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principal infinity-bundle in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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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> <ul> <li><a href='#GeneralPrincBund'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> <li><a href='#connections_on_principal_bundles'>Connections on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> </ul> <li><a href='#concrete_realizations'>Concrete realizations</a></li> <ul> <li><a href='#in_topological_spaces'>In topological spaces</a></li> <li><a href='#in_simplicial_sets__kan_complexes'>In simplicial sets / Kan complexes</a></li> <li><a href='#in_a_petit_topos'>In a petit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</a></li> <li><a href='#in_a_gros_topos'>In a gros <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</a></li> <ul> <li><a href='#smooth_principal_bundles'>Smooth principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#ordinary_principal_bundles'>Ordinary principal bundles</a></li> <li><a href='#circle_bundles'>Circle <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>-bundles</a></li> <li><a href='#bundle_gerbes'>Bundle gerbes</a></li> <li><a href='#normal_morphisms_of_groups'>Normal morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle</em> is a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">categorification</a> of <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> from <a class="existingWikiWord" href="/nlab/show/groups">groups</a> and <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>, or rather from parameterized groupoids (generalized spaces called <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>) to parameterized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids (generalized spaces called <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>).</p> <p>For motivation, background and further details see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a></li> <li><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a></li> <li><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></li> <li><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</li> </ul> <p>A model for principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles is given by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+principal+bundle">simplicial principal bundle</a>s.</li> </ul> <p>See also</p> <ul> <li> <p><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/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a>.</p> </li> </ul> <h2 id="definition">Definition</h2> <p>We define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles in the general context of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">group object in the (∞,1)-topos</a>.</p> <p>Recall that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> an object equipped with a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>pt</mi> <mi>A</mi></msub><mo>:</mo><mo>*</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">pt_A : {*} \to A </annotation></semantics></math>, its corresponding <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\Omega A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Ω</mi><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega A &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& A } \,. </annotation></semantics></math></div> <p>Conversely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">G \in \mathbf{H}</annotation></semantics></math> we say an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> if it has an essentially unique point and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>≃</mo><mi>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">G \simeq \Omega \mathbf{B}G</annotation></semantics></math>. We call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></strong>. More in detail, its structure as a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a> is exhibited by the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+nerve">Čech nerve</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>*</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo></mtd> <mtd><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover></mtd> <mtd><mo>*</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mo>→</mo></mover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover></mtd> <mtd><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mo>→</mo></mover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ &\cdots& {*} \times_{\mathbf{B}G} {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\stackrel{\to}{\to}}& {*} \times_{\mathbf{B}G} {*} &\stackrel{\to}{\to}& {*} } \right) \simeq \left( \array{ &\cdots& G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& G &\stackrel{\to}{\to}& {*} } \right) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">{*} \to \mathbf{B}G</annotation></semantics></math>.</p> <h3 id="GeneralPrincBund"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h3> <p>To every cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g : X \to \mathbf{B}G</annotation></semantics></math> is canonically associated its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G \,. } \,. </annotation></semantics></math></div> <p>We discuss now that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> canonically has the structure of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fine+moduli+space">fine moduli space</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p><strong>(principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">group object in the (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. A <strong>principal <a class="existingWikiWord" href="/nlab/show/action">action</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">(P \to X) \in \mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G</annotation></semantics></math> that sits over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">*//G</annotation></semantics></math> in that we have a morphism of <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial diagram</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\Delta^{op} \to \mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ P \times G \times G &\stackrel{(p_2, p_3)}{\to}& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>;</p> <p>and such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> exhibits the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \simeq \lim_\to (P//G : \Delta^{op} \to \mathbf{H}) </annotation></semantics></math></div> <p>called the <strong>base space</strong> over which the action takes place.</p> </div> <p>We may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G</annotation></semantics></math> as the <strong><a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a></strong> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>. For us it <em>defines</em> this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal action as defined above satisfies the <strong>principality condition</strong> in that we have an equivalence of groupoid objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>P</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{\simeq}{\to}& P } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This principality condition asserts that the groupoid object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">effective</a>. By <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">Giraud's axioms characterizing (∞,1)-toposes</a>, every groupoid object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is effective.</p> </div> <div class="un_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> any morphism, its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is canonically equipped with a principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action with base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>First we show that we have a morphism of simplicial diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots && \vdots \\ P \times_X P \times_X P &\stackrel{\simeq}{\to}& P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times_X P &\stackrel{\simeq}{\to}& P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,, </annotation></semantics></math></div> <p>with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.</p> <p>The defining <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> square for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex">P \times_X P</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P &\to& P \\ \downarrow && \downarrow \\ P &\to& X } </annotation></semantics></math></div> <p>To compute this, we may attach the defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pullback square of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> to obtain the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P &\to& P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. } </annotation></semantics></math></div> <p>and use the <a href="http://ncatlab.org/nlab/show/pullback#Pasting">pasting law for pullbacks</a>, to conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex">P \times_X P</annotation></semantics></math> is the pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& X &\to& \mathbf{B}G \,. } </annotation></semantics></math></div> <p>By definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> as the homotopy fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math>, the lower horizontal morphism is equivbalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">P \to {*} \to \mathbf{B}G</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mrow><annotation encoding="application/x-tex">P \times_X P</annotation></semantics></math> is also equivalent to the pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P &\to& &\to& {*} \\ \downarrow && && \downarrow \\ P &\to& {*} &\to& \mathbf{B}G \,. } </annotation></semantics></math></div> <p>This finally may be computed as the pasting of two pullbacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>P</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P &\simeq& P \times G &\to& G &\to& {*} \\ &&\downarrow && \downarrow && \downarrow \\ &&P &\to& {*} &\to& \mathbf{B}G \,. } </annotation></semantics></math></div> <p>of which the one on the right is the defining one for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and the remaining one on the left is just an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-product">(∞,1)-product</a>.</p> <p>Proceeding by induction from this case we find analogously that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></msup><mo>≃</mo><mi>P</mi><mo>×</mo><msup><mi>G</mi> <mrow><msub><mo>×</mo> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">P^{\times_X^{n+1}} \simeq P \times G^{\times_n}</annotation></semantics></math>: suppose this has been shown for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>, then the defining pullback square for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">P^{\times_X^{n+1}}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P^{\times_X^n} &\to& P \\ \downarrow && \downarrow \\ P^{\times_X^n}&\to& X } \,. </annotation></semantics></math></div> <p>We may again paste this to obtain</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P \times_X P^{\times_X^n} &\to& P &\to& * \\ \downarrow && \downarrow && \downarrow \\ P^{\times_X^n}&\to& X &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>and use from the previous induction step that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup><mo>→</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>P</mi> <mrow><msubsup><mo>×</mo> <mi>X</mi> <mi>n</mi></msubsup></mrow></msup><mo>→</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (P^{\times_X^n} \to X \to \mathbf{B}G) \simeq (P^{\times_X^n} \to * \to \mathbf{B}G) </annotation></semantics></math></div> <p>to conclude the induction step with the same arguments as before.</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>. It remains to show that indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mrow><msub><mi>lim</mi> <mo>→</mo></msub></mrow> <mi>n</mi></msub><mi>P</mi><mo>×</mo><msup><mi>G</mi> <mrow><msup><mo>×</mo> <mi>n</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X = {\lim_\to}_n P \times G^{\times^n}</annotation></semantics></math>. For this notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">* \to \mathbf{B}G</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category">effective epimorphism in an (infinity,1)-category</a>. Hence so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>. This proves the claim, by definition of effective epimorphism.</p> <p>using this we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>X</mi></mtd> <mtd><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><msup><mi>G</mi> <mrow><msup><mo>×</mo> <mi>n</mi></msup></mrow></msup><mo>)</mo></mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>G</mi> <mrow><msup><mo>×</mo> <mi>n</mi></msup></mrow></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo>×</mo><msup><mi>G</mi> <mrow><msup><mo>×</mo> <mi>n</mi></msup></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X \\ & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X \\ & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X ) \\ & \simeq {\lim_\to}_n ( P\times G^{\times^n} ) \\ & \simeq {\lim_\to} P//G \end{aligned} \,. </annotation></semantics></math></div></div> <p>We have established that every <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> canonically induced a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal action on the homotopy fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>. The following definition declares the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</em> to be those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal actions that do arise this way.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>We say a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> if the colimit over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G \to *//G</annotation></semantics></math> produces a pullback square: the bottom square in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>P</mi><mo>×</mo><msup><mi>G</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><msup><mi>G</mi> <mo>•</mo></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ P \times G \times G &\to& G \times G \\ \downarrow\downarrow\downarrow && \downarrow\downarrow\downarrow \\ P \times G &\stackrel{p_2}{\to}& G \\ \downarrow\downarrow && \downarrow\downarrow \\ P &\stackrel{}{\to}& {*} \\ \downarrow && \downarrow \\ X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& \mathbf{B}G = \lim_\to( G^\bullet) } \,. </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/infinity-group">infinity-group</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> any object, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mrow><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow></mrow><annotation encoding="application/x-tex"> G Bund(X) \subset Grpd(\mathbf{H})/{*//G} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/sub-%28infinity%2C1%29-category">sub-(infinity,1)-category</a> on the <a class="existingWikiWord" href="/nlab/show/over-%28infinity%2C1%29-category">over-(infinity,1)-category</a> of the groupoid objects over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">*//G</annotation></semantics></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles as above.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>We have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^I</annotation></semantics></math> is still an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a> and hence the Giraud-Lurie axioms still hold. This means that by the discussion at <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object in an (infinity,1)-category</a> (using the statement below <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, cor. 6.2.3.5</a>) we have an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup><msubsup><mo stretchy="false">)</mo> <mi>eff</mi> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex"> Grpd(\mathbf{H}^I) \simeq (\mathbf{H}^{I})^{I}_{eff} </annotation></semantics></math></div> <p>between groupoid objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^I</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/effective+epimorphism+in+an+%28infinity%2C1%29-category">effective epimorphisms</a> in the arrow category.</p> <p>Notice that groupoid objects and effective epis in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^I</annotation></semantics></math> are given objectwise over the two objects of the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I = \Delta[1]</annotation></semantics></math>.</p> <p>Restricting this equivalence along the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{B}G) \hookrightarrow (\mathbf{H}^I)^I </annotation></semantics></math></div> <p>given by sending a cocycle to its homotopy fiber diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>↦</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> (X \to \mathbf{B}G) \mapsto \left( \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \right) </annotation></semantics></math></div> <p>therefore yields precisely the equivalence in question</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>Grpd</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↪</mo><mi>hofib</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>I</mi></msup><msup><mo stretchy="false">)</mo> <mi>I</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I) \\ \downarrow^\simeq && \downarrow^\simeq \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{hofib}{\hookrightarrow}& (\mathbf{H}^I)^I } \,. </annotation></semantics></math></div></div> <p>In words this says that the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> classified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles, and in fact does so on the level of <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s.</p> <h3 id="connections_on_principal_bundles">Connections on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h3> <p>For some comments on the generalization of the notion of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> to principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles see <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos+--+survey">differential cohomology in an (∞,1)-topos – survey</a>.</p> <h2 id="concrete_realizations">Concrete realizations</h2> <p>We discuss realizations of the general definition in various <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <h3 id="in_topological_spaces">In topological spaces</h3> <p>The following general construction was originally due to Quillen and defines principal <em>groupoid</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> in its <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a small <a class="existingWikiWord" href="/nlab/show/category">category</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>P</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> \rho_P : C \to SSet </annotation></semantics></math></div> <p>be a functor with values in <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a> such that it sends all morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to weak equivalences in <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a> (<a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">weak homotopy equivalences</a> of simplicial sets).</p> <p>Consider first the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has a single object, so that it is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> or <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Then</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>:</mo><mo>=</mo><msub><mi>ρ</mi> <mi>P</mi></msub><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P := \rho_P(\bullet) </annotation></semantics></math></div> <p>be the simplicial set assigned to this single object and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><mo>=</mo><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>:</mo><mo>=</mo><mi>hocolim</mi><msub><mi>ρ</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex"> X := P//G := hocolim \rho_P </annotation></semantics></math></div> <p>be the corresponding <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> (see there for the description as a weak colimit).</p> <p>Notice that, as every <a class="existingWikiWord" href="/nlab/show/action+group">action group</a>, this comes with a canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">P//G \to \mathbf{B}G</annotation></semantics></math>.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>Given the above, the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } </annotation></semantics></math></div> <p>is a homotopy pullback (i.e. defines a <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>).</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>This is originally due to</p> <ul> <li>D. Quillen, <em>Higher algebraic K-theory I</em>, Springer Lecture notes in Math. 341 (1973) 85–147.</li> </ul> <p>The statement is reproduced in section IV of</p> <ul> <li>P. G. Goerss and J. F. Jardine, 1999, <em>Simplicial Homotopy Theory</em>, number 174 in Progress in Mathematics, Birkhauser. (<a href="http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html">ps</a>)</li> </ul> </div> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>For the simple case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/group">group</a>, in which case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\rho_C</annotation></semantics></math> necessarily takes values not just in weak equivalences but is isomorphisms of simplicial sets, this says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle. In particular the <em>principality</em> of the action is manifestly exhibited by the fact that the base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the (weak) quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> by the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The above reproduces manifest the description of ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal topological bundles in the incarnation as groupoids as described in detail at <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a>.</p> <p>More generally, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is just a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> the above descibes something a bit more general than an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> (as then the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on the total space may be by weak equivalences that are not isomorphisms).</p> </div> <p>Quillen’s original construction is more general than this, concerning in fact 1-<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles:</p> <div class="un_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> and for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>P</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex"> \rho_P : C \to SSet </annotation></semantics></math></div> <p>a functor that sends all morphisms to weak equivalences of simplicial sets.</p> <p>Let now for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>c</mi></msub><mo>:</mo><mo>=</mo><msub><mi>ρ</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P_c := \rho_C(c) </annotation></semantics></math></div> <p>be the “bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>-fibers”.</p> <p>Then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>P</mi> <mi>c</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mo>*</mo><mo>↦</mo><mi>c</mi></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P_c &\to& {*} \\ \downarrow && \downarrow^{{*} \mapsto c} \\ X &\stackrel{g}{\to}& C } </annotation></semantics></math></div> <p>is a homotopy pullback (i.e. defines a <a class="existingWikiWord" href="/nlab/show/fibration+sequence">fibration sequence</a>).</p> </div> <p>This classical construction is recalled in the introduction of</p> <ul> <li>Jardine, <em>Diagrams and torsors</em> (<a href="http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf">pdf</a>)</li> </ul> <h3 id="in_simplicial_sets__kan_complexes">In simplicial sets / Kan complexes</h3> <p>See <a class="existingWikiWord" href="/nlab/show/simplicial+principal+bundle">simplicial principal bundle</a>.</p> <h3 id="in_a_petit_topos">In a petit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = Op(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = Sh_{(\infty,1)}(X)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos</a> incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>In its <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentation</a> by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> this is the context in which princpal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles are discussed in</p> <ul> <li>Jardine, <em>Diagrams and torsors</em> (<a href="http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf">pdf</a>)</li> </ul> <h3 id="in_a_gros_topos">In a gros <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/site">site</a> of test <a class="existingWikiWord" href="/nlab/show/space">space</a>, – for instance duals of <a class="existingWikiWord" href="/nlab/show/algebras+over+a+Lawvere+theory">algebras over a Lawvere theory</a> as described at <a class="existingWikiWord" href="/nlab/show/function+algebras+on+infinity-stacks">function algebras on infinity-stacks</a> – let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = Sh_{(\infty,1)}(C)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. This is a <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a>.</p> <h4 id="smooth_principal_bundles">Smooth principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h4> <p>Smooth principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles are realized in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> as described in some detail at <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a>.</p> <p>In this context there is a notion of <a class="existingWikiWord" href="/nlab/show/connection+on+a+principal+%E2%88%9E-bundle">connection on a principal ∞-bundle</a>.</p> <h2 id="examples">Examples</h2> <h3 id="ordinary_principal_bundles">Ordinary principal bundles</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundle in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a> is an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>.</p> <h3 id="circle_bundles">Circle <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>-bundles</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">G = \mathbf{B}^{n-1} U(1) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9ELieGrpd">∞LieGrpd</a>, the <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1">circle Lie n-group</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle is a <strong>circle <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>-bundle</strong>.</p> <p>See <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>.</p> <p>Classes of examples include</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a>.</li> </ul> <h3 id="bundle_gerbes">Bundle gerbes</h3> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> is a concrete model for the total space <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of the total space of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> <p>More generally, a <a class="existingWikiWord" href="/nlab/show/nonabelian+bundle+gerbe">nonabelian bundle gerbe</a> is a concrete model for the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of the total space of a general <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a> is a concrete model for the total space <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of the total space of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1)</annotation></semantics></math>-principal 3-bundle.</p> <p>More generally, a <a class="existingWikiWord" href="/nlab/show/nonabelian+bundle+2-gerbe">nonabelian bundle 2-gerbe</a> is a concrete model for the <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> of the total space of a general principal 3-bundle.</p> <p>Classes of examples include</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/Chern-Simons+bundle+2-gerbe">Chern-Simons bundle 2-gerbe</a>.</li> </ul> </li> </ul> <h3 id="normal_morphisms_of_groups">Normal morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</h3> <p>A principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle over a <a class="existingWikiWord" href="/nlab/show/0-connected">0-connected</a> object / <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">mathf</mo><mi>B</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">\mathf{B}K</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/normal+morphism+of+%E2%88%9E-groups">normal morphism of ∞-groups</a></em>. See there for more details.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> / <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> / <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a> / <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</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> / <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a> / <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a> / <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></p> </li> <li> <p><strong>principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle</strong> / <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupoid-principal+%E2%88%9E-bundle">groupoid-principal ∞-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/%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> <p>The notion of principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle (often addressed in the relevant literature in the language of <a class="existingWikiWord" href="/nlab/show/torsors">torsors</a>) appears in the context of the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model</a> for generalized spaces in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>, Z. Luo: <em>Higher order principal bundles</em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>140</strong> 2 (2006) 221-243 [<a href="https://doi.org/10.1017/S0305004105008911">doi:10.1017/S0305004105008911</a>, web.archive:<a href="https://web.archive.org/web/20050212221131id_/http://www.math.uwo.ca:80/~jardine/papers/cocycles8.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rick+Jardine">Rick Jardine</a>: <em>Cocycle categories</em>, in <em>Algebraic Topology</em> Abel Symposia <strong>4</strong> (2009) [<a href="http://arxiv.org/abs/math.AT/0605198">arXiv:math.AT/0605198</a>, <a href="https://doi.org/10.1007/978-3-642-01200-6_8">doi:10.1007/978-3-642-01200-6_8</a>]</p> </li> </ul> <p>An earlier description in terms of simplicial objects is</p> <ul> <li>P. Glenn, <em>Realization of cohomology classes in arbitrary exact categories</em>, J. Pure Appl. Algebra <strong>25</strong> 1 (1982) 33-105 [<a href="https://doi.org/10.1016/0022-4049(82)90094-9">doi:10.1016/0022-4049(82)90094-9</a>]</li> </ul> <p>In that article not the total space of the bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> is axiomatized, but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on it.</p> <p>See the remarks at <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> <p>See also</p> <ul> <li id="Wendt10"><a class="existingWikiWord" href="/nlab/show/Matthias+Wendt">Matthias Wendt</a>, <em>Classifying spaces and fibrations of simplicial sheaves</em>, J. Homotopy Relat. Struct. <strong>6</strong> 1 (2011) 1-38 [<a href="http://arxiv.org/abs/1009.2930">arXiv:1009.2930</a>]</li> </ul> <p>on <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundles">associated ∞-bundles</a>.</p> <p>The fully general abstract formalization in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> as discussed here was first indicated in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em> (2009)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Zoran+%C5%A0koda">Zoran Škoda</a>, <a href="https://arxiv.org/pdf/1004.2472.pdf#page=20">§7.1</a> of: <em>Categorified symmetries</em> [<a href="https://arxiv.org/abs/1004.2472">arXiv:1004.2472</a>]</p> </li> </ul> <p>with more comprehensive accounts in:</p> <ul> <li id="NSS12"> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+models+and+general+theory">Principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-bundles – General theory</a></em>, J. Hom. Rel. Struc. <strong>10</strong> 4 (2015) 749-801 [<a href="https://arxiv.org/abs/1207.0248">arXiv:1207.0248</a>, <a href="http://link.springer.com/article/10.1007/s40062-014-0083-6">doi:10.1007/s40062-014-0083-6</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> (2013)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+principal+infinity-bundles">Equivariant Principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-bundles</a></em> [<a href="https://arxiv.org/abs/2112.13654">arXiv:2112.13654</a>]</p> </li> </ul> <p>For some additional developments and applications to ∞-group extensions, see also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <em>Principal ∞-bundles and smooth String group models</em>, Mathematische Annalen <strong>387</strong> 689–743 [<a href="https://doi.org/10.1007/s00208-022-02462-0">doi:10.1007/s00208-022-02462-0</a>, <a href="http://arxiv.org/abs/2008.12263">arXiv:2008.12263</a>]</li> </ul> <p>A comparison of smooth principal ∞-bundles and diffeological principal bundles for diffeological groups is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emilio+Minichiello">Emilio Minichiello</a>, <em>Diffeological Principal Bundles and Principal Infinity Bundles</em>, [<a href="http://arxiv.org/abs/2202.11023">arXiv:2202.11023</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> for a large class of principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles are discussed in</p> <ul> <li id="RobertsStevenson"> <p><a class="existingWikiWord" href="/nlab/show/David+Roberts">David Roberts</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em>Simplicial principal bundle in parameterized spaces</em> [<a href="http://arxiv.org/abs/1203.2460">arXiv:1203.2460</a>]</p> </li> <li id="Stevenson"> <p><a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em>Classifying theory for simplicial parametrized groups</em> [<a href="http://arxiv.org/abs/1203.2461">arXiv:1203.2461</a>]</p> </li> </ul> <p>A fairly comprehensive account of the literature is also in the introduction of <a href="#NSS12">NSS 12, “Presentations”</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H}= \infty Grpd</annotation></semantics></math> the statement that homotopy types over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math> are equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-actions">infinity-actions</a> is maybe due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emmanuel+Dror+Farjoun">Emmanuel Dror Farjoun</a>, <a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Equivariant maps which are self homotopy equivalences</em>, Proc. Amer. Math. Soc. <strong>80</strong> 4 (1980) 670672 [<a href="http://www.jstor.org/stable/2043448">jstor:2043448</a>]</li> </ul> <p>This is mentioned for instance as exercise 4.2in</p> <ul> <li id="Dwyer2008"><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <em>Homotopy theory of classifying spaces</em>, Lecture notes Copenhagen (June, 2008) <a href="http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf">pdf</a></li> </ul> <p>Closely related discussion of homotopy fiber sequences and homotopy action but in terms of <a class="existingWikiWord" href="/nlab/show/Segal+spaces">Segal spaces</a> is in section 5 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Matan+Prezma">Matan Prezma</a>, <em>Homotopy normal maps</em> (<a href="http://arxiv.org/pdf/1011.4708v7.pdf">arXiv</a>)</li> </ul> <p>There, conditions are given for a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet \to B_\bullet</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/reduced+Segal+space">reduced Segal space</a> to have a fixed homotopy fiber, and hence encode an action of the loop group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> on that fiber.</p> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> of <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> along <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>, relating to <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>, <a class="existingWikiWord" href="/nlab/show/T-folds">T-folds</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+T-duality">non-abelian T-duality</a>, <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a>, <a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, <em>Global Double Field Theory is Higher Kaluza-Klein Theory</em> (<a href="https://arxiv.org/abs/1912.07089">arXiv:1912.07089</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, <em>The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective</em> (<a href="https://arxiv.org/abs/2007.04969">arXiv:2007.04969</a>)</p> </li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Bundles</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em> [<a href="https://arxiv.org/abs/2308.04196">arXiv:2308.04196</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2025 at 22:08:08. 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