CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> differential cohomology diagram in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> differential cohomology diagram </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/9168/#Item_11" 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="stable_homotopy_theory">Stable Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="cohesive_toposes">Cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Toposes</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definitionconstruction'>Definition/Construction</a></li> <ul> <li><a href='#differential_coefficients_and_maurercartan_forms'>Differential coefficients and Maurer-Cartan forms</a></li> <li><a href='#ChernCharacterAndFractureSquares'>Chern character and the differential fracture squares</a></li> <li><a href='#TheHexagonDiagram'>The hexagon diagram</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#OrdinaryDifferentialCohomology'>Ordinary differential cohomology</a></li> <ul> <li><a href='#DeRhamCoefficients'>De Rham coefficients</a></li> <li><a href='#DeligneCoefficients'>Deligne coefficients</a></li> </ul> <li><a href='#HopkinsSingerCoefficients'>Hopkins-Singer coefficients</a></li> <li><a href='#DifferentialKTheory'>Differential K-theory</a></li> <ul> <li><a href='#HSDifferentialKU'>Hopkins-Singer differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ku</mi></mrow><annotation encoding="application/x-tex">ku</annotation></semantics></math></a></li> <li><a href='#SoothVectorBundlesWithConnectionAndEInvariant'>Algebraic K-theory of smooth manifolds and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>-invariant</a></li> <li><a href='#ViaSmoothSnaithTheorem'>Via smooth Snaith’s theorem</a></li> </ul> <li><a href='#complexanalytic_differential_generalized_cohomology'>Complex-analytic differential generalized cohomology</a></li> <li><a href='#arithmetic_geometry'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-Arithmetic geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>In abstract generality, a <em>differential cohomology diagram</em> is the hexagonal <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> formed by the two <a class="existingWikiWord" href="/nlab/show/fracture+squares">fracture squares</a> of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> and <a class="existingWikiWord" href="/nlab/show/counit+of+a+comonad">counit</a> of, respectively, the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>. The exactness properties of this diagram for any <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math> in the corresponding <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesive (∞,1)-topos</a> express <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math> as being the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> for a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> refinement by differential form data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\flat_{dR} \hat E</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology+theory">generalized (Eilenberg-Steenrod) cohomology theory</a> which is <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>≔</mo><mi>Π</mi><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \coloneqq \Pi(\hat E)</annotation></semantics></math>.</p> <p>Historically, it had been shown in (<a href="#SimonsSullivan07">Simons-Sullivan 07</a>) for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> and in (<a href="#FreedLott10">Freed-Lott 10</a>, <a href="#SimonsSullivan08">Simons-Sullivan 08</a>) for <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> that these cohomology theories are characterized as sitting in the middle of a hexagonal <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of interlocking <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a> of <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a>, which expresses, on the one hand, how every differential cohomology class has underlying it a non-differential cohomology class (“of a <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>”) as well as a <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> datum, and, on the other hand, how the special cases of trivial underlying classes equipped with differential form datum and of <a class="existingWikiWord" href="/nlab/show/flat+connection">flat</a> differential form data sit inside the differential cohomology classes.</p> <p>At this schematic conceptual level a differential cohomology diagram looks as follows (where all unlabelled arrows are meant to be read as “evident 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></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>bundle</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><mrow><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>bundles</mi></mrow></mrow><mrow><mrow><mi>with</mi><mspace width="thickmathspace"></mspace><mi>flat</mi><mspace width="thickmathspace"></mspace><mi>connection</mi></mrow></mrow></mfrac></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>comparison</mi><mo stretchy="false">/</mo><mi>regulator</mi><mspace width="thickmathspace"></mspace><mi>map</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{ && {{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{bundle}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison/regulator\;map}{\longrightarrow} && {{shape}\atop{of\;bundle}} } </annotation></semantics></math></div> <p>One characteristic property is that the two outer sequences are <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a>. This expresses (at this rough schematic level) for instance (for the upper part) that the connections <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> on trivial bundles whose curvature vanishes in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>A</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{d}A = 0</annotation></semantics></math>, are exactly the flat connections; as well as (for the lower part) that bundles with <a class="existingWikiWord" href="/nlab/show/flat+connections">flat connections</a> have <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a> Chern-clases.</p> <p>So a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> theory would be one whose <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a>/<a class="existingWikiWord" href="/nlab/show/cohomology+classes">cohomology classes</a> have the interpretation of (stable) <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection</a> such as in the middle of this diagram.</p> <p>The characterization/construction of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> via <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a> (of <a class="existingWikiWord" href="/nlab/show/mapping+spectra">mapping spectra</a> with <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> data) due to (<a href="#HopkinsSinger02">Hopkins-Singer 02</a>) provides an incarnation of this kind of diagram genuinely in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, so that the outer parts are indeed <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a> and the two squares are homotopy cartesian (are <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a>).</p> <p>In (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13</a>) it was observed (see <a href="#Schreiber13">Schreiber 13, section 4.1.2</a> for the generality in which we present this here) that every <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> canonically sits inside a diagram of this form, being formed from the <a class="existingWikiWord" href="/nlab/show/fracture+squares">fracture squares</a> of the units and counits of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>, which in the right part are interpreted as the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\theta_E</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">ch_E</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></mtd> <mtd></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>θ</mi> <mover><mi>E</mi><mo stretchy="false">^</mo></mover></msub></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>ch</mi> <mi>E</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>Π</mi><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \Pi_{dR} {\hat E} && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}{\hat E} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_{\hat E}}} && \searrow \\ \Pi_{dR} \flat {\hat E} && && {\hat E} && && \Pi \flat_{dR} \hat E \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_E}} \\ && \flat {\hat E} && \longrightarrow && \Pi \hat E } \,. </annotation></semantics></math></div> <p>This is theorem <a class="maruku-ref" href="#TheDifferentialDiagram"></a> below.</p> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><mi>T</mi><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{H} = T Smooth\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a> (<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>) this subsumes the cases mentioned above. But there are many examples of cohomology theories not of the form of (<a href="#HopkinsSinger02">Hopkins-Singer 02</a>) but represented by <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable homotopy types</a> in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> (see <a href="#Schreiber13">Schreiber 13</a>, <a href="#BunkeNikolausVoelkl13">BunkeNikolausVölkl 13</a>) and hence fitting into such a diagram, where the interpretation of the pieces of the diagram is just as it should be.</p> <p>Therefore it makes sense to define generally that a <em>differential cohomology diagram</em> is the above combined <a class="existingWikiWord" href="/nlab/show/fracture+squares">fracture squares</a> with its outer <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a> for <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> and <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> in any <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>.</p> <h2 id="definitionconstruction">Definition/Construction</h2> <h3 id="differential_coefficients_and_maurercartan_forms">Differential coefficients and Maurer-Cartan forms</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T\mathbf{H}</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesive (∞,1)-topos</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">T_\ast \mathbf{H} \hookrightarrow T \mathbf{H}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> inside it.</p> <p>As usual, we use the following notation.</p> <div class="num_defn" id="DifferentialCoefficients"> <h6 id="definition_notation">Definition (Notation)</h6> <p>Write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a></p> </li> </ul> <p>restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}</annotation></semantics></math>, respectively.</p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">E \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a> we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat E \in Stab(\mathbf{H})</annotation></semantics></math> with an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \Pi(\hat E) \simeq E </annotation></semantics></math></div> <p>is a <em>cohesive refinement</em> or <em>differential refinement</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>Write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/counit+of+a+comonad">counit</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_{dR}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> of the looping of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>id</mi><mo>⟶</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\theta \;\colon\; id \longrightarrow \flat_{dR}</annotation></semantics></math> for the canonical morphism (itself the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mo>→</mo><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat_{dR} \to \flat</annotation></semantics></math>) which has the interpretation of being the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>.</p> </li> </ul> <p>Notice that on <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable homotopy types</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi><mo>≃</mo><mi>cofib</mi><mo stretchy="false">(</mo><mo>♭</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo>♭</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \flat_{dR}A \simeq cofib(\flat A \to A) = A/(\flat A) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Warning: Elsewhere we often write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} \Sigma</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math> above and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Pi_{dR}\Omega</annotation></semantics></math> for what is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_{dR}</annotation></semantics></math> here. That other convention has its advanages in the context of unstable cohesion. Here with stable cohesion the present convention is more natural.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>As discussed at <em><a href="cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology">structures in a cohesive ∞-topos – de Rham cohomology</a></em> there (and as is discussed below at <a href="#DeRhamCoefficients">de Rham coefficients</a>) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>T</mi></mstyle> <mo>*</mo></msub><mi>H</mi></mrow><annotation encoding="application/x-tex">A \in \mathbf{T}_\ast H</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type">cohesive homotopy type</a> then we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\Pi_{dR} A</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} A</annotation></semantics></math> as being the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> 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><mi>Π</mi><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(\flat_{dR} \Sigma A)</annotation></semantics></math>, truncated to negative degree and to non-negative degree, respectively; the canonical 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><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Pi_{dR}A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}A \\ & \searrow && \nearrow \\ && A } </annotation></semantics></math></div> <p>interpreting as the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a>.</p> <p>Beware that this is a very general conceptualization of de Rham coefficients. In standard examples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math> does come from traditional <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> data (see at <em><a href="#DeligneCoefficients">Deligne coefficients</a></em> and at <em><a href="#HopkinsSingerCoefficients">Hopkins-Singer coefficients</a></em> below), but generally it may have quite different looking models. But in any case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} \Sigma A</annotation></semantics></math> always has the interpretation of the home of the <em><a class="existingWikiWord" href="/nlab/show/curvature+forms">curvature forms</a></em> of cohomology with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, which makes thinking of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math> as producing generalized form data useful.</p> </div> <h3 id="ChernCharacterAndFractureSquares">Chern character and the differential fracture squares</h3> <div class="num_defn" id="GeneralChernCharacter"> <h6 id="definition">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> of the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math>, def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>, we call the <em><a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mo>≔</mo><mi>Π</mi><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ch \coloneqq \Pi \theta \,. </annotation></semantics></math></div> <p>Its component on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\hat E\in T_\ast \mathbf{H}</annotation></semantics></math> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mover><mi>E</mi><mo stretchy="false">^</mo></mover></msub></mrow><annotation encoding="application/x-tex">ch_{\hat E}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">ch_E</annotation></semantics></math>, for short.</p> </div> <div class="num_prop" id="TheDifferentialFractureSquare"> <h6 id="proposition">Proposition</h6> <p><strong>(differential fracture square)</strong></p> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\hat E \in T_\ast \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality square</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><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ch</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(\hat E) &\stackrel{ch_E}{\longrightarrow}& \Pi \flat_{dR} \hat E } </annotation></semantics></math></div> <p>(of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> applied to the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> of the <a class="existingWikiWord" href="/nlab/show/counit+of+a+comonad">counit</a> of the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a>) is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> square (hence also an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushout">(∞,1)-pushout</a>).</p> <p>Dually, also the square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E \\ \downarrow && \downarrow \\ \flat \hat E & \longrightarrow & \hat E } </annotation></semantics></math></div> <p>is homotopy cartesian.</p> </div> <p>This fact was observed in (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, prop. 3.5</a>). It may be thought of as an incarnation of the concept of a <em><a class="existingWikiWord" href="/nlab/show/fracture+theorem">fracture theorem</a></em>.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> and <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stability</a> we have the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Π</mi><mo stretchy="false">(</mo><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat \hat E &\longrightarrow & \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E \\ \downarrow^{\mathrlap{\simeq}} && \downarrow && \downarrow \\ \Pi(\flat \hat E) &\longrightarrow& \Pi(\hat E) &\stackrel{}{\longrightarrow}& \Pi(\flat_{dR} \hat E) } </annotation></semantics></math></div> <p>where both rows are <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a>. By <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> the left vertical map is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a>. The claim now follows with the <a href="homotopy+pullback#HomotopyFiberCharacterization">homotopy fiber characterization</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>.</p> <p>The second statement follows dually:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></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> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Pi_{dR} \flat \hat E &\longrightarrow& \Pi_{dR} \hat E &\longrightarrow& \Pi_{dR} \flat_{dR} \hat E \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ \flat \hat E & \longrightarrow & \hat E &\longrightarrow&\flat_{dR} \hat E } \,. </annotation></semantics></math></div></div> <p>Notice that generally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\flat_{dR} \hat E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> “<a class="existingWikiWord" href="/nlab/show/anti-modal+type">anti-modal type</a>” in that it is annihilated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</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>♭</mo><mo stretchy="false">(</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \flat (\flat_{dR} \hat E) \simeq 0 \,. </annotation></semantics></math></div> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-anti-modal types are called “pure” in (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, prop. 3.5</a>)).</p> <p>The following says that not only does, by prop.<a class="maruku-ref" href="#TheDifferentialFractureSquare"></a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mo>♭</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Pi \dashv \flat)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fracture+square">fracture square</a> exhibit any stable cohesive homotopy type as the pullback of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/anti-modal+type">anti-modal type</a> along a map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-modal types into its <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a>, but that conversely all homotopy pullbacks of this form are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mo>♭</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Pi \dashv \flat)</annotation></semantics></math>-fracture squares.</p> <div class="num_lemma" id="ReconstructionOfFractureSquare"> <h6 id="lemma">Lemma</h6> <p>For</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a>-<a class="existingWikiWord" href="/nlab/show/modal+type">modal type</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Omega^{\bullet \geq 0}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/flat+modality">flat</a>-<a class="existingWikiWord" href="/nlab/show/anti-modal+type">anti-modal type</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>></mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">E \stackrel{f}{\longrightarrow} \Pi \Omega^{\bullet \gt 0}</annotation></semantics></math> a morphism;</p> </li> </ol> <p>then the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> square for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo>≔</mo><mi>E</mi><munder><mo>×</mo><mrow><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow></munder><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \hat E \coloneqq E \underset{\Pi\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} </annotation></semantics></math></div> <p>is equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mo>♭</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Pi\dashv \flat)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fracture+square">fracture square</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math> according to prop. <a class="maruku-ref" href="#TheDifferentialFractureSquare"></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><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>≃</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>E</mi><mo>≃</mo></mtd> <mtd><mi>Π</mi><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo>≃</mo><msub><mi>ch</mi> <mover><mi>E</mi><mo stretchy="false">^</mo></mover></msub></mrow></mover></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>≃</mo><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \hat E &\longrightarrow& \flat_{dR} \hat E & \simeq \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow \\ E \simeq & \Pi \hat E &\stackrel{f \simeq ch_{\hat E}}{\longrightarrow}& \Pi \flat_{dR} \hat E & \simeq \Pi \Omega^{\bullet \geq 0} } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>First we observe that indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>≃</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\Omega^{\bullet\geq 0}\simeq \flat_{dR} \hat E</annotation></semantics></math>. For that consider the following morphisms of homotopy pullback diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn><mo>≃</mo></mtd> <mtd><mo>♭</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>♭</mo><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>♭</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>E</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><mo>←</mo></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo></mtd> <mtd><mo>♭</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 \simeq & \flat \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0} &\longrightarrow& \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow && \downarrow \\ & \flat \Pi \Omega^{\bullet \geq 0} &\stackrel{\simeq}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0 \\ & \uparrow && \uparrow^{\mathrlap{f}} && \uparrow \\ & \flat E &\stackrel{\simeq}{\longrightarrow}& E &\longrightarrow& 0 \\ \\ \underset{\leftarrow}{\lim}(-) \simeq& \flat \hat E &\longrightarrow& \hat E &\longrightarrow& \flat_{dR}\hat E } </annotation></semantics></math></div> <p>Here in the middle column we are showing the homotopy fiber product defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math>. On the left we have its image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>, with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-counits running horizontally and using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Pi \Omega^{\bullet \geq 0}</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modal+types">modal types</a> by assumption and by <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Omega^{\bullet \geq 0}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/anti-modal+type">anti modal</a> by assumption. This and using that by <a class="existingWikiWord" href="/nlab/show/stability">stability</a> all <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limits">finite (∞,1)-limits</a> and <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-colimits">finite (∞,1)-colimits</a> commute produces the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> morphisms going to the right.</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>≃</mo><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\Omega^{\bullet \geq 0} \simeq \flat_{dR} \hat E</annotation></semantics></math> and that the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><munder><mo>×</mo><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow></munder><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>→</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">E \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to \Omega^{\bullet \geq 0}</annotation></semantics></math> is equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math>.</p> <p>Applying the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> to the right half of this diagram, using that by stability it preserves the homotopy pullbacks gives</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><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>E</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><mo>←</mo></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo></mtd> <mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ch</mi> <mover><mi>E</mi><mo stretchy="false">^</mo></mover></msub></mrow></mover></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \Pi \Omega^{\bullet \geq 0} &\stackrel{\simeq}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \downarrow^{\mathrlap{\simeq}} && \downarrow \\ & \Pi \Omega^{\bullet \geq 0} &\longrightarrow& 0 \\ & \uparrow^{\mathrlap{f}} && \uparrow \\ & E &\longrightarrow& 0 \\ \\ \underset{\leftarrow}{\lim}(-) \simeq & E &\stackrel{ch_{\hat E}}{\longrightarrow}& \Pi \flat_{dR}\hat E } \,. </annotation></semantics></math></div> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ch</mi> <mover><mi>E</mi><mo stretchy="false">^</mo></mover></msub><mo>≃</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">ch_{\hat E} \simeq f</annotation></semantics></math>.</p> <p>Finally, the dual argument</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><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Π</mi><mi>E</mi></mtd> <mtd><mo>≃</mo><mi>E</mi></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><mo>←</mo></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo></mtd> <mtd><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \Omega^{\bullet \geq 0} &\stackrel{}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \downarrow && \downarrow^{\mathrlap{\simeq}} \\ & \Pi \Omega^{\bullet \geq 0} &\stackrel{}{\longrightarrow}& \Pi \Omega^{\bullet \geq 0} \\ & \uparrow && \uparrow \\ & E &\stackrel{}{\longrightarrow}& \Pi E & \simeq E \\ \\ \underset{\leftarrow}{\lim}(-) \simeq & \hat E &\longrightarrow& \Pi(\hat E) & \simeq E } </annotation></semantics></math></div> <p>shows that the other <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><munder><mo>×</mo><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mrow></munder><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E \underset{\Pi_{dR}\Omega^{\bullet \geq 0}}{\times} \Omega^{\bullet \geq 0} \to E</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a>-<a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat E</annotation></semantics></math>.</p> </div> <p>The following proposition says that the construction in lemma <a class="maruku-ref" href="#ReconstructionOfFractureSquare"></a> extends to a decomposition of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of cohesive stable types as a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of moprhisms of stable <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a>-<a class="existingWikiWord" href="/nlab/show/modal+types">modal types</a> with that of <a class="existingWikiWord" href="/nlab/show/flat+modality">flat</a>-<a class="existingWikiWord" href="/nlab/show/anti-modal+types">anti-modal types</a>:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</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><mrow><mtable><mtr><mtd><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi></mrow></mover></mtd> <mtd><mi>ker</mi><mo stretchy="false">(</mo><mo>♭</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ch</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>Π</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Spectra</mi> <mi>I</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>cod</mi></mover></mtd> <mtd><mi>Spectra</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Stab(\mathbf{H}) &\stackrel{\flat_{dR} \Sigma}{\longrightarrow}& ker(\flat) \\ \downarrow^{\mathrlap{ch}} && \downarrow^{\mathrlap{\Pi}} \\ Spectra^I &\stackrel{cod}{\longrightarrow}& Spectra } \,, </annotation></semantics></math></div> <p>where the bottom map is the <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> restricted to stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/anti-modal+types">anti-modal types</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ch</mi></mrow><annotation encoding="application/x-tex">ch</annotation></semantics></math> assigns the <a class="existingWikiWord" href="/nlab/show/Chern+character+map">Chern character map</a> of def. <a class="maruku-ref" href="#GeneralChernCharacter"></a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\flat_{dR} \Sigma</annotation></semantics></math> assigns de Rham coefficients as in def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>.</p> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, prop. 3.5</a>)</p> <p>Dually:</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>T</mi> <mo>*</mo></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in T_\ast \mathbf{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality square</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><mo stretchy="false">(</mo><msub><mi>Π</mi> <mi>dR</mi></msub><msup><mi>Σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>♭</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \flat(\Pi_{dR} \Sigma^{-1} A) &\longrightarrow & \Pi_{dR}(\Sigma^{-1} A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A } </annotation></semantics></math></div> <p>(of the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> applied to the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a>) is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> square.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>As before but dually, the diagram extends to a morphism of <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> diagrams 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><mo>♭</mo><mo stretchy="false">(</mo><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>♭</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>♭</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \flat(\Pi_{dR} A) &\longrightarrow & \Pi_{dR}(A) \\ \downarrow && \downarrow \\ \flat A &\stackrel{}{\longrightarrow}& A \\ \downarrow && \downarrow \\ \flat \Pi(A) &\stackrel{\simeq}{\longrightarrow}& \Pi(A) } \,, </annotation></semantics></math></div> <p>and by <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> the bottom horizontal morphism is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a>.</p> </div> <h3 id="TheHexagonDiagram">The hexagon diagram</h3> <p>Combining these two statements yields the following (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13</a>).</p> <div class="num_theorem" id="TheDifferentialDiagram"> <h6 id="theorem">Theorem</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> with <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>, then for every <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in Stab(\mathbf{H}) \hookrightarrow T \mathbf{H}</annotation></semantics></math> the canonical hexagon diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>θ</mi> <mi>A</mi></msub></mrow></mpadded></msub></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo>♭</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>a</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>b</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>ch</mi> <mi>A</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>Π</mi><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && \Pi_{dR} A && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR} A \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow \\ \Pi_{dR} \flat A && \Downarrow^{\mathrlap{a}} && A && \Downarrow^{\mathrlap{b}} && \Pi \flat_{dR} A \\ & \searrow & & \nearrow & & \searrow && \nearrow_{\mathrlap{ch_A}} \\ && \flat A && \longrightarrow && \Pi A } \,, </annotation></semantics></math></div> <p>formed from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>-counit – the “differential cohomology hexagon” – is homotopy exact in that</p> <ol> <li> <p>the two squares are <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> squares (“<a class="existingWikiWord" href="/nlab/show/fracture+squares">fracture squares</a>”);</p> </li> <li> <p>the two diagonals are the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a> of the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\theta_A</annotation></semantics></math> and its dual;</p> </li> <li> <p>the bottom morphism is the canonical <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a>;</p> </li> <li> <p>the top and bottom outer sequences are long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Only the last statement remains to be shown, for that use the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>: this gives the following diagram in which every square and every <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting</a> rectangle is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo>♭</mo><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>♭</mo><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mrow><msup><mi>a</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>θ</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mi>b</mi></mpadded></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Π</mi><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Pi_{dR} \flat A &\to & \flat A &\to& 0 \\ \downarrow^{\mathrlap{f_1}} &\swArrow_{\mathrlap{a^{-1}}}& \downarrow && \downarrow \\ \Pi_{dR} A &\stackrel{f_{2}}{\to}& A &\stackrel{\theta_A}{\to}& \flat_{dR} A \\ \downarrow && \downarrow & \swArrow_{\mathrlap{b}} & \downarrow^{\mathrlap{f_3}} \\ 0 &\to& \Pi A &\to& \Pi \flat_{dR} A } \,. </annotation></semantics></math></div> <p>This exhibits the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mi>A</mi></msub><mo>∘</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></mover></mrow><annotation encoding="application/x-tex">\stackrel{f_1}{\to}\stackrel{\theta_A \circ f_{2}}{\longrightarrow} \stackrel{f_3}{\to}</annotation></semantics></math>, which is the top part of the hexagon, as a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a>.</p> <p>The dual argument shows that the bottom part of the hexagon is a <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequence">homotopy cofiber sequence</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Theorem <a class="maruku-ref" href="#TheDifferentialDiagram"></a> in particular implies that for <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable</a> <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+types">cohesive homotopy types</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> there are natural equivalences</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi><mo>≃</mo><msub><mi>Π</mi> <mi>dR</mi></msub><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\Pi_{dR} A \simeq \Pi_{dR} \flat_{dR} A</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi><mo>≃</mo><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\flat \Pi_{dR} A\simeq \Pi \flat_{dR} A</annotation></semantics></math></p> </li> </ul> <p>Given that conceptually, as made explicit in the <a href="#Idea">Idea-section</a> above, we may think of cocycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\Pi \flat_{dR} A</annotation></semantics></math> as the rationalized characteristic classes and of coycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><msub><mi>Π</mi> <mi>dR</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\flat \Pi_{dR} A</annotation></semantics></math> as flat differential forms, this expresses the conceptual content of the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>.</p> </div> <h2 id="examples">Examples</h2> <h3 id="OrdinaryDifferentialCohomology">Ordinary differential cohomology</h3> <p>We discuss differential refinements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, hence of cohomology with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> or equivalently, via the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a>, cohomology <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> underlying <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a>-<a class="existingWikiWord" href="/nlab/show/module+spectra">module spectra</a>.</p> <h4 id="DeRhamCoefficients">De Rham coefficients</h4> <p>We briefly recall the <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a> given by the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> (tensored with any <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>) from <em><a href="smooth+spectrum#ExamplesDeRhamSpectra">smooth spectrum – Examples – De Rham spectra</a></em>.</p> <p>Write <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> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a> (of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, hence over the <a class="existingWikiWord" href="/nlab/show/ring">ring</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> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>). It is convenient to choose 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><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet \in Ch_\bullet</annotation></semantics></math> the grading convention</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></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \vdots \\ \downarrow \\ A_{-1} \\ \downarrow \\ A_0 \\ \downarrow \\ A_1 \\ \downarrow \\ \vdots } </annotation></semantics></math></div> <p>such that under the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Ch</mi> <mo>•</mo></msub><mover><mo>⟶</mo><mrow></mrow></mover><mi>Spectra</mi></mrow><annotation encoding="application/x-tex"> DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of spectra relate to the <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>DK</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_n(DK(A_\bullet)) \simeq H_{-n}(A_\bullet) \,. </annotation></semantics></math></div> <p>In particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A[n]</annotation></semantics></math> denotes the chain complex 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><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">-n</annotation></semantics></math> in this counting.</p> <p>The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, which is the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a> via the discussion at <em><a href="smooth+spectrum#FromChainComplexesOfSmoothModules">smooth spectrum – from chain complexes of smooth modules</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo>∈</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo>,</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo>,</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^{\bullet} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots) </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^0(X) = C^\infty(X, \mathbb{R})</annotation></semantics></math> in degree 0.</p> <p>Throughout we keep the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mover><mo>⟶</mo><mi>DK</mi></mover><mi>Spectra</mi><mover><mo>↪</mo><mi>Disc</mi></mover><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C \in Ch_\bullet \stackrel{DK}{\longrightarrow} Spectra \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H}) </annotation></semantics></math></div> <p>notationally implicit, regarding bare chain complexes as geometrically discrete spectrum objects in smooth differential geometry.</p> <p>We consider 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 truncated sheaves of de Rham complexes with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in a given <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>:</p> <div class="num_defn"> <h6 id="definition_2">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><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo>∈</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo>,</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo>,</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a> given under the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a> by the sheaf of truncated <a class="existingWikiWord" href="/nlab/show/de+Rham+complexes">de Rham complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^{\bullet \geq n} \;\colon\; X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots) </annotation></semantics></math></div> <p>with <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></mrow><annotation encoding="application/x-tex">\Omega^n(X)</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> (the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th stage in the <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a>).</p> </div> <p>More genereally:</p> <div class="num_defn" id="DeRhamComplexesWithCoefficients"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C \in Ch_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>,</p> <p>write</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(\Omega \otimes C)^{\bullet \geq n}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a> given over each manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> followed by truncation as indicated:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>⋯</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><msub><mo>⊕</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>±</mo><msub><mi>d</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><msub><mo>⊕</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>±</mo><msub><mi>d</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>deg</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>n</mi><mo>−</mo><mn>1</mn></mtd> <mtd></mtd> <mtd><mi>n</mi></mtd> <mtd></mtd> <mtd><mi>n</mi><mo>+</mo><mn>1</mn></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{array} (\Omega \otimes C)^{\bullet \geq n} &=& (\cdots &\to& 0 &\to& 0 &\to& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}&\stackrel{\mathbf{d}\pm d_{C}}{\to}& \cdots) \\ \\ deg && && && n-1 && n && n+1 && \end{array} \,. </annotation></semantics></math></div> <p>where the first non-trivial term displayed is in taken to be 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> <p>and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo><</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(\Omega \otimes C)^{\bullet \lt n}</annotation></semantics></math> for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left left"><mtr><mtd></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>±</mo><msub><mi>d</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><msub><mo>⊕</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn><mo>−</mo><mi>k</mi></mrow></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>±</mo><msub><mi>d</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd><msub><mo>⊕</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>−</mo><mi>k</mi></mrow></msub><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>deg</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>n</mi><mo>−</mo><mn>1</mn></mtd> <mtd></mtd> <mtd><mi>n</mi></mtd> <mtd></mtd> <mtd><mi>n</mi><mo>+</mo><mn>1</mn></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array} (\Omega \otimes C)^{\bullet \lt n} &=& ( \cdots &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-2-k} &\stackrel{\mathbf{d} \pm d_{C}}{\to}& \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-1-k} 0 &\to& 0 &\to& \cdots ) \\ \\ deg && && n-1 && n && n+1 && \end{array} </annotation></semantics></math></div> <p>where again the first non-trivial term displayed is taken to be 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_remark"> <h6 id="remark_4">Remark</h6> <p>Beware of the degree conventions in def. <a class="maruku-ref" href="#DeRhamComplexesWithCoefficients"></a>: in the first clause the tensor product of complexes is truncated without any shifting, while in the second case the truncation is shifted by one. This notation turns out to well reflect the way that the hexagon, theorem <a class="maruku-ref" href="#TheDifferentialDiagram"></a>, decomposes these <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a> by prop. <a class="maruku-ref" href="#DifferentialComponentsOfDeRhamTensorC"></a> and remark <a class="maruku-ref" href="#OnDecompositionOfDeRhamWithCoefficients"></a> below.</p> </div> <div class="num_prop" id="DifferentialComponentsOfDeRhamTensorC"> <h6 id="proposition_4">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in Ch_\bullet(\mathbb{R})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/real+number">real</a> <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\Pi ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n})\simeq C^\bullet</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\flat ((\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}) \simeq C^{\bullet \geq n} </annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mi>C</mi> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\flat_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq (\Omega \otimes_{\mathbb{R}} C^{\bullet \leq n-1})^{\bullet \geq n}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Pi_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq (\Omega \otimes_{\mathbb{R}} C )^{\bullet \leq n-1}[-1]</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo>≃</mo><msup><mi>C</mi> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Pi \flat_{dR} (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n} \simeq C^{\bullet \leq n-1}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>C</mi><mo>→</mo><msub><mi>C</mi> <mrow><mo>•</mo><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">ch \;\colon\; C \to C_{\bullet\leq n-1} </annotation></semantics></math> is the canonical projection.</p> </li> </ol> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, lemma 4.4</a>)</p> <div class="num_remark" id="OnDecompositionOfDeRhamWithCoefficients"> <h6 id="remark_5">Remark</h6> <p>Prop. <a class="maruku-ref" href="#DifferentialComponentsOfDeRhamTensorC"></a> means first of all that the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq n}</annotation></semantics></math> is a differential refinement, def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>, of the chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> (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>)</p> <p>Moreover it says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat_{dR} \Sigma (-)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_{dR} \Omega(-)</annotation></semantics></math> are similarly the high degree and low degree truncation, respectively, of the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi \flat_{dR} \Sigma (-)</annotation></semantics></math>.</p> <p>Specifically for <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>, only the connected part</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><msub><mi>C</mi> <mrow><mo>•</mo><mo>≤</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> D C_{\bullet \leq -1} </annotation></semantics></math></div> <p>appears in the de Rham coefficients, and we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Π</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>≃</mo></mtd> <mtd></mtd> <mtd><mo>≃</mo></mtd> <mtd></mtd> <mtd><mo>≃</mo></mtd></mtr> <mtr><mtd><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>D</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo><</mo><mn>0</mn></mrow></msup></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mstyle mathvariant="bold"><mi>d</mi></mstyle> <mi>dR</mi></msub><mo>±</mo><msub><mi>d</mi> <mi>C</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mi>Ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>D</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Pi_{dR}(\cdots) && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}(\cdots) \\ \simeq && \simeq && \simeq \\ \Omega(-,D)^{\bullet \lt 0} && \stackrel{\mathbf{d}_{dR} \pm d_{C}}{\longrightarrow} && \Omega(-,D)^{\bullet \geq 0} } \,. </annotation></semantics></math></div></div> <h4 id="DeligneCoefficients">Deligne coefficients</h4> <p>We discuss the differential cohomology hexagon for smooth <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> and hence smooth <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> realizedin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}= </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in Ch_\bullet(\mathbb{Z})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>. 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> an <a class="existingWikiWord" href="/nlab/show/integer">integer</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>conn</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{conn,-n}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>C</mi> <mrow><mi>conn</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi><mo>⊗</mo><mi>ℝ</mi></mtd> <mtd><mo>≃</mo><mi>Π</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ C_{conn,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n} \\ \downarrow && \downarrow \\ C &\longrightarrow& C \otimes \mathbb{R} & \simeq \Pi((\Omega \otimes_{\mathbb{Z}} C)^{\bullet \geq n}) } \,, </annotation></semantics></math></div> <p>where the right map is the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> according to prop. <a class="maruku-ref" href="#DifferentialComponentsOfDeRhamTensorC"></a>.</p> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, 4.3</a>)</p> <div class="num_prop" id="DecompositionForGeneralSmoothDeligneCoefficients"> <h6 id="proposition_5">Proposition</h6> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mi>conn</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Pi(C_{conn,-n}) \simeq C</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><msub><mi>C</mi> <mrow><mi>conn</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><msub><mi>C</mi> <mrow><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\flat_{dR} C_{conn,-n} \simeq (\Omega \otimes C_{\leq n-1})^{\bullet \geq n}</annotation></semantics></math></p> </li> </ol> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, lemma 4.5</a>)</p> <div class="num_example" id="ExampleOrdinaryDeligneCohomology"> <h6 id="example">Example</h6> <p><strong>(ordinary Deligne cohomology)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C = \mathbb{Z}[n+1]</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo>≃</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq 0} \simeq (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq 0} \simeq \Omega^{n+1}_{cl}(-) </annotation></semantics></math></div> <p>(with the last term being in degree 0) and more generally 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><mi>⋯</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \otimes (\mathbb{Z}[n+1]\otimes \mathbb{R}))^{\bullet \geq -n} \simeq (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n} \simeq (\Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^{n+1}_{cl}(-) ) </annotation></semantics></math></div> <p>(with the last term being in chain degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">-n</annotation></semantics></math> to 0 (hence homotopy group degrees <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> to 0)).</p> <p id="PastingForDeligneTower"> We have a pasting diagram of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</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>B</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <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></mtr> <mtr><mtd></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B^n U(1)_{conn} &\to& \Omega^{n+1}_{cl} \\ \downarrow && \downarrow \\ &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -1} \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{}{\longrightarrow}& (\Omega \otimes \mathbb{R}[n+1])^{\bullet \geq -n} \\ \downarrow && \downarrow \\ B^{n+1}\mathbb{Z} &\to& B^{n+1} \mathbb{R} } </annotation></semantics></math></div> <p>where on the right runs the <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a>, with the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mi>conn</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1) = \mathbb{Z}[n+1]_{conn,n}</annotation></semantics></math> as at <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a> and <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><mo>=</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mi>conn</mi><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^n U(1)_{conn} = \mathbb{Z}[n+1]_{conn,0}</annotation></semantics></math> as at <em><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></em>.</p> <p>Hence for all <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mi>conn</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[n+1]_{conn,n} \simeq \mathbf{B}^n U(1) \simeq (C^\infty(-,U(1)) \to 0 \to \cdots\to 0) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mi>conn</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><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><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[n+1]_{conn,0} \simeq \mathbf{B}^n U(1)_{conn} \simeq (C^\infty(-,U(1))\stackrel{\mathbf{d}log}{\to}\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to}\Omega^n) \,. </annotation></semantics></math></div> <p>is represented by the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a>.</p> <p>The intermediate cases in between these two as in (<a href="#Schreiber13">Schreiber 13, def. 3.9.46</a>, <a href="#FRS13">FRS 13, remark 2.3.15</a>) (discussed also for instance at <em><a href="http://ncatlab.org/nlab/show/Courant+algebroid#RelationToAtiyahGroupoids">Courant algebroid – Relation to Atiyah Lie 2-algebroid</a></em>).</p> <p>Notice that by prop. <a class="maruku-ref" href="#DecompositionForGeneralSmoothDeligneCoefficients"></a> the above homotopy pullback diagrams are all indeed the right part of the differential cohomology hexagon, theorem <a class="maruku-ref" href="#TheDifferentialDiagram"></a>, e.g.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <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></msub></mrow></mover></mtd> <mtd><msubsup><mi>Ω</mi> <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></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>θ</mi> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo>→</mo><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>θ</mi> <mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^n U(1)_{conn} &\stackrel{\theta_{\mathbf{B}^n U(1)_{conn}}}{\longrightarrow}& \Omega^{n+1}_{cl} \\ \downarrow && \downarrow \\ \vdots && \vdots \\ \downarrow && \downarrow \\ \mathbf{B}^n U(1) &\stackrel{\theta_{\mathbf{B}^n U(1)}}{\longrightarrow}& (\Omega^1 \to \cdots\to \Omega^{n+1}_{cl}) \\ \downarrow && \downarrow \\ \mathbf{B}^{n+1}\mathbb{Z} &\stackrel{\Pi(\theta_{\mathbf{B}^n U(1)})}{\longrightarrow}&\mathbf{B}^{n+1}\mathbb{R} } \,. </annotation></semantics></math></div> <p>The differential cohomology hexagon obtained from this is on cohomology classes for each <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and each <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> <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>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><msup><mo>↘</mo> <mpadded width="0"><mi>a</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><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><msup><mi>H</mi> <mn>0</mn></msup><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></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><mo>↘</mo></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><mrow></mrow></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{ && \Omega^{n}(X)/im(\mathbf{d}) && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^0(X,\mathbf{B}^n U(1)_{conn}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && H^{n}(X,U(1)) && \underset{}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) } </annotation></semantics></math></div> <p>It is common (e.g. <a href="#SimonsSullivan07">Simons-Sullivan 07</a>) to display this after quotienting out the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> (which is the group <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><msub><mo stretchy="false">)</mo> <mi>ℤ</mi></msub></mrow><annotation encoding="application/x-tex">\Omega^n(X)_{\mathbb{Z}}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/closed+differential+forms">closed differential forms</a> with <a class="existingWikiWord" href="/nlab/show/integer">integral</a> <a class="existingWikiWord" href="/nlab/show/periods">periods</a>), which is such as to make the top-left to bottom-right diagonal sequence be not just <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a> at the middle term, but be a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>:</p> <div class="maruku-equation" id="eq:OrdinarCohomologyHexagon"><span class="maruku-eq-number">(1)</span><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><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>ℤ</mi></msub></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><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><msup><mi>H</mi> <mn>0</mn></msup><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></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><mo>↘</mo></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></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 & && && && & 0 \\ & \searrow && && && \nearrow \\ && \Omega^{n}(X)/\Omega^n(X)_{\mathbb{Z}} && \stackrel{\mathbf{d}}{\longrightarrow} && \Omega^{n+1}_{cl}(X) \\ & \nearrow && \searrow^{\mathrlap{a}} && \nearrow && \searrow \\ H^{n}(X, \mathbb{R}) && && H^0(X,\mathbf{B}^n U(1)_{conn}) && && H^{n+1}(X,\mathbb{R}) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && H^{n}(X,U(1)) && \underset{\beta}{\longrightarrow} && H^{n+1}(X,\mathbb{Z}) \\ & \nearrow && && && \searrow \\ 0 & && && && & 0 } \,. </annotation></semantics></math></div> <p>Here the diagonals are now the “curvature exact sequence” and the “characteristic class exact sequence” as discussed at <em><a href="ordinary+differential+cohomology#CurvatureAndCharClass">ordinary differential cohomology – Properties – curvature and characteristic class</a></em>.</p> <p>The bottom horizontal map is the <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> of the <a class="existingWikiWord" href="/nlab/show/exponential+sequence">exponential sequence</a> (<a href="Bockstein+homomorphism#Mod2BocksteinAndExponentialExactSequence">this example</a>).</p> </div> <h3 id="HopkinsSingerCoefficients">Hopkins-Singer coefficients</h3> <p>We discuss how the <a class="existingWikiWord" href="/nlab/show/differential+function+complexes">differential function complexes</a> of (<a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2Cand+M-Theory">Hopkins-Singer 05</a>), providing differential refinements of geometrically discrete <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a>, and how they fit into their differential cohomology hexagon (following <a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, section 4.4.</a>).</p> <p>Let again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <p>Throughout, we leave notationally implicit</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DK</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Ch</mi> <mo>•</mo></msub><mo>⟶</mo><mi>Spectra</mi></mrow><annotation encoding="application/x-tex">DK \;\colon\;Ch_\bullet \longrightarrow Spectra</annotation></semantics></math>;</p> </li> <li> <p>the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spectra</mi><mover><mo>↪</mo><mi>Stab</mi></mover><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spectra \stackrel{Stab}{\hookrightarrow} Stab(\mathbf{H})</annotation></semantics></math>;</p> </li> </ol> <p>hence always regard a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> as the corresponding <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> and regard any bare spectrum always as a <a class="existingWikiWord" href="/nlab/show/geometrically+discrete+infinity-groupoid">geometrically discrete</a> <a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a>.</p> <div class="num_defn" id="HopkinsSingerPullback"> <h6 id="definition_5">Definition</h6> <p>For</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>Spectra</mi><mover><mo>↪</mo><mi>Disc</mi></mover><mi>Stab</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in Spectra \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \in Ch_\bullet(\mathbb{R})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/real+number">real</a> <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \colon E \longrightarrow C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> to the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a>;</p> </li> <li> <p><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> </ol> <p>write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">E_{conn_c,-n}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>≃</mo><mi>Π</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mo>⊗</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ E_{conn_c,-n} &\longrightarrow& (\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n} \\ \downarrow && \downarrow \\ E &\longrightarrow& C & \simeq \Pi((\Omega \otimes C)^{\bullet \geq -n}) } \,, </annotation></semantics></math></div> <p>where the map on the right is the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> according to prop. <a class="maruku-ref" href="#DifferentialComponentsOfDeRhamTensorC"></a>.</p> <p>We abbreviate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub></mrow></msub><mo>≔</mo><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mn>0</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{conn_c} \coloneqq E_{conn_c,0} \,. </annotation></semantics></math></div></div> <p>This appears in (<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, 4.4</a>), with an earlier version in (<a href="#BunkeGepner13">Bunke-Gepner 13, def. 2.1</a>).</p> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Specifically when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>E</mi><mo>∧</mo><mi>DK</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \simeq E \wedge DK(\mathbb{R})</annotation></semantics></math> is a model for the rationalization (realification) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \colon E \to C</annotation></semantics></math> is the canonical map, then it is a model for the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> and def. <a class="maruku-ref" href="#HopkinsSingerPullback"></a> is essentially the definition of (<a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2Cand+M-Theory">Hopkins-Singer 05</a>).</p> </div> <div class="num_prop" id="DifferentialComponentsOfHopkinsSingerCoefficients"> <h6 id="proposition_6">Proposition</h6> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo>≃</mo><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>C</mi> <mrow><mo>•</mo><mo>≤</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\flat_{dR} E_{conn_c,-n} \simeq (\Omega \otimes_{\mathbb{R}} C_{\bullet \leq -n -1})^{\bullet\geq n}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\Pi(E_{conn_c,-n}) \simeq E</annotation></semantics></math>.</p> </li> </ol> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, lemma 4.7</a>)</p> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Beware that in prop. <a class="maruku-ref" href="#DifferentialComponentsOfHopkinsSingerCoefficients"></a> the intrinsic de Rham coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\flat_{dR} \Sigma E_{conn_c,-n}</annotation></semantics></math> in general differ by a truncation from the de Rham complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo>•</mo><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(\Omega \otimes_{\mathbb{R}} C)^{\bullet \geq -n}</annotation></semantics></math> which is being pulled back in definition <a class="maruku-ref" href="#HopkinsSingerPullback"></a>.</p> <p>But by prop. <a class="maruku-ref" href="#DifferentialComponentsOfHopkinsSingerCoefficients"></a> and theorem <a class="maruku-ref" href="#TheDifferentialDiagram"></a> composition with the truncation map preserves the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, hence in the differential cohomology hexagon the two differ only in the exactness property in the top right entry.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub></mrow></msub><mo>≔</mo><msub><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">E_{conn_c} \coloneqq E_{conn_c,0}</annotation></semantics></math> as in def. <a class="maruku-ref" href="#HopkinsSingerPullback"></a>, then for each <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (indeed for each <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</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>) the differential cohomology diagram on cohomology classes is 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></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><mrow><mi>im</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">)</mo></mrow></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>C</mi><msubsup><mo stretchy="false">)</mo> <mi>cl</mi> <mn>0</mn></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>E</mi> <mrow><msub><mi>conn</mi> <mi>c</mi></msub></mrow> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><msup><mi>E</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msup><mi>E</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && (\Omega(X) \otimes_{\mathbb{R}} C)^{-1}/{im(\mathbf{d})} && \stackrel{\mathbf{d}}{\longrightarrow} && (\Omega(X)\otimes_{\mathbb{R}} C)^0_{cl} \\ & \nearrow && \searrow && \nearrow && \searrow \\ H^{-1}(X,C) && && E_{conn_c}^0(X) && && H^0(X,C) \\ & \searrow && \nearrow && \searrow && \nearrow \\ && \flat E^0(X) && \longrightarrow && E^0(X) } \,. </annotation></semantics></math></div></div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, (25)-(26)</a>)</p> <h3 id="DifferentialKTheory">Differential K-theory</h3> <p>We discuss several differential refinements, def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>, to <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a>, of the complex <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> spectrum <a class="existingWikiWord" href="/nlab/show/KU">KU</a> or of its <a class="existingWikiWord" href="/nlab/show/connective+cover">connective cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ku</mi></mrow><annotation encoding="application/x-tex">ku</annotation></semantics></math>, hence versions of <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a>.</p> <h4 id="HSDifferentialKU">Hopkins-Singer differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ku</mi></mrow><annotation encoding="application/x-tex">ku</annotation></semantics></math></h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[b]</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> with <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> on a single generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>. With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> regarded as in degree 2, regard this as a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> with vanishing <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a>.</p> <p>Then the complex ordinary <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> on K-theory is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ch</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ku</mi><mo>⟶</mo><mi>ℂ</mi><mo stretchy="false">[</mo><mi>b</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ch\;\colon\; ku \longrightarrow \mathbb{C}[b] \,. </annotation></semantics></math></div> <div class="num_defn" id="HSDifferentialKu"> <h6 id="definition_6">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>ku</mi> <mrow><msub><mi>conn</mi> <mi>ch</mi></msub></mrow></msub><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ku_{conn_ch} \in Stab(Smooth\infty Grpd) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/smooth+spectrum">smooth spectrum</a> induced by def. <a class="maruku-ref" href="#HopkinsSingerPullback"></a>) from the standard <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>≔</mo><mi>ch</mi></mrow><annotation encoding="application/x-tex">c \coloneqq ch</annotation></semantics></math> on connective <a class="existingWikiWord" href="/nlab/show/KU">ku</a>.</p> </div> <p>(<a href="#HopkinsSinger02">Hopkins-Singer 02, section 4.4</a>, <a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, section 6.1</a>)</p> <h4 id="SoothVectorBundlesWithConnectionAndEInvariant">Algebraic K-theory of smooth manifolds and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>-invariant</h4> <p>Write</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mo>⊕</mo></msup><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo><mo>↪</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Vect}^{\oplus} \in Sh(SmthMfd) \hookrightarrow \mathbf{H} = Sh_\infty(SmthMfd)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/stack">stack</a> of (complex) <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> equipped with the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mi>conn</mi> <mo>⊕</mo></msubsup><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Vect}_{conn}^{\oplus} \in Sh(SmthMfd)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/stack">stack</a> of (complex) <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">connection</a>, equipped with direct sum;</p> </li> </ol> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>CMon</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mi>Cat</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spectra</mi></mrow><annotation encoding="application/x-tex"> \mathcal{K} \;\colon\; CMon_\infty(\infty Cat) \longrightarrow Spectra </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> from <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">symmetric monoidal (∞,1)-categories</a> into <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> which produces the <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory+of+symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">algebraic K-theory of symmetric monoidal (∞,1)-categories</a>.</p> <div class="num_defn" id="KTheoryOfStackOfVectorBundles"> <h6 id="definition_7">Definition</h6> <p>Forming objectwise the <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory+of+symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">algebraic K-theory of symmetric monoidal (∞,1)-categories</a> and then <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackifying</a> (which we leave notationally implicit) produces <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mi>conn</mi> <mo>⊕</mo></msubsup><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><mi>𝒦</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mo>⊕</mo></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{K}(\mathbf{Vect}_{conn}^\oplus),\;\mathcal{K}(\mathbf{Vect}^\oplus) \in Stab(Smooth\infty Grpd) \,. </annotation></semantics></math></div> <p>This may be called the <em><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory+of+smooth+manifolds">algebraic K-theory of smooth manifolds</a></em>.</p> </div> <div class="num_prop" id="FlatAndPiForDifferentialKTheory"> <h6 id="proposition_7">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Vect</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{K}(\mathbf{Vect})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{K}(\mathbf{Vect}_{conn})</annotation></semantics></math> are both differential refinements, def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>, of connective <a class="existingWikiWord" href="/nlab/show/KU">ku</a>, both whose underlying <a class="existingWikiWord" href="/nlab/show/geometrically+discrete+infinity-groupoid">geometrically discrete spectrum</a> is the <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">K \mathcal{C}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>𝒦</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mo>⊕</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>𝒦</mi><mo stretchy="false">(</mo><msubsup><mi>Vect</mi> <mi>conn</mi> <mo>⊕</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ku</mi></mrow><annotation encoding="application/x-tex">\Pi(\mathcal{K}(Vect^\oplus)) \simeq \Pi(\mathcal{K}(Vect_{conn}^\oplus))\simeq ku</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo stretchy="false">(</mo><mi>𝒦</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mo>⊕</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mo>♭</mo><mo stretchy="false">(</mo><mi>𝒦</mi><mo stretchy="false">(</mo><msubsup><mi>Vect</mi> <mi>conn</mi> <mo>⊕</mo></msubsup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\flat(\mathcal{K}(Vect^\oplus)) \simeq \flat(\mathcal{K}(Vect_{conn}^\oplus))\simeq K \mathbb{C}</annotation></semantics></math>.</p> </li> <li> <p>in both cases the <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mo>→</mo><mi>Π</mi></mrow><annotation encoding="application/x-tex">\flat \to \Pi</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/comparison+map+between+algebraic+and+topological+K-theory">comparison map between algebraic and topological K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>ℂ</mi><mo>→</mo><mi>ku</mi></mrow><annotation encoding="application/x-tex">K \mathbb{C}\to ku</annotation></semantics></math>.</p> </li> </ol> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, lemma 6.3, corollary 6.5</a>)</p> <div class="num_prop" id="SmoothRegulator"> <h6 id="proposition_8">Proposition</h6> <p>The traditonal <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> induces, via the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> in def. <a class="maruku-ref" href="#HopkinsSingerPullback"></a>, a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold"><mi>Vect</mi></mstyle> <mi>conn</mi> <mo>⊕</mo></msubsup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>ku</mi> <mrow><msub><mi>conn</mi> <mi>ch</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{K}(\mathbf{Vect}_{conn}^\oplus) \longrightarrow ku_{conn_{ch}} </annotation></semantics></math></div> <p>from the differential refinement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ku</mi></mrow><annotation encoding="application/x-tex">ku</annotation></semantics></math> given by def. <a class="maruku-ref" href="#KTheoryOfStackOfVectorBundles"></a>, to that given by def. <a class="maruku-ref" href="#HSDifferentialKu"></a>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a>, hence its image under the <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a>, is in <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> the <a class="existingWikiWord" href="/nlab/show/regulator">regulator</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>K</mi><msub><mi>ℂ</mi> <mi>tor</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>ℚ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>K</mi><mi>ℂ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mrow><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><mi>ℂ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K\mathbb{C}_{tor} &\stackrel{\simeq}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} \\ \downarrow && \downarrow \\ K\mathbb{C} &\stackrel{\pi_{2k-1}}{\longrightarrow}& \mathbb{C}/\mathbb{Z} } </annotation></semantics></math></div> <p>related to the <a class="existingWikiWord" href="/nlab/show/Adams+e-invariant">Adams e-invariant</a> via (<a href="#Bunke11">Bunke 11, section 5.3</a>).</p> </div> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, example 6.9</a>)</p> <h4 id="ViaSmoothSnaithTheorem">Via smooth Snaith’s theorem</h4> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/Snaith+theorem">Snaith theorem</a> realizes <a class="existingWikiWord" href="/nlab/show/KU">KU</a> as the localization of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+%E2%88%9E-ring">∞-group ∞-ring</a> of the <a class="existingWikiWord" href="/nlab/show/circle+2-group">circle 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(1)</annotation></semantics></math> at the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b\in \pi_2(KU)</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>KU</mi><mo>≃</mo><mo stretchy="false">(</mo><msubsup><mi>Σ</mi> <mo>+</mo> <mn>∞</mn></msubsup><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> KU \simeq (\Sigma_+^\infty B U(1))[b^{-1}] \,. </annotation></semantics></math></div> <p>With due care to define smooth geometric looping and taking the <a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a> as an actual smooth bundle over the smooth 2-sphere, this lifts to the smooth circle 2-group <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> and with more care to <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><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)_{conn}</annotation></semantics></math> to produce <a class="existingWikiWord" href="/nlab/show/smooth+spectra">smooth spectra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mo>+</mo></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (\Sigma_+ \mathbf{B}U(1))[b^{-1}] </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mo>+</mo></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (\Sigma_+ \mathbf{B}U(1)_{conn})[b^{-1}] </annotation></semantics></math></div> <p>which both are differential refinements, def. <a class="maruku-ref" href="#DifferentialCoefficients"></a>, of <a class="existingWikiWord" href="/nlab/show/KU">KU</a>.</p> <p>(<a href="#BunkeNikolausVoelkl13">Bunke-Nikolaus-Völkl 13, section 6.3</a>)</p> <h3 id="complexanalytic_differential_generalized_cohomology">Complex-analytic differential generalized cohomology</h3> <p>The above examples all take place in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, modelling <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>. Another fundamental example of a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> is <a class="existingWikiWord" href="/nlab/show/ComplexAnalytic%E2%88%9EGrpd">ComplexAnalytic∞Grpd</a>, modelling <a class="existingWikiWord" href="/nlab/show/complex+analytic+higher+geometry">complex analytic higher geometry</a>.</p> <p>Differential cohomology in the complex-analytic context is discussed in (<a href="#HopkinsQuick12">Hopkins-Quick 12</a>), see at <em><a href="complex+analytic+∞-groupoid#DifferentialCohomology">complex analytic ∞-groupoid – Properties – Differential cohomology</a></em> for more.</p> <h3 id="arithmetic_geometry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-Arithmetic geometry</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> in <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> <a class="existingWikiWord" href="/nlab/show/modality">modality</a></th><th>symbol</th><th>interpretation</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a> at</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ʃ</mi></mrow><annotation encoding="application/x-tex">&#643;</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/torsion+approximation">torsion approximation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ʃ</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">&#643;_{dR}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/localization">localization</a> away</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\flat_{dR}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/adic+residual">adic residual</a></td></tr> </tbody></table> <p>the <strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a>/<a class="existingWikiWord" href="/nlab/show/arithmetic+fracture+squares">arithmetic fracture squares</a></strong>:</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><mi>localization</mi><mspace width="thickmathspace"></mspace><mi>away</mi><mspace width="thickmathspace"></mspace><mi>from</mi><mspace width="thickmathspace"></mspace><mi>𝔞</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><mi>𝔞</mi><mspace width="thickmathspace"></mspace><mi>adic</mi><mspace width="thickmathspace"></mspace><mi>residual</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>Π</mi> <mrow><mi>𝔞</mi><mi>dR</mi></mrow></msub><msub><mo>♭</mo> <mi>𝔞</mi></msub><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>Π</mi> <mi>𝔞</mi></msub><msub><mo>♭</mo> <mrow><mi>𝔞</mi><mi>dR</mi></mrow></msub><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>formal</mi><mspace width="thickmathspace"></mspace><mi>completion</mi><mspace width="thickmathspace"></mspace><mi>at</mi><mspace width="thickmathspace"></mspace><mi>𝔞</mi><mspace width="thickmathspace"></mspace></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mi>𝔞</mi><mspace width="thickmathspace"></mspace><mi>torsion</mi><mspace width="thickmathspace"></mspace><mi>approximation</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,, </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></li> </ul> <h2 id="References">References</h2> <p>The differential cohomology hexagon was maybe first highlighted in the context of <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li id="SimonsSullivan07"><a class="existingWikiWord" href="/nlab/show/James+Simons">James Simons</a>, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Axiomatic Characterization of Ordinary Differential Cohomology</em>, Journal of Topology 1.1 (2008): 45-56. (<a href="http://arxiv.org/abs/math/0701077">arXiv:math/0701077</a>)</li> </ul> <p>and in the context of <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> in</p> <ul> <li id="SimonsSullivan08"><a class="existingWikiWord" href="/nlab/show/James+Simons">James Simons</a>, <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>, <em>Structured vector bundles define differential K-theory</em> (<a href="http://arxiv.org/abs/0810.4935">arXiv:0810.4935</a>)</li> </ul> <p>discussed also in</p> <ul> <li id="FreedLott10"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/John+Lott">John Lott</a>, <em>An index theorem in differential K-theory</em>, Geometry and Topology 14 (2010) (<a href="http://math.berkeley.edu/~lott/gt-2010-14-021p.pdf">pdf</a>)</p> </li> <li> <p>Jiahao Hu, <em>Characterization of differential K-theory by hexagon diagram</em> [<a href="https://arxiv.org/abs/2209.04925">arXiv:2209.04925</a>]</p> </li> </ul> <p>An artistic impression of the differential cohomology hexagon as a dance choreography inspired by a talk by <a class="existingWikiWord" href="/nlab/show/James+Simons">James Simons</a> is at</p> <ul> <li>Kyla Barkin, Aaron Selissen, <em>Differential Cohomology</em>, 2011 (<a href="http://vimeo.com/32903887">full video (37 min)</a>, <a href="http://www.dancemedia.com/v/7044">sequence of “The March Back” (4 min)</a>, <a href="http://www.businessinsider.com/jim-simons-mathematical-genius-inspired-a-ballet-2011-1">press release</a>)</li> </ul> <p>That the differential cohomology hexagon exists not just at the level of <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a> of <a class="existingWikiWord" href="/nlab/show/cohomology+classes">cohomology classes</a> but already at the level of <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a> of <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> spaces was observed for <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> and for <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> in</p> <ul> <li>Man-Ho Ho, <em>Refined hexagons for differential cohomology</em> (<a href="http://arxiv.org/abs/1310.0582">arXiv:1310.0582</a>)</li> </ul> <p>That generally the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a>-construction of differential refinements of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories due to</p> <ul> <li id="HopkinsSinger02"><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a> and <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2Cand+M-Theory">Quadratic Functions in Geometry, Topology,and M-Theory</a></em> (<a href="http://arxiv.org/abs/math/0211216">arXiv:math/0211216</a>)</li> </ul> <p>constitutes the right one of the two squares in the homotopy-theoretic version of the diagram is discussed explicitly for instance in prop 4.57 of</p> <ul> <li id="Bunke12"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <em>Differential cohomology</em> (<a href="http://arxiv.org/abs/1208.3961">arXiv:1208.3961</a>)</li> </ul> <p>That every <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable homotopy type</a> in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> naturally sits in a differential cohomology diagram was observed (focusing on <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>) in</p> <ul> <li id="BunkeNikolausVoelkl13"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Michael+V%C3%B6lkl">Michael Völkl</a>, <em>Differential cohomology theories as sheaves of spectra</em>, Journal of Homotopy and Related Structures <strong>11</strong> (2016) 1–66 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://arxiv.org/abs/1311.3188">arXiv:1311.3188</a>, <a href="https://doi.org/10.1007/s40062-014-0092-5">doi:10.1007/s40062-014-0092-5</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>following</p> <ul> <li id="Schreiber13"><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>)</li> </ul> <p>where in section 4.1.2 a fully general abstract account is given. This in turn follows</p> <ul> <li id="Schreiber09"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Background+fields+in+twisted+differential+nonabelian+cohomology">Background fields in twisted differential nonabelian cohomology</a></em>, talk at Oberwolfach Workshop Strings, Fields, Topology (2009), Oberwolfach Report No. 28/2009 (<a href="https://www.mfo.de/document/0924/OWR_2009_28.pdf">pdf</a>)</li> </ul> <p>Surveys/expositions of this:</p> <ul> <li id="Schreiber14a"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Differential+generalized+cohomology+in+Cohesive+homotopy+type+theory">Differential generalized cohomology in Cohesive homotopy type theory</a></em>, talk at IHP trimester on <em>Semantics of proofs and certified mathematics</em>, <em>Workshop 1: Formalization of Mathematics</em>, Institut Henri Poincaré, Paris, 5-9 May 2014</p> </li> <li id="Schreiber14b"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Differential+cohomology+is+Cohesive+homotopy+theory">Differential cohomology is Cohesive homotopy theory</a></em>, talk at <em><a href="http://www.ru.nl/math/research/vmconferences/2014-higher/">Higher Geometric Structures along the Lower Rhine June 2014</a></em>, 19-20 June 2014</p> </li> <li id="AmabelDebrayHaine21"> <p><a class="existingWikiWord" href="/nlab/show/Araminta+Amabel">Araminta Amabel</a>, <a class="existingWikiWord" href="/nlab/show/Arun+Debray">Arun Debray</a>, <a class="existingWikiWord" href="/nlab/show/Peter+J.+Haine">Peter J. Haine</a> (eds.), Section 2.3 in: <em>Differential Cohomology: Categories, Characteristic Classes, and Connections</em>. Based on <a href="https://math.mit.edu/juvitop/pastseminars/2019_Fall.html">Fall 2019 talks at MIT’s Juvitop seminar</a> by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. (<a href="https://arxiv.org/abs/2109.12250">arXiv:2109.12250</a>)</p> </li> </ul> <p>and from the point of view of <a class="existingWikiWord" href="/nlab/show/modal+homotopy+type+theory">modal homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/differential+cohesion+and+idelic+structure">Fractures, Ideles and the Differential Hexagon</a></em>, talk at <em><a href="http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html">Workshop on differential cohomologies</a></em>, CUNY Graduate Center, August 12-14 2014 (<a href="http://videostreaming.gc.cuny.edu/videos/video/1806/in/channel/55/">video recording</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Differential+cohesion+and+Arithmetic+geometry">Differential cohesion and Arithmetic geometry</a></em>, talk at <a href="http://www.math.kit.edu/iag3/~herrlich/seite/weihnachts-workshops">Karslruher Weihnachts-Workshop 2015</a></p> </li> </ul> <p id="ReferencesFormalizationInCohesiveHoTT"> Formalization in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive</a> <a class="existingWikiWord" href="/nlab/show/modal+homotopy+type+theory">modal homotopy type theory</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li id="Myers21"> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Modal Fracture of Higher Groups</em>, talk at <em><a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html">CMU-HoTT Seminar</a>, 2021 (<a href="http://davidjaz.com/Talks/CMU_March_2021.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MyersModalFracture2021.pdf" title="pdf">pdf</a>)</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Modal Fracture of Higher Groups</em> (<a href="https://arxiv.org/abs/2106.15390">arXiv:2106.15390</a>)</p> </li> </ul> <p>For discussion of the hexagon in an <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> setting, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andreas+K%C3%BCbel">Andreas Kübel</a>, <a class="existingWikiWord" href="/nlab/show/Andreas+Thom">Andreas Thom</a>, <em>Equivariant Differential Cohomology</em>, (<a href="https://arxiv.org/abs/1510.06392">arXiv:1510.06392</a>)</li> </ul> <p>See also</p> <ul> <li id="Bunke11"> <p><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <em>On the topological contents of eta invariants</em> (<a href="http://arxiv.org/abs/1103.4217">arXiv:1103.4217</a>)</p> </li> <li id="HopkinsQuick12"> <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> </li> <li id="FRS13"> <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/Chris+Rogers">Chris Rogers</a>, <em>Higher geometric prequantum theory</em> (<a href="http://arxiv.org/abs/1304.0236">arxiv:1304.0236</a>)</p> </li> <li id="BunkeGepner13"> <p><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory</em> (<a href="http://arxiv.org/abs/1306.0247">arXiv:1306.0247</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 18, 2022 at 14:13:59. See the <a href="/nlab/history/differential+cohomology+diagram" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/differential+cohomology+diagram" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9168/#Item_11">Discuss</a><span class="backintime"><a href="/nlab/revision/differential+cohomology+diagram/63" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/differential+cohomology+diagram" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/differential+cohomology+diagram" accesskey="S" class="navlink" id="history" rel="nofollow">History (63 revisions)</a> <a href="/nlab/show/differential+cohomology+diagram/cite" style="color: black">Cite</a> <a href="/nlab/print/differential+cohomology+diagram" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/differential+cohomology+diagram" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>