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> ordinary homology 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> ordinary homology </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/4122/#Item_3" 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="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</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/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </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='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_homotopy_groups'>Relation to homotopy groups</a></li> <li><a href='#InTermsOfHigherLinearAlgebra'>Description in terms of higher linear algebra</a></li> <li><a href='#ordinary_homology_spectra_split'>Ordinary homology spectra split</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><em>Ordinary homology</em> is <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> with respect to an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">H A</annotation></semantics></math>, and often understood over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math>.</p> <p>Equivalently this is computed by <em><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></em> 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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_homotopy_groups">Relation to homotopy groups</h3> <p>Ordinary homology 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>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> serves to approximate the <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>See for instance at <em><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></em></p> <h3 id="InTermsOfHigherLinearAlgebra">Description in terms of higher linear algebra</h3> <p>We discuss (<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a>) ordinary homology and <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in terms of sections of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundles">(∞,1)-module bundles</a> over the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>k</mi><mo>∈</mo><msub><mi>CRing</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex"> H k \in CRing_\infty </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, canonically regarded as an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> (H k) Mod \in (\infty,1)Cat </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-modules">(∞,1)-category of (∞,1)-modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">H k</annotation></semantics></math>.</p> <div class="num_prop" id="DoldKan"> <h6 id="proposition">Proposition</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi><mo>≃</mo><msub><mi>L</mi> <mi>qi</mi></msub><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (H k)Mod \simeq L_{qi} Ch_\bullet(k Mod) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-modules">(∞,1)-category of (∞,1)-modules</a> over the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">H k</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a> of ordinary (1-categorical) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a>.</p> </div> <p>This is the statement of the <em><a class="existingWikiWord" href="/nlab/show/stable+Dold-Kan+correspondence">stable Dold-Kan correspondence</a></em>, see at <em><a href="%28infinity%2C1%29-module#StableDoldKanCorrespondence">(∞,1)-category of (∞,1)-modules – Properties – Stable Dold-Kan correspondence</a></em>.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">\Pi(X) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> for its underlying <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> (its <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a>). Then we say that an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</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><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \Pi(X) \to (H k) Mod </annotation></semantics></math></div> <p>is (the <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>) a <em><a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or a <em><a class="existingWikiWord" href="/nlab/show/local+system">local system</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">H k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-modules">(∞,1)-modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+correspondence">Riemann-Hilbert correspondence</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a>, then there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> [\Pi(X), (H k)Mod] \simeq (T X)Mod </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundles">(∞,1)-module bundles</a>/<a class="existingWikiWord" href="/nlab/show/local+systems">local systems</a> and <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid+representations">L-∞ algebroid representations</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. From right to left the equivalece is established by sending an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid+representation">L-∞ algebroid representation</a> given (as discussed there) by a flat <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>-graded connection on bundles of chain complexes (via prop. <a class="maruku-ref" href="#DoldKan"></a>), to its <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> defined in terms of <a class="existingWikiWord" href="/nlab/show/iterated+integrals">iterated integrals</a>.</p> </div> <p>This is the main theorem in (<a href="#BlockSmith09">Block-Smith 09</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><mi>Γ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><mo>→</mo></munder><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \Gamma \;\coloneqq\; \underset{\to}{\lim} \;\colon\; [\Pi(X), (H k)Mod] \to (H k) Mod </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> functor.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>We may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> equivalently as</p> <ul> <li> <p>forming flat <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a <a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a>;</p> </li> <li> <p>sending a <a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> to its <em><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></em> (see at <em><a href="Thom+spectrum#ForInfinityModuleBundles">Thom spectrum – For (∞,1)-module bundles</a></em>).</p> </li> </ul> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>k</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>K</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \mathbb{I}_X^{H k} \;\colon\; \Pi(X) \to (K k)Mod </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> which is constant on the chain complex concentrated on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in degree 0, the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>L</mi> <mi>qi</mi></msub><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi(X), (H k)Mod] \simeq [\Pi(X), L_{qi}Ch_\bullet(k Mod)]</annotation></semantics></math>.</p> </div> <div class="num_prop" id="SectionsOfTrivialKABundleIsAHomology"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, we have a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a> (with the identification of prop. <a class="maruku-ref" href="#DoldKan"></a> understood) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>k</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Gamma(\mathbb{I}_X^{H k}) \simeq C_\bullet(X,k) \,, </annotation></semantics></math></div> <p>between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H k)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module">(∞,1)-module</a> of sections of the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H k)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>k</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{I}_X^{H k}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/singular+chain+complex">singular chain complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> for ordinary homology 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>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is a classical basic (maybe <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>) statement. Here is one way to see it in full detail.</p> <p>First notice that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> of functors out of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> and constant on the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">(H k)Mod</annotation></semantics></math> is by definition the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-tensoring">(∞,1)-tensoring</a> operation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">(H k)Mod</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. Now if we find a <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mi>k</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">(H k)Mod</annotation></semantics></math> by a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> the by the dicussion at <em><a href="limit+in+a+quasi-category#ModelsForTensoring">(∞,1)-colomit – Tensoring and cotensoring – Models</a></em> this <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-tensoring">(∞,1)-tensoring</a> is given by the left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> in that simplicial model category.</p> <p>To obtain this, use prop. <a class="maruku-ref" href="#DoldKan"></a> and then the discussion at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a></em> in the section <em><a href="model%20structure%20on%20chain%20complexes#ProjectiveModelStructureOnUnboundedChainComplexes">Projective model structure on unbounded chain complexes</a></em> which says that there is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> structure on the <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+objects">category of simplicial objects</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a> which models <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>qi</mi></msub><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_{qi} Ch_\bullet(k Mod)</annotation></semantics></math>, and whose weak equivalences are those morphisms that produce <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> under the <a class="existingWikiWord" href="/nlab/show/total+chain+complex">total chain complex</a> functor.</p> <p>In summary it follows that with any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mi>in</mi><msub><mi>L</mi> <mi>whe</mi></msub><mi>sSet</mi></mrow><annotation encoding="application/x-tex">(\Pi(X))_\bullet in L_{whe} sSet</annotation></semantics></math> representing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math> (under the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem) we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>→</mo></munder><mo stretchy="false">(</mo><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>k</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>⋅</mo><mi>𝕀</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underset{\to}{\lim} (\mathbb{I}_X^{H k}) \simeq \int_{[n] \in \Delta} (\Pi(X))_n \cdot \mathbb{I} \,, </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> over the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> of the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> (of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> with <a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a>) of the standard cosimplicial simplex with the simplicial diagram constant on the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>.</p> <p>The result on the right is manifestly, by the very definition of <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>, under the ordinary <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> the <a class="existingWikiWord" href="/nlab/show/chain+complex+of+singular+simplices">chain complex of singular simplices</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>⋅</mo><mi>𝕀</mi><mo>≃</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>k</mi><mo>≃</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int_{[n] \in \Delta} (\Pi(X))_n \cdot \mathbb{I} \simeq \Pi(X) \otimes k \simeq C_\bullet(X,k) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \in L_{whe} Top</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality+space">Poincaré duality space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>A</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mi>H</mi><mi>A</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\Gamma(\mathbb{I}_X^{H A}) \in (H A) Mod</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> which is almost self-dual except for a degree <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twist</a> of (itself) degree 0:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mrow><mo>(</mo><mi>Γ</mi><mrow><mo>(</mo><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>A</mi></mrow></msubsup><mo>)</mo></mrow><mo>)</mo></mrow> <mo>∨</mo></msup><mo>≃</mo><msup><mi>Σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>Γ</mi><mrow><mo>(</mo><msubsup><mi>𝕀</mi> <mi>X</mi> <mrow><mi>H</mi><mi>A</mi></mrow></msubsup><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \Gamma\left(\mathbb{I}_X^{H A}\right) \right)^\vee \simeq \Sigma^{-n} \left( \Gamma\left(\mathbb{I}_X^{H A}\right) \right) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Under the identifications of prop. <a class="maruku-ref" href="#DoldKan"></a> and prop. <a class="maruku-ref" href="#SectionsOfTrivialKABundleIsAHomology"></a> this is the theorem discussed at <em><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></em> in the section <em><a href="Poincar%C3%A9+duality#RefinementInHomotopyTheory">Poincaré duality – Refinement to homotopy theory</a></em>.</p> </div> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/Dold-Thom+theorem">Dold-Thom theorem</a></em>.</p> <h3 id="ordinary_homology_spectra_split">Ordinary homology spectra split</h3> <p>see at <em><a class="existingWikiWord" href="/nlab/show/ordinary+homology+spectra+split">ordinary homology spectra split</a></em></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>, <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+homology">Čech homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+homology">cellular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Thom+theorem">Dold-Thom theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin+ring">Pontrjagin ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+homology+theory">stable homotopy homology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pochhammer+loop">Pochhammer loop</a></p> </li> </ul> <h2 id="references">References</h2> <p>A classical account:</p> <ul> <li id="Switzer75"><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, chapter 10 of: <em>Algebraic Topology - Homotopy and Homology</em>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.</li> </ul> <p>In terms of <a class="existingWikiWord" href="/nlab/show/current+%28distribution+theory%29">currents</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Georges+de+Rham">Georges de Rham</a>, Chapter IV of: <em>Differentiable Manifolds – Forms, Currents, Harmonic Forms</em>, Grundlehren <strong>266</strong>, Springer (1984) [<a href="https://doi.org/10.1007/978-3-642-61752-2">doi:10.1007/978-3-642-61752-2</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Hatcher02"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, §2 in: <em>Algebraic Topology</em>, Cambridge University Press (2002) [<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401">ISBN:9780521795401</a>, <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">webpage</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anatoly+Fomenko">Anatoly Fomenko</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Fuchs">Dmitry Fuchs</a>, §2 in: <em>Homotopical Topology</em>, Graduate Texts in Mathematics <strong>273</strong>, Springer (2016) [<a href="https://doi.org/10.1007/978-3-319-23488-5">doi:10.1007/978-3-319-23488-5</a>, <a href="https://www.cimat.mx/~gil/docencia/2020/topologia_diferencial/[Fomenko,Fuchs]Homotopical_Topology(2016).pdf">pdf</a>]</p> </li> </ul> <p>Comprehensive monograph:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyn+Quaintance">Jocelyn Quaintance</a>, <em>Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry</em>, World Scientific (2022) [<a href="https://doi.org/10.1142/12495">doi:10.1142/12495</a>, <a href="https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html">webpage</a>]</li> </ul> <p>With an eye towards application in <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mikio+Nakahara">Mikio Nakahara</a>, Chapter 3 of <em><a class="existingWikiWord" href="/nlab/show/Geometry%2C+Topology+and+Physics">Geometry, Topology and Physics</a></em>, IOP 2003 (<a href="https://doi.org/10.1201/9781315275826">doi:10.1201/9781315275826</a>, <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+correspondence">Riemann-Hilbert correspondence</a>/<a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">H A</annotation></semantics></math>-modules is established in</p> <ul> <li id="BlockSmith09"><a class="existingWikiWord" href="/nlab/show/Jonathan+Block">Jonathan Block</a>, Aaron Smith, <em>A Riemann–Hilbert correspondence for infinity local systems</em> (<a href="http://arxiv.org/abs/0908.2843">arXiv:0908.2843</a>)</li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/Kenzo">Kenzo</a> software for homology computations in <a class="existingWikiWord" href="/nlab/show/constructive+algebraic+topology">constructive algebraic topology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Julio+Rubio">Julio Rubio</a>, <a class="existingWikiWord" href="/nlab/show/Francis+Sergeraert">Francis Sergeraert</a>, Yvon Siret: <em>KENZO – a Symbolic Software for Effective Homology Computation</em> (1999) [<a href="https://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/Kenzo-doc.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 29, 2024 at 12:03:46. See the <a href="/nlab/history/ordinary+homology" 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/ordinary+homology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4122/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/ordinary+homology/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/ordinary+homology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/ordinary+homology" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/ordinary+homology/cite" style="color: black">Cite</a> <a href="/nlab/print/ordinary+homology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/ordinary+homology" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>