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chain homology and cohomology in nLab
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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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#functoriality'>Functoriality</a></li> <li><a href='#RespectForDirectSum'>Respect for direct sums and filtered colimits</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#InTheContextOfHomotopyTheory'>In the context of homotopy theory</a></li> <ul> <li><a href='#preliminaries'>Preliminaries</a></li> <ul> <li><a href='#eilenbergmaclane_objects'>Eilenberg-MacLane objects</a></li> <li><a href='#homotopy_and_cohomology'>Homotopy and cohomology</a></li> </ul> <li><a href='#chain_homology_as_homotopy'>Chain homology as homotopy</a></li> <li><a href='#cohomology_of_cochain_complexes'>Cohomology of cochain complexes</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#early_references_on_cohomology'>Early references on (co)homology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_\bullet \in Ch_\bullet(\mathcal{A})</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></em>, its <em>chain <a class="existingWikiWord" href="/nlab/show/homology+group">homology group</a></em> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is akin to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-th <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of a topological space. It is defined to be the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math>.</p> <p><a class="existingWikiWord" href="/nlab/show/duality">Dually</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>•</mo></msup><mo>∈</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V^\bullet \in Ch^\bullet(\mathcal{A})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a>, its <em>cochain <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a></em> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the quotient of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/coboundaries">coboundaries</a>.</p> <p>Basic examples are the <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> and <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> of a topological space, which are the (co)chain (co)homology of the <a class="existingWikiWord" href="/nlab/show/singular+complex">singular complex</a>.</p> <p>Chain homology and cochain cohomology constitute the basic invariants of (co)chain complexes. A <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> is a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> between chain complexes that induces isomorphisms on all chain homology groups, akin to a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>. A <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> equipped with quasi-isomorphisms as <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> is a presentation for the <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a> of chain complexes.</p> <h2 id="definition">Definition</h2> <p>Let <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">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> such as that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> this is the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</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>R</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">R = k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, this is the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over <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> <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><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(\mathcal{A})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</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">\mathcal{A}</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^\bullet(\mathcal{A})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+cochain+complexes">category of cochain complexes</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">\mathcal{A}</annotation></semantics></math>.</p> <p>We label <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> in a chain complex as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">[</mo><mi>⋯</mi><mo>→</mo><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mrow><msub><mo>∂</mo> <mi>n</mi></msub></mrow></mover><msub><mi>V</mi> <mi>n</mi></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> V_\bullet = [ \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \to \cdots ] </annotation></semantics></math></div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_\bullet \in Ch_\bullet(\mathcal{A})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> and <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>, the <strong>chain homology</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(V)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</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> is the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><mfrac><mrow><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>ker</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>im</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> H_n(V) \coloneqq \frac{Z_n(V)}{B_n(V)} = \frac{ker(\partial_{n-1})}{im(\partial_n)} </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (<a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>) of the group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> by that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="functoriality">Functoriality</h3> <div class="num_prop"> <h6 id="proposition">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>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> forming chain homology extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</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">\mathcal{A}</annotation></semantics></math> to <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">\mathcal{A}</annotation></semantics></math> itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒜</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>One checks that <a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a> (see there) respects <a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> and <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a>.</p> </div> <div class="num_prop" id="ChainHomologyRespectsDirectProduct"> <h6 id="proposition_2">Proposition</h6> <p>Chain homology commutes with <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a> of chain complexes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msup><mi>C</mi> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_n(\prod_i C^{(i)}) \simeq \prod_i H_n(C^{(i)}) \,. </annotation></semantics></math></div> <p>Similarly for <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>.</p> </div> <h3 id="RespectForDirectSum">Respect for direct sums and filtered colimits</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The chain homology functor preserves <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A,B \in Ch_\bullet</annotation></semantics></math> and <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>, the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>⊕</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_n(A \oplus B) \to H_n(A) \oplus H_n(B) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The chain homology functor preserves <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A \colon I \to Ch_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/filtered+category">filtered</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> and <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>, the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><munder><mi>lim</mi><mrow><msub><mo>→</mo> <mi>i</mi></msub></mrow></munder><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_n(\underset{\to_i}{\lim} A_i) \to \underset{\to_i}{\lim} H_n(A_i) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>This is spelled out for instance as (<a href="#HopkinsMathew">Hopkins-Mathew , prop. 23.1</a>).</p> <h2 id="examples">Examples</h2> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">N \mathbb{Z}[Sing X]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> that is degreewise the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math>. The resulting chain homology is the <em><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>N</mi><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_\bullet( N \mathbb{Z}[Sing X]) \simeq H_\bullet(X, \mathbb{Z}) \,. </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul+homology">Koszul homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>, <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></p> </li> </ul> <h2 id="InTheContextOfHomotopyTheory">In the context of homotopy theory</h2> <p>We discuss here the notion of (co)homology of a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> from a more abstract point of view of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, using the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> on <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> as discussed there.</p> <p>A <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> in non-negative degree is, under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> a <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> model for a particularly nice <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> or <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>: namely one with an abelian group structure on it, a <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a>. Accordingly, an unbounded (arbitrary) <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> is a model for a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> with abelian group structure.</p> <p>One consequence of this embedding</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><msub><mi>Ch</mi> <mo>+</mo></msub><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> N : Ch_+ \to \infty Grpd </annotation></semantics></math></div> <p>induced by the Dold-Kan <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> is that it allows to think of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> as objects in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> or equivalently <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. Every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> comes with a notion of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> and so such abstract notions get induced on chain complexes.</p> <p>Of course there is an independent, age-old definition of <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of chain complexes and, by dualization, of cohomology of cochain complexes.</p> <p>This entry describes how these standard definition of chain homology and cohomology follow from the general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> nonsense described at <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> and <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> <p>The main statement is that</p> <ul> <li> <p>the naïve <a class="existingWikiWord" href="/nlab/show/homology">homology</a> groups of a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> are really its <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>, in the abstract sense (i.e. with the chain complex regarded as a model for a space/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid);</p> </li> <li> <p>the naïve cohomology groups of a cochain complex are really the abstract <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> of the dual <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>.</p> </li> </ul> <h3 id="preliminaries">Preliminaries</h3> <p>Before discussing chain homology and cohomology, we fix some terms and notation.</p> <h4 id="eilenbergmaclane_objects">Eilenberg-MacLane objects</h4> <p>In a given <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> there is a notion of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> for every (co-)coefficient object <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> (<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>).</p> <p>The particular case of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <a class="existingWikiWord" href="/nlab/show/homology">homology</a> is only the special case induced from coefficients given by the corresponding <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+objects">Eilenberg-MacLane objects</a>.</p> <p>Assume for simplicity here and in the following that we are talking about non-negatively graded chain complexes of vector spaces over some field <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>. Then for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>k</mi></mtd> <mtd><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>k</mi> <mi>n</mi></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><mover><mo>→</mo><mo>∂</mo></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>k</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>∂</mo></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>k</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><mi>k</mi><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{B}^n k &:= ( \cdots \to \mathbf{B}^n k_n \to \cdots \to \stackrel{\partial}{\to} \mathbf{B}^n k_1 \stackrel{\partial}{\to} \mathbf{B}^n k_0) \\ &= ( \cdots \to k \to \cdots \to 0 \to 0 ) \end{aligned} </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a>.</p> <p>Notice that this is often also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[n]</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[-n]</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(k,n)</annotation></semantics></math>.</p> <h4 id="homotopy_and_cohomology">Homotopy and cohomology</h4> <p>With the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> understood, embedding chain complexes into <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>, for any chain complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">A_\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">B_\bullet</annotation></semantics></math> we obtain</p> <ul> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{\infty Grpd}(X_\bullet, A_\bullet) </annotation></semantics></math></div> <p>whose * objects are the <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>-valued <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; * morphisms are the coboundaries between these <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s; * 2-morphisms are the coboundaries between coboundaries * etc. and where the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \mathbf{H}(X_\bullet,A_\bullet)</annotation></semantics></math> are the cohomology classes</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}_{\infty Grpd}(B_\bullet, X_\bullet) </annotation></semantics></math></div> <p>whose * objects are the <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>-co-valued cycles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>; * morphisms are the boundaries between these cycles; * 2-morphisms are the boundaries between boundaries * etc. and where the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 \mathbf{H}(B_\bullet,X_\bullet)</annotation></semantics></math> are the homotopy classes</p> </li> </ul> <h3 id="chain_homology_as_homotopy">Chain homology as homotopy</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo>:</mo><mo>=</mo><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet := V_\bullet</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(V_\bullet)</annotation></semantics></math> its ordinary chain <a class="existingWikiWord" href="/nlab/show/homology">homology</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>k</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_n(V_\bullet) \simeq \pi_0 \mathbf{H}(\mathbf{B}^n k_\bullet, V_\bullet) \,. </annotation></semantics></math></div> <p>A cycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msub><mi>k</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">c : \mathbf{B}^n k_\bullet \to V_\bullet</annotation></semantics></math> is a chain 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><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>k</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mi>n</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& k &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& \cdots \\ && \downarrow && \downarrow^{c_n} && \downarrow \\ \cdots &\stackrel{\partial}{\to}& V_{n+1} &\stackrel{\partial}{\to}& V_n &\stackrel{\partial}{\to}& V_{n-1} &\stackrel{\partial}{\to}& \cdots } </annotation></semantics></math></div> <p>Such chain maps are clearly in bijection with those elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>V</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">c_n \in V_n</annotation></semantics></math> that are in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_n \stackrel{\partial}{\to} V_{n-1}</annotation></semantics></math> in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\partial c_n = 0</annotation></semantics></math>.</p> <p>A boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\lambda : c \to C'</annotation></semantics></math> is a chain homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>k</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>λ</mi> <mi>n</mi></msub></mrow></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& k &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& \cdots \\ && & {}^{\lambda_n}\swarrow \\ \cdots &\stackrel{\partial}{\to}& V_{n+1} &\stackrel{\partial}{\to}& V_n &\stackrel{\partial}{\to}& V_{n-1} &\stackrel{\partial}{\to}& \cdots } </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo><mo>=</mo><mi>c</mi><mo>+</mo><mo>∂</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">c' = c + \partial \lambda</annotation></semantics></math>.</p> <p>(…)</p> <h3 id="cohomology_of_cochain_complexes">Cohomology of cochain complexes</h3> <p>The ordinary notion of cohomology of a <a class="existingWikiWord" href="/nlab/show/chain+complex">cochain complex</a> is the special case of cohomology in the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-</a> <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">V^\bullet</annotation></semantics></math> a cochain complex let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>X</mi></mtd> <mtd><mo>:</mo><mo>=</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>•</mo></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>X</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>V</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} X &:= V_\bullet = (V^\bullet)^* \\ &= (\cdots \to X_{n+1} \stackrel{\partial}{\to} X_n \stackrel{\partial}{\to} X_{n-1} \to \cdots) \\ & := (\cdots \to V_{n+1} \stackrel{\partial}{\to} V_n \stackrel{\partial}{\to} V_{n-1} \to \cdots) \end{aligned} </annotation></semantics></math></div> <p>be the corresponding dual <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>A</mi></mtd> <mtd><mo>:</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mi>I</mi><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} A &:= \mathbf{B}^n I \\ &= (\cdots \to A_{n+1} \to A_n \to A_{n-1} \to \cdots ) \\ & = (\cdots \to 0 \to I \to 0 \to \cdots ) \end{aligned} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> with the tensor unit (the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>, say) in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and trivial elsewhere. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>Ch</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H}(X,A) &= Ch(V_\bullet, \mathbf{B}^n I) \end{aligned} </annotation></semantics></math></div> <p>has</p> <ul> <li> <p>as objects chain morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>I</mi></mrow><annotation encoding="application/x-tex">c : V_\bullet \to \mathbf{B}^n I</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mi>n</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>c</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>I</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& V_{n} &\stackrel{\partial}{\to}& V_{n-1} &\to& \cdots \\ && \downarrow^{c_{n+1}} && \downarrow^{c_{n}} && \downarrow^{c_{n-1}} \\ \cdots &\to& 0 &\stackrel{\partial}{\to}& I &\stackrel{\partial}{\to}& 0 &\to& \cdots } \,. </annotation></semantics></math></div> <p>These are in canonical bijection with the elements in the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">d_{n}</annotation></semantics></math> of the dual cochain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>•</mo></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>,</mo><mi>I</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">V^\bullet = [V_\bullet,I]</annotation></semantics></math>.</p> </li> <li> <p>as morphism chain homotopies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mi>c</mi><mo>→</mo><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\lambda : c \to c'</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mi>n</mi></msub></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>V</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mi>λ</mi></msup><mo>↙</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mi>I</mi></mtd> <mtd><mover><mo>→</mo><mo>∂</mo></mover></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& V_{n} &\stackrel{\partial}{\to}& V_{n-1} &\to& \cdots \\ && && &{}^{\lambda}\swarrow& \\ \cdots &\to& 0 &\stackrel{\partial}{\to}& I &\stackrel{\partial}{\to}& 0 &\to& \cdots } \,. </annotation></semantics></math></div></li> </ul> <p>Comparing with the general definition of cocycles and coboudnaries from above, one confirms that</p> <ul> <li> <p>the <strong>cocycles</strong> are the chain maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub><mo>→</mo><mi>I</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> V_\bullet \to I[n]_\bullet </annotation></semantics></math></div></li> <li> <p>the <strong>coboundaries</strong> are the chain homotopies between these chain maps.</p> </li> <li> <p>the <strong>coboundaries of coboundaries</strong> are the second order chain homotopies between these chain homotopies.</p> </li> <li> <p>etc.</p> </li> </ul> <h2 id="references">References</h2> <div> <h3 id="early_references_on_cohomology">Early references on (co)homology</h3> <p>The original references on <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>/<a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> and <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in the form of <a class="existingWikiWord" href="/nlab/show/cellular+cohomology">cellular cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrei+Kolmogoroff">Andrei Kolmogoroff</a>, <em>Über die Dualität im Aufbau der kombinatorischen Topologie</em>, Recueil Mathématique 1(43) (1936), 97–102. (<a href="http://mi.mathnet.ru/msb5361">mathnet</a>)</li> </ul> <p>A footnote on the first page reads as follows, giving attribution to <a href="#Alexander35a">Alexander 35a</a>, <a href="#Alexander35a">35b</a>:</p> <blockquote> <p>Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn <a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">Alexander</a> in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.</p> </blockquote> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Kolmogoroff">Andrei Kolmogoroff</a>, <em>Homologiering des Komplexes und des lokal-bicompakten Raumes</em>, Recueil Mathématique 1(43) (1936), 701–705. <a href="http://mi.mathnet.ru/msb5475">mathnet</a>.</p> </li> <li id="Alexander35a"> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the chains of a complex and their duals</em>, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (<a href="https://doi.org/10.1073/pnas.21.8.50">doi:10.1073/pnas.21.8.50</a>)</p> </li> <li id="Alexander35b"> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the ring of a compact metric space</em>, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 (<a href="https://doi.org/10.1073/pnas.21.8.511">doi:10.1073/pnas.21.8.511</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J.+W.+Alexander">J. W. Alexander</a>, <em>On the connectivity ring of an abstract space</em>, Ann. of Math., 37 (1936), 698–708 (<a href="https://doi.org/10.2307/1968484">doi:10.2307/1968484</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/alexcon.pdf">pdf</a>)</p> </li> </ul> <p>The term “cohomology” was introduced by <a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a> in</p> <ul> <li id="Whitney37"><a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a>, <em>On matrices of integers and combinatorial topology</em>. Duke Mathematical Journal 3:1 (1937), 35–45 (<a href="https://projecteuclid.org/journals/duke-mathematical-journal/volume-3/issue-1/On-matrices-of-integers-and-combinatorial-topology/10.1215/S0012-7094-37-00304-1.short">doi:10.1215/S0012-7094-37-00304-1</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Hassler Whitney</a>, <em>On products in a complex</em>, Annals of Math. 39 (1938) 397–432 (<a href="https://doi.org/10.2307/1968795">doi:10.2307/1968795</a>)</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em>Singular homology theory</em>, Annals of Mathematics 45:3 (1944) (<a href="https://doi.org/10.2307/1969185">doi:10.2307/1969185</a>)</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/monadic+cohomology">monadic cohomology</a> via <a class="existingWikiWord" href="/nlab/show/canonical+resolutions">canonical resolutions</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Jon+Beck">Jon Beck</a>, <em>Homology and Standard Constructions</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, Lecture Notes in Maths. <strong>80</strong>, Springer (1969), Reprints in Theory and Applications of Categories <strong>18</strong> (2008) 186-248 [<a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html">tac:18</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Cartan-Eilenberg cohomology and triples</em>, J. Pure Applied Algebra <strong>112</strong> 3 (1996) 219–238 [<a href="https://doi.org/10.1016/0022-4049(95)00138-7">doi:10.1016/0022-4049(95)00138-7</a>, <a href="https://www.math.mcgill.ca/barr/papers/coho.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Barr-CECohomology.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Algebraic cohomology: the early days</em>, in <em>Galois Theory, Hopf Algebras, and Semiabelian Categories</em>, Fields Institute Communications <strong>43</strong> (2004) 1–26 [<a href="https://doi.org/10.1090/fic/043">doi:10.1090/fic/043</a>, <a href="https://www.math.mcgill.ca/barr/papers/algcohom.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Barr-AlgebraicCohomology.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>The general abstract perspective on <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> (subsuming <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a>, <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a> and indications of <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology">Whitehead-generalized cohomology</a>) was essentially established in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Brown">Kenneth Brown</a>, <em><a class="existingWikiWord" href="/nlab/show/BrownAHT">Abstract homotopy theory and generalized sheaf cohomology</a></em> (1973)</li> </ul> </div> <p>Review of basics</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, Section 1.1: <em><a class="existingWikiWord" href="/nlab/show/An+Introduction+to+Homological+Algebra">An Introduction to Homological Algebra</a></em></p> </li> <li id="HopkinsMathew"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a> (notes by <a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>), <em>algebraic topology – Lectures</em> (<a href="http://people.fas.harvard.edu/~amathew/ATnotes.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 31, 2022 at 01:08:36. 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