<%BANNER%>

The 2-Lien of a 2-Gerbe

Permanent Link: http://ufdc.ufl.edu/UFE0021903/00001

Material Information

Title: The 2-Lien of a 2-Gerbe
Physical Description: 1 online resource (95 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: category, cohomology, gerbes, homological, stacks
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Principal bundles have a well-known description in terms of nonabelian cocycles of degree 1 with values in a sheaf. A more general notion than that of a sheaf on a space X is that of a lien on X. A lien on X is an object that is locally defined by a sheaf of groups, with descent data given up to inner conjugation. Equivalences classes of gerbes with a given lien L are classified by nonabelian degree 2 cocycles. In his work, Lawrence Breen has given a similar classification of 2-gerbes using nonabelian degree 3 cocycles that take values in a family of group stacks. In our work, we defined the notion of a 2-lien on a space X. It is an object that is given locally by a group stack, with 2-descent given up to inner equivalence. We have proved some theorems about 2-liens of 2-gerbes which correspond to well known results about liens of gerbes. Also, Deligne has shown that any strict Picard stack G corresponds to a 2-term complex of abelian sheaves K. In this case we proved that H^3(X,G) is isomorphic to the hypercohomology group H^3(X,K).
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, 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.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Crew, Richard M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021903:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021903/00001

Material Information

Title: The 2-Lien of a 2-Gerbe
Physical Description: 1 online resource (95 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: category, cohomology, gerbes, homological, stacks
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Principal bundles have a well-known description in terms of nonabelian cocycles of degree 1 with values in a sheaf. A more general notion than that of a sheaf on a space X is that of a lien on X. A lien on X is an object that is locally defined by a sheaf of groups, with descent data given up to inner conjugation. Equivalences classes of gerbes with a given lien L are classified by nonabelian degree 2 cocycles. In his work, Lawrence Breen has given a similar classification of 2-gerbes using nonabelian degree 3 cocycles that take values in a family of group stacks. In our work, we defined the notion of a 2-lien on a space X. It is an object that is given locally by a group stack, with 2-descent given up to inner equivalence. We have proved some theorems about 2-liens of 2-gerbes which correspond to well known results about liens of gerbes. Also, Deligne has shown that any strict Picard stack G corresponds to a 2-term complex of abelian sheaves K. In this case we proved that H^3(X,G) is isomorphic to the hypercohomology group H^3(X,K).
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, 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.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Crew, Richard M.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021903:00001


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Iwouldliketothankmyadvisor,RichardCrew,forguidingmethroughtheliterature,forsuggestingthisproblem,andforpatientlyansweringmyquestionsovertheyears.Ihavelearnedagreatdealfromhim.IamalsogratefultoDavidGroisserandPaulRobinson.Bothofthemhelpedmemanytimesandservedonmycommittee.ThanksalsogotocommitteemembersPeterSinandBernardWhitingfortheirfeedbackonthisproject.Thesupportofmyfamilyandfriendshasbeeninvaluabletome.Ithankallofthem. 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2GERBES,LIENSAND2-GERBES ......................... 12 2.1TorsorsandH1 12 2.2FiberedCategoriesandStacks ......................... 13 2.3GerbesandtheirLiens ............................. 17 2.42-Categories,2-Functors,2-NaturalTransformations ............. 21 2.5Fibered2-Categories,2-Stacksand2-Gerbes ................. 23 2.6RepresentableFunctors ............................. 28 2.72-Representability ................................ 32 2.8Giraud'sapproachtoLiensofGerbes ..................... 34 3EQUALIZERSANDCOEQUALIZERS ...................... 39 3.1Equalizers .................................... 39 3.2RepresentabilityofEqualizersinCAT 40 3.3Coequalizers ................................... 43 3.4RepresentabilityofCoequalizersinCAT 44 4GROUPCATEGORIES ............................... 51 4.1InnerEquivalencesofGroupCategories .................... 51 4.2ActionofaGroupCategoryonaCategory .................. 54 5COEQUALIZERSANDQUOTIENTSINSTACKS 58 5.1ActionofaGroupStackonaStack ...................... 58 5.2CoequalizersinSTACKS 59 6THE2-LIENOFA2-GERBE ............................ 71 6.1Denitionofa2-Lien .............................. 71 6.2The2-Lienofa2-Gerbe ............................ 72 6.3CocycleDescriptionofthe2-Lienofa2-Gerbe ................ 79 72-LIENS,PICARDSTACKSANDH3 81 7.12-Gerbesand3-Cocycles ............................ 81 7.22-LiensandStrictPicardStacks ........................ 83 7.3StrictPicardStacksandH3 86 5

PAGE 6

.................................... 92 REFERENCES ....................................... 93 BIOGRAPHICALSKETCH ................................ 95 6

PAGE 7

Principalbundleshaveawell-knowndescriptionintermsofnonabeliancocyclesofdegree1withvaluesinasheaf.AmoregeneralnotionthanthatofasheafonaspaceXisthatofalienonX.AlienonXisanobjectthatislocallydenedbyasheafofgroups,withdescentdatagivenuptoinnerconjugation.EquivalencesclassesofgerbeswithagivenlienLareclassiedbynonabeliandegree2cocycles.Inhiswork,LawrenceBreenhasgivenasimilarclassicationof2-gerbesusingnonabeliandegree3cocyclesthattakevaluesinafamilyofgroupstacks.Inourwork,wedenedthenotionofa2-lienonaspaceX.Itisanobjectthatisgivenlocallybyagroupstack,with2-descentgivenuptoinnerequivalence.Wehaveprovedsometheoremsabout2-liensof2-gerbeswhichcorrespondtowellknownresultsaboutliensofgerbes.Also,DelignehasshownthatanystrictPicardstackGcorrespondstoa2-termcomplexofabeliansheavesK=[K0d// 7

PAGE 8

IngeometryafundamentalconceptisthenotionofaprincipalG-bundle(alsoknownasaG-torsor).IfGisasheafonaspaceX,aG-torsoronXisageometricrealizationofG-valuedCech1-cocycle.Theproblemofdeningnon-abeliandegree2sheafcohomologyleadstotheconceptofagerbe.AgerbeonXmaybethoughtofnaivelyasa\sheafofcategories"withcertaingluingaxiomsforobjectsandarrows.AgerbeonXisageometricrealizationofa2-cocycleonXwithvaluesinanon-abeliansheaf(ormoregenerallyalienonX).ThistheorywasworkedoutbyGiraud[ 15 ]. Brylinski[ 7 ]developedatheoryofdierentialgeometryforgerbes.ByhistheoryonecanrealizeclassesinH3(M;Z)asequivalenceclassesofabeliangerbesviatheexponentialmapH2(M;C)// 21 ].Thetheoryofbundlegerbeshasprovenveryusefultophysicists;forexamplebundlegerbeshavebeenusedtoexploreanomaliesinquantumeldtheory[ 1 ],[ 2 ].Hitchinhasusedthetheoryofgerbesforstudyingmirrorsymmetry[ 17 ].BrylinskihasusedgerbestogiveaninterpretationofBeilinson'sregulatormapsinalgebraicK-theory[ 8 ]. Onecanthenask,whatkindsofobjectsareclassiedbynon-abeliandegree3sheafcohomology?Theanswerisanotionwhichisbuiltupfrom2-categorieswhichisduetoBreen.Hecallssuchobjects2-gerbesonX[ 4 ].BrylinskiandMcLaughlinhavestudiedacertainclassof2-gerbes[ 9 ],[ 10 ],namely2-gerbesboundbytheabeliansheafC,whichgiverisetoclassesinH4(M;Z)viatheexponentialmapH3(M;C)// Thisprovidesthemotivationforourproject.Givena2-gerbeonXtheassociated3-cocycletakesitsvaluesinafamilyofgroupstacksGi.Breenthenpointsoutthatonecouldintroducetheappropriatelydenednotionofa2-lienLonaspaceX.Thenthe 8

PAGE 9

Inthisthesiswedenethenotionofa2-lienonaspaceXandshowthateach2-gerbegivesrisetosucha2-lien.Afterprovingsomeanaloguesoftheoremsaboutliensofgerbes,weestablishsomeresultsabout2-liensthatarestrictPicardstacks,andanassociateddegree3hypercohomologygroup.Wenowprovideamoredetaileddescriptionofthecontentsofthisthesis. Inchapter2,wereviewthenecessarybackgroundmaterialthatformsthebasisofthisarea,namelynonabeliancohomology.WebeginwiththedenitionofaG-torsorforasheafofgroupsGandexplaintheirclassicationbyH1(X;G).Wethendescribetheobjectsthatembodydegree2cohomology:thatis,gerbes.Todoso,werstgivethedenitionofastack,amorphismbetweenstacksandthedenitionofa2-arrowbetweensuchmorphisms.Thisdatagivesusa2-categoryofstacksonaspaceX.Afterthissetup,wegivethedenitionofagerbeandthatofitsassociatedlien.Wethenprovideadescriptionoftheobjectsthatembodydegree3cohomology:thatis,2-gerbes.Aswiththecaseforgerbes,werstreviewthenotionof2-stackspriortodeninga2-gerbe.ThelatterhalfofChapter2isdevotedtoreviewingtwonotionsfromcategorytheorythatwillproveessentialtodeningthenotionofaquotientinthe2-categoryofstacks:thesenotionsarethoseoftherepresentabilityofafunctorandofa2-functor.WeclosebyreviewingtheoriginalapproachtakenbyGiraud[ 15 ]tointroducegerbes.Itisthisapproachthatwewilladaptinordertodenea2-lienofa2-gerbeinchapter6. Thekeytodeninga2-lienistogureouthowtodeneaquotientinthe2-categoryofstacks.Toanswerthisquestionlocally,werstneedtodeneaquotientinthe2-categoryof(small)categoriesCAT.Thisiswhatisaccomplishedinchapter3:weprovetheexistenceofacoequalizerinthe2-categoryof(small)categoriesCAT.Sincethedenitionofanequalizerisusedtodeneacoequalizer,werstprovetherepresentabilityofanequalizerinCAT.WethenconstructacategoryCoeqcorrespondingtoa(small) 9

PAGE 10

Inchapter4,webeginbygivingadenitionofinnerequivalenceforgroupcategoriesandshowthatsuchinnerequivalencesforaxedgroupcategoryformacategoricalgroup.Wethendenewhatitmeansforagroupcategorytoactonacategory,anddenethequotientbysuchanactionusingtheconceptofacoequalizerdevelopedinchapter3. Inchapter5,webeginbydeningwhatitmeansforagroupstacktoactonastack.Sincethegoalistodenethequotientbysuchanaction,wedotheworkofprovingthatcoequalizersarerepresentableinthe2-categorySTACKS.Thisisdoneusingthecoequalizerconstructioninchapter3,andviatheuniversalpropertiesofthecoequalizer.AfterprovingtheexistenceofacoequalizerinSTACKSweareabletodeneaquotientofanactionbyagroupstackonastack. Inchapter6,wedenethenotionofa2-lienonaspaceX,followingtheapproachofGiraudforanordinarylien.Wethendothenecessaryworkofshowinghowtoany2-gerbe,onemayasociatesucha2-lien,andthatthisassociationisgivenuptocanonical2-equivalence.Afterestablishingthis,wegiveamoredown-to-earthdescriptionofthe2-lienofa2-gerbe:itisanobjectthatisgivenlocallybyagroupstack,withdescentdatagivenuptoinnerequivalence. Inchapter7,ourgoalistoprovesometheoremsaboutG-2-gerbes,whose2-lienarisesfromastrictPicardstackG.Forthispurpose,insection1wereviewBreen'streatmentofthe3-cocycledescriptionofaG-2-gerbe[ 4 ].Insection2ofchapter7,weproveresultsaboutG-2-gerbeswhichareanaloguesoftheoremsaboutforliensofgerbes.Insection3,wewishtogiveanicecohomologicaldescriptionofconnectedG-2-gerbeswhereGisstrictPicard.DelignehasshownthatanystrictPicardstackGcorrespondstoa2-termcomplexofabeliansheavesK=[K0d// 10

PAGE 11

Weconcludewithchapter8,wherewesummarizeourresults,andindicatesomedirectionsforfuturework. 11

PAGE 12

5 ],[ 7 ]. RecallthatgivensetsX,YandZwithmapsf:X!Z,g:Y!Z,theberedproductXZYisthesubsetoftheproductXYconsistingof(x;y)suchthatf(x)=g(y). LetGbeasheafofgroupsonaspaceX. Now,givenaspaceXandasheafGofgroups;wewishtoconsiderisomorphismclassesofG-torsorsq:P!X.Givenanopencovering(Ui)i2IandasectionsiofqoverUi,wehaveafunctiongij:Uij!GjUijsuchthatsj=sigij(recallthatGactsontherightonP).Thesetransitionfunctionssatisfytheequalitygik=gijgjkoverUijk.IfthesectionsiisreplacedoverUibysi0=sihi,thengijisreplacedbyh1igijhj. Thisleadstotheappropriatenotionof1-cocyclesandcoboundaries.ForasheafGof(possiblynonabelian)groupsonaspaceX,and(Ui)i2Ianopencovering,wedeneaCech1-cocylewithvaluesinGtoconsistofafamilyaij2(Uij;G)suchthatthecocycle 12

PAGE 13

holds.Next,twoCech1-cocyclesaijanda0ijaresaidtobecohomologousifthereexistsasectionhiofGoverUisuchthat thusdeninganequivalencerelationonthesetofCech1-cocycles. ForasheafGofgroupsonthespaceX,andanopencoveringU=(Ui)i2IofX,therstcohomologysetH1(U;G)isdenedasthequotientofthesetof1-cocycleswithvaluesinGbytheequivalencerelation:\aiscohomologoustoa". 2. TherstcohomologysetH1(X;G)isdenedasthedirectlimitlim!UH1(U;G)wherethelimitistakenoverthesetofallopencoveringsofX,orderedbytherelationofrenement. Thedenitionofrstcohomologymakesthefollowingresultimmediate. 4 ],[ 5 ]relevanttoourdiscussion.Recallthatagroupoidisacategoryinwhicheverymorphismisinvertible. 13

PAGE 15

aboveU. Thedescentcondition(xi;ij;)iseectiveifthereexistsanobjectx2CUtogetherwithisomorphismsf:xjU'xcompatiblewiththemorphismsi.e.thefollowingdiagramcommutes: Byaconstructionbasedonthecorrespondingconstructionforsheaves,givenanyprestackConX,therecorrespondsanassociatedstackC,togetherwithacartesianfunctor whichisuniversalforcartesianfunctorsfromCintostacks(see[18]). 15

PAGE 16

1. PentagonAxiom. TriangleAxiom. X(2{16)

PAGE 17

15 ].Wefollowthepresentationin[ 4 ],[ 5 ]and[ 20 ]. TosaythatastackGislocallynon-emptymeansthereexistsacoveringU=(Ui)ofXforwhichthesetofobjectsofthecategoryGUiisnon-empty.ThelocallyconnectednessconditiononGistherequirementthatforanypairofobjectsxandyinGU,thereexistsanopencoverV=(V)ofUsuchthatthesetofarrowsfromxjVtoyjVisnon-emptyforall. 15 ],isacollection(Gi)ofsheavesofgroupscorrespondingtoanopencover(Ui)ofXwithdescentdatauptoinnerconjugation. InotherwordsalienonXisanobjectwhichisdenedlocallybyasheafofgroups,butinacategorywheremorphismsbetweengroupsdieringbyinnerconjugationareidentied. Wegiveacocycledescriptionofalien.LetU=(Ui)2IbeanopencoverofX,andsupposewehaveafamilyofsheavesGiofgroups,denedontheopensetsUi.Denoteeachsheafbylien(Gi).Thesheaveslien(Gi)andlien(Gj)aregluedontheopensetUijbyasectionijofthequotientsheafOut(Gj;Gi):=Isom(Gj;Gi)=Int(Gi)onUij.ThusthelienLisdeterminedbyafamilyofsectionsofOut(Gj;Gi)onUijsatisfyingthe1-cocycleconditionijjk=ikinOut(Gk;Gi)andthenormalizationconditionii=1foralli. 17

PAGE 18

LetGbeasheafofgroups,andletGi=GjUi.Thenwhenalienislocallyoftheformlien(Gi),eachijisasectiononthesetUijofthesheafOut(G)ofouterautomorphismsofG.Following[ 4 ]wewillcallsuchliens,whicharelocallyisomorphictothelienlien(G),G-liens.Itfollowsfromthisdescriptionthattheyareclassiedbythenon-abeliancohomologysetH1(X;Out(G)). LetGbeagerbe.Itislocallynon-emptysothereexistsanopencover(U)ofXsuchthatG(U)isnon-empty.Sochooseafamilyofobjectsx2G(U).NowAut(x)isasheafonU.NowGisalsolocallyconnectedsoforeach,thereexistsanopencoverof(U)ofUandforeachanarrowf:x!xinG(U).Conjugationbythisarrowdenesanisomorphismofsheavesofgroups:Aut(x)jU!Aut(x)jU.Thisisomorphismdependsonthechoiceofthefbutdierentchoicesdenethesameouterisomorphism.Inparticular,onoverlapsU\Uthetwoisomorphismsanddenethesameouterisomorphism.Thusforxed,thefamilydenesan\outerisomorphism"ofsheavesonU: whichdoesnotdependonthechoiceoff.ThissystemofsheavesofgroupsAut(x)andouterisomorphismsiscalledthelienofthegerbeG.ToanygerbewithlienL,wecanattacha2-cocyclethattakesitsvaluesinL,see[ 20 ]fordetails.

PAGE 19

GjUxzzttttttttty$$JJJJJJJJJAutPU(x)// 19

PAGE 20

AnabelianG-gerbeisaG-gerbebydenition.Infact,wenowshowthatinthissituationthesheafGmustbeabelian.Inparticular,thecommutivityofthegrouplawinGcanbeveriedlocally,forsectionsofasheafGjU.Letgbeasectionofthissheafandconsidertheabovetriangleassociatedtothecorrespondingarrowu=x(g):x!x.Thisisthediagram whereiudenotesinnerconjugationbyuinthesheafAutPU(x).CommutivityofthisdiagramimpliesthatiuistheidentitymapwhencethesheafAutPU(x),andthereforeGjUisabelian. Westatetwopropositions(see[ 4 ]forproofs)toillustratehowthenotionofalienishelpfulincharacterizingaG-gerbe. 15 ])givingtheconnectionbetweengerbeswithG-lienswhereGisanabeliansheaf,andcohomology.

PAGE 21

(q;g)7!(q;qg)(2{27) 1. acollectionofobjectsA,B,C,..., 2. foreachorderedpairofobjects(A;B),asmallcategoryA(A;B), 3. foreachtripleA,B,Cofobjects,abifunctor 21

PAGE 22

foreachobjectA,afunctor TheseelementsofdataarerequiredtosatisfyassociativitylawsforcompositionandtherequirementthatuAprovidesaleftandrightidentityforthiscomposition. 1. foreachobjectA2A,anobjectF(A)2B, 2. foreachpairofobjectsA,A0inA,afunctor Thisdataisrequiredtosatisfythefollowingaxioms: 1. Compatibilitywithcomposition:giventhreeobjectsA,A0,A00inA,thefollowingdiagramcommutes. 22

PAGE 23

Unit:foreveryobjectA2Athefollowingdiagramcommutes. Denition2.5.1. 23

PAGE 25

forwhichthetetrahedraldiagramof2-arrowsinducedbyinCUcommutes: The2-descentcondition(xi;ij;)iseectiveifthereexistsanobjectx2CUtogetherwithisomorphismsf:xjU'xcompatiblewiththemorphismsijand. Forany2-prestackConX,onedenesbythesamesheacationmethodasforsheavesand1-stacks,an\associated2-stack"2-functor 25

PAGE 26

1. Ar(x;y)!Ar(b(x);b(y))(2{47) isanequivalence. 2. everyobjectinDislocallyisomorphictooneintheimageofC. ThenDisanassociated2-stackofC.

PAGE 27

1. Foranypairofcomposablemorphismsf:x!yandg:y!zinPU,thecomposite2-arrowobtainedbypastingfandgisequaltogf. 2. Forany2-arrow:f)gbetweenapairofmorphismsf,g:x!yinPU,f=g,where:f)gisconjugationby. 3. ForeveryxinPU,1x=1

PAGE 28

12 ]. LetCbeacategoryandletSETdenotethecategoryofsets.ThecategoryofpresheavesofsetsonCisthefunctorcategoryHom(Cop;SET).NowchooseanobjectXofCanddeneapresheafhXby: foranyobjectYofCandarrowfinC.ThisfunctoriscalledthefunctorrepresentedbyX.ThisconstructionisfunctorialinX,i.e.givesafunctor denedbyh(X)=hX,andforf2Hom(X;Y), 28

PAGE 29

whichshowshowtocomputefrom.ThuswecanalsosayFisrepresentedbythepair(X;).Yoneda'slemmastatesthat: 29

PAGE 30

Universalproperties(suchasproducts)canbeexpressedbysayingthatacertainpresheafisrepresentable. foreachZ2Ob(C).ThenpluggingPinforZabove,wegetthattheuniversalelementZ(idP)isapairofmorphismsp1:P!X,p2:P!Ywiththepropertythatgivenanypairofmorphismsq1:Z!X,q2:Z!Y,bothfactorthroughauniquemorphismZ!P,seediagrambelow. Thedenitionforarbitraryproductsisanobviousextensionoftheabovedenition. Wecanmakesimilardenitionsforberedproducts,equalizers,andcoequalizers.FirstwelistthesedenitionsforthecategorySET.GivensetsX,YandZwithmaps 30

PAGE 31

WenowmakethisdenitionsforanyarbitrarycategoryC. Givenarrowsf:X!Z,g:Y!ZinC,wedenetheberedproductXZYofXandYtobetheobjectPwhichrepresentsthefunctor 2. Givenarrowsf;g:X!Y,wedenetheirequalizertobetheobjectKwhichrepresentsthefunctor 31

PAGE 32

3. Givenarrowsf;g:X!Y,wedenetheircoequalizertobetheobjectKwhichrepresentsthefunctor 16 ].LetCbea2-category.RecallthatCopisobtainedfromCby\invertingthedirection"ofthe1-cellsofC.i.e.if[X;Y]isthecollectionof1-cellsinCfromXtoYthen[X;Y]inCopis[Y;X].Wewilldene2-representabilityfor2-functorsf:Cop!CAT.LetF(C)bethe2-categoryof2-functorsfromCoptoCAT.Wedenethefollowingstrict2-functor: andwhichisdenedsimilarlyon1-cellsand2-cells.Letfbeany2-functor.i.e.f2Ob(F(C)).ThenforallX2Ob(C),wecandenethefunctor 32

PAGE 33

ThefunctorhX;fisanequivalenceofcategorieswiththequasi-inversegivenbykX;f:f(X)!HomF(C)(hX;f)whereforall2Ob(f(X)),kX;f()=uisthemorphismof2-functorsu:hX!f,u()=f()()andfor:!02Mor(f(X)), Inparticular,lettingf=hX0wegetthefollowingproposition. 16 ]fortheproofsofthefollowingpropositions.

PAGE 34

15 ].Itisthisapproachthatwewillgeneralizetodenethe2-lienofa2-gerbe. LetXbeatopologicalspace,andletFandGbesheavesofgroupsonX.ConsiderthesheafIsom(F;G)(thesheafofisomorphismsofsheavesofgroupsfromFtoG).LetInt(G)denotethesheafofinnerautomorphismsofthesheafG.Denethequotientsheaf whereInt(G)actsonIsom(F;G)fromtherightviacomposition.Now,foreveryopenUX,letLI(X)(U)denotethecategorywhoseobjectsarethesheavesofgroupsonU,andwhosearrowsbetweentwoobjectsAandBaretheglobalsectionsofthesheafOut(A;B)(U)i.e.HomLI(X)(U)(A;B)=(U;Out(A;B)).ThecompositionlawisdenedafterpassingtothequotientsheafOut,andiswell-dened(seethediscussionofalienofagerbeinSection2.3). NowifV,!UisaninclusionofopensetsinX,thentheformationofthequotientOut(A;B)commuteswiththerestrictionofUtoV.SowecandeneaberedcategoryLI(X).LetGrp(X)denotetheberedcategoryofsheavesofgroupsonX.Thenbyconstructionwehaveamorphismofberedcategories NowifAandBareanytwoobjectsofthecategoryLI(X)(U),thenthepresheafHomLI(X)(U)(A;B)ofarrowsfromAtoBisidentiedwiththesheafOut(A;B).ThusLI(X)isaprestack.Thusuponapplyingthestackicationfunctor,weobtaintheassociatedstackLIEN(X).Composingthestackicationfunctor[ 15 ]withthemorphism(1)yieldsamorphismofstacks 34

PAGE 35

ThestackLIEN(X)iscalledthestackofliensonX. 2. WecallaglobalsectionofthisstackalienoverX. 15 ]. NowletGbeastackonX.RecallthatGrp(X)denotesthestackofsheavesofgroupsonX.Wehaveacartesianfunctor wherex2Ob(G(U)).Composingthismorphismwiththemorphismlien(X):Grp(X)!LIEN(X)givesamorphismofstacks whichtoeveryobjectx2G(U)associatesthelien (calledthelienrepresentedbythesheafofU-automorphismsofx)andtoeveryU-isomorphism:x!yofGassociatesthemorphism (whereInt():AutU(x)!AutU(y)denedbyInt()(a)=a1isamorphismofsheavesofgroupsoverX). 35

PAGE 36

sincethefunctorlientransformseveryinnerautomorphismtoanidentityarrow. Nowletm:F!Gbeamorphismofstacks.Thenwehaveamorphismofmorphismsofstacks(i.ea2-arrow): where, Foreveryx2Ob(F(U)),thefunctorminducesamorphismofsheavesofgroups andbydenitionofthemapliau(m),themorphismliau(m(x))isthemorphismofliensrepresentedbyx: XL%%JJJJJJJJJJFf??liau(F)// 36

PAGE 37

1. Let(L;a)bealienoperatingonF.Thefollowingareequivalent: (a) (b) foreverylienL0onXandeveryactionbofL0onFthereexistsauniquemorphismofliensu:L0!Lsuchthatbistheactioninducedbyaandu. 2. Thereexistsalien(L;a)operatingonFthatsatisestheconditionsof(1). 37

PAGE 38

wherex2F(U),UX.Thisfamilyisrequiredtobecompatiblewiththerestrictionofopensetsandalsosatisfytheconditionthatifi:x!yisaU-isomorphismofF,thenthemorphismofsheavesofgroupsInt(i):AutU(x)!AutU(y)representstheidentitymorphismofL(U). 15 ].SinceFisagerbe,theprojectionf:F!XzartotheZariskisiteisfullyfaithful,hencethemapHom(L0;L)!Hom(L0f;Lf)givenbyu7!ufisbijective.Thisforcesb=a(uf)whencewehavethat1(a)implies1(b).Next,weshowthatforeverygerbeF,thereexistsalienLandanisomorphisma:Lf// iscartesian.Moreover,theprojectionI!Xzarisbothfullyfaithfulandessentiallysurjective(thisfollowstriviallyfromthefactthatF!Xisfullyfaithfulandbecauseliau(m)=liau(n)foranytwoU-isomorphismsm,n:x!yofF).Bytheuniversalpropertyoftheassociatedstack,itfollowsthatXisthestackassociatedtoI,andthatthereexistsacartesiansectionL0:X!IofIandanisomorphisma0:L0f//

PAGE 39

Recallthatalienisdenedbyassigningafamilyofsheavesofgroups(Gi)totheopensets(Ui)thatcoverX,andthesesheavesaregluedontheoverlapsUijbyasectionofthequotientsheafOut(Gj;Gi)=Isom(Gj;Gi)=Inn(Gi). Ourprojectinvolvesappropriatelydeningtheactionofagr-stackonastack,andthequotientbysuchanaction.Todothiswerstneedtodenethesenotionsforgroupcategories. Giventwomapsf;g:X!Yofsets,theequalizerEq(f;g)offandgisthesetfx2Xjf(x)=g(x)g. ForanyarbitrarycategoryC,givenarrowsf;g:X!Y,wedenetheirequalizertobetheobjectKwhichrepresentsthefunctor i.e.,Z7!Eq(Hom(Z;X)Hom(Z;f)// Thusthereisacanonicalmorphismi:K!Xsuchthatfi=gi,andanymorphismh:Z!Xsuchthatfh=ghfactorsthroughauniquemorphismZ!K. EqualizersarerepresentableinthecategorySETbytheobjectEq(f;g).Infactwehaveafunctorialisomorphism 39

PAGE 40

1. TheobjectsofEq(f;g)arepairs(x;)whereeachxisanobjectofthecategoryXand:f(x)// 2. Morphisms(x;)!(y;)arethosearrowsh:x!yinXsuchthatthediagrambelowcommutes. Observethatwehaveanarrow:Eq(f;g)!Xthatsendseachobject(x;)tox2X,andistheinclusiononmorphisms.Wenowdeneequalizersinanarbitrary2-category.

PAGE 41

where,Eq(Hom(Z;f);Hom(Z;g)):=Eq[Hom(Z;X)Hom(Z;f)// :GivenF:Z!Eq(f;g),denotethefunctorFasfollows:F(Z)=(F0(Z);f(F0(Z))F1(Z)// :SupposeFandGaretwofunctorsinHom(Z;Eq(f;g)),andletk:F!Gbeamorphismoffunctors.Then'(F):=F0and'(G):=G0areinEq(Hom(Z;X)f// 41

PAGE 42

commutescommutesinHom(Z;X).But'(F)and'(G)arein Thus':Hom(Z;Eq(f;g))!Eq(Hom(Z;X)f// Wenowshowthat'isinfactanequivalenceofcategories. :ConsiderHomHom(Z;Eq(f;g))(F;G)!HomEq(f;g)('(F);'(G))whereEq(f;g):=Eq(Hom(Z;X)f// 42

PAGE 43

:Nowsupposek:'(F)!'(G)isanarrow.Thenkx:(F)(x)!(G)(x)isanarrowforeachx.ButisafunctorfromEq(f;g)toXwhence(F0)(x)!(G0)(x)=(F0(x)x// :Let(G;)beanobjectofEq(f;g).ThenGisanarrowfromZtoXandisamorphismoffunctorsi.e.:f(G)// ConsiderF:Z!Eq(f;g)denedasfollows: 1. Onobjects:forxinOb(Z),F(x):=(G(x);G;x), 2. Onarrows:for:x!yinZ,F():=G(). ThenforanyxinOb(Z),'(F(x))=F0(x)=(F0(x))=((G0(x);G;x))=G(x).Further'(F)=(G;)whereisanisomorphismoffunctorsf// Thus'isanequivalenceofcategories. LetCbeacategory. 43

PAGE 44

IfF:GA!AistheactionofagroupGonasetAthenthequotientbytheactionisjustthecoequalizerofthearrowsGAp2// Firstwedenecoequalizersinanarbitrary2-category.

PAGE 45

WedenethesetofprearrowsPreAr(F;G)tobethedisjointunionofthefollowingtwosets: 1. 2. ThesetSconsistingof,foreveryx2C,aformalisomorphismF(x)// ThesourceandtargetofanelementofAr(D)isjustitsusualsourceandtargetinD.AmemberofSoftheformF(x)// 45

PAGE 46

(r1;r2;;rm)(s1;s2;;sn)=(r1;r2;;rm;s1;s2;;sn)(3{14) Observethatbythisdenition(s1;s2;;sn)=s1s2sn.Wewillhenceforthrefertotheaboveproductascomposition.ThenthereisasmallestequivalencerelationonPreAr(F;G)suchthat 1. Foranys2S,ifx=source(s)andy=target(s),then 2. Foranycomposable,2Ar(D),(;)isintherelationi.e.. 3. foranya,b2Ob(C)andt:a!binAr(C) 4. If,2Ar(D)[Sandsource()=target()then(s1;)(;s2)(s1;();s2)forsi2PreAr(F;G).Further,if1,2and1,2areelementsofPreAr(F;G)suchthat11and22andsuppose21and21isdened,then2121. DeneCoeq(F;G)asfollows:Ob(Coeq(F;G))=Ob(D)andAr(Coeq(F;G))=PreAr(F;G)=. Proof.Let[f]2HomCoeq(F;G)(A;B)and[g]2HomCoeq(F;G)(B;C).Dene[g][f]:=[gf].Toseethatthisiswelldened,supposef1,f22[f]andg1,g22[g].Thenf1f2andg1g2,whichimpliesthatg1f1g2f2whence[g1f1]=[g2f2].Sowehaveacompositionlaw.Similarly,theassociativelawholdssinceitholdsonthelevelofprearrows.Forsuppose[f]2HomCoeq(F;G)(A;B),[g]2HomCoeq(F;G)(B;C),and[h]2HomCoeq(F;G)(C;D).Then[h]([g][f])=[h]([gf])=[h(gf)]=[(hg)f]=([hg])[f]=([h][g])[f]wherethesecondequalityfollowssinceh(gf)isin 46

PAGE 47

ByconstructionofCoeq(F;G)wehaveanarrowD// Weshowthat'isfullyfaithful.Let,:J!KwhereJ,K:Coeq(F;G)!Earefunctors.Letf='(J)andg='(K).Thenfor2HomHom(Coeq(F;G);E)(J;K),itsimageinHomB(f;g)isamap:f!gwhichisamorphismoffunctors.Sobydenition,isdeterminedwhenwegivemapsx:f(x)!g(x)foreachx2Ob(D),compatiblewiththemorphismsinD.ButOb(D)=Ob(Coeq(F;G))andsincef(x)=J(x)andg(x)=K(x) 47

PAGE 48

Case1:Supposeu2Ar(D).Thensinceisamorphismoffunctors,thediagram commutes,whencethediagram commutes. Case2:Supposeu:x!ywherex=F(a)andy=G(a)fora2C.Thenbydenitionofthediagram commutesso commutessincetheverticalarrowsintheabovetwodiagramsarethesamebydenitionofthemap'. 48

PAGE 49

commutesforall[u]2Ar(Coeq(F;G)).ThuswehaveabijectionbetweenHomB(f;g)andHomHom(Coeq(F;G);E)(J;K):thatis,'isfullyfaithful. Wenowshowthat'isessentiallysurjective.Letf:D!EwithisomorphismsfF// and Thenu1u2.Thenwehave and 49

PAGE 50

commuteswhencethediagram alsocommutes.ThusJ(u1)=J(u2).SinceuisacompositionofarrowsoftheabovetypegetthatF(u)iswelldened.Weclaim'(J)// 50

PAGE 51

6 ]hasalsogivenasimilardescriptionofthisnotion. LetCbeagroupcategory.LetY,Z2Ob(C)suchthatZY'I'YZ.DeneFYZ:C!Casfollows: commutes.SoFYZisafunctor.Further,forA,B2Ob(C)wehave: SoFYZrespectsthegroupstructureofC.ForA,B,CinOb(C)wehave: 51

PAGE 52

(F(A)F(B))F(C)F(A)(F(B)F(C))(4{3) 52

PAGE 53

theabovecommutativediagramshowsthatba'fedc. NextwedeneanisomorphismFYZFZY'IdC: (FYZFZY)(A) LetInn(C)denotethecollectionofequivalencesofCwhichareisomorphictoonesoftheformFYZ.Denearrowsbetweenthemtobemorphismsoffunctors,andletthecompositionlawbetheusualverticalcompositionofnaturaltransformations,whichisagainamorphisminInn(C).ThenInn(C)isagroupoid.ForG,H2Ob(Inn(C)),if 53

PAGE 54

(HG)(A) =H(G(A)) =H((ZA)Y) =(Z0((ZA)Y))Y0 wherethesecondlastarrowisjusttheassociativelaw.Thusthecompositionoffunctorsandhorizontalcompositionofnaturaltransformationsdenesatensorfunctor::Inn(C)Inn(C)!Inn(C): Wewillusenotationsimilartothatof[ 22 ]butourdenitionsaremoregeneral.

PAGE 55

where:F(IdGF))F(mIdC)and:F(IIdC))IdCare2-isomorphisms.FurtherandmustbecompatiblewithassociativityinG:thatis,theysitinthepastingdiagramsshownbelow. 55

PAGE 56

Notethat\asc"abovereferstotheassociativelaw.The2-arrowrequiresexplanation.Notethattheexistenceofthe2-isomorphismsandmeansthatwehaveisomorphismsinC,naturalin(g;h;x), and InparticularthereexistsanisomorphisminCnaturaling,1,anda,obtainedfromthefollowingcomposition (g1)a1// whichyieldsa2-isomorphismthatsitsinthediagram 56

PAGE 57

57

PAGE 58

Denition5.1.1. wheremisthemonoidalfunctorassociatedwiththegr-stackG,Iitsidentityobject,and:F(IdGF))F(mIdC)and:F(IIdC))IdCare2-isomorphisms. 58

PAGE 59

FurtherandmustbecompatiblewithassociativityinG:thatis,theysitinthepastingdiagramsshown,whereIreferstotheidentityobjectinG.The2-arrowexistsforthesamereasonasinthecaseforgroupcategories. GiventwomorphismsofberedcategoriesCF// 2. IfCandDareprestacks(resp.stacks),andFandGaremorphismsofprestacks(resp.stacks),wesaythatFandGareasmallpairofmorphismsofprestacks(resp.stacks)iftheyformasmallpairofmorphismsoftheunderlyingberedcategories.

PAGE 60

LetCandDbeberedcategories,andletFandGdenoteasmallpairofmorphismsfromCtoDi.e.wehaveadiagramCF// LetCoeq(F;G)denotethefollowingberedcategory.ToeachopenUX,Coeq(F;G)(U):=Coeq(FU;GU).LetV,!UbeaninclusionofopensetsinthespaceX.TodenetherestrictionfunctorsfforCoeq(F;G)considerthefollowingdiagram: wherecandddenotetherestrictionfunctorsofthestacksCandDrespectively,'Uand'Varethecanonicalmapsintothecoequalizer.Also,aspartofthedata,wehave2-arrows:FVc)dFUand:GVc)dGU.Thedottedarrowfisthemapweneedtoconstruct.Now,fromthedataofacoequalizer,thereisanisomorphism So Buthorizontalcompositionwithyields andhorizontalcompositionwithyields 60

PAGE 61

Thenbytheuniversalpropertyofcoequalizers,thereexistsanarrow ThatfisuniqueuptouniqueisomorphismfollowsfromthefollowingresultofHakimin[ 16 ]. ByconstructionofCoeq(F;G)wehaveanarrowD'// 61

PAGE 62

ofcategories.SinceJ:Coeq(F;G)!EisamapofberedcategoriessoitiscompatiblewiththerestrictionfunctorsofCoeq(F;G)andE.Similarly(J):D!EisamapofberedcategoriessoitiscompatiblewiththerestrictionfunctorsofDandE.Thus isanequivalence. NowletCandDbeprestacks,andletCF// LetFib(F;G)betheberedcategorycoequalizerofFandG.ConstructaprestackHasfollows:foreachopensetUXandx,yinOb(Fib(F;G)(U))letOb(H(U))=Ob(Fib(F;G)(U)),andHomH(U)(x;y)bethesheacationofthepresheafHomFib(F;G)(U)(x;y). Proof.Weneedtospecifytherestrictionfunctorsandcheckthattheyhavetherequiredcompatibilityontripleinclusionsofopensets.SoletVUbeaninclusionofopensets 62

PAGE 63

whereU,VdenotethesheacationmapsandfistherestrictionfunctorfromFib(F;G)(U)!Fib(F;G)(V).ThenVfisanarrowfromthepresheafHomFib(F;G)(U)(x;y)tothesheafHomH(V)(x;y).Sobytheuniversalpropertyofsheacationthereexistsauniquearrowh:HomH(U)(x;y)!HomH(V)(x;y)suchthatthediagram commutes.Theuniquenessofhgivestherequiredcompatibilityfortripleinclusionsofopensets. Let:Fib(F;G)!Hdenotethemorphismofberedcategoriesthat,overeachopensetU,sendstheobjectsofFib(F;G)(U)totheobjectsofH(U),andeachpresheafHomFib(F;G)(U)(x;y)toitsassociatedsheaf.Sothereisadiagram: Let!:D!Hbethearrow'. 63

PAGE 64

Let:D!JbeagivenmorphismofprestackstogetherwithanisomorphismF// Now,bytheuniversalpropertyofcoequalizersinFIBERthereexistsanarrow:Fib(F;G)!J,uniqueuptouniqueisomorphism,suchthatthediagram commutes.ForeachopensetUXthereisadiagram: wheretheverticalarrowUisamapfromapresheaftoasheaf.Thenbytheuniversalpropertyofsheacation,thereexistsauniquearrowU:HomH(U)(x;y)!HomJ(U)(x;y)suchthatthediagram: 64

PAGE 65

commutesand()// Toshowthatisfullyfaithful,weneedtoestablishabijectionbetweenthesetsHomHom(H;J)(a;b)andHomB((a);(b)),whereBistheequalizercategoryEq(Hom(D;J)F// Sincethemap isfullyfaithfulbytheprevioustheorem,itfollowsthatisfullyfaithfulaswell. Wenowcometothecaseofstacks.NowletCandDbestacks,andletCF// 65

PAGE 66

LetJbeanystackand:D!JbeagivenmorphismofstackstogetherwithanisomorphismF// BytheuniversalpropertyofcoequalizersinPRESTACKSthereexistsanarrow:Pre(F;G)!Jsuchthatthediagram commutes.SincetheassociatedstackcartesianfunctorisfullyfaithfulanduniversalforcartesianfunctorsfromPre(F;G)intoSTACKS([ 18 ]Lemma3.2),thereisamorphismofstacks:St(F;G)!J,whichisuniqueuptouniqueisomorphismsuchthatthediagram 66

PAGE 67

Wenowshowisfaithful.Let'1,'2:St(F;G)!Jdenotemorphismsofstacks.Then('1),('2):D!J.Suppose,areinHomHom(St(F;G);J)('1;'2)with()=(),where()and()areinHomEq(();())(hereEqdenotesthecategoryEq(Hom(D;J)F// isthesamearrowas Butthedenitionofis()=!.Sothearrow ('1!)(a)()U// isthesameasthearrow ('1!)(a)()U// i.e.thearrow'1(!(a))U// isthesameasthearrow ThusUandUagreeoverobjectsinSt(F;G)(U)whichareintheimageof!.Now,recallthat!='where':D!Pre(F;G)istheprestackcoequalizer,and:Pre(F;G)!St(F;G)isthecanonicalstackicationfunctor.Themaps'andbothhavethepropertythateveryobjectintheirtargetislocallycontainedintheiressentialimage,andsothispropertyiscarriedbytheircomposition!.Inotherwords,everyobjectinSt(F;G)(U)islocallycontainedintheessentialimageof!.SoletbbeanobjectinSt(F;G)(U)andsuppose(Ui)isanopencoverofUwhere!(ai)ti// 67

PAGE 68

and Byhypothesis,thetwoverticalandthetophorizontalarrowsarethesameforbothdiagramswhencethelowerhorizontalarrowsarethesameinbothdiagramsi.e.andagreeovereachUi.SinceAr(St(F;G)(U))isasheaf,=overU.SincethisistrueforeachopensetU,getthat=.Thusisfaithful. Finally,weshowthatisfull.Again,let'1,'2:St(F;G)!Jdenotemorphismsofstacks.Then('1),('2):D!J.SupposeisanarrowinHomEq(();()).Weneedtoshowthatthereexistsanarrow so ('1!)(a)U// overU,so where!(a)isanobjectinSt(F;G)(U).Let 68

PAGE 69

Further,onoverlapsUij, Considerthediagram Let suchthat( Proof.Letc1,c2beanytwoobjectsofCandg,hbetwoobjectsofGsuchthatgc1=c2andhc1=c2.ThenifkisanyobjectofGsuchthathk// 69

PAGE 70

Proof.SinceCisastack,foranyopensetUX,C(U)isagroupoid.Theresultnowfollowsfromthepreviousproposition.

PAGE 71

15 ]thisisastackonX. LetInn(G)denotetheberedcategoryofinnerequivalencesofthestackG.ThusforeachopenUX,Inn(G)(U)isthegroupcategoryInn(GjU)i.e.thegroupcategoryofinnerequivalencesoftherestrictedgr-stackGjU.TherestrictionfunctorsandcorrespondingnaturaltransformationsarethesameastheonesdenedinthestackEq(G;G).Recallthatanequivalenceisinnerifandonlyifitissolocally.ThussincetheconditionsdeningInn(G)asasub-beredcategoryofEq(G;G)arelocal,Inn(G)isastackonX.LetOut(F;G)denoteaquotientstackEq(F;G)=Inn(G). Wenowdeneabered2-categoryLI2(X)asfollows.ForeveryopenUX,letLI2(X)(U)denotethe2-categorywhoseobjectsarethegr-stacksonU,andwhosearrowsbetweentwoobjectsAandBaretheglobalsectionsofthestackOut(A;B)(U)i.e.HomLI2(X)(U)(A;B)=(U;Out(A;B)).Thecompositionlawisthesameastheonein 71

PAGE 72

LetGr(X)denotethebered2-categoryofgr-stacksonX.Thenbyconstructionwehaveamorphismofbered2-categories NowifAandBareanytwoobjectsofthe2-categoryLI2(X)(U),thentheprestackHomLI2(X)(U)(A;B)ofarrowsfromAtoBisidentiedwiththestackOut(A;B).ThusLI2(X)isapre-2-stack.Thusuponapplyingthe2-stackicationfunctor,weobtaintheassociated2-stackLIEN2(X).SinceGr(X)isalreadya2-stack,composingthe2-stackicationfunctorwiththemorphism(3)yieldsamorphismof2-stacks The2-stackLIEN2(X)iscalledthe2-stackof2-liensonX. 2. Wecallaglobalsectionofthe2-stackLIEN2(X)a2-lienoverX. 15 ]. 4 ],overanyopensetUX,any1-arrowfinG(U)hasaquasi-inverse,deneduptoaunique2-arrow.Further,Breenshowsthatlocalinversesfor1-arrowsalwaysdescendtoglobalones,soforanyobjectx2G(U),theprestackingroupoidsEq(x)ofself-equivalencesofxisagr-stackonX,afterspecicinversesforthe 72

PAGE 73

wherex2Ob(G(U))andEqU(x)denotesthegr-stackofself-equivalencesofx.Composingthismorphismwiththemorphism givesamorphismof2-stacks whichtoeveryobjectx2G(U)associatesthe2-lien (calledthe2-lienrepresentedbythestackofequivalencesofxoverU)andtoeveryequivalence:x!yofG(U)associatesthemorphism inLIEN2(X)(U)where denedby isamorphismofgr-stacksoverX,where 73

PAGE 74

=Inn()(a1) =(a1)1 ~!()a 4 ],page62,thereexistsaunique2-arrow:)0.Soforanyarrowg:y!y,thereisauniquecommutativesquare Thenleftcompositionbyanequivalencea2Eq(x)ofxyieldsanotheruniquecommutativesquare 74

PAGE 75

foreveryg:y!y.Thus:=IIadenesaunique2-isomorphismbetweenaanda0,thatis,wegetacanonicalequivalenceofcategoriesInn()withchoiceofinverseandInn()withchoiceofinverseofinverse0. Bythedenitionofthe2-stackof2-liens,ifm,n:x!yaretwoequivalencesofG(U),wehave sincethe2-functorlien2givesanaturalisomorphismbetweeneveryinnerequivalenceandtheidentityarrow. Ifm:F!Gisamorphismof2-stacks,thecomposition factorsthroughamorphismofmorphismsof2-stacks: Foreveryx2Ob(F(U)),the2-functorminducesamorphismofgr-stacks andbydenitionofthemapliau2(m),themorphismliau(m(x))isthemorphismof2-liensrepresentedbyx: 75

PAGE 76

1. foradoubleinclusionWg// foratripleinclusionZh// 76

PAGE 77

1. Let(L;a)bea2-lienoperatingonF.Thefollowingareequivalent: (a) (b) forevery2-lienL0onXandeveryactionbofL0onFthereexistsauniquemorphismof2-liensu:L0!L(upto2-equivalence)suchthatbisequivalenttotheactioninducedbyaandu. 2. Thereexistsa2-lien(L;a)operatingonFthatsatisestheconditionsof(1). 77

PAGE 78

Moreexplicitly,the2-lienofa2-gerbeFisa2-lienLandafamilyofequivalencesof2-liensoverU, wherex2F(U),UX.Thisfamilyisrequiredtobecompatiblewiththerestrictionofopensetsandalsosatisfytheconditionthatifi:x!yisaU-equivalenceofF,thenthemorphismofgr-stacksInn(i):EqU(x)!EqU(y)isisomorphictotheidentitymorphismofL(U). Proof.LetIbetheimagecategoryofliau2(F).Clearlythisisabered2-categoryofLIEN2(X)andthe2-functorinducedbyliau2(F), is2-cartesian.Moreover,theprojectionI!Xzarisbothfully2-faithfulandessentiallysurjective(thisfollowstriviallyfromthefactthatF!Xisfully2-faithfulandbecauseliau2(m)// 78

PAGE 79

4 ],pg.64-65)andsoEq(a)isagr-stackonUoncespecicinversesforthe1-arrowshavebeenchosen.Inthefollowingwewillagainassumethatthishasbeendone,anddenoteby1thechoseninverseofa1-arrowinG(seepages72-73). Thusgivena2-gerbeGonXxinversesforallthe1-arrowsofG.ChooseanopencoverX=(U),andforeachchooseanobjectainG(U).Foreachand,chooseanopencoverU=(U)andforeacha1-arrowf:a!ainG(U).Thenwecandeneanequivalenceofgr-stacks:

PAGE 80

i.e. wherethereexistsaninvertible2-arrow [(f2)1f1][(f2)1f1]1)Id:(6{34) Thus1and2dierbyaninnerequivalence.InparticulartheydenethesamesectionofOut(Eq(a);Eq(a)).Soforxedand,thefamilyfgdenesan\outer"equivalenceofgr-stacksonU, whichdoesnotdependonthechoiceofthef.Thissystemofgr-stacksEq(a)andouterequivalenceswillbecalledacocyclerepresentationofthe2-lienofthegerbeG. 80

PAGE 81

4 ]and[ 5 ].Thereadermayconsultthesesourcesforthedetails. LetPbeaG-2-gerbeonagivenspaceX.WewillassumethatPisconnected:thatis,foreachpairofobjectsx,yinsomeber2-categoryPU,thesetofarrowsinPUfromxtoyisnon-empty.Givenanopencover(Ui)ofX,wechooseobjectsxi2PUiandpathsij:xj!xiinthe2-groupoidPUij,togetherwithquasi-inversesij:xi!xj.Tothisweassociatethefollowingdata: 1. AnobjectijinEq(GjjUij;GijUij): determinedbyijanditsinverse. 2. Anarrowgijk(determinedbythepathsijandtheirinverses)viewedasanobjectintheberofthestackGiovertheopensetUijk: 3. Anarrow~mijk(anaturaltransformationinducedbythe2-arrowmijkinthediagramabove)inthecategoryEq(Gj;Gi)Uijk: 81

PAGE 82

whereigistheinnerconjugationfunctordenedbyanobjectginthegr-stackG,andfor2Gk, ~mijk():igijkij()!ijjk()(7{5) inGi.Byrightmultiplicationbytheinverseobjectofitssource,thisarrowcorrespondstoanarrow: inGisourcedattheidentity. 4. Anarrowijkl:1xi)gijkgikl(gijl)1ij(gjkl)1inthecategoryGijUijkldeterminedbythedatalistedabove.Itcanalsobeviewedasa2-arrowthatsitsinthediagrambelow. Further,thefollowingidentityisvalidforijkl: ~mijk(gijk~mikl)iijkl=(ij(~mjkl))(ij(gikl)~mijl)(7{8) wheregmdenotestheleftcompositionofmbyg.Inaddition,ijkljUijklmsatisesthefollowingcocycleconditioninGijUijklm: wheregdenotesconjugationofanarrowbyg. 82

PAGE 83

4 ],[ 5 ]),suchaquadrupleofelements(ijkl;~mijk;gijk;ij)iscalledaG-valuednonabelian3-cocycleonthespaceX. Wenowdiscussthecoboundaryrelations.Weagainreviewtherelevantnotionsfrom[ 4 ]and[ 5 ],beforespecializingtothestrictPicardcase. Twononabelian3-cocycles(ijkl;~mijk;gijk;ij)and(0ijkl;~ijk;ijk;0ij)arecohomologousifthereexist: 1. Afamilyofobjects(i)inEq(GUi)andafamilyofobjectshijinGUij. 2. Afamilyofarrows~ijinthecategoryGUij: ~ij:(i)0ij)ihijij(i)(7{10) andafamilyofmorphismsaijk: inGUijk,forwhichthefollowingtwoidentitiesaresatised:

PAGE 84

X(7{12) Proof.IfPisaG-2-gerbe,PislocallyequivalenttoTors(G).Thusits2-lienislocallyequivalenttolien2(Tors(G))=lien2(G).Conversely,letPbea2-gerbewhose2-lienisequivalent,whenrestrictedtosomeopencoverUofXtolien2(G)forsomegivengr-stackG.WecanthenchooseasecondopencoverU0=(Ui)ofXforwhichthereexistsafamilyofobjectsxi2GUi.The2-gerbePisthenlocallyoftheformTors(Gi)forthetautologicallabelingofPdenedbyGi=Eq(xi).Itfollowsthatlien2(P)jUiisequivalenttolien2(Gi),sothat,ontheelementsVofacommonrenementU00ofUandU0,wehaveequivalencesof2-lienslien2(G)jV!lien2(Gi)jV.Suchanequivalenceisdenedbysections[]ontheopensetsVofthestackOut(G;Gi).Itfollowsfromthedenitionofthisstackthatthesesectionslifttosections:GjW!GijWofEq(G;Gi)ontheopensetsWofanappropriaterenementU000ofU00.SinceGi=Eq(xi),thecollectionofmaps()givetheG-2-gerbestructureonP.

PAGE 85

whichisdescribedbyanouterequivalenceEq(x)!GjU.SinceGisnowPicard,inthediagram ofthedenitionofanabelianG-2-gerbe,the2-arrowfdenesanequivalencebetweensuchanouterequivalenceandanordinaryequivalenceofgr-stacksEq(x)!GjU.Thequasi-inversex:GjU!Eq(x)denesanabelianG-2-gerbestructureonG.Therequiredcommutativityoftheabovediagramisequivalenttothecommutativityofthecorrespondingdiagramof2-liens,sinceitinvolvesPicardcategories.Thisinturnfollowsbyapplyingthe2-lien2-functortothediagram where givesanequivalencebetweenGanda2-gerbeoftorsors. 85

PAGE 86

13 ]hasprovedthatanystrictPicardstackGis2-equivalenttoa2-termcomplexofabeliansheavesK=[K0d// WebeginbydescribingDeligne'sconstructionforspeciccasesinonedirection;thatis,weshowhowonecanassociateastrictPicardcategory(respectivelystack)toagiven2-termcomplexofabeliangroups(respectivelysheaves).Wefollowthetreatmentin[ 11 ]and[ 13 ]. LetK:denotea2-termcomplexK0d// 13 ],thestackicationch(K)oftheprestackHKiscalledthestackassociatedtothecomplexK:,wherech(K)isPicard.Moreprecisely,ifC(X)denotesthe2-categoryof2-termcomplexesofabeliansheavesonX,wheremorphismsinC(X)aremorphismsofcomplexes,and2-isomorphismarehomotopiesbetweenmorphisms,thenwehavethefollowingresult. 86

PAGE 87

13 ],1.4.13.1)ForeveryPicardstackPthereexistsacomplexK:2C(X)suchthatP'ch(K). 13 ],1.4.16.1)LetK:2C(X)andassumethatK0isinjective.ThentheprestackHKisalreadyastack. 13 ],1.4.17)Theconstructionchdescribedaboveinducesa2-equivalenceofthe2-categoriesPIC(X)andC0(X). 11 ]and(continueto)refertoDeligne's2-functorPIC(X)!C(X)bych.DelignealsogivesacharacterizationofthePicardstackHom(ch(K);ch(L)),whereK:,L:2C(X).LetShAb(X)denotethecategoryofabeliansheavesonX. 13 ],1.4.18.1)AssumethatL0isinjective.Thenwehaveanequivalence The1abovereferstoacertaintruncationoperation;see[ 13 ]fordetails.FortherestofthissectionwewillusethispointofviewwhilestudyingG-2-gerbes,whereGisstrictPicard.LetPbeaconnectedG-2-gerbewhereGisstrictPicard.LetK:bethe2-termcomplexassociatedtoG.WeformthedoubleCechcomplex: 87

PAGE 88

7 ].Adegreencocycleconsistsofanitefamilycp2Cp(Knp)suchthat BythedenitionofaPicardstack,the3-cocycletermsij,~mijk,andf~mijk;gklmgaretrivial.Theremainingdata(ijkl;gijk)satisfytherelation Also,thediagram becomes ByDeligne'sconstruction,whenweviewthisdataastakingvaluesinK:insteadofG,wegetthatijklaresectionsofK0andthegijkaresectionsofK1.Further,bytheconstructionofK:,thediagramabovesaysthat: 88

PAGE 89

saysthat so thatis,()=0.Thusthepair(ijkl;gijk)giveusahypercocycleinK:. Wenowdiscussthecoboundaryrelations.WhenGisPicard,theserelationssimplifytothestatementthatthecocycles(ijkl;gijk)and(0ijkl;ijk)arecohomologousifthereexist: 1. AfamilyofobjectshijinGUij. 2. Afamilyofmorphismsaijkthatsatisfy: inGUijk,forwhichthefollowingidentityissatised: Asbefore,switchingtoadditivenotationafterviewingthehijandaijkassectionsinK:,therstidentityabovesaysthat So Sod(aijk)(hij)=ijkgijk:Similarly,thesecondidentitysaysthat 89

PAGE 90

Thusthecoboundaryconditionsforthetwococyclepairs(ijkl;gijk)and(0ijkl;ijk)inGtranslateexactlyintohyper-coboundaryconditionsinK:. Thuswegetawell-denedmap: H3(X;G)!H3(X;K:)(7{34) thatsendsthecohomologyclassofeachpair(ijkl;gijk)toitscorrespondingclass(ijkl;gijk). Toseethatthismapgivesanequivalence,observethatanyhypercocycle(ijkl;gijk)inH3(X;K:)satises and SincethiscocycletakesvaluesinK:,byDeligne([ 13 ]),wecanviewitasacocyclepair(ijkl;gijk)thattakesvaluesinitsassociatedPicardstackGandsatisestherelations and ButbyBreen[ 4 ],Theorem5.6,suchacocyclewithvaluesinGhasanassociatedG-2-gerbeP,wherePisconstructedbypreciselyreversingtheconstructionwhichassociatesanonabelian3-cocycle(ijkl;~mijk;gijk;ij)toaG-2-gerbeP,ofwhichthe 90

PAGE 91

91

PAGE 92

Wesummarizetheresultsofthisthesis.Thegoalofthisprojectwastodenethe2-lienofa2-gerbe.Tothisend,weneededtodenethenotionofaquotientbyanactionofagroupstackonastack.Inchapter3wesolvedthisproblemlocally:thatis,weshowedthatcoequalizerswererepresentableinthe2-categoryCATof(small)categoriesbyexplicitlyconstructingacoequalizerfor(small)pairofarrows.Inchapter4wedenedwhatitmeanstohaveaninnerequivalenceofagroupcategory.Wethengaveadenitionfortheactionofagroupcategoryonacategory,anddenedthequotientbysuchanactionusingthecoequalizerconstructedinchapter3.Inchapter5wecarriedoutthestackicationoftheseconcepts:thatis,weshowedtherepresentabilityofcoequalizersinthe2-categorySTACKS,andgaveadenitionfortheactionofagroupstackonastack,andthequotientbysuchanaction.Inchapter6wenallydenedthe2-lienofaspaceX.Weshowedhowevery2-gerbegivesussucha2-lienanddidtheworktoestablishthatthis2-lienisinfactgivenuptocanonical2-equivalence.Wethengaveamoreconcretedescriptionofa2-lienusingcocycles.Finallyinchapter7weprovedsomeresultsaboutG-2-gerbeswhose2-lienarisesfromastrictPicardstack.WerecalledDeligne'scorrespondencebetweenstrictPicardstacksGand2-termcomplexesofabeliansheavesK:.WethenprovedthatthesetofequivalenceclassesH3(X;G)ofconnectedgerbeswith2-lienGisisomorphictothehypercohomologygroupH3(X;K). Therearesomenaturaldirectionsthatonecouldtakeasacontinuanceofthisproject.Theabovecohomologicalresultofchapter7hasonlybeenprovenforconnected2-gerbes,anditwouldbenicetoproveitafterremovingtheconnectednesshypothesis.Theproofwouldinvolveworkingwithhypercovers(opencoversofeachopenset)buttheresultwouldstillseemtobetrueforGPicard.Thenasanextstep,onecouldrelaxthestrictPicardassumptionandonlyassumethatthe2-lienGisbraided,andtrytondanicecohomologicaldescriptionfor2-gerbeswith2-lienG. 92

PAGE 93

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] 93

PAGE 94

[17] [18] [19] [20] [21] [22]

PAGE 95

IwasborninMumbai,India.IgraduatedfromTrumanStateUniversityinMay2001andstartedgraduateschoolatUFinthefall.IstartedworkingwithDr.Crewin2004. 95