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> category of chain complexes (Rev #21) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <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> category of chain complexes (Rev #21) </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/4146/#Item_6" 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"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homological_algebra'>Homological algebra</h4> <div class='hide'> <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' title='schreiber'>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+sheaf'>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/exact+sequence'>short exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/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/projective+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+%28infinity%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-infinity-category'>A-∞-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%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+limit'>homotopy colimit</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/BV-BRST+formalism'>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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#AbelianCategoryStructure'>Abelian structure</a></li><li><a href='#closed_monoidal_structure'>Closed monoidal structure</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_1' xmlns='http://www.w3.org/1998/Math/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/additive+category'>additive category</a>.</p> <p>Recall the notion of <em><a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complex</a></em>, of <em><a class='existingWikiWord' href='/nlab/show/chain+map'>chain map</a></em> between chain complexes and of <em><a class='existingWikiWord' href='/nlab/show/chain+homotopy'>chain homotopy</a></em> between chain maps in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> <p>Call a <a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complex</a> <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>C_\bullet</annotation></semantics></math></p> <ul> <li> <p><strong>bounded below</strong> if there is <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>≤</mo><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>C_{n \leq k} = 0</annotation></semantics></math>;</p> </li> <li> <p><strong>bounded above</strong> if there is <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>≥</mo><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>C_{n \geq k} = 0</annotation></semantics></math>;</p> </li> <li> <p><strong>bounded</strong> if it is bounded below and bounded above. We have</p> </li> </ul> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_8' xmlns='http://www.w3.org/1998/Math/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'>category</a> whose <a class='existingWikiWord' href='/nlab/show/object'>objects</a> are <a class='existingWikiWord' href='/nlab/show/chain+complex'>chain complexes</a> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> and whose <a class='existingWikiWord' href='/nlab/show/morphism'>morphisms</a> are <a class='existingWikiWord' href='/nlab/show/chain+map'>chain maps</a> between these.</p> <p>This is the <strong>category of chain complexes</strong> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> </div> <p>Several variants of this category are of relevance.</p> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>b</mi></mrow></msubsup><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><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'>Ch_\bullet^{+,-,b}(\mathcal{A}) \hookrightarrow Ch_\bullet(\mathcal{A})</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> on the chain complexes which are, respectively, bounded above, bounded below or bounded.</p> </div> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\mathcal{A})</annotation></semantics></math> for the category obtained from <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_13' xmlns='http://www.w3.org/1998/Math/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> by identifying <a class='existingWikiWord' href='/nlab/show/chain+homotopy'>homotopic</a> chain maps.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo>≔</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>D</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>chain</mi><mo>−</mo><mi>homotopy</mi><mspace width='thinmathspace'></mspace><mo>.</mo></mrow><annotation encoding='application/x-tex'> K(\mathcal{A})(C_\bullet, D_\bullet) \coloneqq Ch_\bullet(C_\bullet, D_\bullet)/chain-homotopy \,. </annotation></semantics></math></div> <p>Accordingly <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>K</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>b</mi></mrow></msup><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K^{+,-,b}(\mathcal{A}) \hookrightarrow K(\mathcal{A})</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> on the chain complexes bounded above, bounded below or bounded, respectively.</p> </div> <p>This is sometimes called the <em><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+chain+complexes'>homotopy category of chain complexes</a></em>. But see the warning on terminology there, as this term is also appropriate for the category in the following remark.</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is moreover an <a class='existingWikiWord' href='/nlab/show/abelian+category'>abelian category</a>, then there is also the <strong><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a></strong> <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(\mathcal{A})</annotation></semantics></math>, obtained from <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_18' xmlns='http://www.w3.org/1998/Math/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> or <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\mathcal{A})</annotation></semantics></math> by <a class='existingWikiWord' href='/nlab/show/localization'>universally inverting</a> all <a class='existingWikiWord' href='/nlab/show/quasi-isomorphism'>quasi-isomorphisms</a>. See at <em><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a></em> for more on this.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='AbelianCategoryStructure'>Abelian structure</h3> <div class='num_theorem' id='IsAbelian'> <h6 id='theorem'>Theorem</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/abelian+category'>abelian category</a> also the category of chain complexes <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_21' xmlns='http://www.w3.org/1998/Math/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> is again an abelian category.</p> </div> <p>We discuss the ingredients that go into this statement.</p> <p>(…)</p> <div class='num_prop' id='KernelsOfChainComplexes'> <h6 id='proposition'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>D</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>f : C_\bullet \to D_\bullet</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/chain+map'>chain map</a>,</p> <ul> <li> <p>the complex <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ker</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>ker(f)</annotation></semantics></math> of degreewise <a class='existingWikiWord' href='/nlab/show/kernel'>kernels</a> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is the kernel of <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_26' xmlns='http://www.w3.org/1998/Math/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>;</p> </li> <li> <p>the complex <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>coker</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>coker(f)</annotation></semantics></math> of degreewise <a class='existingWikiWord' href='/nlab/show/cokernel'>cokernels</a> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is the cokernel of <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_30' xmlns='http://www.w3.org/1998/Math/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>.</p> </li> </ul> </div> <div class='num_remark' id='ShortExactSequencesDegreewise'> <h6 id='remark_2'>Remark</h6> <p>A sequence of chain complexes <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/exact+sequence'>short exact sequence</a> in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_32' xmlns='http://www.w3.org/1998/Math/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> precisely if each component <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>0 \to A_n \to B_n \to C_n \to 0</annotation></semantics></math> is a short exact sequence in <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> </div> <p>(…)</p> <p>In fact:</p> <p>\begin{proposition}\label{ChInGrothAbCatIsGrothAbCat} When <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/Grothendieck+category'>Grothendieck abelian category</a> then so is <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_36' xmlns='http://www.w3.org/1998/Math/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>. \end{proposition} (e.g. <a href='#Hovey99'>Hovey (1999), p. 3</a>, see also <a href='Grothendieck+category#ChainComplexesInGrothAbCat'>this example</a> at <em><a class='existingWikiWord' href='/nlab/show/Grothendieck+category'>Grothendieck category</a></em>).</p> <p>Since every <a class='existingWikiWord' href='/nlab/show/Grothendieck+category'>Grothendieck abelian category</a> is <a class='existingWikiWord' href='/nlab/show/locally+presentable+category'>locally presentable</a> (<a href='locally+presentable+category#Beke00'>Beke 2000, Prop. 3.10</a>, see <a href='locally+presentable+category#GrothAbCatsAreLocPresntbl'>this example</a>), it follows that:</p> <p>\begin{corollary} When <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/Grothendieck+category'>Grothendieck abelian category</a> then its category of chain complexes <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_38' xmlns='http://www.w3.org/1998/Math/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> is <a class='existingWikiWord' href='/nlab/show/locally+presentable+category'>locally presentable</a>. \end{corollary}</p> <p>\begin{example} The assumption in Prop. \ref{ChInGrothAbCatIsGrothAbCat} is fulfilled in the usual situation of chain complexes of <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/module'>modules</a> (e.g.: of <a class='existingWikiWord' href='/nlab/show/vector+space'>vector spaces</a>, when <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is a field), since for any <a class='existingWikiWord' href='/nlab/show/ring'>commutative ring</a> <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> the category <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/Mod'>Mod</a> is a <a class='existingWikiWord' href='/nlab/show/Grothendieck+category'>Grothendieck abelian category</a> (by <a href='Grothendieck+category#CateoriesOfModules'>this example</a>). \end{example}</p> <h3 id='closed_monoidal_structure'>Closed monoidal structure</h3> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>Equipped with the standard <a class='existingWikiWord' href='/nlab/show/tensor+product+of+chain+complexes'>tensor product of chain complexes</a> <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math> the category of chain complexes is a <a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>R</mi><mi>Mod</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ch_\bullet(R Mod), \otimes)</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/unit+object'>unit object</a> is the chain complex concentrated in degree 0 on the tensor unit <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>R Mod</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>In fact <math class='maruku-mathml' display='inline' id='mathml_97d6f7a6091ad3a16035ed8392c2765ca748d355_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ch_\bullet, \otimes)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed monoidal category</a>, the <a class='existingWikiWord' href='/nlab/show/internal+hom'>internal hom</a> is the standard <a class='existingWikiWord' href='/nlab/show/internal+hom+of+chain+complexes'>internal hom of chain complexes</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+chain+complexes'>homotopy category of chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+chain+complexes'>model structure on chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/derived+category'>derived category</a>, <a class='existingWikiWord' href='/nlab/show/derived+functor'>derived functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category+of+chain+complexes'>(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/equivariant+chain+complex'>equivariant chain complex</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed monoidal category</a>-structure on chain complexes:</p> <ul> <li id='EilenbergKelly65'><a class='existingWikiWord' href='/nlab/show/Samuel+Eilenberg'>Samuel Eilenberg</a>, <a class='existingWikiWord' href='/nlab/show/Max+Kelly'>G. Max Kelly</a>, §IV.6 of: <em>Closed Categories</em>, in: <em><a class='existingWikiWord' href='/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965'>Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]</li> </ul> <p>Textbook account:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Charles+Weibel'>Charles Weibel</a>, Chapter 1 in: <em><a class='existingWikiWord' href='/nlab/show/An+Introduction+to+Homological+Algebra'>An Introduction to Homological Algebra</a></em>, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), <a href='https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf'>pdf</a>]</li> </ul> <p>Discussion in the context of <a class='existingWikiWord' href='/nlab/show/model+structure+on+chain+complexes'>model structures on chain complexes</a>:</p> <ul> <li id='Hovey99'><a class='existingWikiWord' href='/nlab/show/Mark+Hovey'>Mark Hovey</a>, p. 3 of: <em>Model category structures on chain complexes of sheaves</em> (1999) [[K-theory:0366](https://faculty.math.illinois.edu/K-theory/0366), <a href='https://faculty.math.illinois.edu/K-theory/0366/sheaves.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Hovey-SheavesOfChainComplexesPreprint.pdf' title='pdf'>pdf</a>]</li> </ul> <p><div class='property'> category: <a class='category_link' href='/nlab/list/category'>category</a></div></p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on August 23, 2023 at 08:44:17 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/category+of+chain+complexes" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/4146/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/show/category+of+chain+complexes" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/category+of+chain+complexes/20" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (20 more)</span><a href="/nlab/show/category+of+chain+complexes" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/category+of+chain+complexes/21" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/category+of+chain+complexes" accesskey="S" class="navlink" id="history" rel="nofollow">History (21 revisions)</a><a href="/nlab/rollback/category+of+chain+complexes?rev=21" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/category+of+chain+complexes/21/cite" style="color: black">Cite</a> <a href="/nlab/source/category+of+chain+complexes/21" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>