a homotopy double groupoid of a hausdorff space ii a van kampen theoremnew

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1、AhomotopydoublegroupoidofaHausdorffspaceII:avanKampentheoremR.Brown∗,K.H.Kamps†andT.Porter‡February1,2008UWBMathPreprint04.01AbstractThispaperisthesecondinaseriesexploringthepropertiesofafunctorwhichassignsahomotopydoublegroupoidwithconnectionstoaHausdorffspace.Weshowthatthisfunctorsatisfiesaver

2、sionofthevanKampentheorem,andsoisasuitabletoolfornonabelian,2-dimensional,local-to-globalproblems.ThemethodsareanalogoustothosedevelopedbyBrownandHigginsforsimilartheoremsforotherhigherhomotopygroupoids.Anintegralpartoftheproofisadetaileddiscussionofcommutativecubesinadou-blecategorywithconne

3、ctions,andaproofofthekeyresultthatanycompositionofcommutativecubesiscommutative.TheseresultshaverecentlybeengeneralisedtoalldimensionsbyPhilipHiggins.1IntroductionAclassicalandkeyexampleofanonabelianlocal-to-globaltheoremindimension1isthevanKampentheoremforthefundamentalgroupofaspacewithbasep

4、oint:ifaspaceistheunionoftwoconnectedopensetswithconnectedintersection,thetheoremdeterminesthefundamentalgroupofthewholespace,andsothe‘global’information,intermsofthelocalinformationonthefundamentalgroupsofthepartsandthemorphismsinducedbyinclusions.SuchatheoremarXiv:math/0410398v1[math.AT]18O

5、ct2004thusrelatesaparticularpushoutofspaceswithbasepointtoapushout,orfreeproductwithamalgamation,oftheirfundamentalgroups.vanKampen’s1935paper,[18],gave,infact,aformulaforthecaseofnonconnectedinter-section,asrequiredforthealgebraicgeometryapplicationshehadinmind.Hisformulafollows∗MathematicsD

6、epartment,DeanSt.,Bangor,GwyneddLL571UT,UK.email:r.brown@bangor.ac.uk.BrownhasbeensupportedforapartofthisresearchbyaLeverhulmeEmeritusFellowship(2002-2004).†FachbereichMathematik,FernUniversit¨atinHagen,D-58084Hagen,Germany.email:heiner.kamps@fernuni-hagen.de‡MathematicsDepartment,DeanSt.,Ban

7、gor,GwyneddLL571UT,UK.email:t.porter@bangor.ac.uk1KEYWORDS:doublegroupoid,doublecategory,thinstructure,connections,commutativecube,vanKampentheoremMATHSUBJECTCLASSIFICATION:18D05,20L05,55Q05,55Q351fromtheversionofthetheoremforthefundamentalgr