The Classifying Space of a Topological 2-Group

The Classifying Space of a Topological 2-Group

32 Pages · 2012 · 310 KB · English

The Classifying Space of a Topological 2-Group John C. Baez Danny Stevensony July 27, 2009 Abstract Categorifying the concept of topological group, one obtains the

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TheClassifying SpaceofaTop ological 2Group John CBaez  DannyStev enson y July 27,2009 Abstract Categorifying theconcept oftop ological group,oneobtains thenotion ofa `top ological 2group' Thisinturn allowsatheory of`principal 2bundles' generalizing theusual theory ofprincipal bundlesItiswellkno wnthat under mildconditions onatop ological groupGand aspace M,principal G bundles ov er M are classi ed by either the Cec hcohomology  H 1 (M ;G ) or the setofhomotop yclasses [M ;B G], where BG isthe classifying space of G Here we review work by Bartels, Jurco, Baas{B okstedt{Kro, and others generalizing thisresult totop ological 2groups andeven top ological 2categories Weexplain various viewpointson top ological 2groups and the  Cec hcohomology  H 1 (M ;G ) with coecien tsin atop ological 2group G , also known as`nonab eliancohomology' Thenwe giv ean elemen tary proofthat under mildconditions onM and Gthere isabijection  H 1 (M ;G )  = [ M ;B jG j] where BjG jis the classifying spaceofthe geo metric realization ofthe nerv eof G Applying thisresult tothe `string 2group' String(G ) of asimplyconnected compactsimpleLiegroup G, it follo wsthat principal String(G )2bundles have rational characteristic classes coming fromelemen tsof H (B G; Q)= hc i, where cis an ygenerator of H4 (B G; Q) 1 Intro duction Recen tw ork inhigher gaugetheory hasrevealed theimp ortance ofcategorifying the theory ofbundles andconsidering `2bundles', wherethe berisatop ological category insteadofatop ological space[5 ] These structures showup not only in mathematics, wheretheyform auseful generalization ofnonab eliangerbes [ 10 ],but also inphysics, where theycanbe used todescrib eparallel transport of strings [31 ,32 ] The concepts of` Cec hcohomology' and`classifying space'playawellkno wn and fundamen talrole inthe theory ofbundles For anytop ological groupG, principal Gbundles ov er aspace Mare classi ed by the rst  Cec hcohomology  Departmen tof Mathematics, Universityof California, Riverside, CA92521, USAEmail: [email protected] yF ac hb ereic hMathematik, UniversitatHam burg, Hamburg, 20146, German y, Email: [email protected] rgde 1 ofM with coecien tsin G Furthermore, undersomemildconditions, these  Cec hcohomology classesarein11 corresp ondence withhomotop yclasses of maps fromMtothe classifying spaceBG This letsusde ne characteristic classes forbundles, comingfromcohomology classesforBG All these concepts andresults canbe generalized frombundles to2bundles Bartels [6 ] has de ned principal G2bundles whereGis a`top ological 2group': roughly speaking, acategori ed version ofatop ological groupFurthermore, his w ork shows how principal G2bundles ov er M are classi ed by  H 1 (M ;G ), the rst  Cec hcohomology ofM with coecien tsin G This form ofcohomology , also known as`nonab eliancohomology', isfamiliar fromwork onnonab elian gerb es[9 , 19 ] In fact, under mildconditions onGand M,there isa11 corresp ondence b et ween  H 1 (M ;G ) and thesetofhomotop yclasses ofmaps fromMtoacertain space BjG j: the classifying spaceofthe geometric realization ofthe nerv eof G So, BjG jserv esas aclassifying spaceforthe topological 2group G! This pap erseeks topro vide anintro duction totop ological 2groups andnonab elian cohomology leadinguptoaselfcon tainedproofof this fact In his pioneering work onthis subject, Jurco [22 ]asserted thatacertain space homotop yequiv alentto ours isaclassifying spaceforthe rst  Cec hcohomology with coecien tsin G Ho wev er, there aresome gapsinhis argumen tfor this assertion (seeSection 52fordetails) Later, Baas,Bokstedt andKro [2 ] ga ve the de nitiv etreatmen tof classifying spaces for2bundles For any`go od' top ological 2category C, they construct a classifying spaceBC They thenshowthat foranyspace Mwith thehomotop y t yp eof aCW complex, concordance classesof`charted C2bundles' correspond to homotop yclasses ofmaps fromMtoBC In particular, atop ological 2group is just atop ological 2category withoneobject and with allmorphisms and2 morphisms invertible |and inthis special case, their result almost reduces to the fact mentioned abov e There aresome subtleties, howev er Most importan tly,while their`charted C 2bundles' reduceprecisely toour principal G2bundles, theyclassify these2 bundles uptoconcordance, whilewe classify themuptoasup er cially di erent equiv alence relation TwoG2bundles ov er aspace Xare `concordan t'ifthey are restrictions ofsome G2bundle ov er X[0;1] to the tw o ends Xf0g and Xf1g This makesiteasy tosee that homotopic mapsfromXtothe classifying spacede ne concordan tG2bundles Weinstead consider tw o G2 bundles tobe equiv alentif their de ning  Cec h1co cycles arecohomologous In this approac h,some work isrequired tosho wthat homotopic mapsfromXto the classifying spacede ne equivalentG2bundles Apriori ,it isnot obvious that tw o G2bundles areequiv alentin this  Cec hsense ifand only ifthey are concordan tHowev er, since theclassifying spaceofBaas, Bokstedt andKro is homotop yequiv alentto the one

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