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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></li> </ul> </div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#via_completing_tunles_of_vector_bundles'>Via completing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-tunles of vector bundles</a></li> <li><a href='#AsAlgebraBundlesWithBimoduleBundlesBetweenThem'>As algebra bundles with bimodule bundles between them</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>2-module bundle</em> / <em>2-vector bundle</em> is a <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> whose typical fiber is a <a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>/<a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a>.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, or more generally an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>. By the discussion at <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a> consider the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>Vect</mi> <mi>R</mi></msub><mo>≃</mo><msub><mi>Alg</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex"> 2 Vect_R \simeq Alg_R </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalent</a> to that whose objects are <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> (or generally <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">algebras</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, (being placeholders for the 2-vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> which is the category of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>) whose <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a> between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between those.</p> <p>Under <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> and by the discussion at <em><a href="module#RelationToVectorBundlesInIntroduction">Modules – as generalized vector bundles</a></em> we may think of this 2-category as being that of (generalized) 2-vector bundles over a <a class="existingWikiWord" href="/nlab/show/space">space</a> called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BDR+2-vector+bundle">BDR 2-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="via_completing_tunles_of_vector_bundles">Via completing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-tunles of vector bundles</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/BDR+2-vector+bundle">BDR 2-vector bundle</a></em>.</p> <h3 id="AsAlgebraBundlesWithBimoduleBundlesBetweenThem">As algebra bundles with bimodule bundles between them</h3> <p>The notion of 2-vector bundles based on regarding 2-vector spaces as algebras with bimodules between them (<a href="2-vector+space#ReferencesAsAlgebrasWithBimodules">here</a>) is first discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, §4.4 of: <em>Connections on non-abelian Gerbes and their Holonomy</em>, Theory Appl. Categ., <strong>28</strong> 17 (2013) 476-540 [<a href="https://arxiv.org/abs/0808.1923">arXiv:0808.1923</a>, <a href="http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html">tac:28-17</a>]</li> </ul> <p>and much further developed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Kristel">Peter Kristel</a>, <a class="existingWikiWord" href="/nlab/show/Matthias+Ludewig">Matthias Ludewig</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>The insidious bicategory of algebra bundles</em> [<a href="https://arxiv.org/abs/2204.03900">arXiv:2204.03900</a>]</li> </ul> <p>The example of the <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Kristel">Peter Kristel</a>, <a class="existingWikiWord" href="/nlab/show/Matthias+Ludewig">Matthias Ludewig</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>A representation of the string 2-group</em>, [<a href="https://arxiv.org/abs/2206.09797">arXiv:2206.09797</a>]</li> </ul> <p>Reviewed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>The stringor bundle</em>, talk at <em><a href="https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html">QFT and Cobordism</a></em>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Mar 2023) [<a href="Center+for+Quantum+and+Topological+Systems#WaldorfMar2023">web</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 16, 2023 at 09:54:38. 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