tf 10.758 0.154 Td[(CBSB>im@1C.Sodenegat)]TJ/F94 11.955 Tf 4.551 0 Td[(Ctobeg)]TJ/F94 11.955 Tf 4.552 0 Td[(C)]TJ/F77 11.955 Tf 10.758 0 Td[(@0ACC.Thisiswell-dened.Forsuppose)]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F24 11.955 Tf 11.043 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(C.ThenC)]TJ/F94 11.955 Tf 11.043 0 Td[(CBforsomeB>im@1C.Then@0ACCB)]TJ/F77 11.955 Tf 10.412 0 Td[(@0ACC@0ACB)]TJ/F77 11.955 Tf 10.412 0 Td[(@0ACCasrequiredsinceB>im@1C)]TJ/F15 11.955 Tf 10.412 0 Td[(im@1AC`ker@0AC.Observethatgissurjectivesince@0ACissurjective.WenowdenefH0AC)]TJ/F75 11.955 Tf 13.776 0 Td[(H0C.Let)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1AC>ker@0AC.ThenqC)]TJ/F24 11.955 Tf 10.412 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1AC.Nowker@0AC`ker@0C,soC>ker@0C.Denegat)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1ACtobeg)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@1AC)]TJ/F24 11.955 Tf 9.686 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1C.Thisiswell-denedbecause)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1AC)]TJ/F24 11.955 Tf 11.014 0 Td[()]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@1ACjustincaseC)]TJ/F94 11.955 Tf 11.014 0 Td[(CBforsomeB>im@1AC)]TJ/F15 11.955 Tf 10.919 0 Td[(im@1Cwhichisexactlytheconditionunderwhich)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1C)]TJ/F24 11.955 Tf 10.919 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@1C.Observethatfisinjectivesinceinclker@0AC)]TJ/F15 11.955 Tf 12.642 0 Td[(ker@0Cisinjective.Atlastobservethatimf)]TJ/F15 11.955 Tf 9.279 0 Td[(kerg.Forsuppose)]TJ/F94 11.955 Tf 4.551 0 Td[(C>H0Cisinimf.ThenC>ker@0AC,sog)]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F77 11.955 Tf 11.29 0 Td[(@0ACC)]TJ/F15 11.955 Tf 11.29 0 Td[(0;i.e.)]TJ/F94 11.955 Tf 4.551 0 Td[(C>kerg.Conversely,supposethat)]TJ/F94 11.955 Tf 4.551 0 Td[(C>kerg.Then@0ACC)]TJ/F15 11.955 Tf 9.279 0 Td[(0;i.e.C>ker@0AC.Sof)]TJ/F94 11.955 Tf 4.552 0 Td[(C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F94 11.955 Tf 4.552 0 Td[(C;thus)]TJ/F94 11.955 Tf 4.552 0 Td[(C>imf.39 PAGE 44 Sowehaveconstructedtheshortexactsequence0)]TJ/F75 11.955 Tf 12.642 0 Td[(H0ACf)]TJ/F75 11.955 Tf 11.486 0 Td[(H0Cg)]TJ/F30 11.955 Tf 11.487 0 Td[(Z)]TJ/F75 11.955 Tf 12.643 0 Td[(0aswehaddesired.Sowearedone:thesequencesplitssinceZisaprojectiveZ-module.ThatisH0CH0AC`Z,whichiswhatwehavebeentryingtoprove.Construction4.3.4.GivenachainmapC)]TJ/F75 11.955 Tf 12.683 0 Td[(DfromCtoD,wewouldliketoinduce,inafunctorialfashion",amaponthelevelofassociatedhomologygroupsHHC)]TJ/F75 11.955 Tf 12.642 0 Td[(HD.Suppose)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C>HnC,i.e.thatC>ker@nC.WewouldliketowriteHn)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf -427.856 -23.084 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(Dim@n1C>HnD,i.e.whereD>ker@nD.Wehaveonlyonesymboltoworkwith,namely,toinducethismap.ThissuggestsdeningD)]TJ/F77 11.955 Tf 9.279 0 Td[(C,i.e.deningHn)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf -442.084 -23.084 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(Cim@n1C.ButweneedtoprovethatthisdenesamapofZ-modules.Solet)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@n1C>HnCsuchthat)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf 9.699 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C.WeprovethatHn)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C)]TJ/F75 11.955 Tf 9.699 0 Td[(Hn)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C.Observethatsincethetwoequivalenceclassesareequal,wemusthaveC)]TJ/F94 11.955 Tf 10.725 0 Td[(CBwhereB>im@n1C.ThusHn)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C)]TJ/F75 11.955 Tf 11.107 0 Td[(Hn)]TJ/F94 11.955 Tf 4.551 0 Td[(CBim@n1C.So,bythedenitionofthismap,wegetHn)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf 11.175 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(nCim@n1D)]TJ/F77 11.955 Tf 4.551 0 Td[(nBim@n1D.Butsincethefacemapscommutewithchainmapcomponents,wehavethatnim@n1C`im@n1D.Inparticular,nB>im@n1D,so)]TJ/F77 11.955 Tf 4.552 0 Td[(nBim@n1D)]TJ/F15 11.955 Tf 9.279 0 Td[(0.ThuswehaveobtainedthatHn)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(nCim@n1D)]TJ/F75 11.955 Tf 9.279 0 Td[(Hn)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@n1Casweneededtoshowthatthemapiswell-dened.ThatitisamodulemapfollowsdirectlyfromthefactthatnisaZ-modulemap.Insummary,weconstructedfromC)]TJ/F75 11.955 Tf 13.272 0 Td[(DthemapHHC)]TJ/F75 11.955 Tf 13.271 0 Td[(HDgivenonthen>ZthcoordinatebyHn)]TJ/F94 11.955 Tf 4.552 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(Cim@n1C40 PAGE 45 Denition4.3.5.The-inducedmaponhomology,whereC)]TJ/F75 11.955 Tf 13.625 0 Td[(D,isthemapHHC)]TJ/F75 11.955 Tf 12.642 0 Td[(HDdescribedinconstruction4.3.4above.Observation4.3.6.HomologyisafunctorHChainZ)]TJ/F75 11.955 Tf 12.642 0 Td[(Ab.Proof.Sofar,wehaveonlydenedapseudofunctor"HChainZ)]TJ/F75 11.955 Tf 12.642 0 Td[(Ab21;inotherwords,wehaveveriedthatthediagramC)]TJ/F75 11.955 Tf 11.487 0 Td[(DissenttothediagramHCH)]TJ/F75 11.955 Tf 13.431 0 Td[(HDToprovetheresult,weneedtoshowthatthepseudofunctorpreserveidentitymor-phismsandcompositions.FirstweconsidertheidentitymapidCC)]TJ/F75 11.955 Tf 13.218 0 Td[(CofanarbitrarychaincomplexCunderthefunctorHandshowthatHidC)]TJ/F15 11.955 Tf 10.529 0 Td[(idHC.Infact,thisisimmediate:Forsuppose)]TJ/F94 11.955 Tf 4.552 0 Td[(C>HC;thenbydenitionHidC)]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F15 11.955 Tf 4.551 0 Td[(idC)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F15 11.955 Tf 9.279 0 Td[(idHC)]TJ/F94 11.955 Tf 4.551 0 Td[(CThereforeHidC)]TJ/F15 11.955 Tf 11.742 0 Td[(idHCasrequired;inotherwords,homologypreservesidentitymorphisms.WewanttoshowthatthefunctorHpreservescompositions.Inotherwords,wewanttoshowthatadiagraminChainZoftheformC)]TJ/F75 11.955 Tf 38.104 0 Td[(D)]TJ/F75 11.955 Tf 35.741 0 Td[(E[[C)]TJ/F77 11.955 Tf 35.741 0 Td[()]TJ/F75 11.955 Tf 35.741 0 Td[(E 21TheskepticalreaderwillhavenoticedthatwehaveactuallydenedapseudofunctorwhichhasasitscodomainthecategorywhoseobjectsareZ-gradedorderedsumsofZ-modulesandwhosemorphismsaregrade-preservingmapsofZ-modules.Luckilyforthepoorauthor{asaconsequenceofthefactsthateveryZ-gradedorderedsumofZ-modulesisaZ-moduleand2thateverygrade-preservingmapofZ-modulesisamapofZ-modules{thereisastraightforwardembeddingofthiscategoryintoAbandwearesecretlyfollowingthehomologyfunctorbythisembedding.41 PAGE 46 istakenbythefunctorHtothediagraminAbHCH)]TJ/F75 11.955 Tf 38.105 0 Td[(HDH)]TJ/F75 11.955 Tf 35.741 0 Td[(HE[[HC)]TJ/F75 11.955 Tf 35.741 0 Td[(H)]TJ/F75 11.955 Tf 35.741 0 Td[(HEMoreconcisely,wewanttoshowthatH)]TJ/F75 11.955 Tf 10.185 0 Td[(HH.Butthisisasimmediateasbefore:Forsuppose)]TJ/F94 11.955 Tf 4.551 0 Td[(C>HC.Bysimplycomputing,wendthatH)]TJ/F94 11.955 Tf 4.551 0 Td[(C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(C)]TJ/F75 11.955 Tf 9.279 0 Td[(H)]TJ/F77 11.955 Tf 4.551 0 Td[(C)]TJ/F75 11.955 Tf 9.279 0 Td[(HH)]TJ/F94 11.955 Tf 4.551 0 Td[(CThereforeH)]TJ/F75 11.955 Tf 9.959 0 Td[(HHasrequired;inotherwords,homologypreservescomposi-tion.WehaveproventhatHChainZ)]TJ/F75 11.955 Tf 12.642 0 Td[(Abisapseudofunctorwhichpreservesidentitymorphismsandcompositions;that,inotherwords,Hisafunctorasclaimed.Asmentionedwhenweintroduceditindenition4.3.2,reducedhomologyisnotafunctorfromthecategoryNChainZ.Howeveritisafunctorfromasubcategory,namelytheimage{whichwedenotebyZA1AChainN{ofthedeaugmentfunctorA1ANChainZ)]TJ/F30 11.955 Tf 12.642 0 Td[(NChainZintroducedattheveryendofsection4.2,givenonobjectsACbyA1AC)]TJ/F75 11.955 Tf 10.455 0 Td[(CwhereACisanaugmentationofCandonmorphismsbyA1.Infact,thisisthelargestsubcategoryonwhichreducedhomologycanbedened:thedenitionofthereducedhomologyofachaincomplexinvolvespassingtoanaugmentationofthechaincomplex,andthiscategoryismadeupexactlythoseN-gradedchaincomplexesforwhichanaugmentationexists.Beforewecanprovethatreducedhomologyisafunctoronthiscategory,wemustnishdeningit.Westillneedtodeneitonchainmaps.Construction4.3.7.GivenachainmapC)]TJ/F75 11.955 Tf 12.642 0 Td[(D,inZA1AChainN,weagainwantto,inafunctorialway,induceamapofassociatedreducedhomologygroups42 PAGE 47 HHC)]TJ/F24 11.955 Tf 14.683 3.112 Td[(HD.Luckily,theworkisalmostdonealready.Indimensionsix0,wehavethatHiC)]TJ/F75 11.955 Tf 9.279 0 Td[(HiCandHiD)]TJ/F75 11.955 Tf 9.279 0 Td[(HiD,sowedeneHi)]TJ/F75 11.955 Tf 9.279 0 Td[(Hi.Fortheremainingcase,wewillmakeuseofthemapH0.Butrst,letusx,onceandforall,foreachchaincomplexC,anisomorphismICH0C)]TJ/F24 11.955 Tf 14.325 3.112 Td[(H0C`Z.Thisispossiblesincebyobservation4.3.3,suchanisomorphismexists.Withthatdone,weproceedtodeneH0tobethecompositionH0Cincl1)]TJ/F24 11.955 Tf 16.954 3.112 Td[(H0C`ZI1C)]TJ/F75 11.955 Tf 12.278 0 Td[(H0CH0)]TJ/F75 11.955 Tf 15.257 0 Td[(H0DID)]TJ/F24 11.955 Tf 13.449 3.112 Td[(H0D`Zproj1)]TJ/F24 11.955 Tf 18.017 3.112 Td[(H0DSo,tosummarize,wehaveassociatedtogivenachainmapC)]TJ/F75 11.955 Tf 13.261 0 Td[(DamapontheassociatedreducedhomologyHHC)]TJ/F24 11.955 Tf 14.325 3.112 Td[(HD.Denition4.3.8.The-inducedmaponreducedhomology,whereC)]TJ/F75 11.955 Tf 14.506 0 Td[(DisachainmapinZA1AChainN,isthemapHHC)]TJ/F24 11.955 Tf 16.81 3.112 Td[(HDgivenaboveinconstruction4.3.7.Wenowobtain,asacorollarytoourobservation4.3.6thathomologyisafunctor,thefollowing:Corollary4.3.9.ReducedhomologyHZA1AChainN)]TJ/F75 11.955 Tf 12.642 0 Td[(Abisafunctor.Proof.Clearly,reducedhomologyaswehavedeneditisapseudofunctorZA1AChainN)]TJ/F75 11.955 Tf -460.467 -23.084 Td[(Ab.Again,weneedtoshowthatitpreservesidentitymorphismsandcompositions.SincethemapHiisequaltoHionlevelsix0,weonlyneedtocheckthatthecom-ponentfunctorH0preservesidentitymorphismsandcompositions,sincewealreadyknowthatHidoesforalli>Zbyobservation4.3.6.SupposethatCisachaincomplexinZA1AChainN.WeneedtoprovethatH0preservestheidentitymorphismidC.Thatis,weneedtoshowthatH0idC)]TJ/F15 11.955 Tf 9.442 0 Td[(idH0C.RecallthatwedenedH0idCtobethecompositionH0Cincl1)]TJ/F24 11.955 Tf 16.954 3.112 Td[(H0C`ZI1C)]TJ/F75 11.955 Tf 12.278 0 Td[(H0CH0idC)]TJ/F75 11.955 Tf 19.487 0 Td[(H0CIC)]TJ/F24 11.955 Tf 13.285 3.112 Td[(H0C`Zproj1)]TJ/F24 11.955 Tf 18.016 3.112 Td[(H0C43 PAGE 48 ButthemiddlemappingH0idCistheidentitymapping,sincehomologyisafunctor,thecompositionI1CICistheidentitymapping,andthecompositionincl1proj1istheidentitymapping.ThereforeH0idC)]TJ/F15 11.955 Tf 10.595 0 Td[(idHCasrequired.Soreducedhomologypreservesidentitymorphisms.NowsupposethatC)]TJ/F75 11.955 Tf 11.486 0 Td[(D)]TJ/F75 11.955 Tf 11.486 0 Td[(EisadiagraminZA1AChainNWeshowthatH0)]TJ/F24 11.955 Tf 13.043 3.112 Td[(H0H0.Toseethis,observethatH0H0isdenedtobethecompositionH0CH0D H0DH0Eincl1proj1incl1proj1H0C`ZH0D`ZH0D`ZH0E`ZI1CIDI1DIEH0CH)]TJ/F75 11.955 Tf 46.134 0 Td[(H0DH0DH)]TJ/F75 11.955 Tf 46.134 0 Td[(H0EthemiddlesnakingportionisjusttheidentitymaponH0D.ThusH0H0isgivenbythemoreconcisecompositionH0Cincl1)]TJ/F24 11.955 Tf 16.955 3.112 Td[(H0C`ZI1C)]TJ/F75 11.955 Tf 12.278 0 Td[(H0CH0)]TJ/F75 11.955 Tf 15.257 0 Td[(H0DH0)]TJ/F75 11.955 Tf 15.508 0 Td[(H0EIE)]TJ/F24 11.955 Tf 13.168 3.112 Td[(H0E`Z)]TJ/F24 11.955 Tf 14.325 3.112 Td[(H0EButhomologyisafunctor,soit's0thcomponentpreservescompositions;inparticularH0H0)]TJ/F75 11.955 Tf 9.279 0 Td[(H0.ThuswecanrewriteH0H0onelasttimeasH0Cincl1)]TJ/F24 11.955 Tf 16.955 3.112 Td[(H0C`ZI1C)]TJ/F75 11.955 Tf 12.278 0 Td[(H0CH0)]TJ/F75 11.955 Tf 18.274 0 Td[(H0EIE)]TJ/F24 11.955 Tf 13.168 3.112 Td[(H0E`Z)]TJ/F24 11.955 Tf 14.325 3.112 Td[(H0EwhichisthedenitionofH0.ThereforeH0H0)]TJ/F24 11.955 Tf 12.068 3.112 Td[(H0asrequired;inotherwords,reducedhomologypreservescompositions.SowehaveshownthatreducedhomologyisapseudofunctorHZA1AChainN)]TJ/F75 11.955 Tf -424.871 -23.084 Td[(Abwhichpreservesidentitymorphismsandcompositions;inotherwords,reducedhomologyisafunctor.44 PAGE 49 Sowehavedenedtwofunctors:standardandreducedhomology.Thesefunctorsplayanessentialrolethroughoutthethesis.Insection4.5,wewillcombinethenotionsofchainhomotopy{whichweaddressinthecomingsection4.4{withthenotionsofhomologyandobtainadeepalgebraicresultwhichisfundamentaltohomologicalalgebra:theacycliccarriertheorem.4.4.ChainHomotopy.Inordertoprovetheacycliccarriertheorem,infacttohavemuchofanideaofthemeaningofitsstatement,wemustdiscussatsomelengththenotionofhomotopyinthecategoryChainZofchaincomplexesandmaps.Historically,thesenotionsarosefrominvestigationsintotherelationshipofalgebraicinvariantstotopologicalhomotopy.Sincethattimehowever,homotopicalalgebra,towhichitisfairtoseethenotionsofchainhomotopyasfundamentalexamples,hasseendramaticgrowth.Denition4.4.1.AchainhomotopyHbetweentwochainmaps;C)]TJ/F75 11.955 Tf 12.957 0 Td[(DisasequenceHnCn)]TJ/F75 11.955 Tf 12.642 0 Td[(Dn1n>Zsothatforeveryn>Zwehave@n1DHnHn1@nC)]TJ/F77 11.955 Tf 9.714 0 Td[(.Ifthereisachainhomotopybetweentwochainmaps,wesaythatthosemapsarechainhomotopic.Aside4.4.2.Itispossibletodenea-category"whoseobjectsarechaincomplexes,whosemorphismsarechainmaps,andwhose2-morphisms"arechainhomotopiesbetweenchainmapswhichsatisestheusualaxioms".IfwechoosetwoobjectsCandDinChainZ,thenthecollectionMapsChainZC;DisstraightforwardlyacategorywhoseobjectsarechainmapsC)]TJ/F75 11.955 Tf 13.507 0 Td[(DandwhosemorphismsarechainhomotopiesHbetweenthesechainmaps.Further,thesecategoriescoherenicely.Thefollowingobservationclariesandjustiesthispointofview.Observation4.4.3.ChainhomotopyisanequivalencerelationonthecollectionMapsC;DofchainmapsfromCtoD.Furthermore,thisequivalencerelationisrespectedbycomposition.45 PAGE 50 Proof.Werstcheckthatchainhomotopyisanequivalencerelation.Toshowthatitisreexive,weneedforeach>MapsC;DthatthereisachainhomotopyH.Thatis,weneedasequenceofmapsHnn>Zsothatforeachn>Zwehavenn)]TJ/F15 11.955 Tf 9.525 0 Td[(dDn1HnHn1dCn.Butsincennisthezeromap,wejustneedtodeneHsothatdDn1HnHn1dCnisthezeromap;butthisiseasy:deneHtobezeroateverylevel.Toshowthatitissymmetric,weneedthat,for;C)]TJ/F75 11.955 Tf 13.706 0 Td[(D,ifthereexistsHachainhomotopyfromto,thenthereexistsachainhomotopyH.ButsupposethatH)]TJ/F24 11.955 Tf 9.279 0 Td[(Hnn>Zisachainhomotopyfromto.Theforeachn>Zwehavethatnn)]TJ/F15 11.955 Tf 9.279 0 Td[(dDn1HnHn1dCn.Soifwejustmultiplybothsidesbynegativeone",weshouldbedone.Wegetnn)]TJ/F39 11.955 Tf 9.279 0 Td[(dDn1HnHn1dDn.Butsinceallofthesearehomomorphisms,wecanwritethisasnn)]TJ/F15 11.955 Tf 9.279 0 Td[(dDn1HnHn1dDn.Sincethisholdsforeachn>Z,wehavethatH)]TJ/F24 11.955 Tf 9.279 0 Td[(Hnn>Zisachainhomotopyfromto.Toshowthatitistransitive,let;;C)]TJ/F75 11.955 Tf 13.463 0 Td[(Dbechainmaps.SupposethatandarechainhomotopicbyHCDandthatandarechainhomotopicbyH.Theforeachn>Zwehavethatnn)]TJ/F15 11.955 Tf 10.391 0 Td[(dDn1HnHn1dCn1andthatnn)]TJ/F15 11.955 Tf 9.474 0 Td[(dDn1HnHn1dCn.Bysimplyaddingthesetwoequations,wegetthatnn)]TJ/F77 11.955 Tf 9.88 0 Td[(nnnn)]TJ/F15 11.955 Tf 9.879 0 Td[(dDn1HnHn1dCn1dDn1HnHn1dCn.NowwesimplyfactorthedDn1anddCnmapsandobtainnn)]TJ/F15 11.955 Tf 9.279 0 Td[(dDn1HnHnHn1Hn1dCn.ThusischainhomotopictobyH)]TJ/F24 11.955 Tf 9.278 0 Td[(HnHnn>Zasrequired.Sowehaveproventhatischainhomotopicto"isanequivalencerelationonthecollectionMapsC;D.Wehavestilltoprovethatthisequivalencerelationisrespectedbycomposition:thatif;C)]TJ/F75 11.955 Tf 12.689 0 Td[(Darechainhomotopicchainmapsand;D)]TJ/F75 11.955 Tf 14.285 0 Td[(Earechainhomotopicchainmaps,then;C)]TJ/F75 11.955 Tf 14.285 0 Td[(Earechainhomotopic.LetHbeachainhomotopyandletHbeachainhomotopy.Thenforalln>Zwehavethatnn)]TJ/F15 11.955 Tf 11.546 0 Td[(dDn1HnHn1dCnandthatnn)]TJ/F87 8.966 Tf -214.219 -43.132 Td[(46 PAGE 51 dEn1HnHn1dDn.Wewanttowritennnn)]TJ/F15 11.955 Tf 10.072 0 Td[(dEn1HnHndCnforsomechainhomotopyH.Todothis,weusethestandardtrickofaddingandsubtractingsomethingcloser"toeachoftheothertwotermsandthenregroupingasfollows:nnnn)]TJ/F77 11.955 Tf 9.279 0 Td[(nnnnnn)]TJ/F77 11.955 Tf 9.279 0 Td[(nnnnnWenowusethethechainhomotopiesH,H.)]TJ/F77 11.955 Tf 9.279 0 Td[(ndDn1HnHn1dCndEn1HnHn1dDnn)]TJ/F77 11.955 Tf 9.279 0 Td[(ndDn1HnnHn1dCndEn1HnnHn1dDnnFinally,weusethatthechainmapsandcommutewiththeboundarymaps.)]TJ/F15 11.955 Tf 9.279 0 Td[(dEn1n1HnnHn1dCndEn1HnnHn1n1dCnSowecannowfactortheboundarymapsoutandobtainaformulaforthechainhomotopyH.)]TJ/F15 11.955 Tf 9.279 0 Td[(dEn1n1HnHnnnHn1Hn1n1dCnThuswedeneHn)]TJ/F77 11.955 Tf 9.279 0 Td[(n1HnHnnforalln>ZgivingusthedesiredchainhomotopyH.Aside4.4.4.ForthepurposesofseeingthatChainZisa2-category,thisobserva-tionsaysthatthereisawell-denednotionofcompositionof2-morphismsthatwhichappearsintheproofoftransitivity,thereareidentity2-morphismsnamelythezerohomotopiesusedtoshowreexivity,andthatthe2-categoricalstructureiscoherentinthesensethatthereiswell-denednotionofhorizontalcomposition"of2-morphismswhichappearsintheproofthatchainmapcompositionrespectstheequivalencerelation.Thisobservationshows,moreover,thateachofthesecategories47 PAGE 52 MapsChainZC;Disagroupoid{thatiseverymorphismHisanisomor-phism.Thefollowingisthemainreason,fromapurelyalgebraicpointofview,thatweareinterestedinchainhomotopies:Proposition4.4.5.LetCandDbechaincomplexes,andlet;C)]TJ/F109 11.955 Tf 12.642 0 Td[(Dbechainmaps.Ifandarechainhomotopic,thenH)]TJ/F109 11.955 Tf 9.279 0 Td[(H.Proof.SupposeHisachainhomotopyfromto.WewanttoshowthatHn)]TJ/F75 11.955 Tf 10.862 0 Td[(Hnforeveryn>Z.Solet)]TJ/F94 11.955 Tf 4.551 0 Td[(Cim@n1C>HnCbeatypicalhomologyclass.WecomputeHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1CHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1C.FromthedenitionofthefunctoronmapsHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1CHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1C)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.552 0 Td[(nCim@n1D)]TJ/F77 11.955 Tf 4.552 0 Td[(nCim@n1DButthequotientmapker@nD)]TJ/F75 11.955 Tf 12.642 0 Td[(HnDislinear,sothisisequalto)]TJ/F77 11.955 Tf 4.551 0 Td[(nCnCim@n1D)]TJ/F24 11.955 Tf 9.278 0 Td[(\210nnCim@n1DWenowusethehypothesisthatnn)]TJ/F77 11.955 Tf 9.646 0 Td[(Hn1@nC@n1DHnwhichallowsusrewritethepreviousquantityasHn1@nC@n1DHnCim@n1D)]TJ/F24 11.955 Tf 9.279 0.155 Td[(Hn1@nCCim@n1D@n1DHnCim@n1DButtherstsummandiszerobecauseC>ker@nCsothat@nCC)]TJ/F15 11.955 Tf 10.617 0 Td[(0,andthesecondsummandiszerobecause@n1DHnC>im@n1D.ThisprovesthatHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1CHn)]TJ/F94 11.955 Tf 10.316 0 Td[(Cim@n1C)]TJ/F15 11.955 Tf 10.705 0 Td[(0foreveryC>HnC.ThereforeHn)]TJ/F75 11.955 Tf 10.705 0 Td[(Hnforeveryn>Zasrequired.Aside4.4.6.Wecancontinuealongthelinesofourpreviouscomments4.4.2and4.4.4inthelanguageof2-categories,andstateproposition4.4.5inthisway.WecanthinkofAbasa2-categorybyregardingthecollectionMapsAbHC;HD48 PAGE 53 asadiscretecategory22.Thenproposition4.4.5saysthathomologyisa2-functorfromthe2-categoryofchaincomplexestothe2categoryofabeliangroups;andthatfurthermore,onMapsChainZC;D,itisgivenbycollapsingconnectedcomponentstoasinglepointandallmorphismswithinthatconnectedcomponenttotheidentitymap.Denition4.4.7.AchainequivalenceisachainmapC)]TJ/F75 11.955 Tf 13.925 0 Td[(DforwhichthereexistsachainhomotopyinverseD)]TJ/F75 11.955 Tf 12.642 0 Td[(CsothatischainhomotopictoidCandthatischainhomotopictoidD.Inthiscase,wesaythatthetwochainsarechainhomotopic.Aside4.4.8.ThinkingofChainZasa2-category,wecanseehowanalogousthedenitionofchainhomotopyinverseistothedenitionoftopologicalhomotopyinverse.AchainhomotopyinverseforachainmapC)]TJ/F75 11.955 Tf 15.221 0 Td[(DisachainmapD)]TJ/F75 11.955 Tf 13.633 0 Td[(Csothat1thecompositionisinthesameconnectedcomponentofthecategoryMapsChainZC;CastheidentitymapidCandthattheanalogousstatementholdsfortheothercomposition.Analogously,ahomotopyinverseforamapoftopologicalspacesfX)]TJ/F77 11.955 Tf 13.949 0 Td[(YisamapintheotherdirectiongY)]TJ/F77 11.955 Tf 13.95 0 Td[(XsothatthecompositiongfisinthesameconnectedcomponentofthecategoryMapsTopX;Xastheidentitymapandthattheanalogousstatementholdsfortheothercomposition.SecretlyhereweareregardingTopasa2-categorywhose2-morphismsarehomotopiesbetweencontinuousmaps.Infact,thesetwo2-categoriesaredeeplyinterrelated.Thefollowingfactisparticularlyimportantforthisthesis,andisanimmediatecorollaryofproposition4.4.5:Proposition4.4.9.SupposeC)]TJ/F109 11.955 Tf 12.684 0 Td[(DisachainequivalencewithchainhomotopyinverseD)]TJ/F109 11.955 Tf 12.642 0 Td[(C.ThenHisanisomorphismwithinverseH. 22Adiscretecategoryisonewhoseonlymorphismsaretheidentitymorphisms.Everysetiscanoni-callyassociatedtoadiscretecategoryandviceversa.Infact,thecategoryofalldiscretecategoriesandfunctorsbetweenthemisisomorphictothecategoryofsetsandfunctionsbetweentheminanobviousway.49 PAGE 54 Proof.Supposeandarechainhomotopyinverses.Fromproposition4.4.5,wehavethatH)]TJ/F75 11.955 Tf 9.278 0 Td[(HidC.But,aswesawinobservation4.3.6,homologyisafunctor;sothelefthandsideisequaltoHH,andtherighthandsideisequaltoidHC.ThereforeHH)]TJ/F15 11.955 Tf 10.299 0 Td[(idHC.Symmetrically,HH)]TJ/F15 11.955 Tf 10.299 0 Td[(idHD.ThusHisanisomorphismwithinverseHasclaimed.Proposition4.4.10.LetCandDbechaincomplexes,andlet;C)]TJ/F109 11.955 Tf 14.416 0 Td[(DbechainmapsfromCtoD.LetACandADbeaugmentationsoftherespectivechaincomplexes.Suppose,arechainhomotopic.Thentheaugmentedmapobtainedfromisaugmentation-preservingwithrespecttotheaugmentationsmentionedifandonlyiftheaugmentedmapobtainedfromis.FurthermoreifHisachainhomotopybetweenand,thentheaugmentation"Hdenedindimension1tobethezeromapisachainhomotopybetweentheaugmentedmaps.Proof.LetHbeachainhomotopyfromto.Thenforeachn>Zwehavethatnn)]TJ/F77 11.955 Tf 9.279 0 Td[(@n1DHnHn1@nC.Ofinteresttousisthattheequality00)]TJ/F77 11.955 Tf 9.279 0 Td[(@1DH0@0CH1)]TJ/F77 11.955 Tf 9.279 0 Td[(@1DH0holds.Buti)]TJ/F77 11.955 Tf 9.279 0 Td[(iandi)]TJ/F77 11.955 Tf 9.279 0 Td[(iforix1,and@iAD)]TJ/F77 11.955 Tf 9.279 0 Td[(@iDforix0;sothissaysthat00)]TJ/F77 11.955 Tf 9.279 0 Td[(@1ADH0Thisyieldsthat@0AD0)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0@1ADH0)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0@0AD@1ADH0)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0Andmoretriviallyobservethat1@0AC)]TJ/F15 11.955 Tf 9.278 0 Td[(idZ@0AC)]TJ/F24 11.955 Tf 9.757 3.212 Td[(1@0AC.50 PAGE 55 Nowsupposethatisaugmentation-preserving.ThenthediagramA1C Z1)]TJ/F16 7.97 Tf 4.631 0 Td[(idZ)]TJ/F30 11.955 Tf 45.65 0 Td[(Z A1D@0AC@0ADA0C C00)]TJ/F19 7.97 Tf 4.632 0 Td[(0)]TJ/F75 11.955 Tf 38.928 0 Td[(D0 A0Dcommutes;i.e.1@0AC)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0.Sofromtheaboveconsiderations,1@0AC)]TJ/F77 11.955 Tf 9.279 0 Td[(1@0AC)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD0Thereforeisaugmentation-preservingasclaimed.Clearlythisargumentiscom-pletelysymmetric.DeneHbyHi)]TJ/F24 11.955 Tf 9.279 18.118 Td[(Hiifix1;20ifi)]TJ/F15 11.955 Tf 9.279 0 Td[(1;2WeclaimthatthisisachainhomotopyHfromto.Weneedtoprovethatforalln>Zwehavenn)]TJ/F77 11.955 Tf 10.84 0 Td[(@n1ADHnHn1@nAC.Wegetthisforfreefornx2;1;0sinceawayfromthesevaluesn)]TJ/F77 11.955 Tf 10.656 0 Td[(,n)]TJ/F77 11.955 Tf 10.655 0 Td[(,@n1AD)]TJ/F77 11.955 Tf 10.656 0 Td[(@n1D,Hn)]TJ/F77 11.955 Tf 10.655 0 Td[(Hn,Hn1)]TJ/F77 11.955 Tf 9.758 0 Td[(Hn1,and@nAC)]TJ/F77 11.955 Tf 9.758 0 Td[(@nCandsinceHisachainhomotopy,yieldingthatnn)]TJ/F77 11.955 Tf 9.443 0 Td[(@n1DHnHn1@nCwhichafterthesubstitutionsisexactlywhatwewanted.Considerthecasen)]TJ/F39 11.955 Tf 11.004 0 Td[(2:wewantthat22)]TJ/F24 11.955 Tf 12.526 3.078 Td[(H3@2AC@1ADH2;butthisisimmediatesinceeverymaphereiszero.Considernowthecasewheren)]TJ/F39 11.955 Tf 11.238 0 Td[(1:wewewantthat11)]TJ/F24 11.955 Tf 12.203 3.079 Td[(H2@1AC@0ADH1;thelefthandsideisclearlyzerosince1)]TJ/F15 11.955 Tf 11.418 0 Td[(idZ)]TJ/F24 11.955 Tf 11.896 3.211 Td[(1asistherighthandsidesinceboth@1ACandH1arezeromakingbothsummandsvanish.Atlastconsiderthecasewheren)]TJ/F15 11.955 Tf 9.29 0 Td[(0:wewantthat00)]TJ/F24 11.955 Tf 10.802 3.078 Td[(H1@0AC@1ADH0;buttheleftsummandontherighthandsideiszerosince@0A0C)]TJ/F15 11.955 Tf 10.26 0 Td[(0andwhatremainsissimplyarewritingof00)]TJ/F77 11.955 Tf 10.26 0 Td[(@1DH0whichwehavesinceHisachainhomotopyand@0C)]TJ/F15 11.955 Tf 9.279 0 Td[(0.Proposition4.4.11.LetC,DbechaincomplexesandletC)]TJ/F109 11.955 Tf 12.642 0 Td[(DandCDbechainhomotopyinverses.LetACandADbeaugmentationsofthesetwochain51 PAGE 56 complexes.Thenisaugmentation-preservingwithrespecttotheseaugmentationsifandonlyifis.Inthiscase,theaugmentationsAC)]TJ/F45 11.955 Tf 12.643 0 Td[(ADandACADarechainhomotopyinverses.ConsequentlythesemapsinduceinverseisomorphismonthereducedhomologygroupsofCandD.Proof.Noticethatgiventhesymmetryofthesituation,itwillbeenoughtoprovethatisaugmentation-preservingifis.Toprovethis,supposeisaugmentation-preserving.Then1@0AC)]TJ/F77 11.955 Tf 9.278 0 Td[(@0AD0.Butwehavethat00idD0)]TJ/F77 11.955 Tf 9.292 0 Td[(@1DH0bythedenitionofthechainhomotopyH;werewritethisintermsoftheaugmentedcomplexADas00idA0D)]TJ/F77 11.955 Tf 9.279 0 Td[(@1ADH0Composingbytheboundarymap,weobtain@0AD00idA0D)]TJ/F77 11.955 Tf 9.279 0 Td[(@0AD@1ADH0Thus@0AD00@0ADidA0D)]TJ/F15 11.955 Tf 9.469 0 Td[(0{thatis@0AD00)]TJ/F77 11.955 Tf 9.468 0 Td[(@0ADidA0D.Thus,applyingthefactthatisaugmentation-preserving,wehave1@0AC0)]TJ/F77 11.955 Tf 9.279 0 Td[(@0ADidA0D.So@0AC0)]TJ/F77 11.955 Tf -425.611 -23.083 Td[(@0ADsince1andidA0Dareidentitymaps.Thereforeweobtain@0AC0)]TJ/F24 11.955 Tf 10.032 3.211 Td[(1@0ADsince1)]TJ/F15 11.955 Tf 9.279 0 Td[(idZ.Soisaugmentation-preservingasrequired.NowobservethatclearlyidC)]TJ/F15 11.955 Tf 9.295 0 Td[(idACisaugmentationpreserving.SowecanapplythepreviousobservationandobtainthatH)]TJ/F24 11.955 Tf 11.895 3.211 Td[(idAC)]TJ/F24 11.955 Tf 11.541 3.211 Td[(idCisachainhomotopy.SimilarlywendthatHidADisachainhomotopy.Soandarechainhomotopyinversesasrequired.Therefore,byobservation4.3.3,wendthatHACHADisadiagramofisomorphisms.52 PAGE 57 4.5.TheAcyclicCarrierTheorem.Beforeprovingtheacycliccarriertheorem,wemustintroducetheideaofanacycliccarrierwhichisthecoreofthetheorem.Inordertodothat,wehavetointroducethefollowingpiecesofterminology:Denition4.5.1.AsubcomplexDofachaincomplexCisasequenceD)]TJ/F24 11.955 Tf 9.279 0 Td[(DnBCnn>ZofsubgroupsfromthecorrespondinglevelofCequippedwiththeboundaryoperator@Dwhichisgivenbyrestriction@nD)]TJ/F77 11.955 Tf 9.279 0 Td[(@nCSDntotheappropriatesubgroup.WedenotethefamilyofallsubchainsofachaincomplexCbySC.Denition4.5.2.AbasisforachaincomplexC)]TJ/F24 11.955 Tf 9.279 0 Td[(Cnn>ZisagradedsetB)]TJ/F24 11.955 Tf 9.279 0 Td[(Bnn>ZwhereBn)]TJ/F24 11.955 Tf 9.279 0 Td[(n>JisabasisforCnasamodule.Denition4.5.3.AnacycliccarrierfromafreeaugmentedchaincomplexCwithbasisBtoanarbitraryaugmentedchaincomplexDisasetmapB)]TJ/F30 11.955 Tf 12.82 0 Td[(SDwhichsatisesthefollowing:foreverybasiselement>B,)]TJ/F77 11.955 Tf 4.551 0 Td[(isacyclic@0)]TJ/F19 7.97 Tf 3.293 0 Td[(0)]TJ/F77 11.955 Tf 4.551 0 Td[()]TJ/F15 11.955 Tf 12.877 0 Td[(1)]TJ/F77 11.955 Tf 4.551 0 Td[()]TJ/F30 11.955 Tf 9.514 0 Td[(Zissurjectiveifisabasiselementindimensiongreaterthan1andfurthermorethediagramA0D@0AD)]TJ/F45 11.955 Tf 38.976 0 Td[(A1Dinclincl0)]TJ/F77 11.955 Tf 4.552 0 Td[(@0)]TJ/F15 11.955 Tf 35.741 0 Td[(1)]TJ/F77 11.955 Tf 4.551 0 Td[(commutesandtherightverticalinclusionarrowistheidentity.Where@nCn)]TJ/F21 11.955 Tf 10.59 -0.94 Td[(P>Jn1cn1n1;iftheweightcn1onthethbasiselementn1ofDn1intheexpansionof@nCnwithrespecttothebasisBn1isnon-zero,then)]TJ/F77 11.955 Tf 4.551 0 Td[(n1isasubchaincomplexofnWesaythatagroupmapfCn)]TJ/F75 11.955 Tf 12.642 0 Td[(DmfromthenthlevelCnofCtothemthlevelDmofDiscarriedbyjustincaseftakeseverybasiselementn>Bnintothemthlevelm)]TJ/F77 11.955 Tf 4.551 0 Td[(nofthesubchain)]TJ/F77 11.955 Tf 4.551 0 Td[(nassignedtonbytheacycliccarrier{i.e.justincasefn>m)]TJ/F77 11.955 Tf 4.552 0 Td[(nforeverybasiselement.53 PAGE 58 Morespecically,wesaythatachainmapiscarriedbyjustincaseeverynCn)]TJ/F75 11.955 Tf 12.642 0 Td[(Dniscarriedby.Thatis,wesaythatachainmapC)]TJ/F75 11.955 Tf 12.642 0 Td[(Discarriedbyifntakeseverybasiselementntothenthleveln)]TJ/F77 11.955 Tf 4.551 0 Td[(nofthesubchain)]TJ/F77 11.955 Tf 4.551 0 Td[(nassignedtonbytheacycliccarrier{i.e.ifnn>n)]TJ/F77 11.955 Tf 4.551 0 Td[(nforeverybasiselementn>Bn.Similarly,wesaythatachainhomotopyH)]TJ/F24 11.955 Tf 10.343 0 Td[(Hnn>ZC1iscarriedbyjustincaseeveryHnCn)]TJ/F75 11.955 Tf 12.642 0 Td[(Dn1iscarriedby.Atlonglast,weprovetheacycliccarriertheorem:Theorem4.5.4.LetAC,andADbeaugmentedchaincomplexes.AndletACbefreewithbasisB)]TJ/F24 11.955 Tf 10.588 0.155 Td[(Bn)]TJ/F24 11.955 Tf 9.279 0 Td[(n>Jnn>^N.SupposethatthereexistsanacycliccarrierB)]TJ/F30 11.955 Tf 13.02 0 Td[(SADfromACtoAD.ThenthereexistsachainmapC)]TJ/F109 11.955 Tf 13.02 0 Td[(Dwhoseinfact,augmentation-preservingaugmentationAC)]TJ/F45 11.955 Tf 14.484 0 Td[(ADiscarriedby.Furthermore,anytwochainmaps;C)]TJ/F109 11.955 Tf 13.489 0 Td[(DwhicharecarriedbyhavechainhomotopicaugmentationsbyachainhomotopyHwhichitselfiscarriedby.Proof.WefollowtheproofwhichMunkressuggests,generalizingtheproofthathedoesgiveforthegeometricversionofthetheorem.WeneedtoproveexistenceanduniquenessuptochainhomotopyofachainmapC)]TJ/F75 11.955 Tf 13.212 0 Td[(Dwhoseaugmentationiscarriedby.Weproveexistencerst.WeconstructAC)]TJ/F45 11.955 Tf 12.95 0 Td[(ADasfollows:Firstdene1A1C)]TJ/F30 11.955 Tf 9.586 0 Td[(Z)]TJ/F30 11.955 Tf 12.95 0 Td[(Z)]TJ/F45 11.955 Tf 9.587 0 Td[(A1Dtobetheidentitymap.Todene0A0C)]TJ/F45 11.955 Tf 12.724 0 Td[(A0D,wechasesomearrows.Suppose0isanelementofourchosenbasisB0forAC.Then@0AC0isinA1C,so1@0AC0isinA1D)]TJ/F15 11.955 Tf 9.279 0 Td[(1)]TJ/F77 11.955 Tf 4.552 0 Td[(0.Now@00issurjective,sothereissomeelementx)]TJ/F15 11.955 Tf 9.279 0 Td[(lift@001@0AC0in0)]TJ/F77 11.955 Tf 4.552 0 Td[(0suchthat@00x)]TJ/F24 11.955 Tf 9.416 3.212 Td[(1@0AC054 PAGE 59 Wedene00tobethiselementlift@001@0AC0)]TJ/F77 11.955 Tf 9.279 0 Td[(xinsidethepreimage.Diagrammatically,wehavedonethefollowing:A0C@0AC)]TJ/F45 11.955 Tf 45.206 0 Td[(A1C0)]TJ/F77 11.955 Tf 44.345 0 Td[(@0AC0[10)]TJ/F77 11.955 Tf 4.551 0 Td[(0@000)]TJ/F15 11.955 Tf 42.168 0 Td[(1)]TJ/F77 11.955 Tf 4.551 0 Td[(0x1@0AC0Byconstruction,wehavethat00>0)]TJ/F77 11.955 Tf 4.552 0 Td[(0asrequiredfortobecarriedby.Inaddition,wehave,byconstruction,thatthefollowingdiagramcommutes:0@0AC)]TJ/F45 11.955 Tf 36.038 0 Td[(A1C01A0D@0AD)]TJ/F45 11.955 Tf 35.741 0 Td[(A1DHavingdenedoneverybasiselement0inthisway,andwedene0byextendinglinearly23.Clearly,thismakesthefollowingdiagramcommute,asrequiredfortobeachainmapatlevel0,A0C@0AC)]TJ/F45 11.955 Tf 36.038 0 Td[(A1C01A0D@0AD)]TJ/F45 11.955 Tf 35.741 0 Td[(A1Dsincethepreceedingdiagramcommutesforall>J0.Now,weproceedbyinduction.SupposethatwehavedenedAq1C)]TJ/F45 11.955 Tf 12.746 0 Td[(Aq1DsothatthefollowingdiagramcommutesAq1C@q1AC)]TJ/F45 11.955 Tf 36.038 0 Td[(Aq2Cq1q2Aq1D@q1AD)]TJ/F45 11.955 Tf 35.741 0 Td[(Aq2Dandthatq1q1>q1q1forall>Jq1. 23Indetail,forAC?C)]TJ/F48 9.963 Tf 12.281 -0.775 Td[(P>J0c00wedene0C)]TJ/F48 9.963 Tf 12.281 -0.775 Td[(P>J0c000.55 PAGE 60 TodeneqAqC)]TJ/F45 11.955 Tf 12.643 0 Td[(AqD,wechasesomemorearrows.SupposeqisanelementofourchosenbasisBqforAqC.Wehavethefollowingdiagrammaticsituation:q)]TJ/F77 11.955 Tf 46.378 0 Td[(@qACq)]TJ/F77 11.955 Tf 56.798 0 Td[(@q1AC@qACqxq1@qACq)]TJ/F77 11.955 Tf 35.741 0 Td[(@q1q1qq1@qACqlivinginsideAqC@qAC)]TJ/F45 11.955 Tf 43.543 0 Td[(Aq1C@q1AC)]TJ/F45 11.955 Tf 43.543 0 Td[(Aq2Cq1q2qq@qq)]TJ/F15 11.955 Tf 38.516 0 Td[(q1q@q1q)]TJ/F15 11.955 Tf 38.515 0 Td[(q2qIndetail,wehave@qACq>Aq1C,soq1@qACq>Aq1D.Butinfact,q1@qACq>q1qsincecarriesq1.Toseethis,let@qACq)]TJ/F21 11.955 Tf 17.234 -0.941 Td[(P>Jq1cq1q1;thenq1`qforall>Jq1suchthatcq1x0.Inparticular,q1q1>q1qforallsuch.Thereforeq1@qACq)]TJ/F24 11.955 Tf 10.836 3.211 Td[(q1P>Jq1cq1q1)]TJ/F21 11.955 Tf 18.654 -0.941 Td[(P>Jq1cq1q1q1>q1qasclaimed,sinceitisaweightedsumofelementsofq1q.Now@q1AC@qACq)]TJ/F15 11.955 Tf 9.279 0 Td[(0,soq2@q1AC@qACq)]TJ/F15 11.955 Tf 9.279 0 Td[(0.ButwehavethatthediagramAq1C@q1AC)]TJ/F45 11.955 Tf 43.543 0 Td[(Aq2Cq1q2Aq1D@q1AD)]TJ/F45 11.955 Tf 43.245 0 Td[(Aq2Dinclinclq1q@q1q)]TJ/F15 11.955 Tf 38.515 0 Td[(q2qcommutes.Commutativityofthetopsquareisjusttheinductivehypothesis,andcommuta-tivityofthebottomsquarecomesfromthedenitionofacycliccarrier.Thuswehavethat0)]TJ/F24 11.955 Tf 11.758 3.211 Td[(q2@q1AC@qACq)]TJ/F77 11.955 Tf 11.621 0 Td[(@q1ADq1@qACq)]TJ/F77 11.955 Tf 11.62 0 Td[(@q1qq1@qACq.Thus56 PAGE 61 q1@qACqisacycleinq1q.Butqisacyclic;thusq1qisabound-ary.Thatis,thereexistsanelementx)]TJ/F15 11.955 Tf 11.062 0 Td[(lift@qqq1@qACq>qqwhich@qqmapstoq1@qACq.Sodeneqq)]TJ/F77 11.955 Tf 9.588 0 Td[(x.Thenbyconstruction,wehaveboththatthediagramq@qAC)]TJ/F45 11.955 Tf 36.039 0 Td[(Aq1Cq1q1AqD@qAD)]TJ/F45 11.955 Tf 35.741 0 Td[(Aq1Dcommutesandthatqq>qqforevery>Jq.Asinthebasecase,welinearlyextendthemapqtoallofAqCandobtainthatthediagramAqC@qAC)]TJ/F45 11.955 Tf 36.039 0 Td[(Aq1Cq1q1AqD@qAD)]TJ/F45 11.955 Tf 35.741 0 Td[(Aq1Dcommutesasrequired.ThiscompletestheconstructionofachainmapAC)]TJ/F45 11.955 Tf 13.469 0 Td[(ADwhichiscarriedby.Butwehavestilltoshowthatthismapisuniqueuptochainhomotopy.Sosupposethat;AC)]TJ/F45 11.955 Tf 12.809 0 Td[(ADarecarriedby.Weconstructachainhomo-topyHAC)]TJ/F45 11.955 Tf 13.203 0 Td[(ADasfollows:First,dene,foreveryk@0,HkAkC)]TJ/F45 11.955 Tf 13.204 0 Td[(Ak1Dtobethezeromap.TodeneH0A0C)]TJ/F45 11.955 Tf 16.064 0 Td[(A1D,wechasesomearrows.Suppose0isanele-mentofourchosenbasisB0forA0C.Then00and00arebothin0)]TJ/F77 11.955 Tf 4.551 0 Td[(0sincetheyarecarriedby,andthus0000isalsoin0)]TJ/F77 11.955 Tf 4.551 0 Td[(0.So@000000)]TJ/F77 11.955 Tf 9.279 0 Td[(@0000@0000)]TJ/F77 11.955 Tf 9.278 0 Td[(@0AD00@0AD00isin1)]TJ/F77 11.955 Tf 4.552 0 Td[(0.Butsince;arechainmapsthesediagramsA0C@0AC)]TJ/F45 11.955 Tf 36.038 0 Td[(A1C01A0D@0AD)]TJ/F45 11.955 Tf 35.741 0 Td[(A1DA0C@0AC)]TJ/F45 11.955 Tf 36.039 0 Td[(A1C01A0D@0AD)]TJ/F45 11.955 Tf 35.741 0 Td[(A1D57 PAGE 62 commute.Thus@000000)]TJ/F24 11.955 Tf 9.416 3.212 Td[(1@0AC01@0AC0)]TJ/F15 11.955 Tf 9.279 0 Td[(idZ@0AC0idZ@0AC0)]TJ/F15 11.955 Tf 9.305 0 Td[(0.Thus0000isacyclein0)]TJ/F77 11.955 Tf 4.551 0 Td[(0;since)]TJ/F77 11.955 Tf 4.551 0 Td[(0isacyclic,0000isaboundary.Thusthereexistssomeelementx)]TJ/F15 11.955 Tf 9.279 0 Td[(lift@100000in1)]TJ/F77 11.955 Tf 4.552 0 Td[(0whichistakenbytheboundarymap@10to0000.SodeneH00)]TJ/F15 11.955 Tf 9.279 0 Td[(liftx.Thesituationisoutlinedbythefollowingdiagram0)]TJ/F77 11.955 Tf 77.631 0 Td[(@0AC0x0000)]TJ/F77 11.955 Tf 35.741 0 Td[(@000000livinginsideA0C@0AC)]TJ/F45 11.955 Tf 43.38 0 Td[(A1C00111)]TJ/F77 11.955 Tf 4.552 0 Td[(0@10)]TJ/F15 11.955 Tf 38.516 0 Td[(0)]TJ/F77 11.955 Tf 4.552 0 Td[(0@00)]TJ/F15 11.955 Tf 38.516 0 Td[(1)]TJ/F77 11.955 Tf 4.551 0 Td[(0ThereforewehavethatH00>1)]TJ/F77 11.955 Tf 4.551 0 Td[(0andthat@1ADH00H1@0AC0)]TJ/F24 11.955 Tf -425.474 -19.872 Td[(0000forevery>J0.ThusH0iscarriedby.Weproceedbyinduction.SupposethatHphasbeendenedforallp@qsoeachmapHpiscarriedbyandsothat@p1ADHpHp1@pAC)]TJ/F24 11.955 Tf 11.042 3.211 Td[(pp.WedeneHqasfollows:SupposeqisanelementofourchosenbasisBqforAqC.WewanttodeneHqqtobeequaltod>q1qsothat@q1qd)]TJ/F24 11.955 Tf 10.632 3.212 Td[(qqqqHq1@qACqsincethatwouldmeanthattheequalityrequiredforHtobeachainhomotopyatlevelqwouldhold.Wewilllifttherighthandside{anelementofpqsinceandarecarriedbyandsinceHp1iscarriedbyandallthesummandsof@p1ACwithrespecttoourchosenbasisaremappedintoqqbythedenitionofacycliccarrieraswehaveobservedbefore{toanelementinq1qbymeansoftheacyclicityofq.Wemustcompute:@qqqqqqHq1@qACq58 PAGE 63 Now,byourinductivehypothesis@qADHq1Hq2@q1AC)]TJ/F24 11.955 Tf 10.252 3.212 Td[(p1p1.Therefore,inparticular,@qqHq1@qACq)]TJ/F39 11.955 Tf 9.279 0 Td[(p1@qACqp1@qACqHq2@q1AC@qACqwherethelasttermofthisiszerosince@q1AC@qAC)]TJ/F15 11.955 Tf 9.279 0 Td[(0.Therefore,isequalto@qqqq@qqqqp1@qACqp1@qACqi.e.to@qqqqp1@qACqp1@qACq@qqqqButwehavethatthediagramsAqC@qAC)]TJ/F45 11.955 Tf 36.038 0 Td[(Aq1Cqq1AqD@qAD)]TJ/F45 11.955 Tf 35.741 0 Td[(Aq1DA0C@0AC)]TJ/F45 11.955 Tf 36.039 0 Td[(A1C01A0D@0AD)]TJ/F45 11.955 Tf 35.741 0 Td[(A1Dcommute.Soinparticular,@qqqq)]TJ/F24 11.955 Tf 9.416 3.211 Td[(p1@qACqandp1@qACq)]TJ/F77 11.955 Tf 9.279 0 Td[(@qqqqTherefore,iszero.ThusqqqqHq1@qACqisacycleinqqandhenceisaboundarysinceqisacyclic.Thusthereexistsanelementx)]TJ/F15 11.955 Tf 9.376 0 Td[(lift@q1ADqqqqHq1@qACq>q1qwhichismappedby@q1ADtoqqqqHq1@qACq.SowedeneHqq)]TJ/F77 11.955 Tf 9.838 0 Td[(x>q1q.Bycon-struction,Hqq>q1qand@q1ADHqqHq1@qACq)]TJ/F24 11.955 Tf 10.318 3.211 Td[(qqqqforevery>Jq.WedeneHqbyextendinglinearlyfromitsdenitiononbasis59 PAGE 64 elements.Aswehaveseen,itiscarriedby,and@q1ADHqHq1@qAC)]TJ/F24 11.955 Tf 10.162 3.212 Td[(qqbylinearity.60 PAGE 65 5.TheNaturalIsomorphism5.1.IntroductoryRemarks.Thepurposeofthischapteristobuildupthema-chinerynecessarytostate,thentoprovethemaintheoremofthethesis:thattheorderedandorientedhomologyfunctorsonobjectsinanice"categorydenedviaaxedstandardcosimplicialobject"arenaturallyisomorphic.Thisisasubstan-tialgeneralizationofthewell-knowntheoremstatedinsection2.4andproven,forinstance,in[3]{thattheorientedandorderedhomologiesofsimplicialcomplexesarenaturallyisomorphic.Inthatcase,thenice"categoryisjustSimpComp,thecategoryofsimplicialcomplexesandsimplicialmapsbetweenthem,andthestan-dardcosimplicialobject"isjustthecollectionofstandardn-simplicesequippedtheassociatedcoface,codegeneracy,andcotranspositionmaps.Theproofwegiveisanal-ogoustotheproofofthisrathermorespecicresult;however,ourresultreallyisasignicantgeneralizationofthatresultsinceitmakesexplicitthesuspicionsthatonemighthaveaboutthegeneralityoftheresult.We,inparticular,areabletosimplyapplyourresulttoallofourfavoritecosimplicialobjectsinnice"categorieswhichalmostallcategorieswemightbeinterestedinpracticeare{thenicenessassumptionsaremild.Inordertoputitinitspropercontext,wemustintroduceafairamountofter-minologyandmachinery.Webeginthechapterwithasectiononcosimplicial-typeobjects"{interestingcasesofthegeneralnotionofcoP-objectsintroducedbrieyinexample3.4.7{andsimplicial-typeobjects"{interestingcasesofP-objects.Inthefol-lowingsection,wefunctoriallyconstructchaincomplexesassociatedtosomeoftheseobjects:inparticular,weassociateoriented"chaincomplexestoDT-simplicialsets"andfree"chaincomplexestoD-simplicialsets".Then,insection5.4wedenethegeneralnotionofsingularsetandinparticularthenotionofinjectivesetforcertainnice"categorieswhichcomeequippedwithstandardcosimplicialobjects.Inthecaseofsimplicialhomology,thesearethecollectionsofallandrespectively,non-degeneratesimplicialmappingsintoasimplicialcomplex.Subsequently,insection61 PAGE 66 5.5,weapplythefunctorialconstructionsofchaincomplexesfromsection5.3tofunc-toriallyassociatetoeachobjectinanice"categoryanorientedchaincomplexandanorderedchaincomplex;wethendenetheorientedandordered,respectivelyhomologiesofobjectstobethecompositionoftheorientedandordered,respec-tivelychaincomplexfunctorswedenedwiththehomologyfunctorpresentedin4.3.Inthecaseofsimplicialhomology,theseobjectsareisomorphicto{althoughde-nedinamannerdierentfromthatusuallyusedexempliedbythepresentationin[3]{theorientedandorderedchaincomplexesassociatedtoasimplicialcomplex.Inthefollowingsection,wedeneanaturaltransformationfromtheorderedchaincom-plexfunctortotheorientedchaincomplexfunctorandaninfranaturaltransformationfromtheorientedchaincomplexfunctortotheorderedchaincomplexfunctor;werstobservethatthecompositionoftheoriented-to-orderedinfranaturaltransformationwiththeordered-to-orientednaturaltransformationisequaltotheidentitynaturaltransformationandsecondprovethatthethecompositionoftheordered-to-orientedwiththeoriented-to-ordereddenesaninfranaturaltransformationchainhomotopictotheidentity.Finally,insection5.7,weprovethattheorderedandorientedho-mologyfunctorsdenedinsection5.5arenaturallyisomorphicbypushing-forwardtheordered-to-orientednaturaltransformationandtheoriented-to-orderednaturaltransformationalongthehomologyfunctor.5.2.SimplicialandCosimplicialTypeObjects.Webeginourexpositionbyin-troducingcosimplicial-type"objectsandsimplicial-type"objectsinsomecategoryC.Asstatedintheintroductoryremarkstothischapter,thesearebothspecicsortsofP-objects24introducedat3.4.7inchapter3:inthatexample,wedenedaP-objectinacategoryCtobeacontravariantfunctorP)]TJ/F112 11.955 Tf 12.642 0 Td[(C.Aside:ItmustalsobesaidthatthesespecialcasesofP-objectsaremuchmoreinterestingthanthegeneralcase{thegeneralcaseamountstoastudyofthecollectionMapsCatP;Cwherewemakeno 24HerePisacategorywhichweknowabout".Inthiscontext,we'rereferringtoPasaprobecategory.62 PAGE 67 constraintsonP;inotherwords,itamountstoastudyofallfunctorsintoC.WerefertocovariantfunctorsP)]TJ/F112 11.955 Tf 13.808 0 Td[(CascoP-objectsinC.Inparticular,aP-setoracoP-setisjustaP-objectoracoP-object,resepctivelywhosecodomaincategoryisthecategorySetintroducedinexample3.2.2.Throughtherestofthethesis,wewillbeworkingwithtwocloselyaliatedprobecategories,andbrieywithathirdcategory,insection5.3,allofwhichwewillsayaresimplex-type"categories.Furthermore,thereisanclearcutwayinwhichthesethreearerelatedandinwhichtheyarethreemembersofafamilyofeightcategories.Wewillinfactsaythatalloftheseeightcategoriesaresimplex-like".Sobyacosimplicial-typeobjectinC"and,respectively,byasimplicial-typeobjectinC"wemeanacovariantrespectively,contravariantfunctorwhosedomaincategoryisoneofthesesimplex-type"categoriesandwhosecodomaincategoryisC.Still,aswementioned,twooftheseplayamuchlargerroleinthethesisthananyoftheothers.Forthatreason,wetaketimetodiscusstheparticularsofeachcategory.Infactoneofthesetwocategories,namelythesemi-cardinalnumbercategory"denotedby!sisasubcategoryoftheother,thecardinalnumbercategory",denotedby!.Inparticular,thesemi-cardinalnumbercategoryhasthesameobjectsetasthecardinalnumbercategory.Infact,thesemi-cardinalnumbercategoryistheinjectivesubcategory"ofthecardinalnumbercategory{wewillintroducethisnotionrigorouslyinsection5.4andgivethisasanexample.Togetslightlyaheadofourselves,wemustpointoutthat,asitturnsout,alleightsimplex-like"categoriesaresupportedonthissamecollectionofobjects.Inordertointroducethesesimplex-like"categoriesandtoexploreinparticularthecardinalandthesemi-cardinalcategories,wemustrstintroducetheobjectsonwhichtheyarebothsupported,thestandardcardinals:Denition5.2.1.ThenthstandardcardinalnumberisthenthsectionofZ,thesetn)]TJ/F24 11.955 Tf 9.279 0 Td[(0;:::;nwherenC1;ingeneral,anitestandardcardinalnumberorsimplyacardinalnumberisjustoneofthese.Alternatively,wecoulddenethenthstandardcardinalnumbertobethediscretecategoryD0;:::;nseeexample3.2.4forthe63 PAGE 68 denitionontheobjects0;:::;n.Itisnothardtoseethatthesenotionsarethesame.Ratherthandiscussingbothatonce,wewillintroducethenitestandardcardi-nalnumbercategoryanditspropertiesrstbeforeproceedingtodiscussthesemi-standardcardinalnumbercategory.5.2.1.TheCardinalNumberCategory.Denition5.2.2.Thenitestandardcardinalnumbercategoryorsimplythecar-dinalnumbercategorydenotedby!hasasobjectsthecardinalnumbersandasmorphismsallsetfunctionsbetweencardinalnumbers.Inotherwords,!isthefullsubcategoryofSetonthecardinalnumbersasdenedabove.Or,underthealternativedenition,itisthefullsubcategoryonDCat.Therearethreefundamental{inasensewewillmaketechnicalwhenwedeneasimplex-likecategory{morphismtypesinthecardinalnumbercategory:Denition5.2.3.Theithcofacemapindimensionnwherei)]TJ/F15 11.955 Tf 9.696 0 Td[(0;:::;nisthemapdinn-1)]TJ/F75 11.955 Tf 12.642 0 Td[(ngivenbykkifk@ik1ifkCiTheintuitivenotionofacofacemapisthatitsplitsthedomainatandincludingtheithelement.Denition5.2.4.Theithcodegeneracymapindimensionn,wherei)]TJ/F15 11.955 Tf 10.551 0 Td[(0;:::;nisthemapsin)]TJ/F15 11.955 Tf 9.279 0 Td[(sin+1)]TJ/F75 11.955 Tf 12.642 0 Td[(ngivenonobjectsbykkifkBik1ifkAiTheintuitivenotionofacodegeneracymapisthatitcollapsestheelementsiandi1together.64 PAGE 69 Denition5.2.5.Theithcotranspositionindimensionn,wherei)]TJ/F15 11.955 Tf 9.858 0 Td[(0;:::;n1,isthemaptnin)]TJ/F75 11.955 Tf 12.642 0 Td[(ngivenbykkifk@ii1ifk)]TJ/F77 11.955 Tf 9.279 0 Td[(iiifk)]TJ/F77 11.955 Tf 9.279 0 Td[(i1kifkAi1Theintuitivenotionofacotranspositionmapisthatitswapstheelementsiandi1.Infact{asisdiscussedin,forinstance,[1]{everymorphismin!factorsasasequenceofmorphismsofthesethreefundamentaltypes.Furthermore,theysatisfythefollowingrelations:65 PAGE 70 Theorem5.2.6.TheSymmetricCosimplicialIdentities1djdi)]TJ/F90 11.955 Tf 9.279 0 Td[(didj1ifi@j2sjdi)]TJ/F90 11.955 Tf 9.279 0 Td[(disj1sjdj)]TJ/F90 11.955 Tf 9.279 0 Td[(id)]TJ/F90 11.955 Tf 9.279 0 Td[(sjdj1sjdi)]TJ/F90 11.955 Tf 9.279 0 Td[(di1sjifi@jifiAj13sjsi)]TJ/F90 11.955 Tf 9.279 0 Td[(sisj1ifiBj4titi)]TJ/F15 11.955 Tf 9.279 0 Td[(15tj1tjtj1)]TJ/F90 11.955 Tf 9.279 0 Td[(tjtj1tj6titj)]TJ/F90 11.955 Tf 9.279 0 Td[(tjtiifi@j17tidj)]TJ/F90 11.955 Tf 9.279 0 Td[(djtitidi)]TJ/F90 11.955 Tf 9.279 0 Td[(di1tidj)]TJ/F90 11.955 Tf 9.279 0 Td[(djti1ifi@j1ifiAj8tisj)]TJ/F90 11.955 Tf 9.279 0 Td[(sjtitisi)]TJ/F90 11.955 Tf 9.279 0 Td[(si1titi1tisj)]TJ/F90 11.955 Tf 9.279 0 Td[(sjti1ifi@j1ifiAj9siti)]TJ/F90 11.955 Tf 9.279 0 Td[(si10tisi1)]TJ/F90 11.955 Tf 9.279 0 Td[(siti1ti11tidi1)]TJ/F90 11.955 Tf 9.279 0 Td[(diInfact[1],thecategoryfreelygeneratedbythesemapssubjecttotheserelationsiscanonicallyisomorphicto!.Thismeansthattheorem5.2.6completelyclassiesthecardinalnumbercategoryintermsofthesegeneratorsandtheserelations.Onaccountofthis,wewillrefertothisasthefundamentaltheoremofthecardinalnumbercategory.Asacorollarytothistheorem,weobtainthateverymorphismin!factorsintoasequenceofcofaces,codegeneracies,andcotranspositions.Thefollowing,whichwillbeusefullater,isastrongerresult:66 PAGE 71 Proposition5.2.7.FactoringAlgorithmfortheCardinalNumberCategoryAnymorphismn)]TJ/F109 11.955 Tf 12.642 0 Td[(min!factorsasntinti1n)]TJ/F109 11.955 Tf 20.606 0 Td[(nsiksi1n1)]TJ/F109 11.955 Tf 23.884 0 Td[(kdimdi1k1)]TJ/F109 11.955 Tf 24.398 0 Td[(mProof.Supposen)]TJ/F75 11.955 Tf 13.945 0 Td[(misamorphismin!.Wedeneamorphismn)]TJ/F75 11.955 Tf 13.945 0 Td[(kin!^sothatforanobviousbijectionbijk)]TJ/F15 11.955 Tf 14.41 0 Td[(imwehavethatbij)]TJ/F77 11.955 Tf 11.046 0 Td[(.Firstdenek)]TJ/F15 11.955 Tf 9.279 0 Td[(Sizeim1andthenlabelim)]TJ/F24 11.955 Tf 9.279 0 Td[(i0;:::;iksothati0@i1@@ik1@ik;thisdenestheaformentionedobviousbijectionik)]TJ/F15 11.955 Tf 13.294 0 Td[(im.Nowdenen)]TJ/F75 11.955 Tf 13.295 0 Td[(kbya`wherea)]TJ/F77 11.955 Tf 9.873 0 Td[(i`.Furthermore,thereexistsasequence{almostalwaysnon-unique{ofcofacemapsdm;:::;i1)]TJ/F15 11.955 Tf 9.279 0 Td[(dimdi1k1suchthatdm;:::;i1)]TJ/F77 11.955 Tf 9.279 0 Td[(.Finally,factorsasasequenceoftranspositionsti;:::;i1)]TJ/F15 11.955 Tf 9.279 0 Td[(ti`nti1nfollowedbyasequencesk;:::;n1)]TJ/F15 11.955 Tf 9.279 0 Td[(siksi1n1ofdegeneracies.Therefore,wecanfactorintonti1n)]TJ/F39 11.955 Tf 11.913 0 Td[(tin)]TJ/F75 11.955 Tf 12.491 0 Td[(nsi1n1)]TJ/F75 11.955 Tf 14.724 0 Td[(n1si2n2)]TJ/F39 11.955 Tf 14.724 0 Td[(sik)]TJ/F75 11.955 Tf 12.35 0 Td[(kdi1k1)]TJ/F75 11.955 Tf 15.143 0 Td[(k1di2k2)]TJ/F39 11.955 Tf 15.143 0 Td[(dim)]TJ/F75 11.955 Tf 12.935 0 Td[(mFromthiswehaveimmediatelythat:Corollary5.2.8.Supposen)]TJ/F109 11.955 Tf 13.804 0 Td[(nisnotinjective.Thenthereexistsaminimalk@nsuchthatfactorsthroughk.Wehavenowprovenallthefactsthatwewillneedthroughoutthethesisaboutthecardinalnumbercategory.Rememberthatthecardinalnumbercategoryisoneofthesimplex-typecategorieswhichweareusingasprobecategories.Sowewanttodene!-objectsandco!-objectsinacategoryC.Denition5.2.9.A!-cosimplicialobjectinacategoryCisacovariantfunctor!)]TJ/F112 11.955 Tf 16.745 0 Td[(C.Byabuseoflanguage,werefertotheimagesofthecofacemapsd`,codegeneracymapss`,andcotranspositionmapst`underthefunctorasthecofacemaps,thecodegeneracymaps,andthecotranspositionmaps.Byabuseof67 PAGE 72 notation,wewrited`n)]TJ/F15 11.955 Tf 9.726 0 Td[(d`,s`n)]TJ/F15 11.955 Tf 9.726 0 Td[(s`,andt`n)]TJ/F15 11.955 Tf 9.726 0 Td[(t`forthemapsd`n)]TJ/F15 11.955 Tf 9.726 0 Td[(d`n1)]TJ/F15 11.955 Tf 13.089 0 Td[(n,s`n)]TJ/F15 11.955 Tf 11.2 0 Td[(s`n1)]TJ/F15 11.955 Tf 14.563 0 Td[(n,andt`n)]TJ/F15 11.955 Tf 11.2 0 Td[(t`n1)]TJ/F15 11.955 Tf 14.564 0 Td[(nrespectively.Amapof!-cosimplicialobjectsfromone!-cosimplicialobjecttoanotherisjustanaturaltransformationa.InthecasethatC)]TJ/F75 11.955 Tf 10.601 0 Td[(Set,werefertoa!-cosimplicialobjectas!-cosimplicialset.Inthiscase,werefertoanelementnasann-cosimplex.Remark5.2.10.Onaccountoffunctoriality,the!-simplicialidentitiesoftheorem5.2.6hold,exactlyaswritten,inany!-cosimplicialobject.Duetothefactthatthecardinalnumbercategoryisgeneratedbythecoface,codegeneracy,andcotrans-positionmorphisms,onecangeneratea!-simplicialobjectinsomecategoryCbymeansofacollectionofmorphismswhichwethinkofascofaces,codegeneracies,andcotranspositionswhichsatisfytheseidentities.Example5.2.11.Thefollowingisa!-cosimplicialobjectinTopofcentralim-portanceforsimplicialmethodsinalgebraictopology:!)]TJ/F75 11.955 Tf 14.376 0 Td[(Set.Onobjects,itisgivenbynnwhere,fornC0nisthedirectedtopologicaln-simplexinRn1givenbyn)]TJ/F24 11.955 Tf 9.279 0 Td[(a0;:::;annPi)]TJ/F16 7.97 Tf 4.631 0 Td[(0ai)]TJ/F15 11.955 Tf 9.279 0 Td[(1;aiC0,andforn)]TJ/F39 11.955 Tf 9.279 0 Td[(1,n)]TJ/F39 11.955 Tf 9.279 0 Td[(g.Onmorphisms,itisgivenbyn)]TJ/F75 11.955 Tf 11.487 0 Td[(mz)]TJ/F15 11.955 Tf 19.448 0 Td[(n)]TJ/F15 11.955 Tf 12.481 0 Td[(mwheretakesa0;:::;antob0;:::;bmwherebi)]TJ/F21 11.955 Tf 20.154 -0.941 Td[(Pj>1iaj.Thisfunctorisknownasthecosimplicialobjectofdirectedtopologicalsimplices.Denition5.2.12.A!-simplicialobjectinacategoryCisacontravariantfunctor!op)]TJ/F112 11.955 Tf 13.334 0 Td[(C.Werefertotheimagesofthecofacemapsd`,codegeneracymapss`,andcotranspositionmapst`underthefunctorasthefacemaps,thedegeneracymaps,andthetranspositionmaps.Inaddition,wewriteswappingtop-rightandbottom-rightdecorationsdn`)]TJ/F15 11.955 Tf 10.026 0 Td[(d`,sn`)]TJ/F15 11.955 Tf 10.026 0 Td[(s`,andtn`)]TJ/F15 11.955 Tf 10.025 0 Td[(t`forthemapsd`n)]TJ/F15 11.955 Tf 10.025 0 Td[(d`n1)]TJ/F15 11.955 Tf 13.389 0 Td[(n,s`n)]TJ/F15 11.955 Tf 11.487 0 Td[(s`n1)]TJ/F15 11.955 Tf 14.85 0 Td[(n,andt`n)]TJ/F15 11.955 Tf 11.487 0 Td[(t`n1)]TJ/F15 11.955 Tf 14.851 0 Td[(nrespectively.Amapof!-simplicialobjectsfromone!-simplicialobjecttoanotherisjustanaturaltransformationa.68 PAGE 73 InthecasethatC)]TJ/F75 11.955 Tf 9.278 0 Td[(Set,werefertoa!-simplicialobjectas!-simplicialset.Inthiscase,werefertoanelementnasann-simplex.Remark5.2.13.Asremarkeduponat5.2.10,fora!-cosimplicialobject,the!-cosimplicialidentitiesoftheorem5.2.6holdexactlyaswritten.However,fora!-simplicialobjectthe!-cosimplicialidentitiesholdcontravariantly.Thisisthecasebecauseasimplicialsetisacontravariantfunctorsandhencereversesthedirection"ofmorphisms.Werefertothecontravariantversionofthe!-cosimplicialidentitiesasthe!-simplicialidentities.Inamanneranalogoustothatdiscussedinremark5.2.10,onecanspecifya!-simplicialobjectinsomecategoryCbysimplyspecifyingacollectionofmorphismswhichwethinkofascofaces,codegeneracies,andcotranspositionswhichsatisfythe!-simplicialidentities.Example5.2.14.Moreconcretely,butnotindetail,observethatthecollectionofclassicsimplices{inthecategorySimpCompofsimplicialcomplexesandsimpli-cialmapsbetween{equippedwiththeobviousmapsdenesa!-simplicialobjectinSimpComp.Example5.2.15.Thefollowingisa!-simplicialsetofcentralimportancetosingularhomologytheoryoftopologicalspaces:SXwhereXaxedtopolog-icalspace.Onobjects,itisgivenbynMapsn;XwhereMapsn;XisthesetofallcontinuousmapsfromntoX.Onmorphisms,itisgivenbyn)]TJ/F75 11.955 Tf 11.486 0 Td[(mz)]TJ/F75 11.955 Tf 21.063 0 Td[(Mapsm;XX)]TJ/F75 11.955 Tf 17.029 0 Td[(Mapsn;XwhereXtakesacontinuousmapfm)]TJ/F77 11.955 Tf 12.962 0 Td[(XtofXn)]TJ/F77 11.955 Tf 12.962 0 Td[(X.This!-simplicialsetisknownasthesingularsetofthespaceX.Remark5.2.16.Thistechniquegeneralizesagreatdeal,asthefollowingdiscussion{whichisitselfbynomeansapresentationofthemostgeneralizedversionpossible{shows:Werecallthesituationofexample3.4.7andexpanduponit.GivenacategoryCwhichwewanttogetinformationabout,letPbesomewell-understoodcategory{a69 PAGE 74 probecategory{andletFP)]TJ/F112 11.955 Tf 15.166 0 Td[(Cbeacovariantfunctorwhoseimageisawell-understoodsubcategoryofC{thisfunctoracoP-objectinC.DenetheF-singularsetofanobjectCinCtobethecontravariantfunctor,theP-setSFCPop)]TJ/F75 11.955 Tf 14.657 0 Td[(Set:rst,sendtheobjectPtothesetMapsCFP;CofmorphismsinCfromtheimageFPoftheobjectPunderthefunctorFtothexedobjectCinC;second,sendthemorphismP)]TJ/F77 11.955 Tf 12.68 0 Td[(Ptotheprecompose-by-FmapFMapsCFP;C)]TJ/F75 11.955 Tf 12.643 0 Td[(MapsCFP;C.InfactwhatwehavedescribedistheobjectcomponentoftheF-singularsetfunctorSFC)]TJ/F75 11.955 Tf 14.209 0 Td[(SetPop.RecallthatthesymbolSetPopdenotesthecategoryofcontravariantfunctorsPop)]TJ/F75 11.955 Tf 14.478 0 Td[(Set.WedeneitonthemorphismfC)]TJ/F77 11.955 Tf 14.478 0 Td[(CtobethemorphismSFfSFCSFCeachofwhosecoordinatesisgivenbypost-composing-by-f,themapfXMapsCFP;C)]TJ/F75 11.955 Tf 12.642 0 Td[(MapsCFP;CdescribedbyfXaFP)]TJ/F77 11.955 Tf 12.642 0 Td[(CfXaFP)]TJ/F77 11.955 Tf 12.642 0 Td[(C.Example5.2.15,suggeststhedenition,whichwepresentinsection5.4,ofsingularset"whichweuseinthisthesis.Itislessgeneralthanthepreceedingdiscussion,butitisfarmoreuseful.Sowehavedenedthecardinalnumbercategoryandestablisheditsbasicfacts.Furthermore,wetreateditasaprobecategoryanddened!-simplicialobjectsand!-cosimplicialobjectsinacategoryC.Wenowmoveontoconsidersimplex-likecategorywhichhasacentralroletoplayinthethesis.5.2.2.TheSemi-CardinalCategory.Denition5.2.17.Thenitestandardsemi-cardinalnumbercategory,thesemi-cardinalnumbercategory,ortheinjectivecardinalnumbercategory,denotedby!s,hasasobjectsthecardinalnumbersandasmorphismsallinjectivemapsofcardinalnumbers.Therefore,!s,unlike!,isnotafullsubcategoryofSet.70 PAGE 75 Whereasin!therewerethreefundamentaltypesofmorphism,in!thereareonlytwosinceallmapsareinjectiveornon-degenerate."Werecallthemhereforthereadersconveniencethoughtheyaregivenalreadyatdenitions5.2.3and5.2.5Denition5.2.18.Theithcofacemapindimensionnwherei)]TJ/F15 11.955 Tf 9.279 0 Td[(0;:::;nisthemapdinn-1)]TJ/F75 11.955 Tf 12.642 0 Td[(ngivenbykkifk@ik1ifkCiTheintuitivenotionofacofacemapisthatitsplitsthedomainatandincludingtheithelement.Denition5.2.19.Theithcotranspositionindimensionn,wherei)]TJ/F15 11.955 Tf 9.279 0 Td[(0;:::;n1,isthemaptnin)]TJ/F75 11.955 Tf 12.642 0 Td[(ngivenbykkifk@ii1ifk)]TJ/F77 11.955 Tf 9.279 0 Td[(iiifk)]TJ/F77 11.955 Tf 9.279 0 Td[(i1kifkAi1Theintuitivenotionofacotranspositionmapisthatitswapstheelementsiandi1.Thesemapssatisfyalltheidentitieslistedintheorem5.2.6involvingonlycofacesandcotranspositionssince!sisasubcategoryof!.Werefertothoseidentitiesoftheorem5.2.6whichinvolvethecofaceandcotranspositionmapsthesemi-cardinalnumbercategoryasthenon-degeneratecosimplicialidentities.Forthereader'scon-venience,wecollectthemhere:71 PAGE 76 Theorem5.2.20.TheNon-DegenerateCosimplicialIdentities1djdi)]TJ/F90 11.955 Tf 9.279 0 Td[(didj1ifi@j2titi)]TJ/F15 11.955 Tf 9.279 0 Td[(13tj1tjtj1)]TJ/F90 11.955 Tf 9.279 0 Td[(tjtj1tj4titj)]TJ/F90 11.955 Tf 9.279 0 Td[(tjtiifi@j15tidj)]TJ/F90 11.955 Tf 9.279 0 Td[(djtitidi)]TJ/F90 11.955 Tf 9.279 0 Td[(di1tidj)]TJ/F90 11.955 Tf 9.279 0 Td[(djti1ifi@j1ifiAjObservethatananalogueofthefactorizationalgorithm5.2.7holdsaswellin!s:allmorphismsfactorasasequenceofcotranspositionmapsfollowedbyasequenceofcofacemaps.Thistheoremtogetherwiththatalgorithmestalishthebasicfactsabouttheinjectivecardinalnumbercategory.Now,wewanttothinkabout!sasaprobecategory.Soweproceedtodene!s-objectsandco!s-objectsalthoughtheywillnotbeknownasthatinacategoryC.Denition5.2.21.A!s-cosimplicialobjectinacategoryCisacovariantfunctor!s)]TJ/F112 11.955 Tf 15.205 0 Td[(C.Byabuseoflanguage,werefertotheimagesofthecofacemapsd`,andcotranspositionmapst`underthefunctorasthecofacemaps,thecodegeneracymaps,andthecotranspositionmaps.Byabuseofnotation,wewrited`n)]TJ/F15 11.955 Tf 11.117 0 Td[(d`andt`n)]TJ/F15 11.955 Tf 9.623 0 Td[(t`forthemapsd`n)]TJ/F15 11.955 Tf 9.624 0 Td[(d`n1)]TJ/F15 11.955 Tf 12.987 0 Td[(nandt`n)]TJ/F15 11.955 Tf 9.623 0 Td[(t`n1)]TJ/F15 11.955 Tf 12.987 0 Td[(nrespectively.Amapof!s-cosimplicialobjectsfromone!-simplicialobjecttoanotherisjustanaturaltransformationa.InthecasethatC)]TJ/F75 11.955 Tf 9.839 0 Td[(Set,werefertoa!s-cosimplicialobjectas!s-cosimplicialset.Inthiscase,werefertoanelementnasann-cosimplex.Remark5.2.22.Inexactlythesameveinasremark5.2.10,wehavethatthecofaceandcotranspositionmapsinany!s-cosimplicialobjectsatisfythe!s-cosimplicialidentities,exactlyastheyaredisplayedintheorem5.2.20becausetheseidentitiesarepreservedbysinceisafunctor.72 PAGE 77 Example5.2.23.Since!sisasubcategoryof!,wecanrestricteveryfunctor,i.e.every!-simplicialobject,!)]TJ/F112 11.955 Tf 12.643 0 Td[(Ctoobtaina!s-cosimplicialobject!s0!)]TJ/F112 11.955 Tf 12.643 0 Td[(C.Soforexample,considerthe!s-cosimplicialobjectwhicharisesastherestrictionofthe!-simplicialobjectinTopdiscussedinexample5.2.11.Denition5.2.24.A!s-simplicialobjectinacategoryCisacontravariantfunctor!sop)]TJ/F112 11.955 Tf 14.33 0 Td[(C.Werefertotheimagesofthecofacemapsd`andcotranspositionmapst`underthefunctorasthefacemapsandthetranspositionmaps.Byabuseofnotation,wewritedn`)]TJ/F15 11.955 Tf 11.792 0 Td[(d`andtn`)]TJ/F15 11.955 Tf 11.793 0 Td[(t`forthemapsdn`)]TJ/F15 11.955 Tf 11.792 0 Td[(d`n1)]TJ/F15 11.955 Tf 15.156 0 Td[(nandt`n)]TJ/F15 11.955 Tf 10.561 0 Td[(t`n1)]TJ/F15 11.955 Tf 13.924 0 Td[(nrespectively.Amapof!s-simplicialobjectsfromone!s-simplicialobjecttoanotherisjustanaturaltransformationa.InthecasethatC)]TJ/F75 11.955 Tf 9.837 0 Td[(Set,werefertoa!s-simplicialobjectas!s-simplicialset.Inthiscase,werefertoanelementnasann-simplex.Remark5.2.25.Alongjustthesamelinesasthoselaidoutinremark5.2.22,theidentitieslistedinlemma5.2.20donotholdaswrittenfora!s-simplicialobjectwhiletheyalwaysholdfora!s-cosimplicialobjectaswementionedinremark5.2.22.Again,thisisthecasebecauseasymmetricsemi-simplicialobjectisacon-travariantfunctor!s)]TJ/F112 11.955 Tf 12.642 0 Td[(C;sotheidentitiesaretakentotheircontravariantformbyevery!s-simplicialobject.Werefertotheseidentities,thecontravariantformofthe!s-cosimplicialidentities,asthe!s-simplicialidentities.Example5.2.26.Justasinexample5.2.23above,observeagainthatsince!sisasubcategoryof!,thereisa!s-simplicialobjectgivenbyrestrictionto!sforevery!-simplicialobject.Considertherestrictionofthe!-simplicialobjectgiveninexample5.2.15.Example5.2.27.Moreconcretely,butnotindetail,observethatthecollection,whichweexaminedrstinexample5.2.14,ofclassicsimplices{inthecategorySimpCompofsimplicialcomplexesandsimplicialmapsbetween{equippedwiththeobviousmaps73 PAGE 78 denesa!s-simplicialobjectinSimpComp.Thisisclearlyaninstanceofthere-strictionto!sofa!-simplicialcomplexmentionedinexample5.2.26.Thisconcludesthediscussionofthesemi-cardinalnumbercategory!s.Wenowformallystatethedenitionsgivenatthebeginningofthesection.5.2.3.Simplex-TypeCategories.Atthebeginningofthissection,wediscussedtwoideas:simplicial-typeobject",cosimplicial-typeobject."WedescribedthemasspecialcasesofP-objectsandcoP-objectsrespectively.HoweverthiswasnotquiteanaccuratedescriptionsincethereisnotasinglecategoryPwhosecontravariantandcovariantfunctorsrespectivelyweareconsidering:thereareeightsuchcategories.Twoofthesecategoriesareofspecialinteresttousinthisthesis.Insubsection5.2.1,weintroducedthecardinalnumbercategory!andstatedthefundamentaltheoremaboutit,theorem5.2.6,anddevelopedthebasictheoryofcovariantandcontravariantfunctors{knownas!-simplicialobjectsand!s-simplicialobjects,respectively{fromthiscategoryintoanarbitrarycategoryC.Inthesubsequentportion,subsection5.2.2,wedidthesameforthesemi-cardinalnumbercategory!s:stateditsfundamentaltheoremlemma5.2.20anddiscussedfunctorswithitasdomain.Now,wearepreparedtoreturntothelinguisticpointofviewadoptedatthebeginningofthesection.Weformallystateherethedenitionsweusedlooselythere.Denition5.2.28.LetD,S,andTdenotethecollections,respectively,ofcofaces,codegeneracies,andcotranspositionsinthecardinalnumbercategory!.Choosesomeornoneofthesecollections.Thesimplex-typecategorycorrespondingtothatchoice{denotedbyDTif,forexample,thechoicewas1;0;1{isthesubcategoryofthecardinalnumbercategorywhoseobjectcollectionconsistsofallcardinalnumbersandwhosemorphismsaregeneratedbyfreelycomposingallmorphismsinthechosencollections.Remark5.2.29.Letbeanysimplex-typecategory.Thenalltheidentitiesfromtheorem5.2.6whichinvolvemapsbywhichthecategoryisgeneratedapplyaswell74 PAGE 79 tothecategory.Wecallthiscollectionofidentitiesthecosimplicial-typeidentitiesfororalternativelythe-cosimplicialidentities.Example5.2.30.Wehavealreadyconsideredinsomedetailtwoexamplesofsimplex-typecategories,namelythecardinalnumbercategory!)]TJ/F75 11.955 Tf 10.317 0 Td[(DSTandtheinjectivecardinalnumbercategory!s)]TJ/F75 11.955 Tf 9.279 0 Td[(DT.Denition5.2.31.Acosimplicial-typeobjectinacategoryCisacovariantfunctorfromasimplex-typecategorytoC.Asimplicial-typeobjectinacategoryCisacontravariantfunctorfromasimplex-typecategorytoC.Remark5.2.32.Letbeanysimplex-typecategory.Aswementionedinremark5.2.29,someofthecosimplicialidentities,namelythe-cosimplicialidentities,holdin.Theseidentitiesarepreservedunderfunctorsaswell.Underacovariantfunctor,i.e.a-cosimplicialobject,theidentitiesarepreservedastheyarewritten.Underacontravariantfunctor,i.e.a-simplicialobject,theidentitiesarepreservedcontravariantly.Example5.2.33.Considerthesimplex-typecategoryST.ThenaST-objectinacategoryCisaST-simplicialobject,or,moregenerally,asimplicial-typeobject.Thislinguisticmaneuverwillbeusefulfororganizingtheconceptsinthethesis.Furthermore,thislanguageallowsustoeasilydene,insection5.4,thesingularsetsweareinterestedin.Inthissection,wediscussedcosimplicial-typeandsimplicial-typeobjectsinacat-egoryC.Wementionedbriey{inexamples5.2.14and5.2.27{theobjectsatthislevelwhichcorrespondtothetheoryinthecaseofhomologyofsimplicialcomplexes.Inaddition,wediscussedsingularsets{simplicialsetsarisingfromcosimplicial-typeobjects{atsomelengthandinsomegenerality.Insection5.4,wewilldenemuchmorespecicsingularsetswhichareappropriateforthepurposesofthisthesis.Inthenextsection,webegintodevelopthealgebraicmachinerynecessarytostateandprovethemaintheoremofthethesis.75 PAGE 80 5.3.FreeandOrientedChainComplexes.5.3.1.IntroductoryRemarks.Inthissection,webeginthetransitionfromcombina-torialdatatoalgebraicdata.Wedenetwofunctorsfromtwodierentsimplex-likecategoriestothecategoryChainZ.Tobeginwith,wedenethefreechaincomplexfunctor,"andsubsequently,wedenetheorientedchaincomplexfunctor."Theseobjectsaretoogeneraltohaveaclearanalogueinthecaseofhomologiesofsimplicialcomplexes.Insection5.4,wewilldeneinjectiveandsingularsets,certainsortsofsimplicial-typesets.Thefreechaincomplexfunctorappliedtothesingularsetasso-ciatedtoanobjectCinCcorrespondstotheorderedchaincomplexofasimpicialcomplex,andtheorientedcomplexfunctorappliedtotheinjectivesetassociatedtotheobjectCcorrespondstotheorientedchaincomplex.Infact,insection5.5,wewilldenetheorderedchaincomplex"andtheorientedchaincomplex"ofanobjectCinCinanalogousways.5.3.2.TheFreeChainComplexFunctor.LetDdenotethesimplex-likecategorywhosemorphismsaregeneratedonlybycofacemaps.WedeneafunctorSetopD)]TJ/F75 11.955 Tf -422.247 -23.083 Td[(ChainZfromthecategoryofcontravariantfunctorsD)]TJ/F75 11.955 Tf 15.388 0 Td[(Set{thatis,ofD-simplicialsets{tothecategoryChainZofchaincomplexesofZ-modules.Denition5.3.1.ThefreecomplexofanD-simplicialsetisthechaincomplexZfree@n2)]TJ/F30 11.955 Tf 15.52 0 Td[(Zfreen1@n1)]TJ/F30 11.955 Tf 15.52 0 Td[(Zfreen@n)]TJ/F30 11.955 Tf 11.626 0 Td[(Zfreen1@n1)]TJ/F39 11.955 Tf 15.52 0 Td[(@1)]TJ/F30 11.955 Tf 11.487 0 Td[(Zfree0@0)]TJ/F30 11.955 Tf 11.487 0 Td[(Zfree1@1)]TJ/F75 11.955 Tf 13.22 0 Td[(0ofabeliangroupsZfreen)]TJ/F24 11.955 Tf 11.186 0 Td[(`negeneratedfreelybytheelementsofnwheretheboundaryoperatoris@n)]TJ/F77 11.955 Tf 9.278 0 Td[(@nZfree)]TJ/F19 7.97 Tf 12.942 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01idnithelinearextensionofthealternatingsumofthefacemaps.Weshowthatthisreallyisachaincomplex:76 PAGE 81 Proposition5.3.2.LetbeanD-simplicialsetandletZfreebetheassociatedchaincomplex.Then@@)]TJ/F15 11.955 Tf 9.279 0 Td[(0.Proof.Theproofisanapplicationofanidentityfromtheorem5.2.6.Supposen>nisabasiselementforZfreenfornA0forn)]TJ/F15 11.955 Tf 9.601 0 Td[(0;1,itistrivial.Then@@atnisgivenbyn1Qi)]TJ/F16 7.97 Tf 4.632 0 Td[(0nQj)]TJ/F16 7.97 Tf 4.631 0 Td[(01ijdidjnButwehavethatdidj)]TJ/F15 11.955 Tf 9.377 0 Td[(dj1diwheni@j.Thissuggeststhatwerewritethesumastwosums,oneoverindicessuchthati@jandtheotheroverindicessuchthatiCj:Qi@j1ijdidjnQiCj1ijdidjnNoticethatsincen11)]TJ/F77 11.955 Tf 9.279 0 Td[(n,bothsumsareovernn1 2terms,soitispossiblethattheywillcanceltermbyterm;infact,theywill.Wecanrewritethetwosumsasn1Qi)]TJ/F16 7.97 Tf 4.631 0 Td[(0nQj)]TJ/F19 7.97 Tf 4.632 0 Td[(i11ijdidjnnQj)]TJ/F16 7.97 Tf 4.631 0 Td[(0n1Qi)]TJ/F19 7.97 Tf 4.631 0 Td[(j1ijdidjnButwecanrewritetheleftsumusingthesimplicialidentityandrewritetherightsumbynoticingthattheoutersumisreallyfrom0ton1since,forj)]TJ/F77 11.955 Tf 9.96 0 Td[(nweareincreasingthesumbyn1Pi)]TJ/F19 7.97 Tf 4.632 0 Td[(n1ijdidjn)]TJ/F15 11.955 Tf 9.279 0 Td[(0.Sowegetn1Qi)]TJ/F16 7.97 Tf 4.632 0 Td[(0nQj)]TJ/F19 7.97 Tf 4.631 0 Td[(i11ijdj1dinn1Qj)]TJ/F16 7.97 Tf 4.632 0 Td[(0n1Qi)]TJ/F19 7.97 Tf 4.632 0 Td[(j1ijdidjnWenowreindexbothsums:intheleftsumwereplacejbyj1;intherighthandsumweswaptheindices.Sowegetn1Qi)]TJ/F16 7.97 Tf 4.631 0 Td[(0n1Qj)]TJ/F19 7.97 Tf 4.631 0 Td[(i1ij1djdinn1Qi)]TJ/F16 7.97 Tf 4.631 0 Td[(0n1Qj)]TJ/F19 7.97 Tf 4.631 0 Td[(i1ijdjdinWefactorout1fromthelefthandsumandobtainn1Qi)]TJ/F16 7.97 Tf 4.632 0 Td[(0n1Qj)]TJ/F19 7.97 Tf 4.631 0 Td[(i1ijdjdinn1Qi)]TJ/F16 7.97 Tf 4.632 0 Td[(0n1Qj)]TJ/F19 7.97 Tf 4.631 0 Td[(i1ijdjdin77 PAGE 82 whichisplainlyzero.Tosummarize,we'vejustshownthat@@n)]TJ/F15 11.955 Tf 9.279 0 Td[(0foreverybasiselementn>n.Thisprovesthat@@)]TJ/F15 11.955 Tf 9.279 0 Td[(0asrequired.Furthermore,thisprocedureextendsnaturallytoafunctor:Observation5.3.3.Passing-to-free-complexdenesafunctorZfreeSetD)]TJ/F75 11.955 Tf 12.642 0 Td[(Set.Proof.Toseethis,wedescribethefunctoronmorphismsbetweenD-simplicialsets.Let;betwoD-simplicialsets,andsupposethataisamapofD-simplicialsetsi.e.anaturaltransformation.Thenforeachcardinalnumbern>D,wehaveafunctionann)]TJ/F15 11.955 Tf 14.872 0 Td[(n.Thesefunctionssatisfythatforanymorphismn)]TJ/F75 11.955 Tf 12.642 0 Td[(mthediagramnan)]TJ/F15 11.955 Tf 36.917 0 Td[(nmam)]TJ/F15 11.955 Tf 35.741 0 Td[(mcommutesbythedenitionofnaturaltransformation.Ofparticularinterestarethecofacemapsdinn1)]TJ/F75 11.955 Tf 12.643 0 Td[(nforwhichwegetthatthediagramnan)]TJ/F15 11.955 Tf 40.383 0 Td[(ndnidnin1an1)]TJ/F15 11.955 Tf 35.741 0 Td[(n1commutes.ThissuggestsdeningthemapanZfreen)]TJ/F30 11.955 Tf 14.402 0 Td[(Zfreenbyann)]TJ/F15 11.955 Tf 14.402 0 Td[(nlinearly.Infactthisdenesamorphismofchaincomplexes;thatis,thisdenitionmakesthediagramZfreenan)]TJ/F30 11.955 Tf 36.244 0 Td[(Zfreen@n@nZfreen1an1)]TJ/F30 11.955 Tf 35.741 0 Td[(Zfreen178 PAGE 83 commute.Wecheckthatthisistruewithacomputation:Onabasiselementn,wehavean1@nn)]TJ/F120 11.955 Tf 9.279 0 Td[(an1nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01idnin)]TJ/F19 7.97 Tf 12.942 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01ian1dnin)]TJ/F19 7.97 Tf 12.942 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01idniann)]TJ/F77 11.955 Tf 9.279 0 Td[(@nannwhereweusedthecommutativityarisingfromnaturalityinthediagraminvolvingthefacemapsinthethirdequality.Thereforethediagramcommutesasrequired.SogivenamapbetweenD-simplicialsets,wehavedenedfunctoriallyamapbetweentheirfreecomplexes.Tobeclear,wehaveonlycheckedthatthisisapseudofunctor.However,itisaneasythingtocheckthatthepseudofunctorpreservesidentitymorphismsandcomposition.5.3.3.TheOrientedChainComplexFunctor.Construction5.3.4.WeconsiderthewaythatthesymmetricgroupactsonanDT-simplicialset.LetopDT)]TJ/F75 11.955 Tf 12.758 0 Td[(SetbeanDT-simplicialset.Writethenthsetasn)]TJ/F24 11.955 Tf 9.279 0 Td[(n>Jn.ThesymmetricgroupSn{thegroupofpermutationsontheletters0;:::;n{actsonnviatheidenticationofeachpermutationp>Sn,presentedbytijti1intermsoftranspositionsti`>Sn,withthecompositiontnijtni1oftranspositionmapstni`n)]TJ/F15 11.955 Tf 13.664 0 Td[(n.Inotherwords,thegroupactionYSnn)]TJ/F15 11.955 Tf 13.664 0 Td[(nisgivenongeneratorsbytiYn)]TJ/F15 11.955 Tf 9.279 0 Td[(tnin.Foreachsimplexnin!n,letSnSdenoteitsorbitundertheactionofSn.WedeneanequivalencerelationonSnSbypYnqYnifandonlyifsgnp)]TJ/F15 11.955 Tf 9.279 0 Td[(sgnq.25ObservethatthisisindependentofthechosenrepresentativenofSnS.26ThisdenesapartitiononSnSintotwoequivalenceclasses)]TJ/F77 11.955 Tf 4.551 0 Td[(na,and)]TJ/F77 11.955 Tf 4.551 0 Td[(nb. 25Recallthatthenumberaoftranspositionsinonefactorizationtiati1ofapermutationpiscon-gruentmod2tothenumberboftranspositionsinanotherfactorizationtjbtj1;thesignofthepermutationsgnpisthendenedtobe1ifcmod20and1ifcmod21wherecisthelengthofanychosenfactorizationofpintotranspositions.Then,sgnisahomomorphismS)]TJ/F51 9.963 Tf 8.406 1.494 Td[(1;1.26Todoso,writethepreceedingrelation,whichwasdenedaprioriwithrespectto,as,andxanyotherrepresentativenofSnS{writerYn)]TJ/F11 9.963 Tf 7.888 0 Td[(n{anddenebypYnqYnifandonlyifsgnp)]TJ/F8 9.963 Tf 7.888 0 Td[(sgnq.SupposepYnqYn,i.e.thatsgnp)]TJ/F8 9.963 Tf 7.888 0 Td[(sgnq.ThenpYn)]TJ/F104 9.963 Tf 7.888 0 Td[(pYrYnandqYn)]TJ/F104 9.963 Tf 7.888 0 Td[(qYrn.Butsgnisahomomorphism,sosgnpr)]TJ/F8 9.963 Tf 8.29 0 Td[(sgnpsgnr)]TJ/F8 9.963 Tf 8.29 0 Td[(sgnqsgnr)]TJ/F8 9.963 Tf 8.29 0 Td[(sgnqr.ThereforepYnqYn.Butthisargumentissymmetricinandsincewecanwriten)]TJ/F104 9.963 Tf 7.888 0 Td[(r1Yn.79 PAGE 84 EachoftheseequivalenceclassesisknownasanorientationofSnSorasanorientedn-simplexin.Wedenoteby)]TJ/F77 11.955 Tf 4.552 0 Td[(ntheequivalenceclasswhichactuallycontainsnandby)]TJ/F77 11.955 Tf 4.551 0 Td[(ntheequivalenceclasswhichdoesnotcontainn.Wesaythat)]TJ/F77 11.955 Tf 4.551 0 Td[(nisoppositeto)]TJ/F77 11.955 Tf 4.551 0 Td[(northatthetwotogetherareoppositelyoriented.WedenethenthorientedsimplicialchaingroupZorintobeZorin)]TJ/F24 11.955 Tf 9.279 0 Td[(`)]TJ/F77 11.955 Tf 9.102 0 Td[(nS)]TJ/F77 11.955 Tf 20.871 0 Td[(n1)]TJ/F39 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.552 0 Td[(netheabeliangroupgeneratedbytheorientedn-simplicesofsubjecttotherelationthatanorientedn-simplexisinversetothesimplexoppositeit.WedenotethesetgeneratingZorinbyBorinandrefertoit,byabuseofterminology,astheorientedbasisfor.Itiseasytoseethat,infact,abasisforZorinisobtainedbychoosingoneorientation)]TJ/F77 11.955 Tf 4.552 0 Td[(nchfromthepartition)]TJ/F77 11.955 Tf 10.403 0 Td[(na;)]TJ/F77 11.955 Tf 4.552 0 Td[(nbofeachorbitSnS.Wedenethequotientmapindimensionntobethemapqnn)]TJ/F30 11.955 Tf 12.642 0 Td[(Zoringivenbyqnnz)]TJ/F24 11.955 Tf 20.603 0 Td[()]TJ/F77 11.955 Tf 4.552 0 Td[(nObservethatn>)]TJ/F77 11.955 Tf 4.551 0 Td[(nifandonlyiftnjn>)]TJ/F77 11.955 Tf 4.552 0 Td[(nsinceallelementsof)]TJ/F77 11.955 Tf 4.551 0 Td[(nareanevennumberoftranspositionsfromeachotherbydenition.Thereforeqntnin)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F15 11.955 Tf 4.551 0 Td[(tnin)]TJ/F39 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F39 11.955 Tf 9.279 0 Td[(qnnforalltranspositionstni.Wedenetheithpartialboundaryoperatorindimensionn,fori)]TJ/F15 11.955 Tf 9.967 0 Td[(0;:::;ntobethemap@nin)]TJ/F30 11.955 Tf 12.642 0 Td[(Zorin1tobethecomposition@ni)]TJ/F15 11.955 Tf 9.279 0 Td[(qn1dni.Nowcertainly,thiswillnotdenefunctions@niBorin)]TJ/F30 11.955 Tf 12.884 0 Td[(Zorin1onorientedn-simplicesinasonecanseefromsomeofthesimplestcomputations.However,thealternatingsumofthepartialboundarymapsis.80 PAGE 85 Wedenetheboundarymap27@nn)]TJ/F30 11.955 Tf 12.642 0 Td[(Zorin1tobe@n)]TJ/F19 7.97 Tf 12.942 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01i@nithealternatingsumofpartialboundarymaps.Apriorithismapdoesnotinduceawell-denedfunctionBorin)]TJ/F30 11.955 Tf 12.995 0 Td[(Zorin1onorientedn-simplices;but,aswestatedinthepreceedingparagraph,itdoes,infact.Toseethattheboundarymapdoesinduceawell-denedmaponorientedsimplices,wewillneedtomakeuseofthenon-degeneratesymmetricsimplicialidentitiesforDT-simplicialsetsarising,asdiscussedatremark5.2.25,contravariantlyfromthenon-degeneratecosimplicialidentitiesoftheorem5.2.20.Inparticular,wewillmakeuseofthecontravariantversionofidentity5fromtheorem5.2.20.Itgivesthefollowingcommutativityrelationsbetweenthefacesdjandthetranspositionsti:djti)]TJ/F15 11.955 Tf 9.279 0 Td[(tidjditi)]TJ/F15 11.955 Tf 9.278 0 Td[(di1djti)]TJ/F15 11.955 Tf 9.279 0 Td[(ti1djifi@j1ifiAjItwillbeenoughtoshowthat@nisconstantoneveryorientedn-simplex)]TJ/F77 11.955 Tf 4.551 0 Td[(nin.Thiswewilldobyshowingthat@ntnin)]TJ/F39 11.955 Tf 9.279 0 Td[(@nnSosupposen>nandthati>0;:::;n1.Wecompute@ntnin:Fromthedenitionoftheboundarymapandthepartialboundarymaps,@ntnin)]TJ/F19 7.97 Tf 13.085 12.275 Td[(nQj)]TJ/F16 7.97 Tf 4.632 0 Td[(01j@njtnin)]TJ/F19 7.97 Tf 13.085 12.275 Td[(nQj)]TJ/F16 7.97 Tf 4.632 0 Td[(01jqn1dnjtnin 27Wewillusethewordsboundarymap"torefertothreeintimatelyrelatedmaps.81 PAGE 86 ThecommutativityrelationsbetweenthefacesandtranspositionssuggestwewritethisasQj@i1jqn1dnjtnin1iqn1dnitnin1i1qn1dni1tninQjAi11jqn1dnjtninByrewritingtherstsumaccordingtothethirdidentity,thesecondandthirdtermsaccordingtothesecondidentity,andthefourthtermaccordingtotherstidentity,weobtainQj@i1jqn1tn1i1dnjn1iqn1dn1in1i1qn1dnitnitninQjAi11jqn1tn1idnjnButtranspositionsareinvolutions,sowecanrewritethethirdtermassimply1i1qn1dnin.Wenowmakeuseofthefact,remarkedonabove,thatqktk`k)]TJ/F39 11.955 Tf 9.279 0 Td[(qkkandobtain,aftertransposingandrewriting28{themiddletermsQj@i1jqn1dnjn1iqn1dnin1i1qn1dn1inQjAi11jqn1tn1idnjnButthisisclearlyjustnQj)]TJ/F16 7.97 Tf 4.632 0 Td[(01jqn1dnjn)]TJ/F39 11.955 Tf 9.279 0 Td[(@nnSoinsummary,wecheckedthat@ntnin)]TJ/F39 11.955 Tf 9.279 0 Td[(nQj)]TJ/F16 7.97 Tf 4.632 0 Td[(01j@njn)]TJ/F39 11.955 Tf 9.279 0 Td[(@nnaswehadhoped.Thereforetheboundarymapisconstantoneachorientation)]TJ/F77 11.955 Tf 4.551 0 Td[(nofeveryorbitSnSwithinunderthegroupactionofSn.Thereforetheboundarymap@n)]TJ/F30 11.955 Tf 15.141 0 Td[(Zorin1inducesawell-denedmap@nBorin)]TJ/F30 11.955 Tf 13.7 0 Td[(Zorin1onorientedn-simplices,alsoknownastheboundarymap.Finally,wedenetheboundarymaporboundaryoperatortobetheextensionof@nBorin)]TJET1 0 0 1 108 83.651 cmq[]0 d0 J0.398 w0 0.199 m59.776 0.199 lSQ1 0 0 1 -108 -83.651 cmBT/F16 7.97 Tf 108 75.86 Td[(28Werewritebymatchingthepowerson1totheindicesonthefacemaps.82 PAGE 87 Zorin1toamaponthenthorientedchaincomplex@nZori)]TJ/F77 11.955 Tf 9.279 0 Td[(@nZorin)]TJ/F30 11.955 Tf 12.642 0 Td[(Zorin1givenongeneratorsby)]TJ/F77 11.955 Tf 4.551 0 Td[(nz)]TJ/F19 7.97 Tf 24.266 12.276 Td[(nQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01i@ninwhichisnotcontingentontherepresentativenoftheorientedn-simplex)]TJ/F77 11.955 Tf 4.552 0 Td[(nchosenaswehavejustshown.TheZ-gradedsequenceZori@n1Zori)]TJ/F30 11.955 Tf 18.499 0 Td[(Zorin@nZori)]TJ/F30 11.955 Tf 18.498 0 Td[(Zorin1@n1Zori)]TJ/F39 11.955 Tf 18.498 0 Td[(@0Zori)]TJ/F30 11.955 Tf 18.499 0 Td[(Zori1)]TJ/F75 11.955 Tf 12.642 0 Td[(0)]TJ/F75 11.955 Tf 12.642 0 Td[(0ofabeliangroupsequippedwiththesequenceofboundarymapsisknownastheorientedchaincomplexassociatedtoorsimplytheorientedcomplex.Toseethatitreallyisachaincomplex,wemustshowthat@@)]TJ/F15 11.955 Tf 9.279 0 Td[(0.Todothis,rstrecallthatnisthebasisforthenthgroupinthefreechaincomplexassociatedtotheD-simplicialsetopD0opDT)]TJ/F75 11.955 Tf 14.376 0 Td[(Set,andthenlinearlyextendthequotientqnn)]TJ/F30 11.955 Tf 14.94 0 Td[(ZorintoamapqnZfreen)]TJ/F30 11.955 Tf 13.206 0 Td[(Zorin.ItisclearfromthedenitionoftheboundaryoperatorsontheorientedZoriandfreeZfreechaincomplexesthatthediagramZfreenqn)]TJ/F30 11.955 Tf 35.741 0 Td[(Zorin@nZfree@nZoriZfreen1qn1)]TJ/F30 11.955 Tf 35.741 0 Td[(Zorincommutes.Forsupposen>nisageneratorforZfreen.Thenwendthatqn1@nZfreen)]TJ/F15 11.955 Tf 9.279 0 Td[(qn1nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01idnin)]TJ/F19 7.97 Tf 12.942 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01iqn1dnin)]TJ/F77 11.955 Tf 9.279 0 Td[(@nZori)]TJ/F77 11.955 Tf 4.552 0 Td[(n)]TJ/F77 11.955 Tf 9.279 0 Td[(@nZoriqnnasrequired.Andfurthermore,thesignedquotientmapsaresurjective.SoletCnbeachaininZorin;thenthereexistsCn>ZfreenwhichismappedbythesignedquotienttoCn.ButZfreeisachaincomplexaswe'veproven;thus0)]TJ/F15 11.955 Tf 9.279 0 Td[(qn2@n1Zfree@nZfreeCn)]TJ/F77 11.955 Tf 9.279 0 Td[(@n1Zoriqn1@nZfreeCn)]TJ/F77 11.955 Tf 9.279 0 Td[(@n1Zori@nZoriqnCn)]TJ/F77 11.955 Tf 9.278 0 Td[(@n1Zori@nZoriCn83 PAGE 88 ThereforeZoriisachaincomplex,asclaimed.Observation5.3.5.Passing-to-augmented-oriented-complexdenesafunctorZoriDT)]TJ/F75 11.955 Tf 12.642 0 Td[(Set.Proof.Toseethis,wedescribethefunctoronmapsofDT-simplicialsets.LetandbetwoDT-simplicialsets,andletabeamorphismbetweenthemi.e.anaturaltransformation.Thenforeachsemi-cardinalnumberninDTwehaveamorphismann)]TJ/F15 11.955 Tf 13.138 0 Td[(nsuchthatforeverymapn)]TJ/F75 11.955 Tf 13.138 0 Td[(mwehavethatthediagramnan)]TJ/F15 11.955 Tf 36.917 0 Td[(nmam)]TJ/F15 11.955 Tf 35.741 0 Td[(mcommutes.Inparticular,wehavethecofacemapsdinn-1)]TJ/F75 11.955 Tf 12.655 0 Td[(nandthecotransposi-tionmapstinn)]TJ/F75 11.955 Tf 12.642 0 Td[(n;thusthediagramsn1an1)]TJ/F15 11.955 Tf 35.741 0 Td[(n1dnidninan)]TJ/F15 11.955 Tf 40.383 0 Td[(nnan)]TJ/F15 11.955 Tf 35.741 0 Td[(ntnitninan)]TJ/F15 11.955 Tf 35.741 0 Td[(ncommute.WedeneanqBorin)]TJ/F45 11.955 Tf 12.642 0 Td[(Borinby)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F120 11.955 Tf 4.552 0 Td[(annApriorithisisnotwell-dened;toseethatitisinfactwell-dened,supposen>)]TJ/F77 11.955 Tf 4.551 0 Td[(n,i.e.thatn)]TJ/F15 11.955 Tf 9.279 0 Td[(tni2ktni1n.Thenann)]TJ/F120 11.955 Tf 9.279 0 Td[(antni2ktni1n)]TJ/F15 11.955 Tf 9.279 0 Td[(tni2ktni1ann>)]TJ/F120 11.955 Tf 4.551 0 Td[(annsinceancommuteswithtni`.SoanqBorin)]TJ/F45 11.955 Tf 14.417 0 Td[(Boringivenbyqnnqnanniswell-denedasclaimed.84 PAGE 89 WeclaimthatthisinducesachainmapaZori)]TJ/F30 11.955 Tf 12.882 0 Td[(Zori.Inordertoprovethis,weshowthatforeverypartialboundarymap29@niwehavethatnan)]TJ/F15 11.955 Tf 45.91 0 Td[(n@ni@niZorin1anq)]TJ/F30 11.955 Tf 35.741 0 Td[(Zorin1commutes.Toseethis,recallthat@ni)]TJ/F15 11.955 Tf 9.279 0 Td[(qn1dni.Sosupposen>n;thenan1q@nin)]TJ/F120 11.955 Tf 9.279 0 Td[(an1qqn1dnin)]TJ/F15 11.955 Tf 9.279 0 Td[(qn1an1dnin)]TJ/F15 11.955 Tf 9.279 0 Td[(qn1dniann)]TJ/F77 11.955 Tf 9.279 0 Td[(@niannbythecommutativityofthelefthanddiagramobtainedbynaturality,asrequired.SowecanextendthiscommutativitytotheboundarymapandobtainthatthediagramZorinanq)]TJ/F30 11.955 Tf 38.009 0 Td[(Zorin@n@nZorin1anq)]TJ/F30 11.955 Tf 35.741 0 Td[(Zorin1commutes,demonstratingthataqisachainmap.Forsuppose)]TJ/F77 11.955 Tf 4.551 0 Td[(n>Borin.Thenan1q@n)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F120 11.955 Tf 9.279 0 Td[(an1qnQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01i@nin)]TJ/F19 7.97 Tf 12.942 12.276 Td[(nQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01ian1q@ninforanyn>)]TJ/F77 11.955 Tf 4.551 0 Td[(nch.Butaswe'vejustseenan1qcommuteswith@ni,sowecanwritethisasnQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01i@niann)]TJ/F77 11.955 Tf 9.279 0 Td[(@nann)]TJ/F77 11.955 Tf 9.279 0 Td[(@nannch)]TJ/F77 11.955 Tf 9.279 0 Td[(@nanqnch)]TJ/F77 11.955 Tf 9.279 0 Td[(@nanq)]TJ/F77 11.955 Tf 4.551 0 Td[(nchThereforeanqisachainmapasclaimed.Thuswehavedescribedafunctorial30constructionofamaponchaincomplexesarisingfromamaponDT-simplicialsets.5.4.InjectiveandSingularSets.Wearenowcomingclosetothespecicsituationtowhichthetheoremofthethesisapplies.Inthissectionwewillrstdiscussageneral 29Forthedenition,seethethetopofthethirdparagraphofconstruction5.3.4.30Itiseasytocheckthatthisisinfactthecase.85 PAGE 90 construction,thesingularsetconstruction.ThenwewillusethisconstructiontoassociatetoanobjectCinanice"categoryCtwosimplicial-typesetsinafunctorialway:theinjective-setofmapsintothatobjectandthesingular-setofmapsintothatobject.Inthecaseofsimplicialcomplexes,thesetwosimplicial-typesetsareisomorphictoalthoughconstructeddierentlyfromthenormalconstructionforthesimplicial-typesetoforientedsimplicesinthecomplexandthesimplicial-typesetoforderedsimplicesinthecomplex,respectively.Wewillthenbeable,inthesubsequentsection5.5,toapplyourmachineryfromsection5.3inordertoassociatetoobjectsCinacategoryCtwochaincomplexes:theorientedandtheorderedchaincomplexesoftheobject.Tobeginwith,wedescribethegeneralprocedureofpassingfromacosimplicial-typesettoasingularset:Construction5.4.1.LetCbesomecategory,andlet)]TJ/F112 11.955 Tf 13.011 0 Td[(Cbeacosimplicial-typeobjectinC,i.e.justacovariantfunctorfromsomesimplex-typecategorytoC.Weconstructthe-singularsetfunctorSC)]TJ/F75 11.955 Tf 12.642 0 Td[(SetonCasfollows:ForeachobjectCinC,thereisasimplicial-typesetarisingfrom:the-singularsetofC.ItisthecontravariantfunctorSCop)]TJ/F75 11.955 Tf 15.275 0 Td[(SetgivenonobjectsbysendingntoMapsCn;Candonmorphismsbysendingn)]TJ/F75 11.955 Tf 13.104 0 Td[(mtoXMapsCm;C)]TJ/F75 11.955 Tf 12.642 0 Td[(MapsCn;C.AndforeachmapfC)]TJ/F77 11.955 Tf 14.769 0 Td[(Cthereisamap31SfSC)]TJ/F104 11.955 Tf 14.769 0 Td[(SCofsimplicial-typesets:Letnbeacardinal-typenumber,i.e.anobjectofDST;thenSfisdened,onthiscoordinate"n,tobefXSCn)]TJ/F75 11.955 Tf 9.279 0 Td[(MapsCn;C)]TJ/F75 11.955 Tf 12.642 0 Td[(MapsCn;C)]TJ/F104 11.955 Tf 9.279 0 Td[(SCn 31Recallthatamapofsimplicial-typesetsisanaturaltransformationandthatanaturaltrans-formationcanbedescribedbyitsbehaviouroncoordinates{i.e.onobjectsinthedomainofbothfunctors.86 PAGE 91 whichisgivenbyMapsCn;C?an)]TJ/F77 11.955 Tf 12.642 0 Td[(CfXan)]TJ/F77 11.955 Tf 12.643 0 Td[(CMapsCn;CToseethatthisreallydenesamapofsimplicial-typesets,i.e.toseethatitdenesanaturaltransformationbetweenthefunctors,weneedtocheckthatforanymorphismn)]TJ/F75 11.955 Tf 12.642 0 Td[(mthediagramMapsCn;CfX)]TJ/F75 11.955 Tf 35.741 0 Td[(MapsCn;CXXMapsCn;CfX)]TJ/F75 11.955 Tf 37.283 0 Td[(MapsCn;Ccommutes{i.e.thatXfX)]TJ/F24 11.955 Tf 9.78 0 Td[(fXX.ButthisjustadierentwayofstatingtheassociativityofcertaincompositionsinC:Forleta>MapsCn;C,i.e.letan)]TJ/F77 11.955 Tf 14.34 0 Td[(C.Thetwocompositions{XfX,andfXX{evaluatedataareequaltothemorphismsfXaXandfXaXrespectivelyinC;andthesemorphismsareequalsinceCisacategory{inparticular,sinceCisassociative.Sofar,wehaveseenthatSasdenedisapseudofunctor.Thatitisafunctor,thatitpreservesidentitymorphismsandcomposition,isbasicallyaconsequenceofthefactthatCisacategory.Toseethatitpreservesidentitymorphisms,letCbeanobjectofC.TheninfactSidCistheidentitynaturaltransformation:Itisgivenoneachcomponentnbypost-composition-with-f,bypost-composition-with-idC.Butpost-composition-with-idCistheidentitymapMapsCn;C)]TJ/F75 11.955 Tf 14.488 0 Td[(MapsCn;Csinceforanymorphisma>MapsC;CinCwithcodomainC,idCXa)]TJ/F77 11.955 Tf 9.714 0 Td[(asinceCisacategory.SoSdoesindeedpreserveidentitymorphisms.Toseethatitalsopreservescomposition,consideritsbehaviouronthecommutativediagramCf)]TJ/F77 11.955 Tf 48.48 0 Td[(Cf)]TJ/F77 11.955 Tf 35.741 0 Td[(CidCidCC)]TJ/F24 11.955 Tf 35.74 0 Td[(fXf)]TJ/F77 11.955 Tf 35.741 0 Td[(C87 PAGE 92 inC.Inparticular,itwillbeenoughtoconsidertherestrictionoftheimagetosomexedcoordinaten>{thatis,considerthediagramunderthecompositionofSwithevaluateatn".Thisimageistheapriorinot-necessarilycommutativediagramSCnSfn)]TJ/F104 11.955 Tf 50.932 0 Td[(SCnSfn)]TJ/F104 11.955 Tf 46.023 0 Td[(SCnSidCnSidCnSCn)]TJ/F104 11.955 Tf 39.568 0 Td[(SfXfn)]TJ/F104 11.955 Tf 40.882 0 Td[(SCnwhichwecanrewritemorecomprehensiblyasMapsCn;CfX)]TJ/F75 11.955 Tf 35.741 0 Td[(MapsCn;CfX)]TJ/F75 11.955 Tf 36.32 0 Td[(MapsCn;CididMapsCn;C)]TJ/F24 11.955 Tf 43.86 0 Td[(fXfX)]TJ/F75 11.955 Tf 36.031 0 Td[(MapsCn;Cusingthedentionsandthefactthatidentitymorphismsarepreserved.Thediagramdoesinfactcommute,however,sinceCisacategoryhencehasassociativecomposi-tion.Forleta>MapsCn;C,i.e.letan)]TJ/F77 11.955 Tf 12.916 0 Td[(C.Thenaistakenbythebottompath,i.e.bythecompositionidfXfXidtofXfXa)]TJ/F24 11.955 Tf 9.894 0 Td[(fXfXa;andaistakenbythetoppath,bythecompositionfXfXtofXfXa.BytheassociativityofcompositioninC,theseareequal.Therefore,Spreservescomposi-tions.ThisprovesthatSC)]TJ/F75 11.955 Tf 12.643 0 Td[(SetonCisafunctor.32Denition5.4.2.AconcretecategoryCisacategoryequippedwithaninitialob-ject330C,aterminalobject341CandafaithfulfunctorFC)]TJ/F104 11.955 Tf 9.984 0 Td[(FC)]TJ/F75 11.955 Tf 13.347 0 Td[(Set,knownastheforgetfulfunctorwhichtakesallinitialobjects0CtogandallterminalobjectstoterminalobjectsinSeti.e.ittakesterminalobjectsinCtosingletons. 32Noticethattheproofdoesnotdependontheprobingfunctor)]TJ/F115 9.963 Tf 11.019 0 Td[(Corindeedontheprobecategory.Thissuggestsamassivegeneralizationmentionedpreviouslyinremark5.2.16.33Seedenition3.2.12.34Seedenition3.2.13.88 PAGE 93 Amongtheotherthings35thatcanbedonewhenwexafunctortoSet,wecantalkaboutthosemorphismswhoseimagesunderthefunctorareinjective.Inparticular,thisallowsustospecifyasubcategorywhichwecanthinkofasbeingthesubcategoryofinjective"maps.Denition5.4.3.AmorphismfinaconcretecategoryCisinjectivejustincaseitsimageFCfundertheforgetfulfunctorisinjective.Wewillrefertothosemorphismswhicharenotinjectiveasdegenerate.ThesubcategoryofinjectivemapsInjCofaconcretecategoryCthenisjustthesubcategorywhoseobjectsaretheobjectsofCandwhosemorphismsaretheinjectivemorphisms.SincewedemandedthateveryterminalobjectinCissenttoasingletoninSet,wehavethateverymorphismfromaterminalobjectisinjectivesinceeveryfunctionfromasingletonsetisinjective.Inthesameway,wehavethateverymorphismfromaninitialobjectisinjective.Furthermore,thelabeling,asitisfunctorial,hasothernicepropertieswhichcoincidewithourintuitionaboutinjectivity:forinstancethecompositionoftwoinjectivemorphismsisinjectivesincefunctorspreservecom-positionandeveryidentitymorphismisinjectivesincefunctorspreserveidentitymorphisms.Denition5.4.4.Astandardcosimplicialobject,denotedbyinaconcretecate-goryCisaDST-cosimplicialobject)]TJ/F15 11.955 Tf 9.279 0 Td[(DST)]TJ/F112 11.955 Tf 12.642 0 Td[(Cwhichsatisesthat1isinitialand0isterminalinCallcotranspositionandcofacemapsareinjective,butallcodegeneracymapsaredegenerateeverydegeneratemorphismi)]TJ/F77 11.955 Tf 12.768 0 Td[(CfactorsasadegeneratemorphismwithinfollowedbyaninjectivemorphismtoC. 35Infact,choosingaparticularfunctortoSetisasomewhatstrongmove,andisnotnecessaryforourpurposes.However,describingthelabelinjective"thatisappendedtosomemorphismsinthecategoryismucheasierthisway.89 PAGE 94 Remark5.4.5.Observethatamorphismn)]TJ/F15 11.955 Tf 12.882 0 Td[(minisinjectiveifandonlyifitspreimagen)]TJ/F77 11.955 Tf 12.642 0 Td[(misinjective,sincethegeneratingmapsforwhichwedemandareinjectivedegenerateareexactlythosewhichcorrespondtothegeneratingmapsofDSTwhichareinjectivedegenerate.Thiscorrespondenceremainsforarbitrarygeneratedmapsinbythefunctorialityoftheadjectiveinjective."Example5.4.6.Thefollowingcosimplicial-typeobjectsarestandard:TheinclusionfunctorDST)]TJ/F75 11.955 Tf 12.642 0 Td[(Set.TheclassicsimplexfunctorDST)]TJ/F75 11.955 Tf 12.642 0 Td[(SimpComp.Lemma5.4.7.LetCbeaconcretecategoryandletbeastandardDST-cosimplicialobject.Supposen)]TJ/F77 11.955 Tf 13.639 0 Td[(Cisdegenerate.Thenthereexistsaminimalk@nsuchthatfactorsthroughk.Furthermore,canbewrittenasnti`nti1n)]TJ/F15 11.955 Tf 19.751 0 Td[(nsi`ksi1n1)]TJ/F15 11.955 Tf 23.194 0 Td[(kkinj)]TJ/F77 11.955 Tf 13.459 0 Td[(CProof.ThisisanimmediateconsequenceofthefactoringalgorithmforDSTofproposition5.2.7.Fromthedenitionofastandardcosimplicialobject,wecanfactor)]TJ/F77 11.955 Tf 9.327 0 Td[(minjwhereminjm)]TJ/F77 11.955 Tf 12.69 0 Td[(Cisinjectiveandisadegeneratemapwithin.Thus,bythatalgorithmweobtainafactorizationntinti1n)]TJ/F75 11.955 Tf 21.056 0 Td[(nsiksi1n1)]TJ/F75 11.955 Tf 23.727 0 Td[(kdimdi1k1)]TJ/F75 11.955 Tf 24.73 0 Td[(mminj)]TJ/F77 11.955 Tf 13.499 0 Td[(CThen,sinceisstandard,dimdi1k1isinjective,sowecandenekinj)]TJ/F77 11.955 Tf 9.672 0 Td[(minjdimdi1k1whichworks.Denition5.4.8.LetCbeaconcretecategory,andletDST)]TJ/F112 11.955 Tf 12.642 0 Td[(Cbeastandardcosimplicialobject.Wedenethe-injectivesetfunctor,denotedbyI,tobesimplythe-singularsetfunctorSInjC)]TJ/F75 11.955 Tf 12.643 0 Td[(SetDTassociatedtoInjC.Whenitisclearwhichstandardcosimplicialobjectwearedeningtheinjectivesetwithrespectto,wedenoteitsimplybyI.90 PAGE 95 ObviouslyIisdenedforeveryobjectinCsincetheobjectsofInjCareexactlythoseofC.However,weneedtobeclearthat,thiscannotextendtoafunctorfromC:LetfC)]TJ/F77 11.955 Tf 13.104 0 Td[(Cbeanon-injectivemorphisminC.ThenwemighthopetodeneIftobetheprecompose-by-fmapaswehadbefore.HowevertheimageofthismapisnotcontainedinICsincefisnotinjective.Infactthereisnonon-arbitrarywaytoextendthedomainofItoallmorphismsinC.Denition5.4.9.Asusual,wereferto,asinjectiven-simplicesinC,theelementsnofthenth-gradedsetInCoftheinjectivesetfunctorevaluatedatanobjectC.Inaddition,werefertotheelements0)]TJ/F77 11.955 Tf 9.688 0 Td[(ofI0CastheverticesofC.Foreveryinjectiven-simplexn,thereareexactlyn1verticesi0)]TJ/F77 11.955 Tf 12.643 0 Td[(Cwhichfactor0)]TJ/F15 11.955 Tf 12.777 0 Td[(nn)]TJ/F77 11.955 Tf 11.722 0 Td[(C.Theseareknownastheverticesofn;thecollectionoftheseverticesofnisknownasthevertexsetofnandisdenotedbyVn.ObservethatthereisanaturalorderingonthevertexsetVn.Asetwiseaccountviatheorderstructureonncanbeproduced,butthatisratherunnaturalandinvolvesrepeatedlyenlistingsettheoreticmachinerybymeansofpullingbacktoDSTandthenpushingforwardagaintoC.Insteadofpursuingthispath,wegiveanaccountmoreathomeinthecontextofastandardcosimplicialobjectinaconcretecategory.Toobtainthis,werstorderthecofacemapsn1)]TJ/F15 11.955 Tf 12.642 0 Td[(n:wesaythatdin@djnifandonlyifi@j.Wenoworderlength-2sequencesofcofacemapsn2)]TJ/F15 11.955 Tf 13.766 0 Td[(n:wesaythatdi2ndi1n1@dj2ndj1n1wherewehaverewrittenbothsequencessothati2@i11andj2@j11bymeansoftherelevantidentitiesifandonlyifoneofi1@j1,2i1)]TJ/F77 11.955 Tf 9.279 0 Td[(j1,i2@j2.Supposethatlength-k1sequencesofcofacemapshavebeenordered.Weorderlength-ksequencesofcofacemapsnk)]TJ/F39 11.955 Tf 14.594 0 Td[()]TJ/F15 11.955 Tf 14.593 0 Td[(nasfollows:wesaythatdikndik1n1di1nk1@djkndjk1n1dj1nk1wherewehaverewrittenbothsequencessothatforall`,i`1@i`1andj`1@j`1byrepeatedapplicationoftherelevantidentitiesifandonlyifoneofdik1n1di1nk1@djk1n1dj1nk1,dik1n1di1nk1)]TJ/F15 11.955 Tf 9.879 0 Td[(djk1n1dj1nk1anddikn@djkn.Thislinearlyordersthesetofallnite-lengthsequencesofcoface91 PAGE 96 mapsm)]TJ/F15 11.955 Tf 13.049 0 Td[(ninadictionary-orderstyle."Inparticular,thisorderssequencesofcofacemaps0)]TJ/F15 11.955 Tf 12.642 0 Td[(n.Atlast,wecanproducethenaturalorderingonvertexsets.Supposethatnn)]TJ/F77 11.955 Tf 12.905 0 Td[(Cisaninjectiven-simplexinC.Everyvertexiofnfactorsasasequencedinndi11ofcofacemapsfollowedbynasacorollaryoflemma5.4.7.ThusweobtainalinearorderonVnwhichhas,forinstancethesequencerepresentedbyd0nd01asminimalandhasthesequencednnd11asmaximal.Denition5.4.10.Wealsodenethe-singularsetfunctorSInjC)]TJ/F75 11.955 Tf 12.89 0 Td[(SetDSTtobetherestrictiontoInjCofthestandard-singularsetfunctorSC)]TJ/F75 11.955 Tf 12.643 0 Td[(SetDST.ThereisaninjectivenaturaltransformationinclISfromItoS36whichisgivenoncomponents,asthenotationindicates,byinclusion.Thismakessensebecausethesetofinjectivemorphismsbetweentwoobjectsisasubsetofthesetofallmorphismsbetweenthem.SinceIcanbedenedonlyforthecategoryInjC,thisisthecategorywhichwillbethecodomainoftheorientedandorderedchaincomplexfunctors.Inthecomingsection,section5.5,wewilldenethesefunctorstobethecompositionoftheinjectivesetandsingularsetfunctorswiththeorientedandfreechaincomplexfunctors,denedinsections5.3.5.5.OrientedandOrderedChainComplexes.Inthissection,weapplythealgebraicmachineryweconstructedinsection5.3tothesimplicial-typesetswejustconstructedinsection5.4.Inthecaseofhomologyofsimplicialcomplexes,thiscorrespondstodeningtheorientedandorderedchaincomplexes.Oncewehavedenedthechaincomplexfunctors,wewilldenetheorientedandorderedhomologyfunctorssimplybymeansofcomposingwiththehomologyfunctordenedinsection4.3.Then,withthesedenitionsoutoftheway,wecangettotheworkofthetheoreminsection5.6. 36Thisisalmosttrue.Infact,thecodomainisthecomponent-wiserestrictionto!s.92 PAGE 97 Denition5.5.1.LetCbeaconcretecategory,andlet!s)]TJ/F75 11.955 Tf 15.09 0 Td[(InjCbetherestrictionto!sofastandardDST-cosimplicialset.Theorientedchaincomplexfunctor37ZoriInjC)]TJ/F75 11.955 Tf 15.316 0 Td[(ChainZisthecompositionZoriIofthe-injectivesetfunctorwiththeorientedcomplexfunctor38.Denition5.5.2.LetCbeaconcretecategory,andletbeastandardDST-cosimplicialsetinC.Theorderedchaincomplexfunctor39ZordInjC)]TJ/F75 11.955 Tf 13.306 0 Td[(ChainZisthecompositionZfreeSofthe-singularsetfunctorwiththefreecomplexfunctor.40Wenowdenetheorientedandorderedhomologiessimplybycomposing:Denition5.5.3.LetCbeaconcretecategory,andlet!s)]TJ/F75 11.955 Tf 15.09 0 Td[(InjCbetherestrictionto!sofastandardDST-cosimplicialset.TheorientedhomologyfunctorHoriInjC)]TJ/F75 11.955 Tf 13.446 0 Td[(AbisthecompositionHZorioftheorientedcomplexfunctorjustdenedatdenition5.5.1withthehomologyfunctor,denedinsection4.3.Denition5.5.4.LetCbeaconcretecategory,andletbeastandardDST-cosimplicialsetinC.TheorderedhomologyfunctorHordInjC)]TJ/F75 11.955 Tf 13.838 0 Td[(Abisthecom-positionHZordoftheorderedcomplexfunctordenedindenition5.5.2withthehomologyfunctordenedinsection4.3.Finally,allterminologyisinplaceforthetheoremtobeproven.Theheartoftheproofisinthecomingsection.5.6.NaturalandInfranaturalTransformationsonChainComplexes.Inthissection,wewillbeworkingtoestablishanaturaltransformationTZordZoriandaninfranaturaltransformationTZoriZordsuchthatthecompositionTTistheidentitynaturaltransformationZoriZoriandthecompositionTT,whilenotequaltotheidentityinfranaturaltransformationZordZord,ischainhomotopictoit. 37Ifnecessary,wesaythatthisistheorientedchaincomplexfunctorassociatedto.38Introducedinsection5.3.39Ifnecessary,wesaythatthisistheorderedchaincomplexfunctorassoicatedto.40Seedenition5.3.1.93 PAGE 98 Onceweestablishthesefacts,asimpleapplicationoftheacycliccarriertheoremofsection4.5willprovethemainresult.Throughoutthissection,weworkinaxedconcretecategoryCwithaxedDST-cosimplicialset!^.Construction5.6.1.Wewouldliketocomparetheorientedandorderedchaincomplexfunctorsintroducedinsection5.5,soweconstructnaturaltransformations41betweenthemasfollows.First,wedenethenaturaltransformationTZordZori.Todoso,wedenemapsTCZordCZoriCforeachCinCandcheckthattheysatisfythecommutativityrequirements.LetCbeanobjectinC,andwriteZordnC)]TJ/F30 11.955 Tf -425.611 -23.083 Td[(ZfreenS!C)]TJ/F24 11.955 Tf 9.279 0 Td[(`MapsC!sn;Ce)]TJ/F24 11.955 Tf 9.278 0 Td[(`n!sn)]TJ/F77 11.955 Tf 12.642 0 Td[(Ce.WedeneTCn)]TJ/F24 11.955 Tf 9.279 18.118 Td[()]TJ/F77 11.955 Tf 4.552 0 Td[(nifninjective0otherwiseThisisinfactachainmap.Inthecasen)]TJ/F15 11.955 Tf 9.279 0 Td[(0,itistrivialthattheboundaryoperatorcommuteswiththemap.Forthecasewherenisinjective,consultconstruction5.3.4whereweshowedthatthequotientcommuteswiththeboundaryoperatoronthefreecomplexgeneratedbyI!Cwhichisasubcomplexoftheorderedcomplexbecausethesetofinjectivemapsisasubsetofthesetofallmaps.SowehavedenedTtobeaninfranaturaltransformation.Wecheckthatitisactuallyanaturaltransformation.SupposefC)]TJ/F77 11.955 Tf 13.507 0 Td[(CisamorphisminInjC.WecheckthatthediagramZordCTC)]TJ/F30 11.955 Tf 37.284 0 Td[(ZoriCZordfZorifZordCTC)]TJ/F30 11.955 Tf 35.741 0 Td[(ZoriCcommutes.SupposenisabasiselementofZordC{thenitissomemorphisminMapsC!n;C.Considerthecasewherenisnotinjective:wehaveZorifTCn)]TJ/F30 11.955 Tf -425.611 -23.083 Td[(Zorif0)]TJ/F15 11.955 Tf 9.919 0 Td[(0andsimilarlyTCZordfn)]TJ/F77 11.955 Tf 9.919 0 Td[(TCfn)]TJ/F15 11.955 Tf 9.92 0 Td[(0sincecompositiondoesnot 41Infact,twoinfranaturaltransformations,oneofwhichreallyisanaturaltransformation.94 PAGE 99 alterwhetherafunctionisdegenerate.Theothercaseismoreinvolved.ComputethatZorifTCn)]TJ/F30 11.955 Tf 9.278 0 Td[(Zorif)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 9.279 0 Td[()]TJ/F77 11.955 Tf 4.551 0 Td[(fn)]TJ/F15 11.955 Tf 9.279 0 Td[(qfn)]TJ/F77 11.955 Tf 9.279 0 Td[(TCfn)]TJ/F77 11.955 Tf 9.279 0 Td[(TCZorifnThiscomputationshowsthatthetwoquantitiesofinterestareequalasrequired.ThusTreallyisanaturaltransformation.Infact,thereisnonaturaltransformationTZoriZordofinterest;moreover,everyinfranaturaltransformation42involvesanarbitraryspecicationofonepartic-ularn !sn)]TJ/F77 11.955 Tf 12.672 0 Td[(CforeachorbitSnSundertheaction43ofSn.Butinsteadoflettingthatstopus,wesimplyx,onceandforall,onen ineachTnTforeveryobjectCinC.Butwewillnotdothisbymeansofanarbitrarychoiceforeachorbit.Instead,wechooseapartialorderingonthesetVC)]TJ/F104 11.955 Tf 9.278 0 Td[(I0CofverticesinCwhichinducesalinearorderingonthevertexsetVnofeverysimplexinC.Clearly44everysimplexintheorbitofnhasthesamevertexsetasn;wechoosethesimplexn >SnSwhosenaturalvertexordering45coincideswiththeinducedorderingonVn ,whichispossiblesincethereisasimplexn>SnSforeachpossibleordering.Consequently,everyfacen d10isitselfequaltosomechosensimplexn1 sincebothorderingsboththatarisingnaturallyfromtheorderingofthesequencesofcoface 42Aninfranaturaltransformationisanaturaltransformationlackingthecommutativitycondition.43Thisisdiscussedatsomelengthinconstruction5.3.4.44Toseethis,recallthatthereareexactlyn1morphisms,whichfactorassequencesofcofacemaps,din;1ndi1;11;:::;din;n1ndi1;n1n0)]TJ/F8 9.963 Tf 10.759 0 Td[(nTheverticesofnn)]TJ/F11 9.963 Tf 10.76 0 Td[(Carepreciselythesecofacesequencesfollowedbyn.Supposethatnisinthesameorbitasn.Thenn)]TJ/F11 9.963 Tf 7.888 0 Td[(nti`nti1n.Sinceti`nti1nisanisomorphisminC,wehavethatti`nti1ndin;1ndi1;11;:::;ti`nti1ndin;n1ndi1;n1naren1distinctmorphisms.Andthesefollowedbynareverticesofn.Butn)]TJ/F11 9.963 Tf 9.096 0 Td[(nti`nti1n.Thereforetheverticesofn,whichwejustobservedcanbewrittennti`nti1ndin;1ndi1;11;:::;nti`nti1ndin;n1ndi1;n1n,areexactlythesameastheverticesofnunderthissubstitution.45Seethediscussionfollowingimmediatelyafterdenition5.4.9formoredetail.95 PAGE 100 mapsandthatinducedbythepartialorderonVConVn d10aregivenbyrestriction-to-a-subsetoftheorderingsonVn .However,fromtheotherdirection,simplythatasimplexn isselected,thattheorderingonVn inducedbytheorderingonsequencesofcofacemapsisthesameastheorderingonVn inducedbythepartialorder,doesnotprovethatthereexistsasimplexn1^ andacofacemapdin1suchthatn1^ isachosensimplexandthatn1^ din1)]TJ/F77 11.955 Tf 9.902 0 Td[(n sincethereisnotevenanyguaranteethatthereisaninjectivemorphismn1^whichhasn^asaface.WenowdenetheinfranaturaltransformationTZoriZord.LetCbeanobjectinC.ThenwedeneTCZoriC)]TJ/F30 11.955 Tf 12.642 0 Td[(ZordConbasiselementsby)]TJ/F77 11.955 Tf 4.551 0 Td[(nchn ifn >)]TJ/F77 11.955 Tf 4.552 0 Td[(nn ifn )]TJ/F77 11.955 Tf 4.552 0 Td[(nInordertobetheC-componentofaninfranaturaltransformationZoriC)]TJ/F30 11.955 Tf 12.787 0 Td[(ZordC,itmustbeamorphisminthecodomaincategoryofbothfunctors,namelyChainZ.Inotherwords,thismustbeachainmap;sowemusthavethatthediagramZorinCTC)]TJ/F30 11.955 Tf 36.833 0 Td[(ZordnC@nZoriCZordCZorin1CTC)]TJ/F30 11.955 Tf 35.741 0 Td[(Zordn1Ccommutes.Suppose)]TJ/F77 11.955 Tf 4.551 0 Td[(nisageneratorofZorinC.Wecompute@nZordCTC)]TJ/F77 11.955 Tf 4.552 0 Td[(n)]TJ/F77 11.955 Tf 9.279 0 Td[(@nZordCn )]TJ/F19 7.97 Tf 12.942 12.276 Td[(nQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01idnin wherethesignisdeterminedbywhethern >)]TJ/F77 11.955 Tf 4.551 0 Td[(n.AndalsoTC@nZoriC)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F77 11.955 Tf 9.279 0 Td[(TCnQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01i@nin)]TJ/F77 11.955 Tf 9.279 0 Td[(TCnQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01iqn1dnin)]TJ/F77 11.955 Tf 9.279 0 Td[(TCnQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01iqn1dnin 96 PAGE 101 wherethesigncomesfromthechangeofrepresentativefromnton .46WecanrewritethisasnQi)]TJ/F16 7.97 Tf 4.631 0 Td[(01iTCqn1dnin )]TJ/F19 7.97 Tf 12.941 12.275 Td[(nQi)]TJ/F16 7.97 Tf 4.632 0 Td[(01idnin sincethecompositionTCqn1ineectmultipliesbythesamesigntwice.Thusthetwomapsareequalonbasiselements,sothediagramcommutes.ThusTCisachainmapasrequired.Ideally,wewouldhavethatTisanaturaltransformation.Whydowenotexpectittobeone?LetfC)]TJ/F77 11.955 Tf 12.643 0 Td[(CbeamorphisminC;thatTisanaturaltransformationisthestatementthatthisdiagramZoriCTC)]TJ/F30 11.955 Tf 37.283 0 Td[(ZordCZorifZordfZoriCTC)]TJ/F30 11.955 Tf 35.741 0 Td[(ZordCcommutes.Butsuppose)]TJ/F77 11.955 Tf 4.552 0 Td[(n>ZoriC.ThenwecomputeZordfTC)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 9.279 18.118 Td[(Zordfn ifn >)]TJ/F77 11.955 Tf 4.551 0 Td[(nZordfn ifn >)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 9.279 18.118 Td[(fn ifn >)]TJ/F77 11.955 Tf 4.552 0 Td[(nfn ifn >)]TJ/F77 11.955 Tf 4.552 0 Td[(nAndontheotherhandTCZorif)]TJ/F77 11.955 Tf 4.552 0 Td[(n)]TJ/F77 11.955 Tf 9.279 0 Td[(TC)]TJ/F77 11.955 Tf 4.552 0 Td[(fn)]TJ/F24 11.955 Tf 9.279 18.118 Td[(n1 ifn1 >n1n1 ifn1 n1wherefn>TnT.Ingeneral,wecannotexpectthemtobeequal;infact,thereisnowaytodeneTsothatareequal.Observation5.6.2.ThecompositionTTZoriZoriistheidentitynaturaltrans-formation. 46Inconstruction5.3.4,weshowedthatfor,n;n>SnS,@nn)]TJ/F66 9.963 Tf 7.888 0 Td[(@nnwherethesignispositivejustincasenisanevenpermutationfromn.Thusthissign,justlikethelastone{andinthesameway{isdeterminedbywhethern >)]TJ/F11 9.963 Tf 3.874 0 Td[(n.97 PAGE 102 Proof.LetCbeanobjectinC.WecomputethecompositionTTontheCcompo-nent:Suppose)]TJ/F77 11.955 Tf 4.551 0 Td[(nisabasiselementofZoriC.FromthedenitionofTwehaveTC)]TJ/F77 11.955 Tf 4.551 0 Td[(n)]TJ/F24 11.955 Tf 9.279 18.118 Td[(n ifn >)]TJ/F77 11.955 Tf 4.551 0 Td[(nn ifn )]TJ/F77 11.955 Tf 4.551 0 Td[(nAndfromthedenitionofTwehaveTCn )]TJ/F24 11.955 Tf 9.279 18.118 Td[(n ifn >n n ifn >n Ifn >n ,thenTCTCn )]TJ/F77 11.955 Tf 9.278 0 Td[(TCn )]TJ/F24 11.955 Tf 9.279 0.155 Td[(n wherethelastequalitycomesfromourassumptionthatn isequivalentton.Sim-ilarlyifn n ,thenTCTCn )]TJ/F77 11.955 Tf 9.279 0 Td[(TCn )]TJ/F39 11.955 Tf 9.279 0 Td[(TCn )]TJ/F24 11.955 Tf 9.279 0 Td[(12n )]TJ/F24 11.955 Tf 9.279 0.154 Td[(n ThereforeTCTC)]TJ/F15 11.955 Tf 9.279 0 Td[(idC.Andso,asclaimed,TT)]TJ/F15 11.955 Tf 9.279 0 Td[(id.Itisclearthattheothercomposition,whichisonlyaninfranaturaltransformation,TTisnottheidentity.Infactittakeseverybasiselementnton ,thechoiceofrepresentativefromtheorbitSnSxedaheadoftimewherethesignisthesignofanypermutationfromnton andeverynon-injectivebasiselementto0.However,somethingweakeristrue:Theorem5.6.3.TheinfranaturaltransformationTTZordZordischainho-motopictotheidentitynaturaltransformation.Thatis,foreveryCinC,TCTCZordC)]TJ/F30 11.955 Tf 12.642 0 Td[(ZordCischainhomotopictoidZordC.Notabene:ThistheoremsaysnothingaboutmorphismsC)]TJ/F77 11.955 Tf 13.363 0 Td[(Csincethetrans-formationTTisonlyinfranatural.98 PAGE 103 Inordertoprovethis,weconstruct,foreachobjectCinC,anacycliccarrierBordC)]TJ/F30 11.955 Tf 13.726 0 Td[(SZordCfromZordCtoitselfwhichcarriesbothidZordCandTCTCandthenapplytheacycliccarriertheorem.However,anaiveconstruction,directlyfromthedenitionofmapsbeingcarriedbyanacycliccarrier,is,althoughpossible,verymessyandagainstthespiritofthetheorem.Soinstead,wefollowamuchmoreeleganttrajectory:Lemma5.6.4.Letkbeak-simplexinC,andletVkbethevertexsetofk.Sup-poseBFisthegradedsubsetofBordCgivenbyBkF)]TJ/F77 11.955 Tf 4.552 0 Td[(k)]TJ/F24 11.955 Tf 9.847 0.154 Td[(nn)]TJ/F15 11.955 Tf 12.642 0 Td[(k)]TJ/F77 11.955 Tf 12.642 0 Td[(C`MapsCn;C.Inwords,BnF)]TJ/F77 11.955 Tf 4.552 0 Td[(kisthesetofallmapsntoCwhichfactorthroughkbyamorphismintheimageof.ThenthesubcomplexF)]TJ/F77 11.955 Tf 4.551 0 Td[(kisacyclic.WerefertoF)]TJ/F77 11.955 Tf 4.552 0 Td[(kasthefundamentalacyclicsubcomplexofk.Proof.ToseethatF)]TJ/F77 11.955 Tf 4.552 0 Td[(kisacyclic,wedeneamapDn1Fn)]TJ/F77 11.955 Tf 4.551 0 Td[(k)]TJ/F45 11.955 Tf 14.375 0 Td[(Fn1)]TJ/F77 11.955 Tf 4.551 0 Td[(kwhichsatisesthat@n1F)]TJ/F19 7.97 Tf 3.292 0 Td[(kDn1Dn@nF)]TJ/F19 7.97 Tf 3.292 0 Td[(k)]TJ/F15 11.955 Tf 9.279 0 Td[(idF)]TJ/F19 7.97 Tf 3.292 0 Td[(ksothatinparticularann-chainCn>Fn)]TJ/F77 11.955 Tf 4.551 0 Td[(kinthekernelof@nF)]TJ/F19 7.97 Tf 3.293 0 Td[(kisliftedtoann1chainCn1^>Fn1)]TJ/F77 11.955 Tf 4.551 0 Td[(kwhoseboundary@n1F)]TJ/F19 7.97 Tf 3.293 0 Td[(kCn1^isCn.Choosetobeanyxedvertexink.WedeneDninductivelyonbasiselements.Forthecasen)]TJ/F15 11.955 Tf 9.279 0 Td[(0,deneD01)]TJ/F77 11.955 Tf 9.279 0 Td[(.Observethatthissatisesthatd00D01)]TJ/F77 11.955 Tf 9.279 0 Td[(1.Forthecasen)]TJ/F15 11.955 Tf 9.567 0 Td[(1,deneD10)]TJ/F77 11.955 Tf 9.567 0 Td[(1^tobetheunique1-simplexsuchthat1^d01)]TJ/F77 11.955 Tf 9.567 0 Td[(0andthat1^d11)]TJ/F77 11.955 Tf 11.492 0 Td[(.Observethatthissatisesthatd10D00)]TJ/F77 11.955 Tf 11.492 0 Td[(0andd11D00)]TJ/F77 11.955 Tf -425.611 -23.084 Td[(D00d00.SupposethatDdenedform@n;wedeneDnn1)]TJ/F77 11.955 Tf 11.404 0 Td[(n^tobetheuniquen-simplexsuchthatdn0n^)]TJ/F77 11.955 Tf 9.279 0 Td[(n1andthatdnin^)]TJ/F77 11.955 Tf 9.279 0 Td[(Dn1dn1i1m^.99 PAGE 104 Wenowcheckthattheequationholds.Supposen>Fn)]TJ/F77 11.955 Tf 4.551 0 Td[(k.Wecompute:@n1F)]TJ/F19 7.97 Tf 3.293 0 Td[(kDn1nDn@nF)]TJ/F19 7.97 Tf 3.292 0 Td[(kn)]TJ/F21 11.955 Tf 16.352 -0.941 Td[(Pn1i)]TJ/F16 7.97 Tf 4.631 0 Td[(01idn1iDn1nDnPni)]TJ/F16 7.97 Tf 4.631 0 Td[(01idnin)]TJ/F15 11.955 Tf 16.352 0 Td[(dn10Dn1nPni)]TJ/F16 7.97 Tf 4.632 0 Td[(01i1dn1i1Dn1nPni)]TJ/F16 7.97 Tf 4.632 0 Td[(01iDndnin)]TJ/F15 11.955 Tf 16.352 0 Td[(dn10Dn1nPni)]TJ/F16 7.97 Tf 4.632 0 Td[(01i1Dndnim^Pni)]TJ/F16 7.97 Tf 4.631 0 Td[(01iDndnin)]TJ/F77 11.955 Tf 16.352 0 Td[(nPni)]TJ/F16 7.97 Tf 4.631 0 Td[(01i11iDndnin)]TJ/F77 11.955 Tf 16.352 0 Td[(nThus,theidentityholds.WecannowprovethatF)]TJ/F77 11.955 Tf 4.551 0 Td[(kisacyclic.First,observethat@0F)]TJ/F19 7.97 Tf 3.292 0 Td[(kissurjectiveifkx0:every0-simplex0inF0)]TJ/F77 11.955 Tf 4.552 0 Td[(kismappedby@0F)]TJ/F19 7.97 Tf 3.292 0 Td[(kto1.ThusF0k)]TJ/F45 11.955 Tf 12.642 0 Td[(F1k)]TJ/F15 11.955 Tf 12.642 0 Td[(0isexact.NowsupposeCn>Fn)]TJ/F77 11.955 Tf 4.551 0 Td[(kisinthekerneloftheboundarymap.WriteCn1^)]TJ/F77 11.955 Tf 9.844 0 Td[(Dn1Cn.Then@n1F)]TJ/F19 7.97 Tf 3.292 0 Td[(kDn1Cn1^)]TJ/F94 11.955 Tf 9.844 0 Td[(CnDn@nF)]TJ/F19 7.97 Tf 3.292 0 Td[(kCn)]TJ/F94 11.955 Tf 9.845 0 Td[(Cnsince@nF)]TJ/F19 7.97 Tf 3.293 0 Td[(kCn)]TJ/F15 11.955 Tf 9.844 0 Td[(0.ItfollowsthatFn1k)]TJ/F45 11.955 Tf 12.642 0 Td[(Fnk)]TJ/F45 11.955 Tf 12.642 0 Td[(Fn1kisexact.InotherwordsF)]TJ/F77 11.955 Tf 4.551 0 Td[(kisacyclic.Observation5.6.5.ThefunctionBordC)]TJ/F30 11.955 Tf 15.143 0 Td[(SZordCgivenbynF)]TJ/F77 11.955 Tf 4.552 0 Td[(nisanacycliccarrier.Proof.Fromlemma5.6.4,F)]TJ/F77 11.955 Tf 4.551 0 Td[(nisacyclicforeveryn.Whatremains,then,toproveisthat@0F)]TJ/F19 7.97 Tf 3.292 0 Td[(kF0)]TJ/F77 11.955 Tf 4.551 0 Td[(k)]TJ/F45 11.955 Tf 12.643 0 Td[(F1)]TJ/F77 11.955 Tf 4.551 0 Td[(kissurjectiveifkx0andmoreoverthatthediagramZord0C@0ZordC)]TJ/F30 11.955 Tf 46.001 0 Td[(Zord1CinclinclF0)]TJ/F77 11.955 Tf 4.551 0 Td[(k@0Fk)]TJ/F45 11.955 Tf 40.194 0 Td[(F1)]TJ/F77 11.955 Tf 4.551 0 Td[(kcommutesandtherightverticalinclusionarrowistheidentity.100 PAGE 105 everyn1-simplexn1whichappearswithnon-zerocoecientintheboundaryofnsatisesthatFn1isasubcomplexofFn.Letkbeann-simplexinC.Wealreadyhaveproventhat@0F)]TJ/F19 7.97 Tf 3.292 0 Td[(kF0)]TJ/F77 11.955 Tf 4.551 0 Td[(k)]TJ/F45 11.955 Tf 12.643 0 Td[(F1)]TJ/F77 11.955 Tf 4.551 0 Td[(kissurjectiveintheproofoflemma5.6.4.Inordertoseethat1holds,justobservethattheinclusionF1)]TJ/F77 11.955 Tf 4.551 0 Td[(k)]TJ/F30 11.955 Tf 15.249 0 Td[(ZordCistheidentitysince1isthegeneratorofF1)]TJ/F77 11.955 Tf 4.552 0 Td[(k.Inordertoprovethat2holds,itisenoughtocheckthatthesubcomplexFk1associatedtoeveryfaceofkisasubcomplexofF)]TJ/F77 11.955 Tf 4.551 0 Td[(k.Butthisisimmediate:thefundamentalacyclicsubcomplexF)]TJ/F77 11.955 Tf 4.551 0 Td[(kofkisgeneratedbyallsimplicesnwhichfactorthroughk,andthefundamentalacyclicsubcomplexFk1ofk1isgeneratedbyallsimpliceswhichfactorthroughk1.Butk1isafaceofk.Soeverysimplexwhichfactorsthroughk1factorsthroughksincek1itselffactorsthroughk.ThusthegeneratingsetofFk1isasubsetofthegeneratingsetofF)]TJ/F77 11.955 Tf 4.551 0 Td[(k.ThereforeFk1isasubcomplexofF)]TJ/F77 11.955 Tf 4.551 0 Td[(kasrequired.WenowobtainTheorem5.6.3asacorollaryofthisobservationtogetherwiththeacycliccarriertheorem:Proof.Byobservation5.6.5,FisanacycliccarrierZordC)]TJ/F30 11.955 Tf 14.227 0 Td[(ZordC.Furthermore,itclearlycarriesbothidZordCandTCTC:foreveryk-simplexk,wehavethatk;TCTCk>F)]TJ/F77 11.955 Tf 4.552 0 Td[(ksinceTCTCkfactorsthroughkaskti`kti1k.Therefore,bytheorem4.5.4,TCTCischainhomotopictoidZordC,asclaimed.Asacorollary,weobtainthemaintheoremofthethesis:5.7.NaturalandInfranaturalIsomorphismsonHomology.Theorem5.7.1.OrientedandOrderedHomologyareNaturallyIsomorphic:Theorderedhomologyfunctorisnaturallyisomorphicandinfranaturallyisomorphictotheorientedhomologyfunctor.101 PAGE 106 Proof.WenowhavetheinfranaturaltransformationTZoriZord.ForeveryobjectCinC,wehaveTCZoriC)]TJ/F30 11.955 Tf 12.93 0 Td[(ZordCwhichsatisesthatTCTC)]TJ/F15 11.955 Tf 9.566 0 Td[(idZoriC,aswecheckedin5.6.2,and,ashavejustestablishedintheorem5.6.3,thatTCTCZordC)]TJ/F30 11.955 Tf -422.247 -23.084 Td[(ZordCisachainhomotopy.Therefore,wehavethatHTCTC)]TJ/F75 11.955 Tf 9.334 0 Td[(HidZoriCandthat,byproposition4.4.5,thatHTCTC)]TJ/F75 11.955 Tf 10.515 0 Td[(HidZordC.Buthomologyisafunctor,sowehavethatHTCHTC)]TJ/F75 11.955 Tf 9.279 0 Td[(HidZoriCandthatHTCHTC)]TJ/F75 11.955 Tf 9.279 0 Td[(HidZordC.Thus,HTCisanisomorphism.ThereforeHTisaninfranaturalisomorphism.Furthermore,HTCisanisomorphism.WeconstructedthenaturaltransformationTZordZoriinsection5.6.ForeverymorphismfC)]TJ/F77 11.955 Tf 14.296 0 Td[(C,wehavethatthediagramZordCTC)]TJ/F30 11.955 Tf 37.284 0 Td[(ZoriCZordfZorifZordCTC)]TJ/F30 11.955 Tf 35.741 0 Td[(ZoriCcommutes.Andhomologyisafunctoraswesawinsection4.3.ThereforethiscommutativediagramgivesrisetothefollowingdiagramHZordCHTC)]TJ/F75 11.955 Tf 37.283 0 Td[(HZoriCHZorifHZordfHZordCHTC)]TJ/F75 11.955 Tf 35.741 0 Td[(HZoriCwhichalsocommutesasaconsequenceoffunctoriality.Buttheverticalmorphismsinthisdiagramareisomorphisms.Therefore,bycommutativity,thehorizontalmor-phismsareisomorphisms.102 PAGE 107 References1.M.Grandis,FiniteSetsandSymmetricSimplicialSets,TheoryandApplicationsofCategories8,244{252.2.S.MacLane,CategoriesfortheWorkingMathematician,GraduateTextsinMathematics,vol.5,Springer-Verlag,1998.3.J.Munkres,ElementsofAlgebraicTopology,WestviewPress,1993.103 |