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On Infinity Topoi

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arXiv:math/0306109v2 [math.CT] 8 Jul 2003ON∞-TOPOIJACOBLURIE

LetXbeatopologicalspaceandGanabeliangroup.TherearemanydifferentdefinitionsforthecohomologygroupHn(X,G);wewillsingleoutthreeofthemfordiscussionhere.Firstofall,onehasthesingularcohomologyHnsing(X,G),whichisdefinedasthecohomologyofacomplexofG-valuedsingularcochains.Alternatively,onemayregardHn(•,G)asarepresentablefunctoronthehomotopycategoryoftopologicalspaces,andtherebydefineHnrep(X,G)tobethesetofhomotopyclassesofmapsfromXinto

anEilenberg-MacLanespaceK(G,n).AthirdpossibilityistousethesheafcohomologyHnsheaf(X,GonX.IfXisasufficientlynicespace(forexample,aCWcomplex),thenallthreeofthesedefinitionsagree.Ingeneral,however,allthreegivedifferentanswers.ThesingularcohomologyofXisconstructedusingcontinuousmapsfromsimplices∆kintoX.IftherearenotmanymapsintoX(forexampleifeverypathinXisconstant),thenwecannotexpectHnsing(X,G)totellusverymuchaboutX.Similarly,thecohomologygroupHnrep(X,G)isdefinedusingmapsfromXintoasimplicialcomplex,which(ultimately)reliesontheexistenceofcontinuousreal-valuedfunctionsonX.IfXdoesnotadmitmanyreal-valuedfunctions,weshouldnotexpectHnrep(X,G)tobeausefulinvariant.However,thesheafcohomologyofXseemstobeagoodinvariantforarbitraryspaces:ithasexcellentformalpropertiesingeneralandsometimesyieldsgoodresultsinsituationswheretheotherapproachesdonotapply(suchasthe´etaletopologyofalgebraicvarieties).WeshalltakethepointofviewthatthesheafcohomologyofaspaceXgivestherightanswerinallcases.Weshouldthenaskforconditionsunderwhichtheotherdefinitionsofcohomologygivethesameanswer.WeshouldexpectthistobetrueforsingularcohomologywhentherearemanycontinuousfunctionsintoX,andforEilenberg-MacLanecohomologywhentherearemanycontinuousfunctionsoutofX.Itseemsthatthelatterclassofspacesismuchlargerthantheformer:itincludes,forexample,allparacompactspaces,andconsequentlyforparacompactspacesonecanshowthatthesheafcohomologyHnsheaf(X,G)coincideswiththeEilenberg-MacLanecohomologyHnrep(X,G).Oneofthemainresultsofthispaperisageneralizationoftheprecedingstatementtonon-abeliancohomology,andtothecasewherethecoefficientsystemGisnotnecessarilyconstant.Classically,thenon-abeliancohomologyH1(X,G)ofXwithcoefficientsinapossiblynon-abeliangroupGisunderstoodasclassifyingG-torsorsonX.WhenXisparacompact,suchtorsorsareagainclassifiedbyhomotopyclassesofmapsfromXintoanEilenberg-MacLanespaceK(G,1).NotethatthegroupGandthespaceK(G,1)areessentiallythesamepieceofdata:GdeterminesK(G,1)uptohomotopyequivalence,andconverselyGmayberecoveredasthefundamentalgroupofK(G,1).Tomakethiscanonical,weshouldsaythatspecifyingGisequivalenttospecifyingthespaceK(G,1)togetherwithabasepoint;thespaceK(G,1)aloneonlydeterminesGuptoinnerautomorphisms.However,innerautomorphismsofGinducetheidentityonH1(X,G),sothatH1(X,G)isreallyafunctorwhichdependsonlyonK(G,1).Thissuggeststhepropercoefficientsfornon-abeliancohomologyarenotgroups,but“homotopytypes”(whichweregardaspurelycombinatorialentities,representedperhapsbysimplicialsets).Wemaydefinethenon-abeliancohomologyHrep(X,K)ofXwithcoefficientsinanysimplicialcomplexKtobethecollectionofhomotopyclassesofmapsfromXintoK.ThisleadstoagoodnotionwheneverXisparacompact.Moreover,wegainagreatdealbyallowingthecasewhereKisnotanEilenberg-MacLanespace.Forexample,ifK=BU×ZistheclassifyingspaceforcomplexK-theoryandXisacompactHausdorffspace,thenHrep(X,K)istheusualcomplexK-theoryofX,definedastheGrothendieckgroupofthemonoidofisomorphismclassesofcomplexvectorbundlesonX.

12JACOBLURIEWhenXisnotparacompact,weareforcedtoseekabetterwayofdefiningH(X,K).Giventheappar-entpowerandflexibilityofsheaf-theoreticmethods,itisnaturaltolookforsomegeneralizationofsheafcohomology,usingascoefficients“sheavesofhomotopytypesonX.”Inotherwords,wewantatheoryof∞-stacks(ingroupoids)onX,whichwewillhenceforthrefertosimplyasstacks.OneapproachtothistheoryisprovidedbytheJoyal-JardinehomotopytheoryofsimplicialpresheavesonX.Accordingtothisapproach,ifKisasimplicialset,thenthecohomologyofXwithcoefficientsinKshouldbedefinedasHJJ(X,K)=π0(F(X)),whereFisafibrantreplacementfortheconstantsimplicialpresheafwithvalueKonX.WhenKisanEilenberg-MacLanespaceK(G,n),thenthisagreeswiththesheaf-cohomologygroup(orset)Hnsheaf(X,G).ItfollowsthatifXisparacompact,thenHJJ(X,K)=Hrep(X,K)wheneverKisanEilenberg-MacLanespace.However,itturnsoutthatHJJ(X,K)=Hrep(X,K)ingeneral,evenwhenXisparacompact.Infact,onecangiveanexampleofacompactHausdorffspaceforwhichHJJ(X,BU×Z)isnotequaltothecomplexK-theoryofX.WeshallproceedontheassumptionthatK(X)isthe“correct”answerinthiscase,andgiveanalternativetotheJoyal-Jardinetheorywhichcomputesthisanswer.OuralternativeisdistinguishedfromtheJoyal-Jardinetheorybythefactthatwerequireourstackstosatisfyadescentconditiononlyforcoverings,ratherthanforarbitraryhypercoverings.Asidefromthispointweproceedinthesameway,settingH(X,K)=π0(F′(X)),whereF′isthestackwhichisobtainedbyforcingthe“constantprestackwithvalueK”tosatisfythisweakerformofdescent.Ingeneral,F′willnotsatisfydescentforhypercoverings,andconsequentlyitwillnotbeequivalenttothesimplicialpresheafFusedinthedefinitionofHJJ.Theresultingtheoryhasthefollowingproperties:•IfXisparacompact,H(X,K)isthesetofhomotopyclassesfromXintoK.•IfXisaparacompactspaceoffinitecoveringdimension,thenourtheoryofstacksisequivalenttotheJoyal-Jardinetheory.(Thisisalsotrueforcertaininductivelimitsoffinitedimensionalspaces,andinparticularforCWcomplexes.)•ThecohomologiesHJJ(X,K)andH(X,K)alwaysagreewhenKis“truncated”,forexamplewhenKisanEilenberg-MacLanespace.Inparticular,H(X,K(G,n))isequaltotheusualsheafcohomologyHnsheaf(X,G).Inaddition,ourtheoryof∞-stacksenjoysgoodformalpropertieswhicharenotalwayssharedbytheJoyal-Jardinetheory;weshallsummarizethesituationin§2.10.However,thegoodpropertiesofourtheorydonotcomewithouttheirprice.Itturnsoutthattheessentialdifferencebetweenstacks(whicharerequiredtosatisfydescentonlyforordinarycoverings)andhyperstacks(whicharerequiredtosatisfydescentforarbitraryhypercoverings)isthattheformercanfailtosatisfytheWhiteheadtheorem:onecanhave,forexample,apointedstack(E,η)forwhichπi(E,η)isatrivialsheafforalli≥0,suchthatEisnot“contractible”(forthedefinitionofthesehomotopysheaves,see§2.8).InordertomakeathoroughcomparisonofourtheoryofstacksonXandtheJoyal-JardinetheoryofhyperstacksonX,itseemsdesirabletofitbothofthemintosomelargercontext.Theproperframeworkisprovidedbythenotionofan∞-topos,whichisintendedtobean∞-categorythat“lookslike”the∞-categoryof∞-stacksonatopologicalspace,justasanordinarytoposissupposedtobeacategorythat“lookslike”thecategoryofsheavesonatopologicalspace.ForanytopologicalspaceX(or,moregenerally,anytopos),the∞-stacksonXcomprisean∞-topos,asdothe∞-hyperstacksonX.However,itistheformer∞-toposwhichenjoysthemoreuniversalpositionamong∞-topoirelatedtoX(seeProposition2.7.4).Letusnowoutlinethecontentsofthispaper.Wewillbeginin§1withaninformalreviewofthetheoryof∞-categories.Therearemanyapproachestothefoundationofthissubject,eachhavingitsownparticularmeritsanddemerits.Ratherthansingleoutoneofthosefoundationshere,weshallattempttoexplaintheideasinvolvedandhowtoworkwiththem.Thehopeisthatthiswillrenderthispaperreadabletoawideraudience,whileexpertswillbeabletofillinthedetailsmissingfromourexpositioninwhateverframeworktheyhappentoprefer.Section2isdevotedtothenotionofan∞-topos.Wewillbeginwithanintrinsiccharacterization(analogoustoGiraud’saxiomswhichcharacterizeordinarytopoi:see[5]),andthenarguethatany∞-categorysatisfyingouraxiomsactuallyarisesasan∞-categoryof“stacks”onsomething.Wewillthenshow

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westlife 部分歌曲歌词

westlife 部分歌曲歌词

as long as u love me.只要你爱我就好although loneliness has always been a friend of mine.孤独一直是我的朋友i'm leaving my life in ur hands.自从你离开我的生活people say i'm crazy that i am blind.朋友说我疯了太盲目risking it all in a glance.激情总是短暂的how you got me blind is still a mystery.你为何能使我如此盲目仍是个谜i can't get u out of my head.我就是无法忘了你don't care what is written in ur history.我不在乎你过去的种种as long as u're here with me.只要你陪在我身边i don't care who u are.我不在乎你是怎样个人where u're from.你从那里来what u did.你做过什么as long as u love me.只要你爱我就好who u are.你是怎样个人where u're from.你从那里来don't care what u did.我不在乎你做过什么as long as u love me.只要你爱我就好every little thing that u have said done.所有你说过的话和做过的事feels like it's deep within me.都深深的烙印在我心里doesn't really matter if u're on the run.我甚至于不在乎你是否就要逃开it seems like we're meant 2 be.我以为我们是一对的i've tried 2 hide it so that no one knows.我试着把感情隐藏起来不让任何人知道but i guess it shows.但我无法不流露真情when u look in 2 my eyes.当你凝视着我时what u did where u're comin from.你做过什么从那里来i don't care,as long as u love me,baby!我不在乎,只要你爱我就好,宝贝!Evergreen天长地久EyesLike a sunriseLike a rainfallDown my soulAnd I wonderI wonder why you look at me like that What you're thinkingWhat's behindDon't tell meBut it feels like love 你的双眸犹如朝阳好似雨珠灌注我的灵魂我想知道想知道你为何这样看着我你再想些什么对我隐瞒了什么算了还是别告诉我可是我觉得那是爱I'm gonna take this momentAnd make it last foreverI'm gonna give my heart away And pray we'll stay together Cause you're the one good reason You're the only girl that I need Cause you're more beautiful than I have ever seenI'm gonna take this nightAnd make it Evergreen我要保我这个片刻让他永远停留我要献上我的真心祈祷彼此紧紧相拥因为你是我最好的理由你是我唯一需要的女孩以为你比任何人都美丽我要把握今晚让记忆地久天长TouchLike a angelLike velvet to my skinAnd I wonderWonder why you wanna stay the night What you're dreamingWhat's behindDon't tell meBut it feels like love像天使般的抚摸浪丝绒般的触感我想知道今夜为何你想停留你在梦想什么对我隐瞒了什么算了还是别告诉我可是我觉得那是爱I lay my love on youjust a smile and the rain is gone你用仅仅一个微笑，让雨天放晴。

高级阅读No Signposts in the Sea海上无航标 原文+翻译+生词注解+修辞赏析

高级阅读No Signposts in the Sea海上无航标 原文+翻译+生词注解+修辞赏析

No Signposts in the Sea一、ur.sit.som.wa.of.wit..marrie.coupl.an.thei.daughter..ca .observ.he.withou.he.knowin.an.thi.give.m.pleasure.fo.i.i.a.i..movin.pictur.tha..ca.not.th.grac.o.he.ges tures,①whethe.sh.raise..glas.o.win.t.he.lip.or.turn.wit..remar.t.on.o.he.neighbor.o.take..cigarett.fro.he.casewit .thos.slende.fingers（loos.sentense松散句）ur.i.alway.i.gre.an. whit.b.day.lookin.coo.whe.othe.peopl.ar.flushe.an.shinyi.th.tropica.heat.i.th.evening.sh.wear.sof.ric.c olors（metonymy借代）.dar.red.oliv.green.midnigh.blue.alway.o.th.mos.suppl.flowin.texture..venture.t.sa.somethin.o.th. plimen.an.said..ha.bette.tak.t.writin.fashio.article.instea.o.polit ica.leaders.PS: ①soft: AD.Somethin.tha.i.sof.i.ver.gentl.an.ha.n.force.Fo.example..sof.soun.o.voic.i.quie.an.no.harsh..sof.ligh.o.colou.i.pleasan.t.loo.a.becaus.i.i.no.bright.(声音、光线或色彩.柔和的；②rich:Ric.smell.ar.stron.an.ver.pleasant.Ric.colour.an.sound.ar.dee.an.ver.pleasant.浓郁.(气味).浓厚.(色彩)因此soft和rich这里的修辞手法是：oxymoro.n.（修词中的）矛盾修饰法在餐厅里我同另外三个男人围坐在一张桌子旁, 而劳拉同一对夫妇及他们的女儿一块儿坐在离我不远的地方。

阳朔攀岩路书

阳朔攀岩路书

WEAR A HELMET as climber AND belayer! TAKE GRADES with CAUTIOUNESS!
注意： GETU是一片开阔的全新自然攀爬地点。虽然专业开线队员已经在这里做 了非常细致的工作，但是这里的岩壁仍然有可能出现落石，多尘和/或松 脱的抓点。因此在你攀爬这里的线路时一定要谨记这些内容。因为你的伙 伴可能随后攀爬这些线路，因此在下降时如果有可能的话，花一两分钟把 那些松脱或碍事的野草进行清理。如果有任何安全问题请及时向组织方或 其它工作人员汇报。只有在你明确知道采用何种技术和方法的情况下才可 以尝试去解决这些问题。最后：在攀爬时你并不是只对自己的安全负责， 同时你也要对你脚下正在攀爬和徘徊的岩友负责。 攀爬者和保护员必须戴头盔！务必注意安全！
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LAZy dragon’s
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MIAO SWORD
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岩壁最长路线的高度（米）。请选择 适合的绳索长度。
Number of quickdraws needed for the longest route at the cliff.
攀爬长距离大岩壁路线所 需的最多快挂数量。。
Estimated approach time to hike to the base of the cliff from Getu Village center.
星际台词

星际台词

一、T e r r a n人族的兵Marine（出场00）You want a piece of me, boy?想要我的一部分吗，小子？（闲聊00）We go move! 我们走吧！（闲聊01）Are you gonna give me orders? 你是不是想给我发命令？（闲聊02）Oh my god! He's whacked! 我的天，他困了！（闲聊03）I vote we frag this commander. 我建议我们干掉这个指挥官。

（闲聊04）How do I get out of this chicken （BEEP）outfit?! 我怎么才能从这个垃圾盔甲中出来？（闲聊05）You want a piece of me, boy? 想要我的一部分吗，小子？（闲聊06）If it weren't for these damned neural implants you'd be a smoldering crater by now! 如果不是因为这个混账的心灵灌输，我早把你变成弹坑了。

（Sti00）Ahh...That's the stuff! 啊，就是它（Sti01）Ahh...Yea! 啊，耶！（选择00）Commander. 老大！（选择01）Standing' by. 待命中（选择02）Checked up and good to go. 检查完毕，准备动身（选择03）Give me something to shoot. 给我一个活靶子（行动00）Go! Go! Go!（行动01）Let's move! 大家前进吧（行动02）Outstanding! 非常明智！（行动03）Rock 'n roll! 一起摇滚吧！Firebat（出场00）Need a light? 需要打火机吗？（闲聊00）Is something burning? 有烧着了的东西吗？（闲聊01）Haha, that's what I thought. 哈哈，正如我想。