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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> Deligne cohomology </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/6013/#Item_26" 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="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="topos_theory"><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 Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><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></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><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> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-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/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><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">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#PreliminariesOnSheafCohomology'>Preliminaries on sheaf cohomology</a></li> <li><a href='#TheDeligneComplex'>The Deligne complex</a></li> <li><a href='#CechDeligneComplex'>Cech-Deligne complex</a></li> <li><a href='#cup_product_in_deligne_cohomology'>Cup product in Deligne cohomology</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#CharacteristicMaps'>Curvature and characteristic classes</a></li> <li><a href='#TheChernCharacter'>The Chern character</a></li> <li><a href='#ExactSequenceForCurvatureAndCharacteristicClass'>The exact sequences for curvature and characteristic classes</a></li> <li><a href='#TheExactDifferentialCohomologyHexagon'>The exact differential cohomology hexagon</a></li> <li><a href='#GAGA'>GAGA</a></li> <li><a href='#moduli_and_deformation_theory'>Moduli and deformation theory</a></li> <li><a href='#interpretation_in_terms_of_higher_parallel_transport'>Interpretation in terms of higher parallel transport</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p><em>Deligne cohomology</em> – or <em><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Deligne</a>-<a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Beilinson</a> cohomology</em> – is an <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> that models <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, a refinement of the <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in a <a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant</a> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> (modeling <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>) by <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> data.</p> <p>The <em>Deligne complex</em> is like a truncated <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> but, crucially, with the sheaf of 0-forms – the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> - replaced by the <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒪</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{O}^\times</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/exponential+map">exponential map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><msup><mi>𝒪</mi> <mo>×</mo></msup><mover><mo>⟶</mo><mrow><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mover><mo>⟶</mo><mi>d</mi></mover><mi>⋯</mi><mover><mo>⟶</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left[ \mathcal{O}^\times \stackrel{d log}{\longrightarrow} \Omega^1 \stackrel{d}{\longrightarrow} \Omega^2 \stackrel{d}{\longrightarrow} \cdots \stackrel{d}{\longrightarrow} \Omega^n \right] \,. </annotation></semantics></math></div> <p>Deligne cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{conn}(X, \mathbb{Z})</annotation></semantics></math> 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> is the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in this <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/sheaves+of+abelian+groups">sheaves of abelian groups</a> (“<a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a>”).</p> <p>This was introduced in (<a href="#Deligne71">Deligne 71, Section 2.2</a>) in the context of <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> (hence using <a class="existingWikiWord" href="/nlab/show/holomorphic+differential+forms">holomorphic differential forms</a>) as a <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge-filtered</a> version of <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>, designed to be a target for the <a class="existingWikiWord" href="/nlab/show/Beilinson+regulator">Beilinson regulator</a> from <a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a>. But the form of the definition applies more generally, in particular also in smooth <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, a fact amplified and popularized in (<a href="#Brylinski93">Brylinski 93</a>).</p> <p>In smooth <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> the typical minor variant has the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>U</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</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></mrow><annotation encoding="application/x-tex">\underline{U}(1) = C^\infty(-,U(1))</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-valued <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</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><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mover><mo>⟶</mo><mi>d</mi></mover><mi>⋯</mi><mover><mo>⟶</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left[ C^\infty(-,U(1)) \stackrel{d log}{\longrightarrow} \Omega^1 \stackrel{d}{\longrightarrow} \Omega^2 \stackrel{d}{\longrightarrow} \cdots \stackrel{d}{\longrightarrow} \Omega^n \right] \,. </annotation></semantics></math></div> <p>Given any <a class="existingWikiWord" href="/nlab/show/manifold">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>, then the resulting complex of abelian groups is, under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a>, the <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> whose underlying <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle (n-1)-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal infinity-bundle</a> is trivialized. Passing to the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> implicitly corresponds to considering the <a class="existingWikiWord" href="/nlab/show/infinity-stackification">infinity-stackification</a> of this <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>-groupoid valued presheaf, and in this way Deligne cohomology computes equivalence classes of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>. Another way to say this is that under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> and <a class="existingWikiWord" href="/nlab/show/infinity-stackification">infinity-stackification</a>, the above Deligne complex defines a <a class="existingWikiWord" href="/nlab/show/smooth+infinity-stack">smooth infinity-stack</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>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn}</annotation></semantics></math> which is the universal <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli infinity-stack</a> for <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>, and Deligne cohomology computes the <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps (of <a class="existingWikiWord" href="/nlab/show/infinity-stacks">infinity-stacks</a>) into this (<a href="#FSS10">FSS 10</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></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><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^{n+1}_{conn}(X,\mathbb{Z}) \simeq \pi_0(X \to \mathbf{B}^n U(1)_{conn}) \,. </annotation></semantics></math></div> <p>In this way Deligne cohomology, or rather the collection of Deligne <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the Deligne complex that defines it, is considerably richer than other models for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> such as <a class="existingWikiWord" href="/nlab/show/Cheeger-Simons+differential+characters">Cheeger-Simons differential characters</a>, which see only the <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a>, but not the full <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli n-stack</a>.</p> <p>Explicitly, computing the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> via <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> gives that a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{A}</annotation></semantics></math> on some <a class="existingWikiWord" href="/nlab/show/space">space</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 represented with respect to a suitable <a class="existingWikiWord" href="/nlab/show/open+cover">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> by a collection of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> and functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo>¯</mo></mover><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mi>A</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo>}</mo></mrow> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></msubsup><mo>∪</mo><mo stretchy="false">{</mo><msub><mi>g</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>∈</mo><msup><mi>𝒪</mi> <mo>×</mo></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</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><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \overline{A} = \left\{ A_{i_0, \cdots, i_k} \in \Omega^{n-k}(U_{i_0, \cdots i_k}) \right\}_{k = 0}^{n} \cup \{ g_{i_0, \cdots, i_n} \in \mathcal{O}^\times(U_{i_0, \cdots, i_{n+1}}) \} </annotation></semantics></math></div> <p>such that the failure 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><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k+1)</annotation></semantics></math>-forms to glue on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-fold intersections of charts is given by the de Rham differential 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><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k)</annotation></semantics></math>-forms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow> <mi>k</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>j</mi></msup><msub><mi>A</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>i</mi> <mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><msub><mi>A</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sum_{j = 0}^k (-1)^j A_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} = d_{dR} A_{i_0, \cdots, i_{k+1}} \,. </annotation></semantics></math></div> <p>This evidently generalizes the familiar Cech cocycle data for traditional <a class="existingWikiWord" href="/nlab/show/line+bundles+with+connection">line bundles with connection</a>.</p> <p>As the notation indicates, Deligne cohomology is a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> refinement of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, exhibited by a canonical forgetful map</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><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \searrow \\ && H^{n+1}(X,\mathbb{Z}) } </annotation></semantics></math></div> <p>which is induced by the evident morphism of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>. This is one map in an exact <a class="existingWikiWord" href="/nlab/show/differential+hexagon">differential hexagon</a> which exhibits Deligne cohomology as the differential refinement of ordinary integral cohomology by <a class="existingWikiWord" href="/nlab/show/closed+differential+form">closed</a> <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> data.</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><mn>0</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msubsup><mi>Ω</mi> <mi>int</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></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></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>a</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>DD</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mi>n</mi></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></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>β</mi></munder></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mrow><mi>connection</mi><mspace width="thickmathspace"></mspace><mi>forms</mi></mrow></mrow><mrow><mrow><mi>on</mi><mspace width="thickmathspace"></mspace><mi>trivial</mi><mspace width="thickmathspace"></mspace><mi>bundles</mi></mrow></mrow></mfrac></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>de</mi><mspace width="thickmathspace"></mspace><mi>Rham</mi><mspace width="thickmathspace"></mspace><mi>differential</mi></mrow></mover></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>curvature</mi></mrow><mrow><mi>forms</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>curvature</mi></mpadded></msub></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>de</mi><mspace width="thickmathspace"></mspace><mi>Rham</mi><mspace width="thickmathspace"></mspace><mi>theorem</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mfrac linethickness="0"><mrow><mi>flat</mi></mrow><mrow><mrow><mi>differential</mi><mspace width="thickmathspace"></mspace><mi>forms</mi></mrow></mrow></mfrac></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mrow><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>bundles</mi></mrow></mrow><mrow><mrow><mi>with</mi><mspace width="thickmathspace"></mspace><mi>connection</mi></mrow></mrow></mfrac></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>rationalized</mi></mrow><mrow><mi>bundles</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>topol</mi><mo>.</mo><mspace width="thickmathspace"></mspace><mi>class</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mi>Chern</mi><mspace width="thickmathspace"></mspace><mi>character</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>flat</mi></mrow><mrow><mi>connections</mi></mrow></mfrac></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>Bockstein</mi><mspace width="thinmathspace"></mspace><mi>homomorphism</mi></mrow></munder></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>shape</mi></mrow><mrow><mrow><mi>of</mi><mspace width="thickmathspace"></mspace><mi>bundle</mi></mrow></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 & && && && & 0 \\ & \searrow && && && \nearrow \\ && \Omega^{n}(X)/\Omega^n_{int}(X) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && {}^{\mathllap{F_{(-)}}}\nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^{n+1}_{conn}(X,\mathbb{Z}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow^{\mathrlap{DD}} && \nearrow \\ && H^{n}(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) \\ & \nearrow && && && \searrow \\ 0 & && && && & 0 \\ \\ && {{connection\;forms} \atop {on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature} \atop {forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat} \atop {differential\;forms}} && && {{geometric\;bundles} \atop {with \;connection}} && && {{rationalized} \atop {bundles}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{flat} \atop {connections}} && \underset{Bockstein\,homomorphism}{\longrightarrow} && {{shape} \atop {of\;bundle}} } </annotation></semantics></math></div> <h2 id="definition">Definition</h2> <p>In any context where these symbols make the evident sense, the Deligne complex of 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> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒪</mi> <mo>×</mo></msup><mover><mo>→</mo><mrow><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo>→</mo><mi>⋯</mi><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{O}^\times \stackrel{d log}{\to}\Omega^1 \stackrel{d}{\to} \Omega^2 \to \cdots \to \Omega^n</annotation></semantics></math>, and Deligne cohomology 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> is the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in this complex.</p> <p>More generally one considers any <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow \mathcal{O}</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a>, then the corresponding Deligne complex is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>𝒪</mi><mover><mo>→</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo>→</mo><mi>⋯</mi><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">A \hookrightarrow \mathcal{O} \stackrel{d}{\to} \Omega^1 \to \cdots \to \Omega^n</annotation></semantics></math>.</p> <p>For definiteness we consider here in detail the Deligne complex in the context of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> modeled on <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. All variants work essentially directly analogously, but it may be useful to have a specific case in hand. This discussion overlaps with and is put into a broader context at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+principal+connections">geometry of physics – principal connections</a></em>.</p> <h3 id="PreliminariesOnSheafCohomology">Preliminaries on sheaf cohomology</h3> <p>In order to be somewhat self-contained, this section reviews some elements of <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> specified to the context that we need. It also sets up some notation. The definition of the Deligne complex itself is <a href="#TheDeligneComplex">below</a> in def. <a class="maruku-ref" href="#TheSmoothDeligneComplex"></a>.</p> <div class="num_defn" id="Smooth0Types"> <h6 id="definition_2">Definition</h6> <p>Write <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> for the <a class="existingWikiWord" href="/nlab/show/site">site</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</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><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> between these;</p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> is given by “differentially <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a>”, those <a class="existingWikiWord" href="/nlab/show/open+covers">open covers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>s all whose finite non-empty intersections are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, hence again to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> </li> </ul> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(CartSp) = Func(CartSp^{op},Set)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> over this site. Write</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>0</mn><mi>Type</mi><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Smooth0Type \coloneqq Sh(CartSp) </annotation></semantics></math></div> <p>for its <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, also called the <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+spaces">smooth spaces</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Instead of the site <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of def. <a class="maruku-ref" href="#Smooth0Types"></a> one could use the site <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> of all smooth manifolds. All of the statements and constructions in the following go through in that case just as well. In fact <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> is a <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a> of <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a>. On the one hand this implies that the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> is the same for both sites, but on the other hand means that it is convenient to restrict to the much “smaller” site of Cartesian spaces. In fact since the <a class="existingWikiWord" href="/nlab/show/stalks">stalks</a> of sheaves over smooth manifolds are evaluations on small <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> and since every open ball is <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to a Cartesian space, many statements that are true (only) stalkwise over <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> are actually true globally over <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>. It is the “<a class="existingWikiWord" href="/nlab/show/descent">descent</a>” or “<a class="existingWikiWord" href="/nlab/show/infinity-stackification">infinity-stackification</a>” which is implicit in abelian sheaf cohomology that takes care of these global statements over <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> to translate into the same local statements as one gets over <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a>.</p> </div> <div class="num_example" id="RepresentableSheavesAndTheirConstantVersion"> <h6 id="example">Example</h6> <p>The assignment <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>ℝ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,\mathbb{R}) \colon \mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n,\mathbb{R})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> with values in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>. Since this is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> we are entitled to identify this with the <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>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>) itself, and just write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∈</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \in Smooth0Type</annotation></semantics></math>.</p> <p>Similarly 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> any other <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, it represents a sheaf on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and we just write <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>0</mn><mi>Type</mi></mrow><annotation encoding="application/x-tex">X \in Smooth0Type</annotation></semantics></math> for this.</p> <p>Of particular interest below is the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = S^1 = \mathbb{R}/\mathbb{Z} = U(1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>, to be regarded as the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>.</p> <p>Notice that traditionally the sheaf represented by <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></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> is indicated by an underline as in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>ℝ</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{\mathbb{R}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>U</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{U}(1)</annotation></semantics></math>, but we do not follow this tradition here.</p> <p>Instead, if we consider the other sheaf that might deserve to be denoted by <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>, namely the <a class="existingWikiWord" href="/nlab/show/constant+sheaf">constant sheaf</a> on <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>, which sends each <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> to the set underlying <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>, then we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\flat \mathbb{R}</annotation></semantics></math> for that. Similarly</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi></mrow><annotation encoding="application/x-tex"> \flat U(1) \in Smooth0Type </annotation></semantics></math></div> <p>is the sheaf sending each test manifold to the set of points in the circle, and each smooth function between Cartesian spaces to the identity function on that set.</p> </div> <div class="num_example" id="SheavesOfFormsAndDeRhamDifferential"> <h6 id="example_2">Example</h6> <p>For <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> write</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>Ω</mi></mstyle> <mi>k</mi></msup><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^k \in Sh(CartSp) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>↦</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^k \colon U \mapsto \Omega^k(U)</annotation></semantics></math> of smooth <a class="existingWikiWord" href="/nlab/show/differential+n-forms">differential k-forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> extends to a morphism of sheaves</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>d</mi></mstyle><mo lspace="verythinmathspace">:</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mo>→</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d} \colon \mathbf{\Omega}^k \to \mathbf{\Omega}^{k+1} \,. </annotation></semantics></math></div> <p>For positive <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> its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is the <a class="existingWikiWord" href="/nlab/show/subfunctor">sub-sheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mi>k</mi></msubsup><mo>↪</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> 0 \to \mathbf{\Omega}^k_{cl} \hookrightarrow \mathbf{\Omega}^k \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1} </annotation></semantics></math></div> <p>of closed differential forms; and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math> its kernel is the <a class="existingWikiWord" href="/nlab/show/subfunctor">sub-sheaf</a> of constant functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mo>♭</mo><mi>ℝ</mi><mo>↪</mo><mi>ℝ</mi><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to \flat \mathbb{R} \hookrightarrow \mathbb{R} \stackrel{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^1 \,. </annotation></semantics></math></div></div> <p>In the background, what plays a role for the following is the full <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>. This receives a map from the following coarse <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a>, which is all that is necessary for the present purpose.</p> <div class="num_defn" id="FibrantOnjectStructureOnChSmooth"> <h6 id="definition_3">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo>,</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ch_+(Smooth0Type) = Ch_+(Sh(CartSp)) = Sh(CartSp,Ch_+) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> in the smooth sheaves of def. <a class="maruku-ref" href="#Smooth0Types"></a>, hence for the <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> whose objects are <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> on <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>.</p> <p>Regard this as equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> induced by the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+chain+complexes">projective model structure on chain complexes</a>, hence with classes of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> labeled as follows: a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f_\bullet \colon A_\bullet \to B_\bullet</annotation></semantics></math> is called</p> <ul> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a></em> if it is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>, hence if it induces <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on (<a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of) <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups;</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a></em> if it is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> (of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a>) in positive degree.</p> </li> </ul> </div> <div class="num_remark" id="HomotopyFibersViaFactorizationLemma"> <h6 id="remark_2">Remark</h6> <p>For our purpose the main use of this structure is to compute <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> via the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a>. Namely</p> <ol> <li> <p>every chain map may be replaced, up to weak equivalence of its domain, by a fibration;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of a chain map is the ordinary fiber of any of its fibration replacements.</p> </li> </ol> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>That the properties in def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a> are interpreted in sheaves simply means that they apply <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>-wise. For instance a morphism of chain complexes of presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f_\bullet \colon A_\bullet \to B_\bullet</annotation></semantics></math> is a weak equivalence precisely if the underlying presheaf of chain complexes becomes a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> for each point <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 each Cartesian space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> after restricting (via the presheaf structure maps) to a small enough <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> of that point. Similarly for epimorphisms.</p> </div> <div class="num_remark" id="HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes"> <h6 id="remark_4">Remark</h6> <p>There is a canonical map of <a class="existingWikiWord" href="/nlab/show/homotopy+theories">homotopy theories</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math> to the full <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> which is given by applying the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> followed by <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a>. The key point is that this map preserves <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a>, which is the <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a> that already captures most of the relevant properties of the Deligne complex. In this way it is sufficient to concentrate on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math> for much of the theory.</p> </div> <p>When writing out the components of chain complexes we will use square brackets always denote the group in degree-0 to the far right, and the group in degree <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> being <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> steps to the left from that.</p> <div class="num_example" id="DeloopingsOfAbelianSheaf"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo>,</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Ab(Smooth0Type) = Sh(CartSp,Ab)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> and 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> we 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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>≔</mo><mi>A</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mrow><mo>[</mo><mi>A</mi><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n A)_{\bullet} \coloneqq A[-n] = \left[ A \to 0 \to \cdots \to 0 \right] </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of sheaves concentrated on <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> 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>.</p> </div> <div class="num_example" id="LogDiffInSmoothContext"> <h6 id="example_4">Example</h6> <p>There is a weak equivalence, def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><mo>]</mo></mrow><mover><mo>⟶</mo><mo>≃</mo></mover><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \left[\mathbb{Z}\to \mathbb{R} \right] \stackrel{\simeq}{\longrightarrow} U(1) </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mi>ℤ</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><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{ \mathbb{Z} &\hookrightarrow& \mathbb{R} \\ \downarrow && \downarrow^{\mathrlap{mod\,\mathbb{Z}}} \\ 0 &\to& U(1) } </annotation></semantics></math></div> <p>(which is just the <a class="existingWikiWord" href="/nlab/show/exponential+sequence">exponential sequence</a> regarded as a chain map). That this is a weak equivalence is the statement that every smooth <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 function is locally the quotient of a smooth <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>-valued function by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-valued function. In fact on <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> this is of course true even globally.</p> <p>The de Rham differential <a class="existingWikiWord" href="/nlab/show/extension">extends</a> through this equivalence to produce a morphism denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow><annotation encoding="application/x-tex">\mathbf{d} log</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{d}}{\longrightarrow}& \mathbf{\Omega}^1 \\ \downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{\mathbf{d} log}} \\ U(1) \,. } </annotation></semantics></math></div> <p>On a given <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 function this is given by representing the function by a smooth <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>-valued function under mod-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-reduction (which is always possible over a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>) and applying the de Rham differential to that.</p> <p>The <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of that is the constant sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat U(1)</annotation></semantics></math> of example <a class="maruku-ref" href="#RepresentableSheavesAndTheirConstantVersion"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mo>♭</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><mover><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to \flat U(1) \hookrightarrow U(1) \stackrel{\mathbf{d} log}{\longrightarrow} \mathbf{\Omega}^1 \,. </annotation></semantics></math></div></div> <p>Under addition of differential forms, the sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^k</annotation></semantics></math> of example <a class="maruku-ref" href="#SheavesOfFormsAndDeRhamDifferential"></a> becomes <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a>, and we will implicitly understand them this way now.</p> <div class="num_defn" id="DeRhamResolutionOfConstantFunctions"> <h6 id="definition_4">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub><mo>∈</mo><mi>Ch</mi><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \in Ch(Smooth0Type)</annotation></semantics></math> for the complex of sheaves given by the truncated <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub><mo>≔</mo><mrow><mo>[</mo><mi>ℝ</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \coloneqq \left[ \mathbb{R} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^2 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n+1}_{cl} \right] \,. </annotation></semantics></math></div></div> <div class="num_prop" id="TheDeRhamResolutionOfConstantFunctions"> <h6 id="proposition">Proposition</h6> <p>The morphism</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><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>⟶</mo><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet \longrightarrow \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet </annotation></semantics></math></div> <p>given by the canonical <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</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><mo>♭</mo><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></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><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat \mathbb{R} &\stackrel{}{\to}& 0 &\stackrel{}{\to}& 0 &\stackrel{}{\to}& \cdots &\stackrel{}{\to}& 0 \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1}_{cl} } </annotation></semantics></math></div> <p>is a weak equivalence in the sense of def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+lemma">Poincaré lemma</a>. This <em>is</em> the Poincaré Lemma.</p> </div> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_\bullet \in Ch_+(Smooth0Type)</annotation></semantics></math> serves as the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> for an <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> theory on smooth manifolds. Abelian sheaf cohomology has a general abstract characterization (see at <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em>) in terms of <a class="existingWikiWord" href="/nlab/show/derived+hom-spaces">derived hom-spaces</a>. For definiteness, we recall the model for this construction given by <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> .</p> <div class="num_defn" id="CechComplex"> <h6 id="definition_5">Definition</h6> <p><strong>(Čech complex)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_\bullet \in Ch_+(Smooth0Type)</annotation></semantics></math> be a sheaf of chain complexes. 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> 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>, i.e. an open cover such that each finite non-empty intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i_0, \cdots, i_k}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>/<a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>.</p> <p>The <strong>Čech cochain complex</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet((X,\{U_i\}),A_\bullet)</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> with respect 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> and with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> is in degree <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> given by the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></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> <mi>k</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≔</mo><msub><mo>⊕</mo> <mfrac linethickness="0"><mrow><mrow><mi>l</mi><mo>,</mo><mi>n</mi></mrow></mrow><mrow><mrow><mi>k</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>l</mi></mrow></mrow></mfrac></msub><msub><mo>⊕</mo> <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><msub><mi>A</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><msub><mi>U</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^k((X,\{U_i\}),A_\bullet) \coloneqq \oplus_{{l,n} \atop {k = n-l}} \oplus_{i_0, i_1, \cdots, i_n} A_l(U_{i_0, \cdots, i_n}) </annotation></semantics></math></div> <p>which is the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> on the given intersections as indicated; and whose <a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>C</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \colon C^{k}((X,\{U_i\}),A_\bullet) \longrightarrow C^{k+1}((X,\{U_i\}),A_\bullet) </annotation></semantics></math></div> <p>is defined componentwise (see at <a class="existingWikiWord" href="/nlab/show/matrix+calculus">matrix calculus</a> for conventions on maps between <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>) by</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>d</mi><mi>a</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd> <mtd><mo>≔</mo><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>A</mi></msub><mi>a</mi><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mi>δ</mi><mi>a</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><msub><mo>∂</mo> <mi>A</mi></msub><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mn>0</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>j</mi></msup><msub><mi>a</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>i</mi> <mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} (d a)_{i_0, \cdots, i_{k+1}} & \coloneqq (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}} \\ & \coloneqq \partial_A a_{i_0, \cdots, i_{k+1}} + (-1)^k \sum_{0 \leq j \leq k+1} (-1)^{j} a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} |_{U_{i_0, \cdots, i_{k+1}}} \end{aligned} </annotation></semantics></math></div> <p>where on the right the sum is over all components 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> obtained via the canonical restrictions obtained by discarding one of the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math> subscripts.</p> <p>The <strong>Cech cohomology</strong> groups 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> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> <strong>relative to the given cover</strong> are the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> groups of the Cech complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Cech</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{Cech}^k((X,\{U_i\}), A_\bullet) \coloneqq H^k(C^\bullet((X,\{U_i\}),A_\bullet)) \,. </annotation></semantics></math></div> <p>The <strong>Cech cohomology</strong> groups as such are the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> (“<a class="existingWikiWord" href="/nlab/show/direct+limit">direct limit</a>”) of these groups over refinements of covers</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Cech</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≔</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow></msub><msubsup><mi>H</mi> <mi>Cech</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^k_{Cech}(X, A_\bullet) \coloneqq \underset{\longrightarrow}{\lim}_{\{U_i \to X\}} H_{Cech}^k((X,\{U_i\}), A_\bullet) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Often Cech cohomology is considered for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> is concentrated in a single degree, in which case the first term in the sum defining the differential in def. <a class="maruku-ref" href="#CechComplex"></a> disappears. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> is not concentrated in a single degree, then for emphasis one sometimes speaks of <em><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></em>. This is the case of relevance for Deligne cohomology.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>The Cech chain complex in def. <a class="maruku-ref" href="#CechComplex"></a> is the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of the <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> whose vertical differential is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> and whose horizontal differential is the <em>Cech differential</em> <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> given by alternating sums over restrictions along patch inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∂</mo> <mi>A</mi></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∂</mo> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msub><mo>⊕</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∂</mo> <mi>A</mi></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∂</mo> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msub><mo>⊕</mo> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><msub><mi>A</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_1(U_{i_0}) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_0, i_1} A_1(U_{i_0, i_1}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_0(U_i) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_0, i_1} A_0(U_{i_0, i_1}) &\stackrel{\delta}{\longrightarrow}& \cdots } </annotation></semantics></math></div></div> <div class="num_remark" id="GoodCoverCechCohomologyIsHomotopicallyGood"> <h6 id="remark_7">Remark</h6> <p>For analyzing the properties of Deligne cohomology <a href="#Properties">below</a>, all one needs is the following fact about <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a>, which is discussed for instance at <em><a class="existingWikiWord" href="/nlab/show/infinity-cohesive+site">infinity-cohesive site</a></em>:</p> <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> (in particular <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a>),</p> <ol> <li> <p><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> admits a <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> (by <a class="existingWikiWord" href="/nlab/show/charts">charts</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> all whose finite non-empty intersections are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>/<a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>);</p> </li> <li> <p>for any such good open cover the <a href="&#268;ech+cohomology#CechComplex">Cech complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet((X,\{U_i\}),A_\bullet)</annotation></semantics></math> already computes Cech cohomology (i.e. there is no further need to form the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of Cech complexes over refinements of covers);</p> </li> <li> <p>the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ch</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">C^\bullet((X,\{U_i\}),-) \colon Ch_+(Smooth0Type) \to Ch_+</annotation></semantics></math> preserves weak equivalences and fibrations.</p> </li> </ol> <p>This means in particular that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>Z</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet \to Y_\bullet \to Z_\bullet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math>, then also</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> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Y</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C^\bullet((X,\{U_i\}), X_\bullet) \to C^\bullet((X,\{U_i\}), Y_\bullet) \to C^\bullet((X,\{U_i\}), Z_\bullet) </annotation></semantics></math></div> <p>is a homotopy fiber sequence of chain complexes, and therefore the <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> sit in the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+homology">long exact sequence in homology</a> of this sequence of chain complexes.</p> </div> <div class="num_example" id="OrdinaryCohomologyFromCechCohomology"> <h6 id="example_5">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet = (\mathbf{B}^{n+1}\mathbb{Z})_\bullet</annotation></semantics></math> as in example <a class="maruku-ref" href="#DeloopingsOfAbelianSheaf"></a>, then 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Cech</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^0_{Cech}(X, (\mathbf{B}^{n+1}\mathbb{Z})_\bullet) \simeq H^{n+1}(X,\mathbb{Z}) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</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> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, the cohomology which is also computed as the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> of the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</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>.</p> <p>Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>=</mo><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet = (\flat \mathbf{B}^n U(1))_{\bullet}</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>Cech</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>n</mi></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^0_{Cech}(X, (\flat\mathbf{B}^{n}U(1))_\bullet) \simeq H^{n}(X,U(1)) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</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> with <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, the cohomology which is also computed as the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> of the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</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> 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>-coefficients.</p> </div> <div class="num_example" id="DeRhamTheorem"> <h6 id="example_6">Example</h6> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> (e.g. via def. <a class="maruku-ref" href="#CechComplex"></a>), then prop. <a class="maruku-ref" href="#TheDeRhamResolutionOfConstantFunctions"></a> is the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>.</p> </div> <p>We will have need to give names to truncations of the de Rham complex. One is this:</p> <div class="num_defn" id="TruncatedDeRham"> <h6 id="definition_6">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> write</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>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{\bullet \leq n} \in Ch_+(Smooth0Type) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of the form</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>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup><mo>≔</mo><mrow><mo>[</mo><mi>ℝ</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{\bullet \leq n} \coloneqq \left[ \mathbb{R} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n-1} \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^{n} \right] </annotation></semantics></math></div> <p>with all <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, not just the closed ones, in degree 0.</p> </div> <div class="num_example" id="CohomologyWithCoefficientsInTruncatedDeRham"> <h6 id="example_7">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of the truncated de Rham complex in def. <a class="maruku-ref" href="#TruncatedDeRham"></a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n(X)/im(\mathbf{d})</annotation></semantics></math>.</p> </div> <h3 id="TheDeligneComplex">The Deligne complex</h3> <div class="num_defn" id="TheSmoothDeligneComplex"> <h6 id="definition_7">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> the <strong>smooth Deligne complex</strong> of 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></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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n U(1)_{conn})_\bullet \in Ch_+(Smooth0Type) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> given by</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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n U(1)_{conn})_\bullet \; \coloneqq \; \left[ U(1) \stackrel{\mathbf{d} log}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^n \right] </annotation></semantics></math></div> <p>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> 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> and with the differentials as in def. <a class="maruku-ref" href="#SheavesOfFormsAndDeRhamDifferential"></a> and example <a class="maruku-ref" href="#LogDiffInSmoothContext"></a>.</p> <p>We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mi>H</mi> <mi>Cech</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{n+1}_{conn}(X,\mathbb{Z}) \coloneqq H^0_{Cech}(X,(\mathbf{B}^n U(1)_{conn})_\bullet) </annotation></semantics></math></div> <p>for its <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>.</p> </div> <div class="num_remark" id="RModZResolutionOfU1DeligneComplex"> <h6 id="remark_8">Remark</h6> <p>By example <a class="maruku-ref" href="#LogDiffInSmoothContext"></a> the obvious <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>ℤ</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow^{\mathrlap{(-)/\mathbb{Z}}} && \downarrow^{id} && && \downarrow^{id} \\ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } </annotation></semantics></math></div> <p>is a weak equivalence, def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a>, and one could define the top chain complex here as “the” Deligne complex, just as well. In the context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, all that matters is the complex up to <a class="existingWikiWord" href="/nlab/show/zig-zags">zig-zags</a> of weak equivalences.</p> </div> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>In def. <a class="maruku-ref" href="#TheSmoothDeligneComplex"></a> the de Rham complex is truncated to the right by discarding what would be the next differentials, without passing to their <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, i.e. in degree 0 the Deligne complex has all 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>-forms, not just the closed <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>-fomrs. This simple point is the key aspect of the Deligne complex. If one instead truncates while preserving the chain homology in the lowest degree, then one obtains the following complex with the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n}_{cl}</annotation></semantics></math> of <em>closed</em> forms in lowest degree, which gives <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>.</p> </div> <div class="num_defn" id="FlatSmoothDeligneComplex"> <h6 id="definition_8">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> the <strong>flat smooth Deligne complex</strong> of 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></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><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>flat</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n U(1)_{flat})_\bullet \in Ch_+(Smooth0Type) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> given by</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><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>flat</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mi>n</mi></msubsup><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n U(1)_{flat})_\bullet \; \coloneqq \; \left[ U(1) \stackrel{\mathbf{d} log}{\to} \mathbf{\Omega}^1 \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \mathbf{\Omega}^n_{cl} \right] </annotation></semantics></math></div> <p>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> 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> and with the differentials as in def. <a class="maruku-ref" href="#SheavesOfFormsAndDeRhamDifferential"></a> and example <a class="maruku-ref" href="#LogDiffInSmoothContext"></a>, and with the <em>closed</em> <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 on the right.</p> </div> <div class="num_defn" id="FlatBnU1"> <h6 id="definition_9">Definition</h6> <p>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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\flat\mathbf{B}^n U(1))_\bullet = (\mathbf{B}^n \flat U(1))_\bullet \in Ch_+(Smooth0Type) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> given by</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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow></mrow></mover><mn>0</mn><mover><mo>→</mo><mrow></mrow></mover><mi>⋯</mi><mover><mo>→</mo><mrow></mrow></mover><mn>0</mn><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> (\flat \mathbf{B}^n U(1))_\bullet \; \coloneqq \; \left[ \flat U(1) \stackrel{}{\to} 0 \stackrel{}{\to} \cdots \stackrel{}{\to} 0 \right] </annotation></semantics></math></div> <p>with the constant sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat U(1)</annotation></semantics></math> of example <a class="maruku-ref" href="#RepresentableSheavesAndTheirConstantVersion"></a> 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>.</p> </div> <div class="num_prop" id="TwoEquivalentModelsForFlatBnU1"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(\flat \mathbf{B}^n U(1))_\bullet</annotation></semantics></math> as in def. <a class="maruku-ref" href="#FlatBnU1"></a>, then the morphism</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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mover><mo>⟶</mo><mo>≃</mo></mover><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>flat</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\flat \mathbf{B}^n U(1))_\bullet \stackrel{\simeq}{\longrightarrow} (\mathbf{B}^n U(1)_{flat})_\bullet </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</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><mo>♭</mo><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><mi>⋯</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mi>n</mi></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat U(1) &\to& 0 &\to& \cdots &\to& 0 \\ \downarrow && \downarrow && \cdots && \downarrow \\ U(1) &\stackrel{\mathbf{d}log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n_{cl} } </annotation></semantics></math></div> <p>(with the vertical morphism on the left being the inclusion of example <a class="maruku-ref" href="#LogDiffInSmoothContext"></a>) is a weak equivalence, def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+lemma">Poincaré lemma</a>, this is just an immediate variant of prop. <a class="maruku-ref" href="#TheDeRhamResolutionOfConstantFunctions"></a>.</p> </div> <h3 id="CechDeligneComplex">Cech-Deligne complex</h3> <p>The Cech complex, def. <a class="maruku-ref" href="#CechComplex"></a>, for Deligne cohomology of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of a <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> of the following form</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><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><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><mi>δ</mi></mover></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><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><mi>δ</mi></mover></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>k</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><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><mi>δ</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\longrightarrow& 0 &\longrightarrow& 0 &\longrightarrow & \cdots \\ \uparrow && \uparrow && \uparrow \\ \Omega^{p+1}(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^{p+1}(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \vdots && \vdots && \vdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^2(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^2(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \uparrow^{\mathrlap{d}} && \\ \Omega^1(\coprod_i U_i) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j} U_{ i j}) &\stackrel{\delta}{\longrightarrow}& \Omega^1(\coprod_{i,j, k} U_{i j k}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \uparrow^{\mathrlap{d log}} && \\ C^\infty(\coprod_i U_i, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j} U_{i j}, U(1)) &\stackrel{\delta}{\longrightarrow}& C^\infty(\coprod_{i,j,k} U_{i j k}, U(1)) &\stackrel{\delta}{\longrightarrow}& \cdots } </annotation></semantics></math></div> <p>where vertically we have the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> and horizontally the Cech differential given by alternating sums of <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> has 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> the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the entries in this double complex which are on the <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 nw-se off-diagonal and has the total differential</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mi>d</mi><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>deg</mi></msup><mi>δ</mi></mrow><annotation encoding="application/x-tex"> D = d + (-1)^{deg} \delta </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi></mrow><annotation encoding="application/x-tex">deg</annotation></semantics></math> denoting form degree.</p> <div class="num_example"> <h6 id="example_8">Example</h6> <p>A Cech-Deligne cocycle in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/bundle+gerbe+with+connection">bundle gerbe with connection</a>”) is data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>B</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\{B_{i}\}, \{A_{i j}\}, \{g_{i j k}\})</annotation></semantics></math> such that</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><mo stretchy="false">{</mo><msub><mi>B</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mrow><mrow><mo stretchy="false">{</mo><msub><mi>B</mi> <mi>j</mi></msub><mo>−</mo><msub><mi>B</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><mo>=</mo><mrow><mi>d</mi><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">{</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</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><mo>−</mo><msub><mi>A</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>d</mi><mi>log</mi><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>δ</mi></mover></mtd> <mtd><mo stretchy="false">{</mo><msub><mi>g</mi> <mrow><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>l</mi></mrow></msub><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">}</mo><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \{B_i\} &\stackrel{\delta}{\longrightarrow}& {{\{B_j - B_i\}} = {d A_{i j}}} && && \\ && \uparrow^{\mathrlap{d}} && && \\ && \{A_{i j}\} &\stackrel{\delta}{\longrightarrow}& \{-A_{ j k} + A_{i k} - A_{i j}\} = \{d log g_{i j k}\} && \\ && && \uparrow^{\mathrlap{d log}} && \\ && && \{g_{i j k}\} &\stackrel{\delta}{\longrightarrow}& \{g_{j k l} g_{i k l}^{-1} g_{i j l} g_{i j k}^{-1} \} = 1 } </annotation></semantics></math></div></div> <h3 id="cup_product_in_deligne_cohomology">Cup product in Deligne cohomology</h3> <p>The <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> on <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> refines to Deligne cohomology.</p> <p>For more on this see at <em><a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup-product">Beilinson-Deligne cup-product</a></em>.</p> <h2 id="Properties">Properties</h2> <h3 id="CharacteristicMaps">Curvature and characteristic classes</h3> <p>We discuss the construction of two canonical morphisms out of Deligne cohomology, and two canonical morphisms into it. Below these are shown to form two interlocking <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a> and in fact an exact <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> which accurately characterizes Deligne cohomology as the <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> extension of integral <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> by <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>.</p> <p>Throughout, for ease of notation, we assume <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> to be positive,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \geq 1 \,. </annotation></semantics></math></div> <p>The remaining case <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> describes “circle 0-bundles with connection”, which are just <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 functions, and is hence essentially trivial in itself.</p> <p>In the following <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 any <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>.</p> <div class="num_defn" id="UnderlyingBundleMap"> <h6 id="definition_10">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}^{n+1}\mathbb{Z})_\bullet \in Ch_+(Smooth0Type)</annotation></semantics></math> be as in example <a class="maruku-ref" href="#DeloopingsOfAbelianSheaf"></a>. 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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mover><mo>⟵</mo><mo>≃</mo></mover><mover><mo>⟶</mo><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mover><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^n U(1)_{conn})_{\bullet} \stackrel{\simeq}{\longleftarrow} \stackrel{DD_\bullet}{\longrightarrow} (\mathbf{B}^{n+1} \mathbb{Z})_{\bullet} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of chain complexes where the left weak equivalence is that of remark <a class="maruku-ref" href="#RModZResolutionOfU1DeligneComplex"></a>, i.e. for the <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a> 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><mi>ℤ</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></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><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mi>ℤ</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\to& 0 &\to& 0 &\to& \cdots &\to& 0 \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{}} && \uparrow^{\mathrlap{}} && \cdots && \uparrow^{\mathrlap{}} \\ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{mod\,\mathbb{Z}}} && \downarrow^{\mathrlap{id}} && \cdots && \downarrow^{\mathrlap{id}} \\ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } </annotation></semantics></math></div> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> this gives, by def. <a class="maruku-ref" href="#TheSmoothDeligneComplex"></a> and example <a class="maruku-ref" href="#OrdinaryCohomologyFromCechCohomology"></a>, a morphism</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><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>DD</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \searrow^{\mathrlap{DD}} \\ && H^{n+1}(X,\mathbb{Z}) } </annotation></semantics></math></div> <p>from Deligne cohomology to <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</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>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>∇</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\nabla] \in H^{n+1}_{conn}(X,\mathbb{Z})</annotation></semantics></math> we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DD</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">DD(\nabla) \in H^{n+1}(X,\mathbb{Z})</annotation></semantics></math><br />the <a class="existingWikiWord" href="/nlab/show/Dixmier-Douady+class">Dixmier-Douady class</a> of the <em>underlying <a class="existingWikiWord" href="/nlab/show/circle+n-bundle">circle n-bundle</a></em>.</p> </div> <div class="num_defn" id="CurvatureMap"> <h6 id="definition_11">Definition</h6> <p>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><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mover><mo>⟶</mo><mrow></mrow></mover><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> (F_{(-)})_\bullet \colon (\mathbf{B}^n U(1)_{conn})_\bullet \stackrel{}{\longrightarrow} \mathbf{\Omega}^{n+1}_{cl} </annotation></semantics></math></div> <p>for the morphism given by the chain map which is just the de Rham differential in degree 0</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>d</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& U(1) &\stackrel{\mathbf{d} log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\mathbf{d}}} \\ 0 &\to& 0 &\to& 0 &\to& \cdots &\stackrel{}{\to}& 0 &\to& \mathbf{\Omega}^{n+1}_{cl} } </annotation></semantics></math></div> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> this gives a morphism of the form</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><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></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msup><mo>↗</mo></mtd></mtr> <mtr><mtd><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \Omega^{n+1}_{cl}(X) \\ & {}^{\mathllap{F_{(-)}}}\nearrow \\ H^{n+1}_{conn}(X,\mathbb{Z}) } </annotation></semantics></math></div> <p>We call this the <em><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> map</em>, i.e. for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>∇</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\nabla] \in H^{n+1}_{conn}(X,\mathbb{Z})</annotation></semantics></math> the class of a Deligne cocycle, we call</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>∇</mo></msub><mo>∈</mo><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><annotation encoding="application/x-tex"> F_\nabla \in \Omega^{n+1}_{cl}(X) </annotation></semantics></math></div> <p>its <em><a class="existingWikiWord" href="/nlab/show/curvature+form">curvature form</a></em>.</p> </div> <div class="num_defn" id="InclusionOfGloballyDefinedConnectionForms"> <h6 id="definition_12">Definition</h6> <p>Consider the <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a></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>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup><mover><mo>⟶</mo><mrow></mrow></mover><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{\bullet\leq n} \stackrel{}{\longrightarrow} (\mathbf{B}^n U(1)_{conn})_\bullet </annotation></semantics></math></div> <p>out of the complex of def. <a class="maruku-ref" href="#TruncatedDeRham"></a>, given by the <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi mathvariant="normal">id</mi></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow^{\mathrm{id}} && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ \mathbb{Z} &\hookrightarrow& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \downarrow && \downarrow && \downarrow^{\mathrm{id}} && \cdots && \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}} \\ 0 &\to& U(1) &\stackrel{\mathbf{d}log}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n } </annotation></semantics></math></div> <p>where the bottom <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> is from remark <a class="maruku-ref" href="#RModZResolutionOfU1DeligneComplex"></a>.</p> <p>On passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> this gives, by example <a class="maruku-ref" href="#CohomologyWithCoefficientsInTruncatedDeRham"></a>, a morphism</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>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega^n(X)/im(\mathbf{d}) \\ & \searrow \\ && H^{n+1}_{conn}(X,\mathbb{Z}) } </annotation></semantics></math></div></div> <div class="num_defn" id="InclusionOfFlatConnections"> <h6 id="definition_13">Definition</h6> <p>Consider the canonical morphism</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><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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mover><mo>⟶</mo><mo>≃</mo></mover><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>flat</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>⟶</mo><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\flat \mathbf{B}^n U(1))_\bullet \stackrel{\simeq}{\longrightarrow} (\mathbf{B}^n U(1)_{flat})_\bullet \longrightarrow (\mathbf{B}^n U(1)_{conn})_\bullet </annotation></semantics></math></div> <p>via def. <a class="maruku-ref" href="#FlatSmoothDeligneComplex"></a>, prop. <a class="maruku-ref" href="#TwoEquivalentModelsForFlatBnU1"></a>.</p> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> this induces a morphism</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><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mi>n</mi></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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \nearrow \\ H^n(X,U(1)) } </annotation></semantics></math></div> <p>We call this map the <em>inclusion of the <a class="existingWikiWord" href="/nlab/show/flat+infinity-connections">flat infinity-connections</a></em> into all <a class="existingWikiWord" href="/nlab/show/circle+n-connections">circle n-connections</a>.</p> </div> <p>Combining what we have so far:</p> <div class="num_prop" id="BocksteinHomomorphism"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composite</a> of the morphisms of def. <a class="maruku-ref" href="#InclusionOfGloballyDefinedConnectionForms"></a> and of the curvature morphism of def. <a class="maruku-ref" href="#CurvatureMap"></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><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></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></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega^n(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \searrow && \nearrow_{\mathrlap{F_{(-)}}} \\ && H^{n+1}_{conn}(X,\mathbb{Z}) } </annotation></semantics></math></div> <p>is given by the de Rham differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math> on differential forms.</p> <p>The composite of the morphisms of def. <a class="maruku-ref" href="#InclusionOfFlatConnections"></a> and def. <a class="maruku-ref" href="#UnderlyingBundleMap"></a> is the <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</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></mtd> <mtd></mtd> <mtd><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>DD</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mi>n</mi></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></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>β</mi></munder></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && H^{n+1}_{conn}(X,\mathbb{Z}) \\ & \nearrow && \searrow^{\mathrlap{DD}} \\ H^n(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) } </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By composing the defining zig-zags of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a> the statement is immediate.</p> </div> <h3 id="TheChernCharacter">The Chern character</h3> <p>While the explicit definition of the Deligne complex in def. <a class="maruku-ref" href="#TheSmoothDeligneComplex"></a> is easy enough, all its good abstract properties are best understood by realizing that it is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of a kind of higher abelian <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> map with the closed differential forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n+1}_{cl}</annotation></semantics></math>. This is the content of prop. <a class="maruku-ref" href="#HomotopyFiberProductCharacterization"></a> below.</p> <div class="num_defn" id="ChernCharacterMap"> <h6 id="definition_14">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>⟶</mo><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>ℝ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> ch_\bullet \colon (\mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet </annotation></semantics></math></div> <p>for the morphism given as the composite</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><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>=</mo><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>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>⟶</mo><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>ℝ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^{n+1}\mathbb{Z})_\bullet = (\flat \mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet </annotation></semantics></math></div> <p>where the second morphism is induced by the canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \hookrightarrow \mathbb{R}</annotation></semantics></math>.</p> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> this induces a morphism</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>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>ch</mi></mpadded></msub></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && H^{n+1}(X,\mathbb{R}) \\ & \nearrow_{\mathrlap{ch}} \\ H^{n+1}(X,\mathbb{Z}) } </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_10">Remark</h6> <p>The morphism in def. <a class="maruku-ref" href="#ChernCharacterMap"></a> is just the traditional map from <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> coefficients to that with <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> coefficients given for instance via <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> simply by forming the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> with <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>. In the broader context of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> however it is useful to think of this map as the <em><a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></em>, whence the notation.</p> <p>While this is easy enough to construct in itself, the underlying <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> here is not a fibration in the sense of def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a> (having as non-trivial component the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \hookrightarrow \mathbb{R}</annotation></semantics></math>, which is evidently not an epimorphism). But the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this map plays a crucial role in the theory, and so in view of remark <a class="maruku-ref" href="#HomotopyFibersViaFactorizationLemma"></a> we consider now a fibration <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of this map</p> </div> <div class="num_defn" id="DifferentialResolutionOfIntegralCoefficients"> <h6 id="definition_15">Definition</h6> <p>Consider the chain complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub><mo>≔</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>↪</mo><mrow></mrow></mover></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>±</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>∓</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>ℝ</mi></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><mi>⋯</mi></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet \coloneqq \left[ \array{ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \oplus &\nearrow_{-\mathrlap{id}}& \oplus &\nearrow_{+\mathrlap{id}}& \oplus &\nearrow_{-\mathrlap{id}}& \cdots &\nearrow_{\pm\mathrlap{id}}& \oplus &\nearrow_{\mp\mathrlap{id}}& \\ \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} && } \right] </annotation></semantics></math></div> <p>where we use <a class="existingWikiWord" href="/nlab/show/matrix+calculus">matrix calculus</a>-notation as for <a class="existingWikiWord" href="/nlab/show/mapping+cones">mapping cones</a> (see at <em><a href="mapping+cone#InChainComplexes">mapping cone – Examples – In chain complexes</a></em>).</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub><mover><mo>⟶</mo><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mover><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet \stackrel{\widehat{ch}_\bullet}{\longrightarrow} \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet </annotation></semantics></math></div> <p>for the morphism to the chain complex of def. <a class="maruku-ref" href="#DeRhamResolutionOfConstantFunctions"></a> which is given by the chain map that in positive degree projects onto the lower row in the above <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> expression and in degree 0 is given by the de Rham differential:</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>ℤ</mi></mtd> <mtd><mover><mo>↪</mo><mrow></mrow></mover></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>±</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd><mo>⊕</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>∓</mo><mpadded width="0"><mi>id</mi></mpadded></mrow></msub></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>ℝ</mi></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><mi>⋯</mi></mtd> <mtd><munder><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>d</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>2</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n-1} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^n \\ \oplus &\nearrow_{-\mathrlap{id}}& \oplus &\nearrow_{+\mathrlap{id}}& \oplus &\nearrow_{-\mathrlap{id}}& \cdots &\nearrow_{\pm\mathrlap{id}}& \oplus &\nearrow_{\mp\mathrlap{id}}& \\ \mathbb{R} &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\underset{\mathbf{d}}{\to}& \cdots &\underset{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} && \\ \downarrow && \downarrow && \downarrow && \cdots && \downarrow && \downarrow^{\mathrlap{\mathbf{d}}} \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^2 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1} } </annotation></semantics></math></div> <p>Finally 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><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>⟶</mo><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> (\mathbf{B}^{n+1}\mathbb{Z})_\bullet \longrightarrow \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet </annotation></semantics></math></div> <p>for the morphism given by the chain map which 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> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{R} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n \mapsto (n,n) \,. </annotation></semantics></math></div></div> <div class="num_lemma" id="ChernCharacterResolution"> <h6 id="lemma">Lemma</h6> <p>The construction in def. <a class="maruku-ref" href="#DifferentialResolutionOfIntegralCoefficients"></a> gives a fibration resolution of the Chern character morphism of def. <a class="maruku-ref" href="#ChernCharacterMap"></a> in that it gives a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</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><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ch</mi> <mo>•</mo></msub></mrow></mover></mtd> <mtd><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>ℝ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mover></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\mathbf{B}^{n+1}\mathbb{Z})_\bullet &\stackrel{ch_\bullet}{\longrightarrow}& (\flat\mathbf{B}^{n+1} \mathbb{R})_\bullet \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \widehat {(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat {ch}_\bullet}{\longrightarrow}& \widehat {(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } </annotation></semantics></math></div> <p>with, on the right, the weak equivalence of prop. <a class="maruku-ref" href="#TheDeRhamResolutionOfConstantFunctions"></a>, where</p> <ol> <li> <p>the left vertical morphism is a weak equivalence;</p> </li> <li> <p>the bottom horizontal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\widehat{ch}_\bullet</annotation></semantics></math> is a fibration.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>That the diagram commutes is a straightforward inspection, unwinding the definitions. That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\widehat{ch}_\bullet</annotation></semantics></math> is a fibration according to def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a> is by its very construction, being a <a class="existingWikiWord" href="/nlab/show/projection">projection</a> in positive degree. That the left morphism is a weak equivalence comes down to the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+lemma">Poincaré lemma</a>, in a slight variant of the simple argument that proves prop. <a class="maruku-ref" href="#TheDeRhamResolutionOfConstantFunctions"></a>.</p> </div> <div class="num_defn" id="MapFromClosedFormsToHyperDeRhamCoefficients"> <h6 id="definition_16">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mover><msub><mover><mrow><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mrow><mo>^</mo></mover> <mo>•</mo></msub><mover><mo>⟵</mo><mo>≃</mo></mover><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>ℝ</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^{n+1}_{cl} \stackrel{\iota_\bullet}{\longrightarrow} \widehat{\flat \mathbf{B}^{n+1}\mathbb{R}}_\bullet \stackrel{\simeq}{\longleftarrow} (\flat \mathbf{B}^{n+1}\mathbb{R})_\bullet </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> whose right morphism is the weak equivalence of prop. <a class="maruku-ref" href="#TheDeRhamResolutionOfConstantFunctions"></a> and whose left morphism is given by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</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><mn>0</mn></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> <mtd><mo>→</mo></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℝ</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& 0 &\to& \cdots &\to& 0 &\to& \mathbf{\Omega}^{n+1}_{cl} \\ \downarrow && \downarrow && \vdots && \downarrow && \downarrow^{\mathrlap{id}} \\ \mathbb{R} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^1 &\stackrel{\mathbf{d}}{\to}& \cdots &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n} &\stackrel{\mathbf{d}}{\to}& \mathbf{\Omega}^{n+1}_{cl} } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="HomotopyFiberProductCharacterization"> <h6 id="proposition_4">Proposition</h6> <p>The chain maps</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(F_{(-)})_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#CurvatureMap"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">DD_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#UnderlyingBundleMap"></a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\iota_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#MapFromClosedFormsToHyperDeRhamCoefficients"></a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\widehat{ch}_\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#DifferentialResolutionOfIntegralCoefficients"></a></p> </li> </ul> <p>fit into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</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></mtd> <mtd></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{\Omega}^{n+1}_{cl} \\ & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/pullback+diagram">pullback diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math>. This exhibits the Deligne complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(\mathbf{B}^n U(1)_{conn})_\bullet</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of the inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n+1}_{cl}</annotation></semantics></math> along the Chern character map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">ch_\bullet</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>The first statement follows straightforwardly by inspection, using that pullbacks of chain complexes are computed componentwise. From this the second statement follows then since by <a class="maruku-ref" href="#ChernCharacterResolution"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\widehat{ch}_\bullet</annotation></semantics></math> is a fibration resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">ch_\bullet</annotation></semantics></math>.</p> </div> <h3 id="ExactSequenceForCurvatureAndCharacteristicClass">The exact sequences for curvature and characteristic classes</h3> <div class="num_defn" id="IntegralDifferentialForms"> <h6 id="definition_17">Definition</h6> <p>Write</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>int</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><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><annotation encoding="application/x-tex"> \Omega^{n+1}_{int}(X) \hookrightarrow \Omega^{n+1}_{cl}(X) </annotation></semantics></math></div> <p>for the inclusion of those closed <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> whose <a class="existingWikiWord" href="/nlab/show/periods">periods</a> (<a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration</a> over <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>-cycles) takes values in the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>.</p> </div> <div class="num_prop" id="ImageOfCurvatureInIntegralDifferentialForms"> <h6 id="proposition_5">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/image">image</a> of the curvature map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo lspace="verythinmathspace">:</mo><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>⟶</mo><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><annotation encoding="application/x-tex">F_{(-)} \colon H^{n+1}_{conn}(X,\mathbb{Z}) \longrightarrow \Omega^{n+1}_{cl}(X)</annotation></semantics></math> of def. <a class="maruku-ref" href="#CurvatureMap"></a> are the integral forms of def. <a class="maruku-ref" href="#IntegralDifferentialForms"></a>.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> characterization of of prop. <a class="maruku-ref" href="#HomotopyFiberProductCharacterization"></a> implies that the image consists of precisely those closed differential forms which under the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, remark <a class="maruku-ref" href="#DeRhamTheorem"></a>, represent <a class="existingWikiWord" href="/nlab/show/real+cohomology">real cohomology</a> classes that are in the image of integral cohomology classes. These are the differential forms with integral periods.</p> </div> <div class="num_prop" id="CurvatureExactSequence"> <h6 id="proposition_6">Proposition</h6> <p><strong>(curvature exact sequence)</strong></p> <p>The Deligne cohomology group fits into a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> (of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>H</mi> <mi>n</mi></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><mo>⟶</mo><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mover><msubsup><mi>Ω</mi> <mi>int</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><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to H^n(X,U(1)) \longrightarrow H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{F_{(-)}}{\longrightarrow} \Omega^{n+1}_{int}(X) \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><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">F_{(-)}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> map of def. <a class="maruku-ref" href="#CurvatureMap"></a>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By prop. <a class="maruku-ref" href="#ImageOfCurvatureInIntegralDifferentialForms"></a> the morphism on the right is indeed an epimorphism. It remains to determine its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>.</p> <p>To that end, consider the <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> obtained form the homotopy pullback in prop. <a class="maruku-ref" href="#HomotopyFiberProductCharacterization"></a>. Using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> and the fact that the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of a <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> object such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^{n+1}_{cl}</annotation></semantics></math> is trivial, this is of the form</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><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></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><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo></mo><mo>•</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mover></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0 \\ \downarrow && \downarrow && \downarrow^{} \\ 0 &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}& \mathbf{\Omega}^{n+1}_{cl} \\ && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}& \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } \,. </annotation></semantics></math></div> <p>Passing to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> and applying the induced <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+homology">long exact sequence in homology</a>, in view of remark <a class="maruku-ref" href="#GoodCoverCechCohomologyIsHomotopicallyGood"></a>, implies the claim.</p> </div> <div class="num_prop" id="CharacteristicClassExactSequence"> <h6 id="proposition_7">Proposition</h6> <p><strong>(characteristic class exact sequence)</strong></p> <p>The Deligne cohomology group fits into a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> (of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msubsup><mi>Ω</mi> <mi>int</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>DD</mi></mover><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \Omega^{n}(X)/\Omega^n_{int}(X) \stackrel{}{\longrightarrow} H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{DD}{\longrightarrow} H^{n+1}(X,\mathbb{Z}) \to 0 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DD</mi></mrow><annotation encoding="application/x-tex">DD</annotation></semantics></math> is the characteristic class map of def. <a class="maruku-ref" href="#UnderlyingBundleMap"></a>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> that represents the <a class="existingWikiWord" href="/nlab/show/Dixmier-Douady+class">Dixmier-Douady class</a> by def. <a class="maruku-ref" href="#UnderlyingBundleMap"></a> is manifestly a fibration in the sense of def. <a class="maruku-ref" href="#FibrantOnjectStructureOnChSmooth"></a>. Therefore its ordinary <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> is already its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>. That ordinary fiber is evidently the domain of the morphism constructed in def. <a class="maruku-ref" href="#InclusionOfGloballyDefinedConnectionForms"></a>, in its second weakly equivalent incarnation as displayed there.</p> <p>Therefore the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+homology">long exact sequence in homology</a>, induced by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">DD_\bullet</annotation></semantics></math> under passage to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> (in view of remark <a class="maruku-ref" href="#GoodCoverCechCohomologyIsHomotopicallyGood"></a>) goes as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>⟶</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>DD</mi></mover><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cdots \longrightarrow H^n(X,\mathbb{Z}) \longrightarrow \Omega^n(X)/im(\mathbf{d}) \stackrel{}{\longrightarrow} H^{n+1}_{conn}(X,\mathbb{Z}) \stackrel{DD}{\longrightarrow} H^{n+1}(X,\mathbb{Z}) </annotation></semantics></math></div> <p>As in the proof of prop. <a class="maruku-ref" href="#ImageOfCurvatureInIntegralDifferentialForms"></a> it follows that the rightmost morphism is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>. Hence we get a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> by dividing out the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X,\mathbb{Z})</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n/im(\mathbf{d})</annotation></semantics></math>. That image is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>int</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n_{int}(X)</annotation></semantics></math>. Since this image contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im(\mathbf{d})</annotation></semantics></math> (as the closed differential forms all whose periods are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{N}</annotation></semantics></math> ) the resulting quotient is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msubsup><mi>Omega</mi> <mi>int</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n(X)/Omega^n_{int}(X)</annotation></semantics></math> and the claim follows.</p> </div> <div class="num_remark"> <h6 id="remark_11">Remark</h6> <p>In words the statement of prop. <a class="maruku-ref" href="#CurvatureExactSequence"></a> and prop. <a class="maruku-ref" href="#CharacteristicClassExactSequence"></a> is that Deligne <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{conn}(X,\mathbb{Z})</annotation></semantics></math> constitute a <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> <ol> <li> <p>of integral <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> by 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>-forms modulo integral forms;</p> </li> <li> <p>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 by <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 ordinary cohomology.</p> </li> </ol> <p>The first statement is what gives the name to “<a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>” as it makes precise how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>conn</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{n+1}_{conn}(X)</annotation></semantics></math> is a combination of ordinary integral cohomology with <em><a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></em>-data.</p> <p>The second statement is secretly of the same flavor, if maybe not as manifestly so: 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>-valued ordinary cohomology is really what classifies <em><a class="existingWikiWord" href="/nlab/show/flat+infinity-connections">flat</a></em> <a class="existingWikiWord" href="/nlab/show/circle+n-connections">circle n-connections</a> on <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a> <a class="existingWikiWord" href="/nlab/show/principal+infinity-bundles">principal infinity-bundles</a> (either, depending on perspective, by def. <a class="maruku-ref" href="#FlatSmoothDeligneComplex"></a> or else, more intrinsically, by the very statement of prop. <a class="maruku-ref" href="#CurvatureExactSequence"></a>) and hence again describes a combination of underlying bundles with differential form data.</p> </div> <h3 id="TheExactDifferentialCohomologyHexagon">The exact differential cohomology hexagon</h3> <p>Summing up, the homotopy pullback square of prop. <a class="maruku-ref" href="#HomotopyFiberProductCharacterization"></a> together with the maps of prop. <a class="maruku-ref" href="#BocksteinHomomorphism"></a> form a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math> of the form.</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>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msub><mover><mrow><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 stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>β</mi></munder></mtd> <mtd></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{\Omega}^{\bullet \leq n} && \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl} \\ &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ && (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ \widehat{(\flat \mathbf{B}^n U(1))}_\bullet && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } </annotation></semantics></math></div> <div class="num_prop" id="TheDifferentialHexagonForDeligne"> <h6 id="proposition_8">Proposition</h6> <p>This extends to a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math> of the form</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><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mover><mrow><mo stretchy="false">(</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><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 stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mi>β</mi></munder></mtd> <mtd></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{\Omega}^{\bullet \leq n} && \stackrel{\mathbf{d}}{\longrightarrow} && \mathbf{\Omega}^{n+1}_{cl} \\ & \nearrow& &\searrow& & {}^{\mathllap{(F_{(-)})_\bullet }}\nearrow && \searrow^{\mathrlap{\iota_\bullet}} \\ \widehat{(\flat \mathbf{B}^n \mathbb{R})}_{\bullet} && && (\mathbf{B}^n U(1)_{conn})_\bullet && && \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet \\ &\searrow & &\nearrow& & {}_{\mathllap{DD_\bullet}}\searrow && \nearrow_{\mathrlap{\widehat{ch}_\bullet}} \\ && \widehat{(\flat \mathbf{B}^n U(1))}_\bullet && \underset{\beta}{\longrightarrow} && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet } </annotation></semantics></math></div> <p>such that</p> <ol> <li> <p>both square are <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> squares;</p> </li> <li> <p>both diagonals are <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a>;</p> </li> <li> <p>the two outer sequences are <a class="existingWikiWord" href="/nlab/show/long+homotopy+fiber+sequences">long homotopy fiber sequences</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>For the first statement consider the pasting of homotopy pullback diagrams as in the proof of prop. <a class="maruku-ref" href="#CurvatureExactSequence"></a>, now extended to the left, via the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, as</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><mover><mrow><mo stretchy="false">(</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></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><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><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>conn</mi></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>F</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><msub><mo></mo><mo>•</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>DD</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mover><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>ch</mi><mo>^</mo></mover> <mo>•</mo></msub></mrow></mover></mtd> <mtd><msub><mover><mrow><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>ℝ</mi><mo stretchy="false">)</mo></mrow><mo>^</mo></mover> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \widehat{(\flat \mathbf{B}^n \mathbb{R})}_\bullet &\longrightarrow& \widehat{(\flat \mathbf{B}^{n}(\mathbb{R}/\mathbb{Z}))}_\bullet &\longrightarrow& 0 \\ \downarrow && \downarrow && \downarrow^{} \\ \mathbf{\Omega}^{\bullet \leq n} &\longrightarrow& (\mathbf{B}^n U(1)_{conn})_\bullet &\stackrel{(F_{(-)}_\bullet)}{\longrightarrow}& \mathbf{\Omega}^{n+1}_{cl} \\ && \downarrow^{\mathrlap{DD_\bullet}} && \downarrow^{\mathrlap{\iota_\bullet}} \\ && \widehat{(\mathbf{B}^{n+1}\mathbb{Z})}_\bullet &\stackrel{\widehat{ch}_\bullet}{\longrightarrow}& \widehat{(\flat \mathbf{B}^{n+1}\mathbb{R})}_\bullet } \,. </annotation></semantics></math></div> <p>That the NE-diagonal is a homotopy fiber sequence is the statement in the proof of prop. <a class="maruku-ref" href="#CurvatureExactSequence"></a>. That the SE-diagonal is a homotopy fiber sequence follows by inspection as remarked in the proof of prop. <a class="maruku-ref" href="#CharacteristicClassExactSequence"></a>.</p> <p>From this the last statement now is implied by using the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> yet once more, as show in the proof <a href="differential+cohomology+diagram#TheDifferentialDiagram">here</a>.</p> </div> <div class="num_remark"> <h6 id="remark_12">Remark</h6> <p>The form of the exact hexagon characterizing the Deligne complex via prop. <a class="maruku-ref" href="#TheDifferentialHexagonForDeligne"></a> is in fact a general abstract consequence of the fact that all universal constructions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>+</mo></msub><mo stretchy="false">(</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_+(Smooth0Type)</annotation></semantics></math> considered here indeed may be understood as taking place in the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>, via remark <a class="maruku-ref" href="#HomotopyPullbacksPreservedbyInclusionIntoAllSmoothHomotopyTypes"></a>. This is discussed in detail at <em><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></em>.</p> </div> <h3 id="GAGA">GAGA</h3> <p>The Deligne complex is naturally defined in smooth <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> as well as in <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> as well as in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> over the complex numbers. In the spirit of <a class="existingWikiWord" href="/nlab/show/GAGA">GAGA</a> it is of interest to know how Deligne cohomology in these different settings relates.</p> <p>One useful statement is: given an <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth</a> <a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic variety</a> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, then a sufficient condition for a complex-analytic Deligne cocycle over its <a class="existingWikiWord" href="/nlab/show/analytification">analytification</a> to lift to an algebraic Deligne cocycle is that its <a class="existingWikiWord" href="/nlab/show/curvature+form">curvature form</a> is an <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+form">algebraic form</a> (<a href="#Esnault89">Esnault 89, corollary 1.3</a>).</p> <h3 id="moduli_and_deformation_theory">Moduli and deformation theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of holomorphic Deligne cohomology groups are closely related to <a class="existingWikiWord" href="/nlab/show/intermediate+Jacobians">intermediate Jacobians</a>, see there fore more.</p> <p>The <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a> of Deligne cohomology groups is given by <a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a>, see there for more</p> <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> <h3 id="interpretation_in_terms_of_higher_parallel_transport">Interpretation in terms of higher parallel transport</h3> <p>There is a natural way to understand the Deligne complex of sheaves as a sheaf which assigns to each patch the Lie <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>-groupoid of smooth <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a> <a class="existingWikiWord" href="/nlab/show/n-functors">n-functors</a>.</p> <p>We start by discussing this in low degree.</p> <p>There is <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_1(X)</annotation></semantics></math> whose smooth space of objects is <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> and whose smooth space of morphisms is a space of classes of smooth paths 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>. Every smooth 1-form <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> induces a smooth <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <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>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><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">tra_A : P_1(X) \to \mathbf{B}U(1)</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_1(X)</annotation></semantics></math> to to the smooth <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</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> with one object and <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 smooth space of morphisms by sending each path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : [0,1] \to X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><msup><mi>γ</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp (2 \pi i\int_0^1 \gamma^* A)</annotation></semantics></math>. This map from 1-forms to smooth functors turns out to be bijective: every smooth functor of this form uniquely arises this way. Similarly, one finds that smooth <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>f</mi></msub><mo>:</mo><msub><mi>tra</mi> <mi>A</mi></msub><mo>→</mo><msub><mi>tra</mi> <mrow><mi>A</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\eta_f : tra_A \to tra_{A'}</annotation></semantics></math> between two such functors is in components precisely a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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></mrow><annotation encoding="application/x-tex">f : X \to U(1)</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><mi>d</mi><mi>log</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">A' = A + d log f</annotation></semantics></math>.</p> <p>Since the analogous statements are true for every open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> this defines a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Funct</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</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><mo stretchy="false">)</mo><mo>:</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>LieGrpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Funct^\infty(P_1(-), \mathbf{B}U(1)) : Op(X)^{op} \to LieGrpd \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> this sheaf of groupoids corresponds to a sheaf of complexes of groups. This complex of sheaves is nothing but the degree 2 Deligne complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Funct</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</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><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">(</mo><mn>2</mn><msubsup><mo stretchy="false">)</mo> <mi>D</mi> <mn>∞</mn></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Funct^\infty(\Pi_1(-), \mathbf{B}U(1)) \simeq \mathbb{Z}(2)^\infty_D \,. </annotation></semantics></math></div> <p>This way Deligne cohomology is realized as computing the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-sheafification">stackification</a> of the pre-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Funct</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Funct^\infty(P_1(-), \mathbf{B}(1))</annotation></semantics></math> of smooth <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 parallel transport functors.</p> <p>The identification generalizes: for all <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> there is a <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_n(X)</annotation></semantics></math> whose <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>-morphisms are <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>-dimensional smooth paths 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>. 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>-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>tra</mi> <mi>C</mi></msub><msub><mo>:</mo> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><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">tra_C : _n(X) \to \mathbf{B}^n U(1)</annotation></semantics></math> are canonically identified with 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>-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in \Omega^n(X)</annotation></semantics></math> and under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> the Deligne-complex 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> is identified with the sheaf 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>-groupoids of such 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>-functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><msup><mi>Funct</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>n</mi></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> <mi>n</mi></msup><mo stretchy="false">)</mo><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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n Funct^\infty(P_n(-), \mathbf{B}^n) \simeq \mathbb{Z}(n+1)^\infty_D \,. </annotation></semantics></math></div> <p>See</p> <ul> <li>John Baez, Urs Schreiber, <em>Higher Gauge Theory</em> (<a href="http://arxiv.org/abs/math/0511710">arXiv</a>)</li> </ul> <p>The full proof 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> this is in</p> <ul> <li>Urs Schreiber, Konrad Waldorf, <em>Parallel transport and functors</em> (<a href="http://arxiv.org/abs/0705.0452">arXiv</a>);</li> </ul> <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> in</p> <ul> <li>Urs Schreiber, Konrad Waldorf, <em>Smooth functors versus differential forms</em> (<a href="http://arxiv.org/abs/0802.0663">arXiv</a>)</li> </ul> <p>For more on this see <a class="existingWikiWord" href="/nlab/show/infinity-Chern-Weil+theory+introduction">infinity-Chern-Weil theory introduction</a>.</p> <p>For higher <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> there is as yet no detailed proof in the literature, but the low dimensional proofs have obvious generalizations.</p> <h2 id="examples">Examples</h2> <p>As described in some detail at <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> in abelian higher <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theories</a> the background field naturally arises as a <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech</a>–Deligne cocycle, i.e. a <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cocycle</a> representative with values in the Deligne complex.</p> <ul> <li> <p>Degree 2 Deligne cohomology classifies <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>. The Deligne complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">¯</mo></mover><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar \mathbf{B}U(1)</annotation></semantics></math> in this case coincides with the <a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie-algebra+valued+forms">groupoid of Lie-algebra valued forms</a> for the Lie algebra of <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>.</p> <ul> <li>In physics the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is modeled by a degree 2 Deligne cocycle. See there for a derivation of <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech</a>–Deligne cohomology from physical input.</li> </ul> </li> <li> <p>Degree 3 Deligne cohomology classifies <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a>s with connection.</p> <ul> <li>In <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a>, degree 3 Deligne cocycles (with compact support, possibly) model <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a>. In formal high energy physics the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a> is modeled by a Deligne 3-cocycle.</li> </ul> </li> <li> <p>Degree 4 Deligne cohomology classifies <a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a>s with connection. In particular Chern-Simons bundle 2-gerbes whose degree 4 curvature characteristic class is a multiple of the Pontryagin 4-form on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>.</p> <ul> <li>In formal high energy physics degree 4 Deligne classes model the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>.</li> </ul> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a> (<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Abel-Jacobi+map">Abel-Jacobi map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beilinson+regulator">Beilinson regulator</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 href="differential+cohomology+diagram#DeligneCoefficients">differential cohomology diagram – Examples – Deligne coefficients</a></p> </li> </ul> <h2 id="References">References</h2> <p>Deligne cohomology was introduced in <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> (by a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/holomorphic+differential+forms">holomorphic differential forms</a>) in</p> <ul> <li id="Deligne71"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, Section 2.2 of: <em>Théorie de Hodge II</em>, IHES Pub. Math. (1971), no. 40, 5–57 (<a href="http://www.numdam.org/item/?id=PMIHES_1971__40__5_0">numdam:PMIHES_1971__40__5_0</a>)</li> </ul> <p>with applications to <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a> and <a class="existingWikiWord" href="/nlab/show/intermediate+Jacobians">intermediate Jacobians</a>. The same definition appears in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, <a class="existingWikiWord" href="/nlab/show/William+Messing">William Messing</a>, Section 3.1.7 of: <em>Universal extensions and one-dimensional crystalline cohomology</em>, Springer lecture notes 370, 1974 (<a href="https://link.springer.com/book/10.1007/BFb0061628">doi:10.1007/BFb0061628</a>)</p> </li> <li id="ArtinMazur77"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Artin">Michael Artin</a>, <a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, section III.1 of <em>Formal Groups Arising from Algebraic Varieties</em>, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 (<a href="http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0">numdam:ASENS_1977_4_10_1_87_0</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=56:15663">MR56:15663</a>)</p> </li> </ul> <p>under the name “multiplicative de Rham complex” (and in the context of studying its <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a> by <a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+groups">Artin-Mazur formal groups</a>). The theory was further developed in</p> <ul> <li id="Beilinson85"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+regulators+and+values+of+L-functions">Higher regulators and values of L-functions</a></em>, Journal of Soviet Mathematics 30 (1985), 2036-2070, (<a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng">mathnet (Russian)</a>, <a href="http://dx.doi.org/10.1007%2FBF02105861">DOI</a>)</p> <p>English translation: J Math Sci 30, 2036–2070 (1985) (<a href="https://doi.org/10.1007/BF02105861">doi:10.1007/BF02105861</a>)</p> <blockquote> <p>(reviewed in <a href="#EsnaultViehweg88">Esnault-Viehweg 88</a>)</p> </blockquote> </li> </ul> <p>with the application to <a class="existingWikiWord" href="/nlab/show/Beilinson+regulators">Beilinson regulators</a>. Later the evident version of the Deligne complex in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> over <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> gained more attention and is still referred to as “Deligne cohomology”.</p> <p>Identifying the <a class="existingWikiWord" href="/nlab/show/background+field">background</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> in <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> (<a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>) as a Deligne 3-cocycle (<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> <ul> <li><a class="existingWikiWord" href="/nlab/show/Krzysztof+Gaw%C4%99dzki">Krzysztof Gawędzki</a>, <em>Topological Actions in two-dimensional Quantum Field Theories</em>, in: <em>Nonperturbative quantum field theory</em>, Nato Science Series B <strong>185</strong>, Springer (1986) [<a href="http://inspirehep.net/record/257658">spire:257658</a>, <a href="https://doi.org/10.1007/978-1-4613-0729-7_5">doi:10.1007/978-1-4613-0729-7_5</a>, <a class="existingWikiWord" href="/nlab/files/Gawedzki-TopologicalActions.pdf" title="pdf">pdf</a>]</li> </ul> <p>Surveys and introductions in the context of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> include</p> <ul> <li id="Brylinski93"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, section I.5 of: <em>Loop Spaces, Characteristic Classes and geometric Quantization</em>, Birkhäuser 1993 (<a href="https://link.springer.com/book/10.1007/978-0-8176-4731-5">doi:10.1007/978-0-8176-4731-5</a>)</p> </li> <li id="Bunke12"> <p><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, section 3 of <em>Differential cohomology</em> (<a href="http://arxiv.org/abs/1208.3961">arXiv:1208.3961</a>)</p> </li> </ul> <p>Review with more emphasis on <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> and the theory of (<a href="#Beilinson85">Beilinson 85</a>) with more details spelled out is in</p> <ul> <li id="EsnaultViehweg88"> <p><a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, <a class="existingWikiWord" href="/nlab/show/Eckart+Viehweg">Eckart Viehweg</a>, <em>Deligne-Beilinson cohomology</em>, in: <a class="existingWikiWord" href="/nlab/show/Michael+Rapoport">Michael Rapoport</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Schappacher">Norbert Schappacher</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Schneider">Peter Schneider</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/Beilinson%27s+Conjectures+on+Special+Values+of+L-Functions">Beilinson's Conjectures on Special Values of L-Functions</a></em>, Perspectives in Mathematics <strong>4</strong>, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, <a class="existingWikiWord" href="/nlab/files/EsnaultViehweg-DeligneBeilinsonCohomology.pdf" title="pdf">pdf</a>]</p> </li> <li id="Esnault89"> <p><a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, <em>On the Loday-symbol in the Deligne-Beilinson cohomology</em>, K-theory 3, 1-28, 1989 (<a href="http://www.mi.fu-berlin.de/users/esnault/preprints/helene/16-loday-symbol.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="PetersSteenbrink08"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Peters">Chris Peters</a>, <a class="existingWikiWord" href="/nlab/show/Jozef+Steenbrink">Jozef Steenbrink</a>, section 7.2 of <em><a class="existingWikiWord" href="/nlab/show/Mixed+Hodge+Structures">Mixed Hodge Structures</a></em>, Ergebisse der Mathematik (2008) (<a href="http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck">pdf</a>)</p> </li> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, section 12 of <em><a class="existingWikiWord" href="/nlab/show/Hodge+theory+and+Complex+algebraic+geometry">Hodge theory and Complex algebraic geometry</a> I,II</em>, Cambridge Stud. in Adv. Math. <strong>76, 77</strong>, 2002/3</p> </li> </ul> <p>Discussion of Deligne cohomology as classifying higher <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> <p>Discussion of Deligne cohomology in terms of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> and <a class="existingWikiWord" href="/nlab/show/higher+stacks">higher stacks</a> includes</p> <ul> <li id="FSS10"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Cech+Cocycles+for+Differential+characteristic+Classes">Cech Cocycles for Differential characteristic Classes</a></em>, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (<a href="http://arxiv.org/abs/1011.4735">arXiv:1011.4735</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Extended+higher+cup-product+Chern-Simons+theories">Extended higher cup-product Chern-Simons theories</a></em>, Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 (<a href="http://arxiv.org/abs/1207.5449">arXiv:1207.5449</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>Hodge filtered complex bordism</em>, <a href="http://arxiv.org/abs/1212.2173">arXiv:1212.2173</a></p> <blockquote> <p>(cf. <a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>A higher stacky perspective on Chern-Simons theory</em>, in Damien Calaque et al. (eds.) <em>Mathematical Aspects of Quantum Field Theories</em>, Mathematical Physics Studies, Springer 2014 (<a href="http://arxiv.org/abs/1301.2580">arXiv:1301.2580</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> (<a href="http://arxiv.org/abs/1310.7930">arXiv:1310.7930</a>)</p> </li> </ul> <p>See also the references given at <em><a href="differential+cohomology+diagram#DeligneCoefficients">differential cohomology hexagon – Deligne coefficients</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2025 at 22:26:13. 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