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> model structure on cosimplicial abelian groups in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> model structure on cosimplicial abelian groups </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/1736/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <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>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</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> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <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</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <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</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <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</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <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>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <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>-algebras</em></p> <p><em>general <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>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <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>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <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>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SimplicialEnrichmentOfProjective'>Simplicial enrichment of the projective structure</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^{\Delta}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category+of+cosimplicial+objects">category of cosimplicial objects</a> in the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s – the category of <em><a class="existingWikiWord" href="/nlab/show/cosimplicial+abelian+group">cosimplicial abelian groups</a></em> .</p> <p>This entry discusses structures of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math>.</p> <p>By the dual <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup><mover><munder><mo>→</mo><mi>N</mi></munder><mover><mo>←</mo><mi>Ξ</mi></mover></mover><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab^\Delta \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N}{\to}} Ch^\bullet_+(Ab)</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/category+of+cochain+complexes">category of cochain complexes</a> in non-negative degree. Since <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, we have by general results various <a href="http://ncatlab.org/nlab/show/model+structure+on+chain+complexes#CochainNonNeg">model structures on cochain complexes</a>. Via the Dold-Kan equivalence, all of these induce model structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="SimplicialEnrichmentOfProjective">Simplicial enrichment of the projective structure</h3> <p>Since <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> has all <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, the <a class="existingWikiWord" href="/nlab/show/category+of+cosimplicial+objects">category of cosimplicial objects</a> (as described there) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math> inherits canonically the structure of an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> which is <a class="existingWikiWord" href="/nlab/show/power">power</a>ed and <a class="existingWikiWord" href="/nlab/show/copower">copower</a>ed.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ab</mi> <mi>proj</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex">Ab^\Delta_{proj}</annotation></semantics></math> for the model structure that is induced by the dual <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup><mo>≃</mo><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab^\Delta \simeq Ch^\bullet_+(Ab)</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/model+structure+on+cochain+complexes">model structure on cochain complexes</a> whose fibrations are the <em>degreewise surjections</em> (and weak equivalences the usual <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s). This is described in detail <a href="http://ncatlab.org/nlab/show/model+structure+on+chain+complexes#CochainNonNegProj">here</a>. So this induces the model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ab</mi> <mi>proj</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex">Ab^\Delta_{proj}</annotation></semantics></math> whose fibrations are also the degreewise surjections in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math> (using that the <a class="existingWikiWord" href="/nlab/show/Moore+complex">normalized cochain complex</a>-functor preserves surjections.)</p> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>The canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>-enrichement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math> is compatible with the model category structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ab</mi> <mi>proj</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex">Ab^\Delta_{proj}</annotation></semantics></math> in that the combination gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math> the structure of a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We need check the <a class="existingWikiWord" href="/nlab/show/pushout-product+axiom">pushout-product axiom</a> of an <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> of the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math></p> <p>So we need to show that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">i : C \to C'</annotation></semantics></math> any cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">j : X \to Y</annotation></semantics></math> a fibration of cosimplcial abelian groups (degreewise surjection) the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><msup><mi>X</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo>→</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> k : X^{C'} \to X^C \times_{Y^C} Y^{C'} </annotation></semantics></math></div> <p>induced by the <a class="existingWikiWord" href="/nlab/show/power">power</a>ing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>Ab</mi> <mi>Δ</mi></msup><mo>×</mo><mi>sSet</mi><mo>→</mo><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">(-)^{(-)} : Ab^\Delta \times sSet \to Ab^\Delta</annotation></semantics></math> is a fibration, which is acyclic if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> is.</p> <p>That <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> is a fibration is easily checked. To see acyclicity we first notice the following</p> <p><strong>Lemma.</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">i : C \to C'</annotation></semantics></math> is a weak equivalence then for every cosimplicial abelian group <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> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">A^i</annotation></semantics></math> is a weak equivalence.</p> <p>To see this observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">A^C</annotation></semantics></math> is the diagonal of an evident <a class="existingWikiWord" href="/nlab/show/bisimplicial+object">bisimplicial abelian group</a> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">A^i</annotation></semantics></math> is then in one argument a degreewise <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>. Since forming total complexes preserves degreewise equivalences, the lemma follows.</p> <p>To continue the main proof, notice that we have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo>→</mo><msup><mi>X</mi> <mi>C</mi></msup><mo>⊕</mo><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mover><mo>→</mo><mi>f</mi></mover><msup><mi>Y</mi> <mi>C</mi></msup><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to X^C \times_{Y^C} Y^{C'} \to X^C \oplus Y^{C'} \stackrel{f}{\to} Y^C \to 0 </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>↦</mo><msup><mi>j</mi> <mi>C</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msup><mi>Y</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f : (x,y) \mapsto j^C(x) - Y^i(y)</annotation></semantics></math>. This induces a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">long exact sequence in cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>Y</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to H^p(X^C \times_{Y^C} Y^{C'}) \to H^p(X^C) \oplus H^i(Y^{C'}) \to H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'}) \to \cdots \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a weak equivalence, then by the above lemma we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>Y</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ker(H^p(X^C) \oplus H^p(Y^{C'}) \to H^p(Y^C)) \simeq H^p(X^C) \,. </annotation></semantics></math></div> <p>Inspection of the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> then shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>Y</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(Y^C) \to H^{p+1}(X^C \times_{Y^C} Y^{C'})</annotation></semantics></math> is the 0-map. In total this implies that we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>C</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^p(X^C \times_{Y^C} Y^{C'}) \stackrel{\simeq}{\to} H^p(X^C) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, and hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>C</mi></msup><msub><mo>×</mo> <mrow><msup><mi>Y</mi> <mi>C</mi></msup></mrow></msub><msup><mi>Y</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo>→</mo><msup><mi>X</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex"> X^C \times_{Y^C} Y^{C'} \to X^C </annotation></semantics></math></div> <p>is a weak equivalence. Since by the above lemma also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>i</mi></msup><mo>:</mo><msup><mi>X</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msup><mo>→</mo><msup><mi>X</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">X^i : X^{C'} \to X^C</annotation></semantics></math> is a weak equivalence, it follows by <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">2-out-of-3</a> that the morphism <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> is indeed a weak equivalence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is.</p> <p>An analogous argument shows that <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> is a weak equivalence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> is.</p> <p>This argument is essentially that on page 41 of (<a href="#Toen">Toën</a>)</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+abelian+groups">model structure on simplicial abelian groups</a></li> </ul> <h2 id="references">References</h2> <p>The <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+algebras">model structure on cosimplicial algebras</a> is discussed in detail in</p> <ul> <li id="Toen"><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <em>Champs affines</em>, Selecta Math. new series <strong>12</strong> (2006), no. 1, 39-135 (<a href="https://arxiv.org/abs/math/0012219">arXiv:math/0012219</a>, <a href="https://doi.org/10.1007/s00029-006-0019-z">doi:10.1007/s00029-006-0019-z</a>)</li> </ul> <p>The above proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ab</mi> <mi>proj</mi> <mi>Δ</mi></msubsup></mrow><annotation encoding="application/x-tex">Ab^\Delta_{proj}</annotation></semantics></math> is a simplicial model category mimics the proof on page 41 there. Indeed, the claim is that the model structure on cosimplicial algebras is the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> induced by the above from the evident forgetful functor.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on July 14, 2021 at 10:56:43. See the <a href="/nlab/history/model+structure+on+cosimplicial+abelian+groups" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/model+structure+on+cosimplicial+abelian+groups" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1736/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/model+structure+on+cosimplicial+abelian+groups/4" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/model+structure+on+cosimplicial+abelian+groups" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/model+structure+on+cosimplicial+abelian+groups" accesskey="S" class="navlink" id="history" rel="nofollow">History (4 revisions)</a> <a href="/nlab/show/model+structure+on+cosimplicial+abelian+groups/cite" style="color: black">Cite</a> <a href="/nlab/print/model+structure+on+cosimplicial+abelian+groups" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/model+structure+on+cosimplicial+abelian+groups" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>