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> simplicial localization 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> simplicial localization </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/4839/#Item_10" 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>Simplicial localization</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="locality_and_descent">Locality and descent</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/localization">localization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+object">local object</a>, <a class="existingWikiWord" href="/nlab/show/local+morphism">local morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+localization">reflective localization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/descent">descent</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cover">cover</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+object">descent object</a>, <a class="existingWikiWord" href="/nlab/show/descent+morphism">descent morphism</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/matching+family">matching family</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a>, <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>,<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+descent">cohomological descent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadic+descent">monadic descent</a>, <a class="existingWikiWord" href="/nlab/show/higher+monadic+descent">higher monadic descent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a>, <a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/descent+and+locality+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory"><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>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</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/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <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</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <blockquote> <p>See also <a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a></p> </blockquote> <h1 id="simplicial_localization">Simplicial localization</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#construction'>Construction</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#standard_simplicial_localization'>“Standard” simplicial localization</a></li> <li><a href='#hammock_localization'>Hammock localization</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#basic_properties'>Basic properties</a></li> <li><a href='#simplical_localization_gives_all_categories'>Simplical localization gives all <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</a></li> <li><a href='#equivalences_between_simplicial_localizations'>Equivalences between simplicial localizations</a></li> <li><a href='#simplicial_localization_of_model_categories'>Simplicial localization of model categories</a></li> <li><a href='#PresentationCSS'>Presentation in terms of complete Segal spaces</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>A <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> or <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> equipped with the information that some of its <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>, specifically, a subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊃</mo><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \supset Core(C)</annotation></semantics></math>, are to be regarded as “weakly invertible”. One way to make this notion precise is through the concept of <strong>simplicial localization</strong>, one form of which is known as <strong>Dwyer–Kan localization</strong>.</p> <p>The <em>simplicial localization</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">L C</annotation></semantics></math> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> realized concretely as a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> which is such that the original category injects into it, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↪</mo><mi>L</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">C \hookrightarrow L C</annotation></semantics></math>, such that every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> that is labeled as a weak equivalence becomes an actual equivalence in the sense of morphisms in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">L C</annotation></semantics></math>. And <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">L C</annotation></semantics></math> is in some sense universal with this property.</p> <p>Passing to the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">L C</annotation></semantics></math> then reproduces the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> that can also directly be obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>L</mi><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho_C(a,b) \simeq \Pi_0 (L C(a,b)) </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Pi_0</annotation></semantics></math> gives the <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">0th simplicial homotopy groupoid</a>).</p> <p>If the homotopical structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> extends to that of a (combinatorial) <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, then there is another procedure to obtain a simplicially enriched category from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">(∞,1)-category presented by a combinatorial model category</a>. This <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>-category is equivalent to the one obtained by simplicial localization but typically more explicit and more tractable.</p> <p>See also <span class="newWikiWord">localization of a simplicial model category<a href="/nlab/new/localization+of+a+simplicial+model+category">?</a></span>.</p> <h2 id="construction">Construction</h2> <p>See <a class="existingWikiWord" href="/nlab/show/simplicial+localization+of+a+homotopical+category">simplicial localization of a homotopical category</a>.</p> <h2 id="definition">Definition</h2> <h3 id="standard_simplicial_localization">“Standard” simplicial localization</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mstyle mathvariant="bold"><mi>Cat</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>Grph</mi></mstyle></mrow><annotation encoding="application/x-tex">U : \mathbf{Cat} \to \mathbf{Grph}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> that sends a (small) category to its underlying <em><a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a></em> <a class="existingWikiWord" href="/nlab/show/graph">graph</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mstyle mathvariant="bold"><mi>Grph</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">F : \mathbf{Grph} \to \mathbf{Cat}</annotation></semantics></math> be its <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>. We then get a <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ϵ</mi><mo>,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = (G, \epsilon, \delta)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Cat</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Cat}</annotation></semantics></math>, and as usual this defines a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mo>:</mo><mstyle mathvariant="bold"><mi>Cat</mi></mstyle><mo>→</mo><mo stretchy="false">[</mo><msup><mstyle mathvariant="bold"><mi>Δ</mi></mstyle> <mi>op</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mi>Cat</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">G_\bullet : \mathbf{Cat} \to [\mathbf{\Delta}^{op}, \mathbf{Cat}]</annotation></semantics></math> from <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> to <a class="existingWikiWord" href="/nlab/show/simplicial+objects+in+Cat">simplicial objects in Cat</a> equipped with a canonical augmentation, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>n</mi></msub><mi>C</mi><mo>=</mo><msup><mi>G</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>C</mi></mrow><annotation encoding="application/x-tex">G_n C = G^{n+1} C</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>standard resolution</strong> of a small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is defined to be the simplicial category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">G_\bullet C</annotation></semantics></math>.</p> </div> <p>Note that this is also a <a class="existingWikiWord" href="/nlab/show/simplicial+category">simplicial category</a> in the strong sense, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ob</mi><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">ob G_\bullet C</annotation></semantics></math> is discrete! Thus we may also regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">G_\bullet C</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">category</a>. This is a <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> in the sense that the augmentation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\epsilon : G_\bullet C \to C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Dwyer-Kan+equivalence">Dwyer-Kan equivalence</a>. (In fact, for objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_\bullet C (X, Y) \to C (X, Y)</annotation></semantics></math> admits an extra degeneracy and hence a contracting homotopy.)</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The <strong>standard simplicial localization</strong> of a <a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, W)</annotation></semantics></math> is the simplicial category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_\bullet (C, W)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi> <mi>n</mi></msub><mi>C</mi><mo stretchy="false">[</mo><msup><mrow><msub><mi>G</mi> <mi>n</mi></msub><mi>W</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">L_n (C, W) = G_n C [{G_n W}^{-1}]</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#DwyerKanLocalizations">DwyerKanLocalizations, def. 4.1</a>). Again, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">L_\bullet C</annotation></semantics></math> is a simplicial category in the strong sense, because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mo>•</mo></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">G_\bullet C</annotation></semantics></math> is.</p> <h3 id="hammock_localization">Hammock localization</h3> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X,Y \in C</annotation></semantics></math> any two objects, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> L^H C(X,Y) \in sSet </annotation></semantics></math></div> <p>for the simplicial set defined as follows. For each natural number <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> there is a <a class="existingWikiWord" href="/nlab/show/category">category</a> defined as follows:</p> <ul> <li> <p>its <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are length-<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> zig-zags of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><msub><mi>K</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>K</mi> <mn>2</mn></msub><mover><mo>←</mo><mo>≃</mo></mover><msub><mi>K</mi> <mn>3</mn></msub><mo>→</mo><mi>⋯</mi><mo>→</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{\simeq}{\leftarrow} K_1 \to K_2 \stackrel{\simeq}{\leftarrow} K_3 \to \cdots \to Y \,, </annotation></semantics></math></div> <p>where the left-pointing morphisms are to be in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>;</p> </li> <li> <p>its <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are “natural transformations” between such objects, fixing the endpoints:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>K</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>K</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd> <mtd><mi>⋯</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↖</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>↗</mo> <mrow></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>L</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>L</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ && K_1 &\to& K_2 &\stackrel{\simeq}{\leftarrow}& \cdots \\ & {}^{\mathllap{\simeq}}\swarrow &&&&& && \searrow^{} \\ X && \downarrow^{\simeq}&& \downarrow^{\simeq}& \cdots&& && Y \\ & {}_{\mathllap{\simeq}}\nwarrow &&&&& && \nearrow^{} \\ && L_1 &\to& L_2 &\stackrel{\simeq}{\leftarrow}& \cdots } \; </annotation></semantics></math></div></li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^H C(X,Y)</annotation></semantics></math> is obtained by</p> <ul> <li> <p>taking the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of the <a class="existingWikiWord" href="/nlab/show/nerves">nerves</a> of these categories over all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, and</p> </li> <li> <p>quotienting by the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> generated by inserting or removing identity morphisms and composing composable morphisms.</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X,Y,Z</annotation></semantics></math> three objects, there is an evident compositing morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L^H C(X,Y) \times L^H C(Y,Z) \to L^H C(X,Z) </annotation></semantics></math></div> <p>given by horizontally concatenating hammock diagrams as above.</p> <p>The <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi></mrow><annotation encoding="application/x-tex">L^H C</annotation></semantics></math> obtained this way is the <strong>hammock localization</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#DwyerKanCalculating">DwyerKanCalculating, def. 2.1</a>).</p> <h2 id="properties">Properties</h2> <h3 id="basic_properties">Basic properties</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>∈</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">L^H C \in sSet Cat</annotation></semantics></math> for its hammock localization and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C[W^{-1}] \in Cat</annotation></semantics></math> for its ordinary <a class="existingWikiWord" href="/nlab/show/localization">localization</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Ho(L^H C) \in Cat</annotation></semantics></math> for the category with the same objects as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and morphisms between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 L^H C(X,Y)</annotation></semantics></math>.</p> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>≃</mo><mi>C</mi><mo stretchy="false">[</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho L^H C \simeq C[W^{-1}] \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#DwyerKanCalculating">DwyerKanCalculating, prop. 3.1</a>).</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Hammock localization is clearly a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the category of <a class="existingWikiWord" href="/nlab/show/relative+categories">relative categories</a> to <a class="existingWikiWord" href="/nlab/show/sSet-enriched+categories">sSet-enriched categories</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>RelativeCat</mi><mover><mo>⟶</mo><mrow><msubsup><mi>L</mi> <mi>W</mi> <mi>H</mi></msubsup></mrow></mover><mi>SimplicialCategories</mi></mrow><annotation encoding="application/x-tex"> RelativeCat \overset{L^H_W}{\longrightarrow} SimplicialCategories </annotation></semantics></math></div> <p></p> </div> </p> <p>See also <a href="#BarwickKan12">Barwick & Kan 12, Sec. 1.5</a>, <a href="#Spitzweck10">Spitzweck 10, p. 3</a>.</p> <div class="num_prop" id="PrePostCompositionWithWeakEquivalences"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>, and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>W</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X \overset{f}{\to} Y) \in W \subset Mor(C) </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>. Then for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> we have that the concatenation operation on hammocks induce <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_* \;\colon\; L^H C(U,X) \stackrel{\simeq}{\to} L^H C(U,Y) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^* \;\colon\; L^H C(Y,U) \stackrel{\simeq}{\to} L^H C(X, U) \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#DwyerKanCalculating">DwyerKanCalculating, prop. 3.3</a>).</p> <h3 id="simplical_localization_gives_all_categories">Simplical localization gives all <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</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Up to Dwyer-Kan equivalence –the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a> – every <a class="existingWikiWord" href="/nlab/show/simplicial+category">simplicial category</a> is the simplicial localization of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>.</p> <p>This is (<a href="#DwyerKanEquivalences">DwyerKan 87, 2.5</a>).</p> </div> <p>If the <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> in question happens to carry even the structure of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> there exist more refined tools for computing the <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a> of the simplicial localization. These are described at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a>.</p> <h3 id="equivalences_between_simplicial_localizations">Equivalences between simplicial localizations</h3> <div class="num_prop" id="SimplicialLocalizationOfNaturalTransformation"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>W</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C', W')</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>,</mo><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>′</mo><mo>∈</mo><mi>sSet</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">L^H C, L^H C' \in sSet Cat</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/hammock+localizations">hammock localizations</a>.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F_1, F_2 : C \to C'</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/homotopical+functors">homotopical functors</a> (functors respecting the weak equivalences, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>W</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F_i(W) \subset W'</annotation></semantics></math>) with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>⇒</mo><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \eta : F_1 \Rightarrow F_2 </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> with components in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">W'</annotation></semantics></math>, we have that for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X,Y \in C</annotation></semantics></math>, there is induced a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> between morphisms of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>L</mi> <mi>H</mi></msup><msub><mi>F</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mo>*</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>⇓</mo></mtd> <mtd></mtd> <mtd><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>L</mi> <mi>H</mi></msup><msub><mi>F</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && L^H C'(F_1(X), F_1(Y) ) \\ & {}^{\mathllap{L^H F_1}}\nearrow && \searrow^{\mathrlap{\eta(Y)_*}} \\ L^H C(X,Y) && \Downarrow && L^H C'(F_1(X), F_2(Y)) \\ & {}_{\mathllap{L^H F_2}}\searrow && \nearrow_{\mathrlap{\eta(X)^*}} \\ && L^H C' (F_2(X), F_2(Y)) } \,. </annotation></semantics></math></div></div> <p>This is (<a href="#DwyerKanComputations">DwyerKanComputations, prop. 3.5</a>).</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i : (C_1, W_1) \hookrightarrow (C_2, W_2)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> such that</p> <ol> <li> <p><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 homotopy-essentially surjective: for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">c_2 \in C_2</annotation></semantics></math> there is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">c_1 \in C_1</annotation></semantics></math> and a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>2</mn></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>i</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_2 \stackrel{\simeq}{\to} i(c_1)</annotation></semantics></math>;</p> </li> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>W</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q : (C_2,W_2) \to (C_1, W_1)</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∘</mo><mi>Q</mi><mo>⇒</mo><msub><mi>Id</mi> <mrow><msub><mi>C</mi> <mn>2</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i \circ Q \Rightarrow Id_{C_2} \,. </annotation></semantics></math></div></li> </ol> <p>Then we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><msub><mi>C</mi> <mn>1</mn></msub><mo>≃</mo><msup><mi>L</mi> <mi>H</mi></msup><msub><mi>C</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L^H C_1 \simeq L^H C_2 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We have to check that <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 an <a class="existingWikiWord" href="/nlab/show/essentially+surjective+%28%E2%88%9E%2C1%29-functor">essentially surjective (∞,1)-functor</a> and a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functor">full and faithful (∞,1)-functor</a>.</p> <p>The first condition is immediate from the first assumption. The second follows with prop. <a class="maruku-ref" href="#SimplicialLocalizationOfNaturalTransformation"></a> (using prop. <a class="maruku-ref" href="#PrePostCompositionWithWeakEquivalences"></a>) from the second assumption.</p> </div> <h3 id="simplicial_localization_of_model_categories">Simplicial localization of model categories</h3> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">C^\circ</annotation></semantics></math> for the full <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>-subcategory on the fibrant and cofibrant objects.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">C^\circ</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi></mrow><annotation encoding="application/x-tex">L^H C</annotation></semantics></math> are connected by an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a>.</p> </div> <p>This is one of the central statements in (<a href="#DwyerKan80FunctionComplexes">Dwyer & Kan 80 FuncComp</a>). The weak homotopy equivalence between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>∘</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\circ(X,Y)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^H C(X,Y)</annotation></semantics></math> is in corollary 4.7. The equivalence of <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>-categories stated above follows with this and the diagram of morphisms of simplicial categories in prop. 4.8.</p> <div class="num_prop" id="QuillenEquivalencesInducingDKEquivalences"> <h6 id="proposition_6">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⇆</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D \leftrightarrows C</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> induces a Dwyer-Kan-equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>C</mi><mo>↔</mo><mi>L</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">L C \leftrightarrow L D</annotation></semantics></math> between their <a class="existingWikiWord" href="/nlab/show/simplicial+localizations">simplicial localizations</a>.</p> </div> <p>This is made explicit in <a href="#MazelGee15">Mazel-Gee 15, p. 17</a> to follow from <a href="#DwyerKan80FunctionComplexes">Dwyer & Kan 80 FuncComp, Prop. 4.4 with 5.4</a>.</p> <h3 id="PresentationCSS">Presentation in terms of complete Segal spaces</h3> <p>Using <a class="existingWikiWord" href="/nlab/show/quasicategories">quasicategories</a> as a model of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories">(infinity,1)-categories</a>, there is a construction which computes simplicial localization in terms of <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a>. See <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces#RelationToSimplicialLocalization">complete Segal spaces#RelationToSimplicialLocalization</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localizer">localizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localizing+subcategory">localizing subcategory</a></p> </li> </ul> <h2 id="References">References</h2> <p>The original articles:</p> <ul> <li id="DwyerKanLocalizations"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Simplicial localizations of categories</em>, J. Pure Appl. Algebra <strong>17</strong> 3 (1980), 267-284 [<a href="https://doi.org/10.1016/0022-4049(80)90049-3">doi:10.1016/0022-4049(80)90049-3</a>]</p> </li> <li id="DwyerKanCalculating"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Calculating simplicial localizations</em>, J. Pure Appl. Algebra 18 (1980), 17-35 [<a href="https://doi.org/10.1016/0022-4049(80)90113-9">doi:10.1016/0022-4049(80)90113-9</a>]</p> </li> <li id="DwyerKan80FunctionComplexes"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Function complexes in homotopical algebra</em>, Topology 19 (1980), 427-440 [<a href="https://doi.org/10.1016/0040-9383(80)90025-7">doi:10.1016/0040-9383(80)90025-7</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf">pdf</a>]</p> </li> <li id="DwyerKanEquivalences"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>Equivalences between homotopy theories of diagrams</em>, in: <em>Algebraic topology and algebraic K-theory</em>, Ann. of Math. Stud. <strong>113</strong>, Princeton University Press (1988) [<a href="https://doi.org/10.1515/9781400882113-009">doi:10.1515/9781400882113-009</a>]</p> </li> </ul> <p>and in modernized form:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, around 35.6 of : <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Limit+Functors+on+Model+Categories+and+Homotopical+Categories">Homotopy Limit Functors on Model Categories and Homotopical Categories</a></em>, Mathematical Surveys and Monographs <strong>113</strong>, AMS (2004) [<a href="https://bookstore.ams.org/surv-113-s">ISBN: 978-1-4704-1340-8</a>, <a href="http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf">pdf</a>]</li> </ul> <p>Survey:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, Section 4.1 of: <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-Categories, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-groupoids, Segal categories and quasicategories</em> [<a href="http://arxiv.org/abs/math/0401274">arXiv:math/0401274</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clark+Barwick">Clark Barwick</a>, Section 2 of: <em>On (enriched) Bousfield localization of model categories</em> [<a href="http://arxiv.org/abs/0708.2067">arXiv:0708.2067</a>]</p> </li> </ul> <p>Further development:</p> <ul> <li id="BarwickKan12"> <p><a class="existingWikiWord" href="/nlab/show/Clark+Barwick">Clark Barwick</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <em>A characterization of simplicial localization functors and a discussion of DK equivalences</em>, Indagationes Mathematicae Volume 23, Issues 1–2, March 2012, Pages 69-79 (<a href="https://doi.org/10.1016/j.indag.2011.10.001">doi:10.1016/j.indag.2011.10.001</a>)</p> </li> <li id="Hinich13"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>Dwyer-Kan Localization Revisited</em>, Homology, Homotopy and Applications Volume 18 (2016) Number 1 (<a href="https://arxiv.org/abs/1311.4128">arXiv:1311.4128</a>, <a href="https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3">doi:10.4310/HHA.2016.v18.n1.a3</a>)</p> </li> </ul> <p>See also:</p> <ul> <li id="Lurie09"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section A.3.2 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></p> </li> <li id="Spitzweck10"> <p><a class="existingWikiWord" href="/nlab/show/Markus+Spitzweck">Markus Spitzweck</a>, <em>Homotopy limits of model categories over inverse index categories</em>, Journal of Pure and Applied Algebra Volume 214, Issue 6, June 2010, Pages 769-777 (<a href="https://doi.org/10.1016/j.jpaa.2009.08.001">doi:10.1016/j.jpaa.2009.08.001</a>)</p> </li> <li id="MazelGee15"> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Mazel-Gee">Aaron Mazel-Gee</a>, <em>Quillen adjunctions induce adjunctions of quasicategories</em>, New York Journal of Mathematics Volume 22 (2016) 57-93 (<a href="https://arxiv.org/abs/1501.03146">arXiv:1501.03146</a>, <a href="http://nyjm.albany.edu/j/2016/22-4.html">publisher</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2025 at 21:45:23. See the <a href="/nlab/history/simplicial+localization" 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/simplicial+localization" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4839/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/simplicial+localization/47" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/simplicial+localization" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/simplicial+localization" accesskey="S" class="navlink" id="history" rel="nofollow">History (47 revisions)</a> <a href="/nlab/show/simplicial+localization/cite" style="color: black">Cite</a> <a href="/nlab/print/simplicial+localization" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/simplicial+localization" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>