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A Natural Isomorphism from the Ordered Homology to the Oriented Homology of an Injective Set

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004064/00001

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Title: A Natural Isomorphism from the Ordered Homology to the Oriented Homology of an Injective Set
Physical Description: Book
Language: English
Creator: Chandler, Nathaniel
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2009
Publication Date: 2009

Subjects

Subjects / Keywords: Algebraic Topology
Simplicial Homology
Homological Algebra
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the basic combinatorial topology of simplicial complexes, it has been proven that the ordered homology functor is naturally isomorphic to the oriented homology functor. In this thesis we generalize this result substantially. We prove that there is a natural isomorphism from the ordered homology to the oriented homology of the singular set functor associated to a well-behaved simplicial-type object in a category subject to some mild restrictions. While the author's literature review did not turn up this result, he makes no claims to originality.
Statement of Responsibility: by Nathaniel Chandler
Thesis: Thesis (B.A.) -- New College of Florida, 2009
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick

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Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2009 C4
System ID: NCFE004064:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004064/00001

Material Information

Title: A Natural Isomorphism from the Ordered Homology to the Oriented Homology of an Injective Set
Physical Description: Book
Language: English
Creator: Chandler, Nathaniel
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2009
Publication Date: 2009

Subjects

Subjects / Keywords: Algebraic Topology
Simplicial Homology
Homological Algebra
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In the basic combinatorial topology of simplicial complexes, it has been proven that the ordered homology functor is naturally isomorphic to the oriented homology functor. In this thesis we generalize this result substantially. We prove that there is a natural isomorphism from the ordered homology to the oriented homology of the singular set functor associated to a well-behaved simplicial-type object in a category subject to some mild restrictions. While the author's literature review did not turn up this result, he makes no claims to originality.
Statement of Responsibility: by Nathaniel Chandler
Thesis: Thesis (B.A.) -- New College of Florida, 2009
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: McDonald, Patrick

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2009 C4
System ID: NCFE004064:00001


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ANATURALISOMORPHISMFROMTHEORDEREDHOMOLOGYTOTHEORIENTEDHOMOLOGYOFANINJECTIVESETBYNATHANIELCHANDLERAThesisSubmittedtotheDivisionofNaturalSciencesNewCollegeofFloridainpartialfulllmentoftherequirementsforthedegreeBachelorofArtsUnderthesponsorshipofPatrickMcDonaldSarasota,FloridaApril,2009

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DedicationTomyadvisorPatMcDonald.TomyfriendAdeleFournet.i

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ANATURALISOMORPHISMFROMTHEORDEREDHOMOLOGYTOTHEORIENTEDHOMOLOGYOFANINJECTIVESETNathanielChandlerNewCollegeofFlorida,2009ABSTRACTInthebasiccombinatorialtopologyofsimplicialcomplexes,ithasbeenproventhattheorderedhomologyfunctorisnaturallyisomorphictotheorientedhomologyfunctor.Inthisthesiswegeneralizethisresultsubstantially.Weprovethatthereisanaturalisomorphismfromtheorderedhomologytotheorientedhomologyofthesingularsetfunctorassociatedtoawell-behavedsimplicial-typeobjectinacategorysubjecttosomemildrestrictions.Whiletheauthor'sliteraturereviewdidnotturnupthisresult,hemakesnoclaimstooriginality.PatrickMcDonaldDivisionofNaturalSciencesii

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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thehomologiesofmapsfromonesimplicialcomplextoanother.Itwillbeveryhelpfultoreferbacktothisgeometriccasewhentheabstractionbecomesoverbearing.Wewillnotbediscussingtheabstractionoftheorem2.4.1untilchapter5becausewerstneedtodevelopthelanguageinwhichtostatetheresult.Tobeginwith,weintroducethelanguageofcategorytheoryinchapter3andthenweintroducetheabstractalgebraicnotionofhomologyinchapter4.21

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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elements.Aswehaveseen,itiscarriedby,and@q1ADHqHq1@qAC)]TJ/F24 11.955 Tf 10.162 3.212 Td[(qqbylinearity.60

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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References1.M.Grandis,FiniteSetsandSymmetricSimplicialSets,TheoryandApplicationsofCategories8,244{252.2.S.MacLane,CategoriesfortheWorkingMathematician,GraduateTextsinMathematics,vol.5,Springer-Verlag,1998.3.J.Munkres,ElementsofAlgebraicTopology,WestviewPress,1993.103


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