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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 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> circle n-bundle with connection </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1729/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></li> </ul> </div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</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> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</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/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </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> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="cohesive_toposes">Cohesive <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>-Toposes</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#the_ambient_context'>The ambient context</a></li> <li><a href='#FlatCircleCohomology'>Flat differential cohomology</a></li> <li><a href='#OrdinaryDeRham'>de Rham cohomology</a></li> <li><a href='#DifferentialCohomology'>Differential cohomology</a></li> <ul> <li><a href='#CircleBundlesConnection'>Circle bundles with connection</a></li> <ul> <li><a href='#observations'>Observations</a></li> </ul> <li><a href='#CircleBunlePseudoConnection'>Circle bundles with pseudo-connection</a></li> <li><a href='#U1GroupoidBundle'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U(1)_0</annotation></semantics></math>-groupoid bundles</a></li> <li><a href='#AbGerbesConnection'>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 with connection</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#U1FromLieIntegration'>Models from <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>-Lie integration</a></li> <ul> <li><a href='#convention'>Convention</a></li> <li><a href='#observation_2'>Observation</a></li> </ul> <li><a href='#InHomtopyTypeTheory'>In homotopy type theory</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#moduli'>Moduli</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>We discuss the refinement to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> of the concept of a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> (on a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>). Specifically we indicate how the general abstract definition in terms of <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> reproduces in the context of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth cohesion</a> to the representation of 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>-connections by <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in smooth <em><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></em> (<a href="#dcct">dcct</a>).</p> <p>In every <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,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> there is an <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#DifferentialCohomology">intrinsic notion of differential cohomology</a> with coefficients in an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> <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> that classifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connection</a>.</p> <p>Here we discuss the specific realization 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></mrow><annotation encoding="application/x-tex">\mathbf{H} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = U(1)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>.</p> <p>In this case the intrinsic differential cohomology reproduces <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> and generalizes it to base spaces that may be <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>s, <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>s and generally <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s such as <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>s <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 class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-group</a>s <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>. Differential cocycles on the latter support the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+homomorphism">∞-Chern-Weil homomorphism</a> that sends <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian</a> <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a> to 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 whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> form realizes a <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a> in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>.</p> <h2 id="the_ambient_context">The ambient context</h2> <p>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><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} := </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> be the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>. As usual, write</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>Π</mi><mo>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><mover><mover><munder><mo>←</mo><mi>coDisc</mi></munder><mover><mo>→</mo><mi>Γ</mi></mover></mover><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>→</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : Smooth \infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd </annotation></semantics></math></div> <p>for the terminal <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a> with its extra <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">intrinsic fundamental ∞-groupoid</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>.</p> <p>From this induced is the <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#Paths">path ∞-groupoid adjunction</a></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><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo>⊣</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><mo stretchy="false">)</mo><mo>:</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mo>←</mo></mover><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\mathbf{\Pi} \dashv \mathbf{\flat}) : Smooth \infty Grpd \stackrel{\leftarrow}{\to} Smooth \infty Grpd </annotation></semantics></math></div> <p>and the <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#deRhamCohomology">intrinsic de Rham cohomology</a> adjunction</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><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>dR</mi></msub><mo>⊣</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo>*</mo><mo stretchy="false">/</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mover><munder><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub></mrow></munder><mover><mo>←</mo><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mi>dR</mi></msub></mrow></mover></mover><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR}) : */Smooth \infty Grpd \stackrel{\overset{\mathbf{\Pi}_{dR}}{\leftarrow}}{\underset{\mathbf{\flat}_{dR}}{\to}} Smooth \infty Grpd \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> an abelian group object there for each <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <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> is the <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#CurvatureCharacteristics">universal curvature characteristic form</a>, given by a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>-morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> curv : \mathbf{B}^n A \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1}A \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s for <em>differential cohomology</em> in degree <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> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> are the points in the <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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(-, \mathbf{B}^n A)</annotation></semantics></math> of the morphism on <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mo>*</mo></msub><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> curv_* : \mathbf{H}(-, \mathbf{B}^n A) \to \mathbf{H}_{dR}(-, \mathbf{B}^{n+1}A) </annotation></semantics></math></div> <p>induced by this. Every such cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla \in \mathbf{H}_{diff}(X,\mathbf{B}^n A)</annotation></semantics></math> we may think of as an <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connection</a> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> classified by the underlying cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \mathbf{B}^n A)</annotation></semantics></math>.</p> <p>We consider these constructions in the model <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/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. This is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>:</mo><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><msub><mi>CartSp</mi> <mi>smooth</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Smooth\infty Grpd := Sh_{(\infty,1)}(CartSp_{smooth}) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">{}_{smooth}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s and <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s between them. This is a general <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> context for <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>. For computations we can explicitly <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">present</a> this <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> by a local <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,loc}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex"> Smooth \infty Grpd \simeq ([CartSp^{op}, sSet]_{proj, loc})^\circ </annotation></semantics></math></div> <p>as described at <a class="existingWikiWord" href="/nlab/show/presentations+of+%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-toposes">presentations of (∞,1)-sheaf (∞,1)-toposes</a>.</p> <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><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> a canonical choice for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A := U(1) = \mathbb{R}/\mathbb{Z} \,. </annotation></semantics></math></div> <p>We show how the notion of <em>smooth 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 with connection</em> obtained by applying the general setup above to this case reproduces <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>:</p> <ul> <li> <p>a <em>circle 1-bundle with connection</em> is an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>;</p> </li> <li> <p>a <em>circle 2-bundle with connection</em> is 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> <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">with connection</a>, equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">with connection</a>;</p> </li> <li> <p>a <em>circle 3-bundle</em> with connection if 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>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal 3-bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">with connection</a>, equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a> with connection;</p> </li> <li> <p>generally, a <em>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 with connection</em> is 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> <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></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal n-bundle</a> <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">with connection</a>, equivalently a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math>, equivalently a <a class="existingWikiWord" href="/nlab/show/Cheeger-Simons+differential+character">Cheeger-Simons differential character</a> in that degree.</p> </li> </ul> <p>We assume in the following that the reader is familiar with basics of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s.</p> <h2 id="FlatCircleCohomology">Flat differential cohomology</h2> <p>The <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#FlatDifferentialCohomology">coefficient object for flat differential cohomology</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><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>LConst</mi><mi>Γ</mi><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n U(1) = LConst \Gamma \mathbf{B}^n U(1)</annotation></semantics></math>.</p> <p>The <a href="http://nlab.mathforge.org/nlab/show/cohesive%20(infinity,1)-topos#deRhamCohomology">coefficient object for intrinsic de Rham cohomology</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math>, defined by 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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\flat}_{dR} \mathbf{B}^n U(1) &\to& \mathbf{\flat} \mathbf{B}^n U(1) \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}^n U(1) } \,. </annotation></semantics></math></div> <p>The following proposition provides models for these objects in in terms of ordinary <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A fibrant representative in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,cov}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n U(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><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub><mo>:</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}\mathbf{B}^n U(1)_{chn} := \Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)] \,, </annotation></semantics></math></div> <p>and a fibrant representative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub><mo>:</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">[</mo><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}_{dR}\mathbf{B}^n U(1)_{chn} := \Xi[0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)] \,. </annotation></semantics></math></div></div> <p>Notice that the complex of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}\mathbf{B}^n U(1)</annotation></semantics></math> is that which defines <em>flat</em> <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>, while that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math> is essentially that which defines <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math> (see <a href="#OrdinaryDeRham">below</a>). Also notice that we denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR}</annotation></semantics></math> also the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,U(1)) \stackrel{d_{dR} log}{\to} \Omega^1(-)</annotation></semantics></math>; this is to stress that we are looking at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> as the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}/\mathbb{Z}</annotation></semantics></math>.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> amounts to evaluation on the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math> and since constant simplicial presheaves on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> satisfy <a class="existingWikiWord" href="/nlab/show/descent">descent</a> (on objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math>!), we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n U(1)</annotation></semantics></math> is represented by the complex of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">[</mo><mi>const</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Xi[const U(1) \to 0 \to \cdots \to 0]</annotation></semantics></math>. This is weakly equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Xi[C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{cl}(-)]</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> applied to each <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> (using the same standard logic that proves the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>) in that the degreewise inclusion</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>const</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ const U(1) &\to& 0 &\to& \cdots &\to& 0 \\ \downarrow && \downarrow && && \downarrow \\ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\to& \cdots &\to& \Omega^n_{cl}(-) } </annotation></semantics></math></div> <p>is objectwise a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> <p>Therefore a fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math> representing the counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^n U(1)</annotation></semantics></math> is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi></mrow><annotation encoding="application/x-tex">\Xi</annotation></semantics></math> of</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><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(-,U(1)) &\to& \Omega^1(-) &\to & \cdots &\to& \Omega^n_{cl}(-) \\ \downarrow^{\mathrlap{=}} && \downarrow && && \downarrow \\ C^\infty(-, U(1)) &\to& 0 &\to& \cdots &\to& 0 } \,. </annotation></semantics></math></div> <p>We observe that the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of this morphism to the point</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>Ξ</mi><mo stretchy="false">[</mo><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Ξ</mi><mo stretchy="false">[</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Xi[0 \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)] &\to& \Xi[C^\infty(-,U(1)) \to \Omega^1(-) \to \cdots \to \Omega^n_{cl}(-)] \\ \downarrow && \downarrow \\ \Xi[0 \to 0 \to \cdots \to 0] &\to& \Xi[C^\infty(-,U(1)) \to 0 \to \cdots \to 0] } </annotation></semantics></math></div> <p>is the pullback over a cospan all whose objects are fibrant and one of whose morphisms is a fibration. Therefore this is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math> which models the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>←</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">* \to \mathbf{B}^n U(1) \leftarrow \mathbf{\flat}\mathbf{B}^n U(1)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_{(\infty,1)}(CartSp)</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a> preserves finite <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>-limits this models also the corresponding <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>-limit 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>. Therefore the top left object is indeed a model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math>.</p> </div> <h2 id="OrdinaryDeRham">de Rham cohomology</h2> <p>The intrinsic de Rham cohomology of <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">U(1) = \mathbb{R}/\mathbb{Z}</annotation></semantics></math> coincides with the ordinary <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s and smooth <a class="existingWikiWord" href="/nlab/show/simplicial+manifold">simplicial manifold</a>s in degree greater than 1. This we discuss here. The meaning of the discrepancy in degee 1 and lower is discussed <a href="#U1GroupoidBundle">below</a>.</p> <p>So for this section let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>.</p> <p>Above in <em><a href="#FlatCircleCohomology">Flat U(1)-valued differential cohomology</a></em> we found a fibrant representative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1) \in Smooth\infty Grpd</annotation></semantics></math> to be given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)] </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj, cov}</annotation></semantics></math>.</p> <div class="num_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><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">X \in Smooth\infty Grpd</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> we have in 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><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} = Smooth \infty Grpd</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>H</mi> <mi>dR</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H_{dR}(X,\mathbf{B}^n U(1)) := \pi_0 \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1)) \simeq H_{dR}^n(X) \,, </annotation></semantics></math></div> <p>where on the left we have the <a href="http://nlab.mathforge.org/nlab/show/cohesive%20%28infinity,1%29-topos#deRhamCohomology">intrinsic (∞,1)-topos theoretic notion of de Rham cohomology</a>, and on the right the ordinary notion of <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of a smooth manifold.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>. At <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> is discussed that then the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C(\{U_i\}) \to X</annotation></semantics></math> is a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,cov}</annotation></semantics></math>. Therefore we have</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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mi>Ξ</mi><mo stretchy="false">[</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X,\mathbf{\flat}_{dR} \mathbf{B}^n U(1)) \simeq [CartSp^{op}, sSet](C(\{U_i\}), \Xi[\Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \to \Omega^n_{cl}(-)]) \,. </annotation></semantics></math></div> <p>The right hand is 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>-groupoid of cocylces in the <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a> of the complex of sheaves of differential forms. A cocycle is given by a collection</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><msub><mi>C</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>B</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (C_i, B_{i j}, A_{i j k}, \cdots , Z_{i_0, \cdots, i_n}) </annotation></semantics></math></div> <p>of differential forms, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>i</mi></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_i \in \Omega^n_{cl}(U_i)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_{i j} \in \Omega^{n-1}(U_i \cap U_j)</annotation></semantics></math>, etc. , such that this collection is annihilated by the total differentoal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo>±</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">D = d_{dR} \pm \delta</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">d_{dR}</annotation></semantics></math> is the de Rham differential and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> the alternating sum of the pullbacks along the face maps of the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a>.</p> <p>It is a standard result of <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> that such cocycles represent classes in de Rham cohomology.</p> <p>But for the record and since the details of this computation will show up again at some mildly subtle points in further discussion below, we spell this out in some detail.</p> <p>We can explicitly construct coboundaries connecting such a generic cocycle to one of the form</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><msub><mi>F</mi> <mi>i</mi></msub><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (F_i, 0, 0, \cdots, 0) </annotation></semantics></math></div> <p>by using a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rho_i \in C^\infty(X))</annotation></semantics></math> subordinate to the cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⇒</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \in U_i \Rightarrow \rho_i(x) = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>ρ</mi> <mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum_i \rho_i = 1</annotation></semantics></math>.</p> <p>For consider</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></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>B</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>Y</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>,</mo><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">+</mo></mtd> <mtd><mi>D</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>B</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>Y</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & (C_i, B_{i j}, A_{i j k}, \cdots , Y_{i_1, \cdots, i_{n}}, Z_{i_1, \cdots, i_{n+1}}) \\ + & D (0, 0, \cdots, \sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}},0) \\ = & (C_i, B_{i j}, A_{i j k}, \cdots , Y_{i_1, \cdots, i_{n}} + d_{dR}\sum_{i_0} \rho_{i_0} Z_{i_0, i_1, \cdots, i_{n}}, 0) \end{aligned} \,, </annotation></semantics></math></div> <p>where we use that from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>δ</mi><mi>Z</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(\delta Z)_{i_1, \cdots, i_{n+2}} = 0</annotation></semantics></math> it follows that</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><mo stretchy="false">(</mo><mi>δ</mi><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>ρ</mi><mi>Z</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><mo>,</mo><msub><mover><mi>i</mi><mo stretchy="false">^</mo></mover> <mi>k</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Z</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\delta \sum \rho Z)_{i_1, \cdots, i_{n+1}} &= \sum_{i_0} \rho_{i_0} \sum_{k = 1}^{n+1} (-1)^k Z_{i_0, i_1 \cdots, \hat i_k, \cdots, i_{n+1}} \\ & = \sum_{i_0} \rho_{i_0} Z_{i_1 ,\cdots, i_{n+1}} \\ & = Z_{i_1 ,\cdots, i_{n+1}} \end{aligned} \,. </annotation></semantics></math></div> <blockquote> <p>where I am suppressing some evident signs…</p> </blockquote> <p>By recurseively adding coboundaries this way, we can annihilate all the higher Cech-components of the original cocycle and arrive at a cocycle of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F_i, 0, \cdots, 0)</annotation></semantics></math>.</p> <p>Such a cocycle being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-closed says precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>i</mi></msub><mo>=</mo><mi>F</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">F_i = F|_{U_i}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in \Omega^n_{cl}(X)</annotation></semantics></math> a globally defined closed differential form. Moreover, coboundaries between two cocycles both of this form</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><msub><mi>F</mi> <mi>i</mi></msub><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>F</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>λ</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (F_i, 0, \cdots , 0) = (F'_i, 0, \cdots, 0) + D(\lambda_i, \lambda_{i j}, \cdots) </annotation></semantics></math></div> <p>are necessarily themselves of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>λ</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>λ</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>λ</mi> <mi>i</mi></msub><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\lambda_i, \lambda_{i j}, \cdots) = (\lambda_i, 0 ,\cdots, 0)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>i</mi></msub><mo>=</mo><mi>λ</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_i = \lambda|_{U_i}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda \in \Omega^{n-1}(X)</annotation></semantics></math> a globally defined differential <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>-form and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>F</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>λ</mi></mrow><annotation encoding="application/x-tex">F = F' + d_{dR} \lambda</annotation></semantics></math>.</p> </div> <h2 id="DifferentialCohomology">Differential cohomology</h2> <p>The intrinsic <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos#DiffCohWithGroupalCoeffs">definition of the ∞-groupoid of cocycles for the intrinsic differential cohomology</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><mo>=</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} = Smooth\infty Grpd</annotation></semantics></math> with coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> is the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(X,\mathbf{B}^n U(1))</annotation></semantics></math> in 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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>H</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}_{diff}(X,\mathbf{B}^n U(1)) &\to & H_{dR}(X,\mathbf{B}^{n+1} U(1)) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n U(1)) &\stackrel{curv}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)) } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <p>We show now that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> this reproduces the <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>∞</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,\mathbb{Z}(n+1)_D^\infty)</annotation></semantics></math> 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> <div class="num_theorem" id="DeligneCohomologyTheorem"> <h6 id="theorem">Theorem</h6> <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/paracompact+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>(</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>∞</mn></msubsup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>)</mo></mrow><msub><mo>×</mo> <mrow><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><msubsup><mi>H</mi> <mi>dR</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><msub><mo></mo><mi>int</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{diff}(X,\mathbf{B}^n U(1)) \simeq \left( \;\; H(X,\mathbb{Z}(n+1)_D^\infty) \;\; \right) \times_{\Omega_{cl}^{n+1}(X)} H_{dR}^{n+1}_{int}(X) \,. </annotation></semantics></math></div></div> <p>Here on the right we have the subset of Deligne cocycles that picks for each integral de Rham cohomology class 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> only one curvature form representative.</p> <p>We give the proof <a href="#ProofOfDeligneTheorem">below</a> after some preliminary expositional discussion.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The restriction to single representatives in each de Rham class is a reflection of the fact that in the above <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 diagram the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{dR}(X,\mathbf{B}^{n+1}U(1)) \to \mathbf{H}_{dR}(X,\mathbf{B}^{n+1}U(1))</annotation></semantics></math> by definition picks one representative in each connected component. Using the <a href="#OrdinaryDeRham">above model</a> of the intrinsic de Rham cohomology in terms of globally defined differential froms, we could easily get rid of this restriction by considering instead of the above <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 the homotopy 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><mstyle mathvariant="bold"><mi>H</mi></mstyle><msub><mo>′</mo> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}'_{diff}(X,\mathbf{B}^n U(1)) &\to & \Omega_{cl}^{n+1}(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}^n U(1)) &\stackrel{curv}{\to}& \mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1)) } </annotation></semantics></math></div> <p>where now the right vertical morphism is the inclusion of the set of objects of our concrete model for 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>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{dR}(X, \mathbf{B}^{n+1} U(1))</annotation></semantics></math>. With this definition we get the isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><msub><mo>′</mo> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>∞</mn></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H'_{diff}(X,\mathbf{B}^n U(1)) \simeq H(X,\mathbb{Z}(n+1)_D^\infty) \,. </annotation></semantics></math></div> <p>From the tradtional point of view of differential cohomology this may be what one expects to see, but from the intrinsic <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 theoretic point of view it is quite unnatural – and in fact “<a class="existingWikiWord" href="/nlab/show/evil">evil</a>” – to fix that set of objects of 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>-groupoid. Of intrinsic meaning is only the set of their equivalences classes.</p> </div> <h3 id="CircleBundlesConnection">Circle bundles with connection</h3> <p>Before discussing <a href="#DeligneCohomologyTheorem">the full theorem</a>, it is instructive to start by looking at the special case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math> in some detail, which is about ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">with connection</a>.</p> <p>This contains in it already all the relevant structure of the general case, but the low categorical degree is more transparently written out and will allow us to pause to highlight some maybe noteworthy aspects of the situation, such as the phenomenon of <em>pseudo-connections</em> <a href="#CircleBunlePseudoConnection">below</a>.</p> <p>In terms of the <a href="#DoldKan">Dold-Kan correspondence</a> the 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>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1) \in \mathbf{H}</annotation></semantics></math> is modeled in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]</annotation></semantics></math> by</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>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1) = \Xi(\; C^\infty(-,U(1)) \to 0 \;) \,. </annotation></semantics></math></div> <p>Accordingly we have for the double <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> the model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 U(1) = \Xi( \; C^\infty(-,U(1)) \to 0 \to 0 \;) </annotation></semantics></math></div> <p>and for the <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal 2-bundle</a></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>E</mi></mstyle><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mi>Id</mi></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{E}\mathbf{B}U(1) = \Xi( \; C^\infty(-,U(1)) \stackrel{Id}{\to} C^\infty(-, U(1)) \to 0 \; ) \,. </annotation></semantics></math></div> <p>In this notation we have also the constant presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><mi>const</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat} \mathbf{B}^2 U(1) = \Xi( \; const U(1) \to 0 \to 0 \; ) \,. </annotation></semantics></math></div> <p>Above we already found the model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) = \Xi(0 \to \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-)) \,. </annotation></semantics></math></div> <p>In order to compute the differential cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}_{diff}(-,\mathbf{B}U(1))</annotation></semantics></math> by an ordinary pullback in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> we also want to resolve the curvature characteristic morphism <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><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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}U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^2 U(1)</annotation></semantics></math> by a fibration. We claim that this may be obtained by choosing the resolution <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><mover><mo>←</mo><mo>≃</mo></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B} U(1)_{diff,chn}</annotation></semantics></math> given by</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>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo>:</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>⊕</mo><mi>Id</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1)_{diff} := \Xi( \; C^\infty(-,U(1)) \oplus \Omega^1(-) \stackrel{d_{dR} \oplus Id}{\to} \Omega^1(-) \; ) </annotation></semantics></math></div> <p>with the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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">curv : \mathbf{B}_{diff}U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^2 U(1)</annotation></semantics></math> given by</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><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>+</mo><mi>Id</mi></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(-,U(1)) \oplus \Omega^1(-) &\stackrel{d_{dR} + Id}{\to}& \Omega^1(-) \\ \downarrow^{\mathrlap{p_2}} && \downarrow^{\mathrlap{d_{dR}}} \\ \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2_{cl}(-) } \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> applied to each <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, this is indeed a fibration.</p> <p>In the <a href="#AbGerbesConnection">next section</a> we give the proof of this (simple) claim. Here in the warmup phase we instead want to discuss the geometric interpretation of this resolution, along the lines of the section <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+survey#CurvatureCharacteristicsI">curvature characteristics of 1-bundles</a> in the <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos+--+survey">survey-part</a>.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>We have the following geometric interpretation of the above models:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><mo>:</mo><mi>U</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>U</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><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><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></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><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><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}_{dR} \mathbf{B}^2 U(1) : U \mapsto \left\{ \array{ U &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1) } \right\} = \left\{ \mathbf{\Pi}_2(U) \to \mathbf{B}^2 U(1) \right\} </annotation></semantics></math></div> <p>and</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>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mo>:</mo><mi>U</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>INN</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1)_{diff} : U \mapsto \left\{ \array{ U &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1)) } \right\} \,. </annotation></semantics></math></div> <p>And in this presentation the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><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">curv : \mathbf{B}_{diff}U(1) \to \mathbf{B}^2 U(1)</annotation></semantics></math> is given over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> by forming the pasting composite</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>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>underlying</mi><mspace width="thickmathspace"></mspace><mi>cocycle</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>INN</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>connection</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><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></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>curvature</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ U &\to& \mathbf{B}U(1) &&& underlying\;cocycle \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}INN(U(1)) &&& connection \\ \downarrow && \downarrow \\ \mathbf{\Pi}_2(U) &\to& \mathbf{B}^2 U(1) &&& curvature } </annotation></semantics></math></div> <p>and picking the lowest horizontal morphism.</p> </div> <p>Here the terms mean the following:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>INN</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">INN(U(1))</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ξ</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Xi(U(1) \to U(1))</annotation></semantics></math>, which is a <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for the universal U(1)-principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{E}U(1)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Π</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}_2(U)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/path+2-groupoid">path 2-groupoid</a> with homotopy class of 2-dimensional paths as 2-morphisms</p> </li> <li> <p>the groupoids of diagrams in braces have as objects commuting diagrams in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]</annotation></semantics></math> as indicated, and horizontal 2-morphisms fitting into such diagrams as morphisms.</p> </li> </ul> <p>Using the discussion at <a class="existingWikiWord" href="/nlab/show/2-groupoid+of+Lie+2-algebra+valued+forms">2-groupoid of Lie 2-algebra valued forms</a> (<a href="http://arxiv.org/abs/0802.0663">SchrWalII</a>) we have the following:</p> <ol> <li> <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/smooth+manifold">smooth manifold</a>, morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mn>2</mn><mi>Grpd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, 2Grpd]</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>tra</mi> <mi>A</mi></msub><mo>:</mo><msub><mi>Π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><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">tra_A : \Pi_2(X) \to \mathbf{E}\mathbf{B}U(1)</annotation></semantics></math> are in bijection with smooth 1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^1(X)</annotation></semantics></math>: the 2-functor sends a path in <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> to the the <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> along that path, and sends a surface in <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> to the exponentiated integral of the curvature 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mi>d</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">F_A = d A</annotation></semantics></math> over that surface. The <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>F</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d F_A = 0</annotation></semantics></math> says precisely that this assignment indeed descends to homotopy classes of surfaces, which are the 2-morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_2(X)</annotation></semantics></math>.</p> </li> <li> <p>Moreover <a class="existingWikiWord" href="/nlab/show/k-morphism">2-morphisms</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>tra</mi> <mi>A</mi></msub><mo>→</mo><msub><mo lspace="0em" rspace="thinmathspace">tra</mo> <mrow><mi>A</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(\lambda,\alpha) : tra_A \to \tra_{A'}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mn>2</mn><mi>Grpd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, 2Grpd]</annotation></semantics></math> are in bijection with pairs consisting of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda \in C^\infty(X,U(1))</annotation></semantics></math> and a 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in \Omega^1(X)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>′</mo><mo>=</mo><mi>A</mi><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>λ</mi><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">A' = A + d_{dR} \lambda - \alpha</annotation></semantics></math>.</p> </li> <li> <p>And finally <a class="existingWikiWord" href="/nlab/show/k-morphism">3-morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>′</mo><mo>,</mo><mi>α</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h : (\lambda, \alpha) \to (\lambda', \alpha')</annotation></semantics></math> are in bijection with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h \in C^\infty(X,U(1))</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><mo>=</mo><mi>λ</mi><mo>⋅</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">\lambda' = \lambda \cdot h</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>′</mo><mo>=</mo><mi>α</mi><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">\alpha' = \alpha + d_{dR} h</annotation></semantics></math>.</p> </li> </ol> <p>By the same reasoning we find that the coefficient object for flat <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>-valued differential cohomology is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><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><mo stretchy="false">]</mo><mo>=</mo><mi>Ξ</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}\mathbf{B}^2 U(1) = [\Pi_2(-), \mathbf{B}^2U(1)] = \Xi( C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-) ) \,. </annotation></semantics></math></div> <p>So by the above definition of differential cohomology 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> we find 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>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>-differential cohomology of a <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <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 given by choosing any <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\})</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a>, which is then a cofibrant replacement 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>cov</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj,cov}</annotation></semantics></math> and forming the ordinary 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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>H</mi> <mi>dR</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mo>♭</mo> <mi>dR</mi></msub><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><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H^2_{dR}(X) \\ \downarrow && \downarrow \\ [CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1)) &\stackrel{curv}{\to}& [CartSp^{op},sSet](C(\{U_i\}), \flat_{dR}\mathbf{B}^2 U(1)) } </annotation></semantics></math></div> <p>(because the bottom vertical morphism is a fibration, by the fact that our model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mo>♭</mo> <mi>dR</mi></msub><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}_{diff} U(1) \to \flat_{dR}\mathbf{B}^2 U(1)</annotation></semantics></math> is a fibration, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\})</annotation></semantics></math> is cofibrant and using the axioms of the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math>).</p> <div class="num_prop"> <h6 id="observations">Observations</h6> <p>A cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op},sSet](C(\{U_i\}), \mathbf{B}_{diff}U(1))</annotation></semantics></math> is</p> <ol> <li> <p>a collection of functions</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><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (g_{i j } \in C^\infty(U_i \cap U_j, U(1))) </annotation></semantics></math></div> <p>satsifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>g</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{i j} g_{j k} = g_{i k}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math>;</p> </li> <li> <p>a collection of 1-forms</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><msub><mi>A</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A_i \in \Omega^1(U_i)) </annotation></semantics></math></div></li> <li> <p>a collection of 1-forms</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><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (a_{i j} \in \Omega^1(U_i \cap U_j)) </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> A_j = A_i + d_{dR} g_{i j} + a_{i j} </annotation></semantics></math></div> <p>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>a</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> a_{i j} + a_{j k} = a_{i k} </annotation></semantics></math></div> <p>on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j \cap U_k</annotation></semantics></math>.</p> </li> </ol> <p>The curvature-morphism takes such a cocycle to the cocycle</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>d</mi><msub><mi>A</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (d A_i, a_{i j}, ) </annotation></semantics></math></div> <p>in the <a href="#OrdinaryDeRham">above model</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op},sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^2 U(1))</annotation></semantics></math> for intrinsic de Rham cohomology.</p> <p>Every cocycle with nonvanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_{i j})</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C(\{U_i\}), \mathbf{B}_{diff}U(1)]</annotation></semantics></math> coboundant to one with vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_{i j})</annotation></semantics></math></p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The first statements are effectively the definition and the construction of the above models. The last statement is as in the <a href="#OrdinaryDeRham">above discussion</a> of our model for ordinary de Rham cohomology: given a cocycle with non-vanishing closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i j}</annotation></semantics></math>, pick a partition of unity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rho_i \in C^\infty(X))</annotation></semantics></math> subordinate to the chosen cover and the coboundary given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sum_{i_0} \rho_{i_0} a_{i_0 i})</annotation></semantics></math>. This connects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_i,a_{i j}, g_{i j})</annotation></semantics></math> with the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>,</mo><mi>a</mi><msub><mo>′</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A'_i, a'_{i j}, g_{i j})</annotation></semantics></math> where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>=</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> A'_i = A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i} </annotation></semantics></math></div> <p>and</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>a</mi><msub><mo>′</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd> <mtd><mo>=</mo><mi>A</mi><msub><mo>′</mo> <mi>j</mi></msub><mo>−</mo><mi>A</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>−</mo><mi>d</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo>−</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} a'_{i j} & = A'_j - A'_i - d g_{i j} \\ & = a_{i j} + - \sum_{i_0}( a_{i_0 i} - a_{i_0 j} ) \\ & = 0 \end{aligned} \,. </annotation></semantics></math></div></div> <p>So in total we have found the following story:</p> <ol> <li> <p>In order to compute the curvature characteristic form of a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><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">g : C(\{U_i\}) \to \mathbf{B}U(1)</annotation></semantics></math> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundle, we first lift it</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></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}_{diff}U(1) \\ & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) } </annotation></semantics></math></div> <p>to an equivalent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}_{diff}U(1)</annotation></semantics></math>-cocycle, and this amounts to putting (the Cech-representatitve of) a <em>pseudo-connection</em> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundle.</p> </li> <li> <p>From that lift the desired curvature characteristic is simply projected out</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></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}_{diff}U(1) &\stackrel{curv}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^2 U(1) \\ & {}^{\mathllap{(g,\nabla)}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) } \,, </annotation></semantics></math></div> <p>and we find that it lives in the sheaf <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a> that models ordinary de Rham cohomology.</p> </li> <li> <p>Therefore we find that in each <em>cohomology class</em> of curvatures, there is at least one representative which is an ordinary globally defined 2-form. Moreover, the pseudo-connections that map to such a representative are precisely the <em>genuine</em> connections, those for which the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_{i j})</annotation></semantics></math>-part of the cocycle vaishes.</p> </li> </ol> <p>So we see that ordinary connections on ordinary circle bundles are a means to model the homotopy 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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>H</mi> <mi>dR</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></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>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{H}_{diff}(X,\mathbf{B}U(1)) &\to& H_{dR}^2(X) \\ \downarrow && \downarrow \\ \mathbf{H}(X,\mathbf{B}U(1)) &\to& \mathbf{H}_{dR}(X,\mathbf{B}U(1)) } </annotation></semantics></math></div> <p>in a 2-step process: first the choice of a pseudo-connection realizes the bottom horizontal morphism as an <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>, and then second the restriction imposed by forming the ordinary pullback chooses from all pseudo-connections precisely the genuine connections.</p> <p>The general version of this story is discussed in detail at <a href="#Connections">differential cohomology in an (∞,1)-topos – Local (pseudo-)connections.</a></p> <h3 id="CircleBunlePseudoConnection">Circle bundles with pseudo-connection</h3> <p>In the above discussion of extracting ordinary connections on ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundles from the abstract topos-theoretic definition of differential cohomology, we argued that a certain homotopy pullback may be computed by choosing in the Cech-hypercohomology of the complex of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2_{cl}(-))</annotation></semantics></math> over a manifold <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> those cohomology representatives that happen to be represented by globally defined 2-forms 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>. We saw that the homotopy fiber of <em>pseudo-connections</em> over these 2-forms happened to have connected components indexed by <em>genuine</em> connections.</p> <p>But by the general abstract theory, up to isomorphism the differential cohomology computed this way is guaranteed to be independent of all such choices, which only help us to compute things.</p> <p>To get a feeling for what is going on, it may therefore be useful to re-tell the analgous story with pseudo-connections that are not genuine connections.</p> <p>By the very fact 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>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>←</mo><mo>≃</mo></mover><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1) \stackrel{\simeq}{\leftarrow} \mathbf{B}_{diff}U(1)</annotation></semantics></math> is a weak equivalence, it follows that every pseudo-connection is equivalent to an ordinary connection as cocoycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}_{diff}(G))</annotation></semantics></math>.</p> <p>If we choose a <a class="existingWikiWord" href="/nlab/show/partition+of+unity">partition of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rho_i \in C^\infty(X,\mathbb{R}))</annotation></semantics></math> subordinate to the cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math>, then we can construct the corresponding coboundary explicitly:</p> <p>let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><msub><mi>g</mi> <mi>ij</mi></msub><mo>,</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_i g_{ij}, a_{i j})</annotation></semantics></math> be an arbitrary pseudo-connection cocycle. Consider the Cech-hypercohomology coboundary given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sum_{i_0} \rho_{i_0} a_{i_0 i}, 0)</annotation></semantics></math>. This lands in the shifted cocycle</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>A</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mi>a</mi><msub><mo>′</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (A'_i := A_i + \sum_{i_0} \rho_{i_0} a_{i_0 i}, g_{i j}, a'_{i j}) \,, </annotation></semantics></math></div> <p>and we can find the new pseudo-components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mo>′</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a'_{i j}</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><msub><mo>′</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>A</mi><msub><mo>′</mo> <mi>j</mi></msub><mo>−</mo><mi>A</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>−</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a'_{i j} = A'_j - A'_i - d_{dR} g_{i j} \,. </annotation></semantics></math></div> <p>Using the computation</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo>−</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>j</mi></mrow></msub></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>a</mi> <mrow><mi>i</mi><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo>+</mo><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \sum_{i_0} \rho_{i_0} (a_{i_0 i} - a_{i_0 j} &= - \sum_{i_0} \rho_{i_0} (a_{i i_0} + a_{i_0 j} \\ & = \sum_{i_0} \rho_{i_0} a_{i j} \\ & = a_{i j} \end{aligned} </annotation></semantics></math></div> <p>we find that these indeed vanish.</p> <p>The most drastic example for this is a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> of a cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g = (g_{i j})</annotation></semantics></math> 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></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}_{diff} U(1) \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}U(1) } </annotation></semantics></math></div> <p>is one which takes all the ordinary curvature forms to vanish identically</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>:</mo><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla = (A_i := 0, g_{i j}, a_{i j}) \,. </annotation></semantics></math></div> <p>This fixes the pseudo-components to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i j} = - d g_{i j}</annotation></semantics></math>. By the above discussion, this pseudo-connection with vanishing connection 1-forms is equivalent, as a pseudo-connection, to the ordinary connection cocycle with connection forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><msub><mi>ρ</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mi>d</mi><msub><mi>g</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_i := \sum_{i_0} \rho_{i_0} d g_{i_0 i})</annotation></semantics></math>. This is a <a href="http://ncatlab.org/nlab/show/connection+on+a+bundle#Properties">standard formula</a> for equipping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundles with Cech cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g_{i j})</annotation></semantics></math> with a connection.</p> <h3 id="U1GroupoidBundle"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U(1)_0</annotation></semantics></math>-groupoid bundles</h3> <p>We saw above that the intrinsic coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math> yields ordinary de Rham cohomology in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> we <a href="#FlatCircleCohomology">have</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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{\flat}_{dR} \mathbf{B}U(1)</annotation></semantics></math> is given simply by the 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> sheaf of 1-forms, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\Omega^1(-) : CartSp^{op} \to Set \hookrightarrow sSet</annotation></semantics></math>. Accordingly we have 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 paracompact smooth manifold</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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \mathbf{\flat}_{dR}\mathbf{B}U(1)) = \Omega^1_{cl}(X) </annotation></semantics></math></div> <p>instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1_{dR}(X)</annotation></semantics></math>.</p> <p>There is a good reason for this discrepancy: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math> is the recipient of the intrinsic curvature characteristic morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mrow><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></mrow></msub><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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> curv_{\mathbf{B}^{n-1} U(1)} : \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</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></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^{n-1} U(1)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> (an <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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-2)</annotation></semantics></math>-gerbe without connection), the cohomology class of the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}^{n-1} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)</annotation></semantics></math> is precisely the obstruction to the existence of a flat extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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><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></mrow><annotation encoding="application/x-tex">X \to \mathbf{\flat} \mathbf{B}^{n-1} U(1) \to \mathbf{B}^{n-1} U(1)</annotation></semantics></math> for the original cocycle.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> this is the usual <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form of a <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math> it is curvature 3-form of a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>, etc. But for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> we have that the original cocycle is just a map of spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f : X \to U(1) \,. </annotation></semantics></math></div> <p>This may be understood as a cocycle for a <em>groupoid</em> principal bundle, for the 0-truncated groupoid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> as its space of objects. Such a cocycle extends to a flat cocycle precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <em>constant</em> as a function. The corresponding <strong>curvature 1-form</strong> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">d_{dR} f</annotation></semantics></math> and this is precisely the obstruction to constancy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> already, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is constant if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">d_{dR} f</annotation></semantics></math> vanishes. <em>Not</em> (necessarily) if it vanishes <em>in de Rham cohomology</em> .</p> <p>This is the simplest example of a general statement about curvatures of higher bundles: the curvature 1-form is not subject to gauge transformations.</p> <h3 id="AbGerbesConnection">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 with connection</h3> <p>We now generalize the <a href="#CircleBundlesConnection">above discussion</a> on the derivation of the notion of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections on circle bundles</a> from abstract topos-theory to a proof of the full <a href="#DeligneCohomologyTheorem">theorem above</a> on the derivation of general <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>.</p> <p>The main step is to model the double <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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></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><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mo>♭</mo></mstyle><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></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><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1) &\to& * \\ \downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1) &\to& \mathbf{\flat} \mathbf{B}^{n+1} U(1) \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}^{n+1} U(1) } </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><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> that gives the <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n U(1) \to \mathbf{B}^{n} U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)</annotation></semantics></math> which controls the <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos#GroupalCurvature">obstruction theory for flat connections</a> by a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> realized suitably as an ordinary pullback of fibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">]</mo><mover><mo>↪</mo><mi>Ξ</mi></mover><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, Ch_\bullet] \stackrel{\Xi}{\hookrightarrow} [CartSp^{op}, sSet]_{proj}</annotation></semantics></math>.</p> <div class="num_observation"> <h6 id="observation">Observation</h6> <p>We have commuting 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>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mover><mo>→</mo><mrow></mrow></mover><mfrac linethickness="0"><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>⊕</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mfrac><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>−</mo><mi>Id</mi></mrow></mover><mfrac linethickness="0"><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><mo>⊕</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mfrac><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>+</mo><mi>Id</mi></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>±</mo><mi>Id</mi></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>Id</mi><mo>+</mo><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mfrac linethickness="0"><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>⊕</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mfrac><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>−</mo><mi>Id</mi></mrow></mover><mfrac linethickness="0"><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><mo>⊕</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mfrac><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>+</mo><mi>Id</mi></mrow></mover><mi>⋯</mi><mfrac linethickness="0"><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mrow><mo>⊕</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mfrac><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub><mo>±</mo><mi>Id</mi></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>Id</mi><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mover><mo>→</mo><mrow></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Ξ</mi><mo stretchy="false">(</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Xi(\; 0\stackrel{}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \;) &\to& \Xi(\; C^\infty(-,U(1)) \stackrel{Id + d_{dR}}{\to} {C^\infty(-,U(1)) \atop \oplus \Omega^1(-)} \stackrel{d_{dR} - Id}{\to} {\Omega^1(-) \atop \oplus \Omega^2(-)} \stackrel{d_{dR} + Id}{\to} \cdots { \Omega^{n-1}(-) \atop \oplus \Omega^n(-)} \stackrel{d_{dR} \pm Id}{\to} \Omega^n(-) \;) \\ \downarrow && \downarrow^{\mathrlap{(Id, p_2, p_2, \cdots, p_2,d_{dR})}} \\ \Xi( \; 0 \stackrel{}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega_{cl}^{n+1}(-)) &\to& (C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega_{cl}^{n+1}(-) \;) \\ \downarrow && \downarrow \\ \Xi( \; 0 \to 0 \to 0 \to \cdots \to 0\;) &\to& \Xi( \; C^\infty(-,U(1)) \to 0 \to 0 \to \cdots \to 0 \;) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op},sSet]_{proj}</annotation></semantics></math> where</p> <ul> <li> <p>the objects are fibrant models for the corresponding objects in the above <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 diagram;</p> </li> <li> <p>the two right vertical morphisms are fibrations;</p> </li> <li> <p>the two squares are <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> squares.</p> </li> </ul> <p>Therefore this is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math> that realizes 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>-pullback in question in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_{(\infty,1)}(CartSp)</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a> preserves finite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>s, it therefore also presents the above <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 in <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>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = Sh_{(\infty,1)}(CartSp)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>For the lower square we had discussed this already <a href="#OrdinaryDeRham">above</a>. For the upper square the same type of reasoning applies. The main point is to find the chain complex in the top right such that it is a <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of the point and maps by a fibration onto our model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}\mathbf{B}^n U(1)</annotation></semantics></math>. The top right complex 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></mtd> <mtd></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) \\ &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \cdots & {}^{\mathrlap{id}}\nearrow& \\ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \Omega^2(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) } </annotation></semantics></math></div> <p>and the vertical map out of it into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \stackrel{d_{dR}}{\to} \Omega^{n+1}_{cl}(-)</annotation></semantics></math> is in positive degree the projection onto the lower row and in degree 0 the de Rham differential. This is manifestly surjective (by the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> applied to each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>) hence this is a fibration.</p> <p>The pullback object in the top left is in this notation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub><mo>:</mo><mo>=</mo><mi>Ξ</mi><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⊕</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msup><mrow></mrow> <mpadded width="0"><mi>id</mi></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1)_{diff,chn} := \Xi \left( \array{ C^\infty(-,U(1)) &\stackrel{d_{dR}}{\to}& \Omega^1(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) \\ \oplus &{}^{\mathrlap{id}}\nearrow& \oplus &{}^{\mathrlap{id}}\nearrow& \cdots & {}^{\mathrlap{id}}\nearrow& \\ \Omega^1(-) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^n(-) } \right) </annotation></semantics></math></div> <p>and in turn the top left vertical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msubsup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi> <mi>n</mi></msubsup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex">curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)</annotation></semantics></math> is in positive degree the projection on the lower row and in degree 0 the de Rham differential.</p> </div> <p>Notice that the evident forgetful morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>←</mo><mrow></mrow></mover><msubsup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi> <mi>n</mi></msubsup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1) \stackrel{}{\leftarrow} \mathbf{B}^n_{diff} U(1)</annotation></semantics></math> is indeed a weak equivalence.</p> <p>With this description we now have the proof of the <a href="#DeligneCohomologyTheorem">above theorem</a></p> <div id="ProofOfDeligneTheorem"> <h6 id="proof_equivalence_of_with_deligne_cohomology">Proof (equivalence of with Deligne cohomology)</h6> <p>Since the above model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msubsup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi> <mi>n</mi></msubsup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex">curv : \mathbf{B}_{diff}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)</annotation></semantics></math> is a fibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U_i\})</annotation></semantics></math> is cofibrant, also</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>Cartp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msubsup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>diff</mi> <mi>n</mi></msubsup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Cartp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{B}^n_{diff}U(1)) \to [Cartp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^n U(1)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> by the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>. Therefore the homotopy pullback is computed as an ordinary pullback.</p> <p>By the <a href="#OrdinaryDeRham">above discussion of de Rham cohomology</a> we have that we can assume the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>dR</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{dR}^{n+1}(X) \to [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{\flat}_{dR}\mathbf{B}^{n+1})</annotation></semantics></math> picks only cocylces represented by globally defined closed differential forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in \Omega^{n+1}(X)</annotation></semantics></math>.</p> <p>By the nature of the chain complexes apearing in the above proof, we see that the elements inm the fiber over such a globally defined form are precisely the cocycles with values only in the “upper row complex”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow><msub><mi>d</mi> <mi>dR</mi></msub></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\infty(-,U(1)) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \,. </annotation></semantics></math></div> <p>This is precisely the complex of sheaves that defines <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> in degree <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>.</p> </div> <h3 id="U1FromLieIntegration">Models from <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>-Lie integration</h3> <p>In the <a href="#FlatCircleCohomology">previous section</a> we discussed a model</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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>curv</mi> <mi>chn</mi></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1)_{diff,chn} &\stackrel{curv_{chn}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)_{chn} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^n U(1) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]</annotation></semantics></math> for the canoncal curvature characteristic class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex">curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> with the special property that it did model the abstract <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>-theoretic class under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> precisely in terms of the familiar <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> coefficient complex.</p> <p>There is another model for the curvature class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]</annotation></semantics></math>, one that is useful for constructing the <a href="#InfChernWeil">∞-Chern-Weil homomorphism</a> that maps from <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Smooth \infty Grpd</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-valued differential cohomology. This second model is the one naturally adapted to the construction of the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> from its <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b^{n-1} \mathbb{R}</annotation></semantics></math>. This is described at <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#LieIntegration">∞-Lie groupoid – Lie integration</a>.</p> <p>For distinguishing the two models, we will indicate the former one by the subscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>chn</mi></msub></mrow><annotation encoding="application/x-tex">{}_{chn}</annotation></semantics></math> and the one described now by the subscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex">{}_{simp}</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="convention">Convention</h6> <p>Here and in the following we adopt for differential forms on simplices the following notational convention:</p> <ul> <li> <p>by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(\Delta^n)</annotation></semantics></math> we denote the complex of smooth differential forms on the standard smooth <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>-simplex with <strong>sitting instants</strong>: for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-face of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^n</annotation></semantics></math> has a neighbourhood of its boundary such that the form restricted to that neighbourhood is constant in the direction perpendicular to that boundary.</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^k)_{vert}</annotation></semantics></math> for the complex of <a class="existingWikiWord" href="/nlab/show/vertical+differential+form">vertical differential form</a>s with respect to the trivial simplex bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U \times \Delta^k \to U</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, define the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</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> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{simp} \in [CartSp^{op}, sSet]</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>dgAlg</mi></msub><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1)_{simp} := \mathbf{cosk}_{n+1} ( (U, [k]) \mapsto Hom_{dgAlg}( CE(b^{n-1}\mathbb{R}), \Omega^\bullet(U \times \Delta^k)_{vert}) ) /\mathbf{B}^n \mathbb{Z} \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(b^{n-1}\mathbb{R})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b^{n-1}\mathbb{R}</annotation></semantics></math>, which is simply the graded-commutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>) on a single generator in degree <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> with vanishing differential.</p> <p>Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{cosk}_{n+1}(-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coskeleton">coskeleton</a>-operation and the quotient is by constant <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>-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n_{cl}(U \times \Delta^k)_{vert}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow></msub><mi>ω</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\int_{\Delta^n}\omega \in \mathbb{Z}</annotation></semantics></math>. We take the quotient as a quotient of abelian <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>s (the group operation is the addition of differential forms).</p> </div> <div class="num_lemma"> <h6 id="observation_2">Observation</h6> <p>Under the <a href="#DoldKan">Dold-Kan correspondence</a> the <a class="existingWikiWord" href="/nlab/show/Moore+complex">normalized chain complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>sim</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{sim}</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><msub><mi>N</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub><mo stretchy="false">/</mo><mo>∼</mo><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>k</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msubsup><mo>∂</mo> <mi>k</mi> <mo>*</mo></msubsup></mrow></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub><mo stretchy="false">/</mo><mo>∼</mo><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> N_\bullet(\mathbf{B}^n U(1)_{simp}) = ( \cdots \to \Omega^n_{cl}((-)\times \Delta^{n+1})_{vert}/\sim \stackrel{\sum_k (-1)^k \partial_k^* }{\to} \Omega^n_{cl}((-)\times \Delta^{n})_{vert}/\sim \to 0 \to \cdots \to 0 ) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>k</mi></msub><mo>:</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Δ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\partial_k : \Delta^n \to \Delta^{n+1}</annotation></semantics></math> denotes the embedding of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th face of the smooth <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>.</p> </div> <p>Here and in the following we indicate the homologically trivial part of the normalized chain complex of an <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>-coskeletal simplicial abelian group just by ellipses.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The evident <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> of differential forms over simplices</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mi>n</mi></msup></mrow></msub><mo>:</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_{\Delta^n} : \Omega^\bullet(U \times \Delta^n) \to \Omega^\bullet(U) </annotation></semantics></math></div> <p>yields a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_{\Delta^\bullet} : \mathbf{B}^n U(1)_{simp} \stackrel{\simeq}{\to} \mathbf{B}^n U(1) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math>, which is a weak equivalence.</p> </div> <p>This is discussed at <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Write</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\flat} \mathbf{B}^n \mathbb{R}_{simp}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>dgAlg</mi></msub><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}\mathbf{B}^n \mathbb{R}_{simp} := \mathbf{cosk}_{n+1} ( (U,[k]) \mapsto Hom_{dgAlg}( CE(b^{n-1}\mathbb{R}), \Omega^{\bullet}(U \times\Delta^k) ) ) </annotation></semantics></math></div> <p>and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{simp} \in [CartSp^{op}, sSet]</annotation></semantics></math> for the simplicial presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><msub><mi>Hom</mi> <mi>dgAlg</mi></msub><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>1</mn><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}_{simp} := \mathbf{cosk}_{n+1} ( (U,[k]) \mapsto Hom_{dgAlg}( CE(b^{n-1}\mathbb{R}), \Omega^{\bullet \geq 1, \bullet}(U \times\Delta^k) ) ) \,, </annotation></semantics></math></div> <p>where on the right we have the subcomplex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet(U \times \Delta^k)</annotation></semantics></math> on those forms that are non-vanishing on some vector field tangent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>.</p> </div> <p>At <a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#DifferentialCoefficientsOfLieInt">∞-Lie groupoid – Lie integrated ∞-groups – Differential coefficients</a> the following is shown:</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>The evident <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> over simplices induces morphisms of simplicial presheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub><mo>:</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub></mrow><annotation encoding="application/x-tex"> \int_{\Delta^\bullet} : \mathbf{\flat} \mathbf{B}^n U(1)_{simp} \to \mathbf{\flat} \mathbf{B}^n U(1)_{chn} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub></mrow><annotation encoding="application/x-tex"> \int_{\Delta^\bullet} : \mathbf{\flat}_{dR} \mathbf{B}^n U(1)_{simp} \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)_{chn} </annotation></semantics></math></div> <p>that are weak equivalences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">[CartSp^{op}, sSet]_{proj}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{diff,simp}</annotation></semantics></math> for the simplicial presheaf given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub><mo>:</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>cosk</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><msub><mo stretchy="false">)</mo> <mi>vert</mi></msub></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1)_{diff,simp} := \mathbf{cosk}_{n+1} ( U,[k] \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\leftarrow& CE(b^{n-1}\mathbb{R}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k ) &\leftarrow& W(b^{n-1} \mathbb{R}) } \right\} ) / \mathbf{B}^n \mathbb{Z} \,. </annotation></semantics></math></div> <p>Let the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>curv</mi> <mi>simp</mi></msub><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex"> curv_{simp} : \mathbf{B}^n U(1)_{diff,simp} \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1)_{simp} </annotation></semantics></math></div> <p>be the one given by postcomposition with the square of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</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>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(b^{n-1}\mathbb{R}) &\leftarrow& 0 \\ \uparrow && \uparrow \\ W(b^{n-1}\mathbb{R}) &\leftarrow& CE(b^n \mathbb{R}) } </annotation></semantics></math></div> <p>described at <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The set of square diagrams of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s above is over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,[k])</annotation></semantics></math> the set of <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>-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^k</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <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>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \omega</annotation></semantics></math> has no component with all legs along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^k</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The morphism given by <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a> of differential forms over the simplex factor fits into 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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>curv</mi> <mi>simp</mi></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mpadded width="0"><mo>∫</mo></mpadded> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mpadded width="0"><mo>∫</mo></mpadded> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>curv</mi> <mi>chn</mi></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1)_{diff,simp} &\stackrel{curv_{simp}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{simp} \\ \downarrow^{\mathrlap{\int}_{\Delta^\bullet}} && \downarrow^{\mathrlap{\int}_{\Delta^\bullet}} \\ \mathbf{B}^n U(1)_{diff,chn} &\stackrel{curv_{chn}}{\to}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn} } \,, </annotation></semantics></math></div> <p>where the vertical morphisms are weak equivalences.</p> </div> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>Fiber integration induces a weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub><mover><mo>→</mo><mo>≃</mo></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \int_{\Delta^\bullet} : \mathbf{B}^n \mathbb{R}_{diff,simp} \stackrel{\simeq}{\to} \mathbf{B}^n \mathbb{R}_{diff, chn} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>ℝ</mi> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n \mathbb{R}_{diff,simp}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>ℝ</mi> <mi>simp</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp} \to \mathbf{\flat}\mathbf{B}^{n+1} \mathbb{R}_{simp}</annotation></semantics></math> along the evident forgetful morphism from</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>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">{</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>←</mo><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (U,[k]) \mapsto \{\Omega^\bullet(U \times \Delta^k) \leftarrow W(b^{n-1} \mathbb{R})\} \,. </annotation></semantics></math></div> <p>This forgetful morphism is evidently a fibration (because it is a degreewise surjection under Dold-Kan), hence this pullback models the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo>→</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R} \to \mathbf{\flat} \mathbf{B}^{n+1} \mathbb{R}</annotation></semantics></math>. Since by the above fiber integration gives a weak equivalence of pulback diagrams the claim follows.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>conn</mi><mo>,</mo><mi>simp</mi></mrow></msub><mo>↪</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn,simp} \hookrightarrow \mathbf{B}^n U(1)_{diff,simp}</annotation></semantics></math> for the sub-presheaf which over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,[k])</annotation></semantics></math> is the set of those forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">U \times \Delta^k</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">d \omega</annotation></semantics></math> has no leg along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^k</annotation></semantics></math>.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Under fiber integration over simplices, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>conn</mi><mo>,</mo><mi>simp</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn,simp}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>-complex.</p> </div> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>conn</mi><mo>,</mo><mi>simp</mi></mrow></msub></mtd> <mtd><mover><mrow><msup><mo>→</mo> <mo>≃</mo></msup></mrow><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></mover></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>∞</mn></msubsup></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>connection</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>simp</mi></mrow></msub></mtd> <mtd><mover><mrow><msup><mo>→</mo> <mo>≃</mo></msup></mrow><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>diff</mi><mo>,</mo><mi>chn</mi></mrow></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>pseudo</mi><mo>−</mo><mi>connection</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>simp</mi></msub></mtd> <mtd><mover><mrow><msup><mo>→</mo> <mo>≃</mo></msup></mrow><mrow><msub><mo>∫</mo> <mrow><msup><mi>Δ</mi> <mo>•</mo></msup></mrow></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>chn</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>curvature</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^n U(1)_{conn,simp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& U(1)(n)_D^\infty &&& connection \\ & \nearrow & \downarrow && \downarrow \\ \hat X &\to& \mathbf{B}^n U(1)_{diff,simp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& \mathbf{B}^n U(1)_{diff,chn} &&& pseudo-connection \\ & \searrow & \downarrow && \downarrow \\ && \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{simp} &\stackrel{\int_{\Delta^\bullet}}{\to^\simeq}& \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1)_{chn} &&& curvature } \,. </annotation></semantics></math></div> <p>In summary this gives us the following alternative perspective on connections on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s: such a connection is a cocycle with values in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n \mathbb{Z}</annotation></semantics></math>-quotient of the <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>-coskeleton of the simplicial presheaf which over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,[k])</annotation></semantics></math> is the set of diagrams of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</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><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>underlying</mi><mspace width="thickmathspace"></mspace><mi>cocycle</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>connection</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>curvature</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(U)\otimes \Omega^\bullet(\Delta^k) &\leftarrow& CE(b^{n-1}\mathbb{R}) &&& underlying\;cocycle \\ \uparrow && \uparrow \\ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^k) &\leftarrow& W(b^{n-1}\mathbb{R}) &&& connection \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes C^\infty(\Delta^k) &\leftarrow& CE(b^n \mathbb{R}) &&& curvature } </annotation></semantics></math></div> <p>where the restriction to the top morphism is the underlying cocycle and the restriction to the bottom morphism the curvature form.</p> <p>The generalization to such diagram cocycles from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b^{n-1}\mathbb{R}</annotation></semantics></math> to general <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>s <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">\mathfrak{g}</annotation></semantics></math> we discuss below in <a href="#InfinityLieAlgebraConnection">∞-Lie algebra valued connections</a>.</p> <h3 id="InHomtopyTypeTheory">In homotopy type theory</h3> <p>We discuss the formulation of the above in the <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>-<a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> of the <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><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <p>Given the two functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex"> \Omega^{(n+1)}_{cl} \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1} U(1) </annotation></semantics></math></div> <p>(inclusion of the set of closed <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>-forms into the <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>-groupoid of de Rham cocycles)</p> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex"> curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1) </annotation></semantics></math></div> <p>(the universal curvature class / Maurer-Cartan form of the circle <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>-group)</p> <p>the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of 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 with connection from <a href="#DifferentialCohomology">above</a> is expressed in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≃</mo><mrow><mo>{</mo><mi>P</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>∃</mo><mi>F</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>.</mo><mi>curv</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1)_{conn} \simeq \left\{ P : \mathbf{B}^n U(1) | \exists F \in \Omega^{n+1}_{cl} . curv(P) = F \right\} \,. </annotation></semantics></math></div> <p>Spelled out this expresses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> as</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)</annotation></semantics></math> of</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>cl</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^n_{cl}</annotation></semantics></math> of</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/substitution">substitution</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi></mrow><annotation encoding="application/x-tex">curv</annotation></semantics></math> of</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent</a> <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow><annotation encoding="application/x-tex">\mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)</annotation></semantics></math>.</p> </li> </ul> <p>See the discussion at <em><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></em> for why this is indeed interpreted by the homotopy pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mo>♭</mo></mstyle> <mi>dR</mi></msub><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></mrow></msub><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1) \times_{\mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)} \Omega^{n+1}_{cl}</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> a 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 with connection in the sense discussed here is indeed an ordinary hermitian <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> with connection.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> a circle 2-bundle with connection is equivalent to a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> with connection (at least over a smooth manifold. Over an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> the definition given here does produce the correct <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>, which is different from that of bundle gerbes that are equivariant in the ordinary sense.)</p> <p>Classes of examples of higher circle bundles with connection are provided by <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> which provides homomorphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>conn</mi></msub><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><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{conn}(X,\mathbf{B}G) \to \mathbf{H}_{diff}(X, \mathbf{B}^n U(1)) \,. </annotation></semantics></math></div> <p>See for instance</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a></li> </ul> <p>for the class of circle 3-bundles that arise as differential refinements of degree 4 <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es such as the <a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a>.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle">Chern-Simons circle 7-bundle</a>.</li> </ul> <h2 id="properties">Properties</h2> <h3 id="moduli">Moduli</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles+with+connection">line n-bundles with connection</a> on <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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <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></strong></p> <table><thead><tr><th><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></th><th><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau n-fold</a></th><th><a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/flat+infinity-connection">flat</a>/degree-0 n-bundles</th><th><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation moduli</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a>/<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> in <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>/<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>/<a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">Calabi-Yau 3-fold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+3-bundle">line 3-bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">CY3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a href="differential+cohomology+diagram#DeligneCoefficients">differential cohomology diagram – Examples – Deligne coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+geometry">contact geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+U%281%29-gauge+theory">higher U(1)-gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+infinity-connection">principal infinity-connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curving">curving</a>. <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Super+Gerbes">Super Gerbes</a></p> </li> </ul> <h2 id="references">References</h2> <p>The above discussion is from</p> <ul> <li id="dcct"><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> .</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> as classifying highe <a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a> (<a class="existingWikiWord" href="/nlab/show/bundle+2-gerbes">bundle 2-gerbes</a>, etc.) <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">with connection</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pawel+Gajer">Pawel Gajer</a>, <em>Geometry of Deligne cohomology</em>, Invent. Math., 127(1):155-207 (1997) (<a href="http://arxiv.org/abs/alg-geom/9601025">arXiv:alg-geom/9601025</a>, <a href="https://doi.org/10.1007/s002220050118">doi:10.1007/s002220050118</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 3, 2020 at 15:01:41. 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