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> (infinity,1)-categorical hom-space 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> (infinity,1)-categorical hom-space </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/11510/#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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </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> <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> <h4 id="mapping_space">Mapping space</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></p> <h3 id="general_abstract">General abstract</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>, <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>, <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a></p> </li> </ul> <h3 id="topology">Topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (<a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+of+mapping+spaces">topology of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/evaluation+fibration+of+mapping+spaces">evaluation fibration of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/free+loop+space+of+a+classifying+space">free loop space of a classifying space</a></li> </ul> </li> </ul> <h3 id="simplicial_homotopy_theory">Simplicial homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+mapping+complex">simplicial mapping complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+groupoid">inertia groupoid</a></p> </li> </ul> <h3 id="differential_topology">Differential topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+topology+of+mapping+spaces">differential topology of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-k+topology">C-k topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tangent+spaces+of+mapping+spaces">tangent spaces of mapping spaces</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a></p> </li> </ul> <h3 id="stable_homotopy_theory">Stable homotopy theory</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></li> </ul> <h3 id="geometric_homotopy_theory">Geometric homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+stack">inertia stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+loop+stack">free loop stack</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/mapping+space+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#presentations'>Presentations</a></li> <ul> <li><a href='#ForACategoryWithWeakEquivalences'>For a category with weak equivalences</a></li> <li><a href='#Framings'>For a model category</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#EnrichedHomsCofToFib'>For a simplicial model category</a></li> <li><a href='#comparison'>Comparison</a></li> <li><a href='#InACategoryOfFibrantObjects'>In a category of fibrant objects</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SpacesOfEquivalences'>Hom-spaces of equivalences</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>Where an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> has a <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> has an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of morphisms between any two objects, a <em>hom-space</em>.</p> <p>There are several ways to <em>present</em> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> by an ordinary <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 some extra structure: for instance <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> may be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> or a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> or even a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>. In all of these presentations, given two 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 a way to construct a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}\mathbf{C}(X,Y)</annotation></semantics></math> that presents the hom-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{C}(X,Y)</annotation></semantics></math>. This simplicial set – or rather its <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> – is called the <em>derived hom space</em> or <em>homotopy function complex</em> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{R}Hom(X,Y)</annotation></semantics></math> or similarly.</p> <h2 id="presentations">Presentations</h2> <p>There are many ways to present an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> by <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic data</a>, and for each of these there are corresponding tools for explicitly computing the derived hom spaces.</p> <p>The most basic data is that of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called <em><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></em>. This we discuss below in <em><a href="#ForACategoryWithWeakEquivalences">For a category with weak equivalences</a></em>.</p> <p>Particularly useful extra structure on a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> that helps with computing the derived hom spaces is the structure of a <em><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></em>. Using this one can construct simplicial resolutions of objects – called <em>framings</em> – that generalize <a class="existingWikiWord" href="/nlab/show/cylinder+objects">cylinder objects</a> and <a class="existingWikiWord" href="/nlab/show/path+objects">path objects</a>, and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in <em><a href="#Framings">For a model category</a></em>.</p> <p>Still a bit more helpful structure on top of a bare model category is that of a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/hom+objects">hom objects</a>. This we discuss below in <em><a href="#EnrichedHomsCofToFib">For a simplicial model category</a></em>.</p> <h3 id="ForACategoryWithWeakEquivalences">For a category with weak equivalences</h3> <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>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W \subset Mor(C))</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>.</p> <div class="num_defn" id="ZigZagCategories"> <h6 id="definition">Definition</h6> <p>Fix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. 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>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X,Y \in Obj(C)</annotation></semantics></math>, define a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>wMor</mi> <mi>C</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">wMor_C^n(X,Y)</annotation></semantics></math></p> <ul> <li> <p>whose objects are <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a>s 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> of 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>←</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>←</mo><mi>⋯</mi><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>←</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>=</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X = X_0 \leftarrow X_1 \to X_2 \leftarrow \cdots \to X_{n-1} \leftarrow X_n = Y </annotation></semantics></math></div> <p>such that each morphism going to the left, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub><mo>←</mo><msub><mi>X</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{2k}\leftarrow X_{2k +1}</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, an element 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>morphisms between such objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><mo>,</mo><mi>X</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>,</mo><mi>Y</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,X_i,Y) \to (X',X'_i,Y')</annotation></semantics></math> are collections of weak equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo>′</mo> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_i \to X'_i)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \lt i \lt n </annotation></semantics></math> such that all triangles and squares commute.</p> </li> </ul> </div> <div class="num_defn" id="HammockLocalization"> <h6 id="definition_2">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msubsup><mi>wMor</mi> <mi>C</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(wMor_C^n(X,Y))</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of this category, a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/hammock+localization">hammock localization</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>L</mi> <mi>W</mi> <mi>H</mi></msubsup><mi>C</mi></mrow><annotation encoding="application/x-tex">L_W^H C</annotation></semantics></math> of <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 the <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> with objects those of <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 <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> given by the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over the length of these hammock hom-categories</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><mo>=</mo><munder><mi>lim</mi> <mrow><msub><mo>→</mo> <mi>n</mi></msub></mrow></munder><mi>N</mi><mo stretchy="false">(</mo><msubsup><mi>wMor</mi> <mi>C</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L^H C(X,Y) := \lim_{\to_n} N(wMor_C^n(X,Y)) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/Kan+fibrant+replacement">Kan fibrant replacement</a> of this simplicial set is the derived hom-space 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> of the <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 modeled by <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> <h3 id="Framings">For a model category</h3> <p>The derived hom spaces of a model 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> may always be computed in terms of simplicial resolutions with respect to the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, C]_{Reedy}</annotation></semantics></math>. These resolutions are often called <em>framings</em> (<a href="#Hovey">Hovey</a>). These constructions are originally due to (<a href="#DHK">Dwyer-Hirschhorn-Kan</a>).</p> <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 any <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>.</p> <div class="num_prop"> <h6 id="observation">Observation</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>const</mi><mo>⊣</mo><msub><mi>ev</mi> <mn>0</mn></msub><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mover><mover><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>ev</mi> <mn>0</mn></msub></mrow></mover></mover><mover><mo>⟶</mo><mi>const</mi></mover></mover><mspace width="thinmathspace"></mspace><mo>,</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (const \dashv ev_0 \dashv (-)^{\times^\bullet}) : C \stackrel{\overset{const}{\longrightarrow}}{\stackrel{\overset{ev_0}{\longleftarrow}}{\underset{(-)^{\times^\bullet}}{\longrightarrow}}} \,, [\Delta^{op}, C] \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mi>X</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">const X : [n] \mapsto X</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mn>0</mn></msub><msub><mi>X</mi> <mo>•</mo></msub><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">ev_0 X_\bullet = X_0</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msup><mi>X</mi> <mrow><msup><mo>×</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\times^\bullet} : [n] \mapsto X^{\times^{n+1}}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_remark" id="CoDiscreteIsReedyFibrant"> <h6 id="remark">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> fibrant, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\times^\bullet}</annotation></semantics></math> is fibrant in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, C]_{Reedy}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The matching morphisms are in fact <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</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 model category.</p> <ol> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> any object, a <em>simplicial frame</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mi>X</mi><mo>→</mo><msup><mi>X</mi> <mrow><msup><mo>×</mo> <mo>•</mo></msup></mrow></msup></mrow><annotation encoding="application/x-tex">const X \to X^{\times^\bullet}</annotation></semantics></math> into a weak equivalence followed by a fibration in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, C]_{Reedy}</annotation></semantics></math>.</p> </li> <li> <p>A <em>right framing</em> 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> is a functor <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><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(-)_\bullet : C \to [\Delta^{op}, C]</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(X)_0 \simeq X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> is a simplicial frame on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> </ol> <p>Dually for <em>cosimplicial frames</em>.</p> </div> <p>This appears as (<a href="#Hovey">Hovey, def. 5.2.7</a>).</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>By remark <a class="maruku-ref" href="#CoDiscreteIsReedyFibrant"></a> a simplicial frame <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> in the above is in particular fibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, C]_{Reedy}</annotation></semantics></math>.</p> </div> <div class="num_prop" id="SimplicialFunctionComplexes"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \in C</annotation></semantics></math> fibrant, there are weak equivalences 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>diag</mi><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mover><mo>←</mo><mo>≃</mo></mover><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_C(X^\bullet, A) \stackrel{\simeq}{\to} diag Hom_C(X^\bullet, A_\bullet) \stackrel{\simeq}{\leftarrow} Hom_C(X, A_\bullet) \,, </annotation></semantics></math></div> <p>(where in the middle we have the diagonal of the <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mo>•</mo></msup><mo>,</mo><msub><mi>A</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X^\bullet, A_\bullet)</annotation></semantics></math>).</p> </div> <p>This appears as (<a href="#Hovey">Hovey, prop. 5.4.7</a>).</p> <p>Either of these simplicial sets is a model for the derived hom-space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(X,A)</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>By developing these constructions further, one obtains a canonical <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>-resolution of (left proper and combinatorial) model categories <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>, such that the simplicial resolutions given by framings are just the cofibrant<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math>fibrant <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>-hom objects as discussed <a href="#EnrichedHomsCofToFib">below</a>.</p> <p>This is discussed at <em><a href="http://ncatlab.org/nlab/show/simplicial+model+category#SimpEquivMods">Simplicial Quillen equivalent models</a></em>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">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 model category, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">c</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathrm{c}_\mathrm{w} C</annotation></semantics></math> be the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta, C]</annotation></semantics></math> spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">s</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathrm{s}_\mathrm{w} C</annotation></semantics></math> be the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta^{op}, C]</annotation></semantics></math> spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>:</mo><mi>C</mi><mo>→</mo><msub><mi mathvariant="normal">c</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">const : C \to \mathrm{c}_\mathrm{w} C</annotation></semantics></math> is the right half of an adjoint homotopical equivalence of <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical categories</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>:</mo><mi>C</mi><mo>→</mo><msub><mi mathvariant="normal">s</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">const : C \to \mathrm{s}_\mathrm{w} C</annotation></semantics></math> is the left half of an adjoint homotopical equivalence of homotopical categories.</li> <li>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">diag</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo>:</mo><mo stretchy="false">(</mo><msub><mi mathvariant="normal">c</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><msub><mi mathvariant="normal">s</mi> <mi mathvariant="normal">w</mi></msub><mi>C</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\operatorname{diag} Hom_C : (\mathrm{c}_\mathrm{w} C)^{op} \times \mathrm{s}_\mathrm{w} C \to sSet</annotation></semantics></math> admits a right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>.</li> <li>The induced functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">Ho</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><mo lspace="0em" rspace="thinmathspace">Ho</mo><mi>C</mi><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Ho</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">(\operatorname{Ho} C)^{op} \times \operatorname{Ho} C \to \operatorname{Ho} sSet</annotation></semantics></math> is the derived hom-space functor.</li> </ol> </div> <h3 id="EnrichedHomsCofToFib">For a simplicial model category</h3> <p>We describe here in more detail properties of <a class="existingWikiWord" href="/nlab/show/derived+hom-functors">derived hom-functors</a> (see there for more) in a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>.</p> <p>The crucial axiom used for this is the axiom of an <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model 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> which says that</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/copower">tensor operation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><mo>:</mo><mi>C</mi><mo>×</mo><mi>SSet</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \cdot : C \times SSet \to C </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a>;</p> </li> <li> <p>or equivalently that 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></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> a cofibration and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> a fibration the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(Y, A) \to C(X,A) \times_{C(X,B)} C(Y,B) </annotation></semantics></math></div> <p>is a fibration, which is acyclic if either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> is.</p> </li> </ul> <p>First of all the first statement directly implies that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\emptyset \in C</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \in C</annotation></semantics></math> any object, the simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C(\emptyset,A) = {*}</annotation></semantics></math> is the terminal simplicial set (see also <a href="powered+and+copowered+category#InTensoredCotensoredCategoryInitialObjectIsEnrichedInitial">this Prop.</a>): because for any simplicial set <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>SSet</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>∅</mi><mo>⋅</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>colim</mi> <mi>∅</mi></msub><mo>⋅</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} SSet(S,C(\emptyset, A)) & = Hom_C(\emptyset \cdot S, A) \\ & = Hom_C(colim_{\emptyset} \cdot S, A) \\ & = Hom_C(\emptyset, A) \\ &= {*} \end{aligned} \,, </annotation></semantics></math></div> <p>where we use that the <a class="existingWikiWord" href="/nlab/show/copower">tensor</a> <a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a> is required to respect <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s and that the empty colimit is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>. (All equality signs here denote <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, to distinguish them from weak equivalences.)</p> <p>Similarly one has for all <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> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C(X,{*}) = {*}</annotation></semantics></math>.</p> <p>Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model 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>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \in C</annotation></semantics></math> fibrant, the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We apply the <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> axiom to the cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\emptyset \to X</annotation></semantics></math> and the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">A \to {*}</annotation></semantics></math> to obtain a fibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(X,A) \to C(\emptyset, A) \times_{C(\emptyset,{*})} C(X,{*}) \,. </annotation></semantics></math></div> <p>The right hand is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the terminal simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>Δ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">{*} = \Delta^0</annotation></semantics></math> to itself, hence is itself the point. So we have a fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> C(X,A) \to {*}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,A)</annotation></semantics></math> is a fibrant object in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>, hence a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>. .</p> </div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model 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>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> cofibrant and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> a fibration, the morphism of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,f) : C(X,A) \to C(X,B)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> that is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is acyclic.</p> <p>Dually, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">i : X \to Y</annotation></semantics></math> a cofibration and <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> fibrant, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(i,A) : C(X,A) \to C(Y,A)</annotation></semantics></math> is a cofibration of simplicial sets.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This is as before. Explicity, consider the first case, the second one is the formal dual of that:</p> <p>We enter the enrichment axiom with the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\emptyset \to X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> and find for the required <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>∅</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo><msub><mo>×</mo> <mo>*</mo></msub><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(\emptyset,A) \times_{C(\emptyset, B)} C(X,B) = {*} \times_{*} C(X,B) = C(X,B) </annotation></semantics></math></div> <p>and hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,A) \to C(X,B)</annotation></semantics></math> is, indeed, a fibration, which is acyclic if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> is.</p> </div> <div class="num_proposition"> <h6 id="proposition_3">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>.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a cofibant object and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> f : A \stackrel{\simeq}{\to} B </annotation></semantics></math></div> <p>a weak equivalence between fibrant objects, the <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched</a> <a class="existingWikiWord" href="/nlab/show/hom-object">hom-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(X,f) : C(X,A) \to C(X,B) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> <p>Similarly, for <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> a fibrant object 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><mover><mo>→</mo><mo>≃</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">j : X \stackrel{\simeq}{\to} Y</annotation></semantics></math> a weak equivalence between cofibrant objects, the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(j,A) : C(X,A) \to C(Y,A) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The second case is formally dual to the first, so we restrict attention to the first one.</p> <p>By the above, the axioms of an <a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> ensure that the above statement is true when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is in addition a fibration. So we reduce the situation to that case.</p> <p>This is possible because both <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> are assumed to be fibrant. This allows to apply the <em>factorization lemma</em> that is described in great detail at <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>. By this lemma, for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> between fibrant objects there is a commutative diagram</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><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mi>f</mi></msub><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>fib</mi><mo>∩</mo><mi>W</mi></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{E}_f B \\ & {}^{\mathllap{\in fib \cap W}}\swarrow && \searrow^{\mathrlap{\in fib}} \\ A &&\stackrel{\simeq}{\to}&& B } </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is assumed a weak equivalence it follows by <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">2-out-of-3</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mi>f</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}_f B</annotation></semantics></math> is also a weak equivalence.</p> <p>Therefore by the above properties of simpliciall enriched categories we obtain a <a class="existingWikiWord" href="/nlab/show/span">span</a> of acyclic fibrations of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mi>f</mi></msub><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(X,A) \stackrel{\simeq}{\leftarrow} C(X, \mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B) \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> every weak equivalence of Kan complexes is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, hence there is a weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>E</mi></mstyle> <mi>f</mi></msub><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(X,A) \stackrel{\simeq}{\to} C(X,\mathbf{E}_f B) \stackrel{\simeq}{\to} C(X,B) </annotation></semantics></math></div> <p>that is homotopic to our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,f)</annotation></semantics></math>. Therefore this is also a weak equivalence.</p> </div> <h3 id="comparison">Comparison</h3> <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/model+category">model category</a>. We discuss how its simplicial function complexes from prop. <a class="maruku-ref" href="#SimplicialFunctionComplexes"></a> are related to the simplicial localization from def. <a class="maruku-ref" href="#ZigZagCategories"></a> and def. <a class="maruku-ref" href="#HammockLocalization"></a>.</p> <p>Suppose now that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Q : C \to C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement+functor">cofibrant replacement functor</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">R : C \to C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/fibrant+replacement+functor">fibrant replacement functor</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mo>•</mo></msup><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">(</mo><mi>cC</mi><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma^\bullet : C \to (cC)_c</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/cosimplicial+resolution+functor">cosimplicial resolution functor</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Λ</mi> <mo>•</mo></msub><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">(</mo><mi>sC</mi><msub><mo stretchy="false">)</mo> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\Lambda_\bullet : C \to (sC)_f</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simplicial+resolution+functor">simplicial resolution functor</a> in the <a class="existingWikiWord" href="/nlab/show/model+category">model 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>.</p> <div class="num_theorem" id="DKTheorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Dwyer–Kan)</strong></p> <p>There are natural weak equivalences between the following equivalent realizations of this <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a> <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</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><msub><mi>Mor</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msup><mi>Γ</mi> <mo>•</mo></msup><mi>X</mi><mo>,</mo><mi>R</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>diag</mi><msub><mi>Mor</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msup><mi>Γ</mi> <mo>•</mo></msup><mi>X</mi><mo>,</mo><msub><mi>Λ</mi> <mo>•</mo></msub><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>Mor</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>Q</mi><mi>X</mi><mo>,</mo><msub><mi>Λ</mi> <mo>•</mo></msub><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>hocolim</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>×</mo><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msub><msub><mi>Mor</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msup><mi>Γ</mi> <mi>p</mi></msup><mi>X</mi><mo>,</mo><msub><mi>Λ</mi> <mi>q</mi></msub><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>N</mi><msubsup><mi>wMor</mi> <mi>C</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mo>≃</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>Mor</mi> <mrow><msup><mi>L</mi> <mi>H</mi></msup><mi>C</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Mor_C(\Gamma^\bullet X, R Y) &\stackrel{\simeq}{\to}& diag Mor_C(\Gamma^\bullet X, \Lambda_\bullet Y) &\stackrel{\simeq}{\leftarrow}& Mor_C(Q X, \Lambda_\bullet Y) \\ && \uparrow^\simeq \\ && hocolim_{p,q \in \Delta^{op} \times \Delta^{op}} Mor_C(\Gamma^p X, \Lambda_q Y) \\ &&\downarrow^\simeq \\ &&N wMor_C^3(X,Y) \\ &&\downarrow^\simeq \\ &&Mor_{L^H C}(X,Y) } \,. </annotation></semantics></math></div></div> <p>The top row weak equivalences are those of prop. <a class="maruku-ref" href="#SimplicialFunctionComplexes"></a></p> <h3 id="InACategoryOfFibrantObjects">In a category of fibrant objects</h3> <p>There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>. This is described in (<a href="#Cisinski10">Cisinksi 10</a>) and (<a href="#NSS12">Nikolaus-Schreiber-Stevenson 12, section 3.6.2</a>).</p> <h2 id="properties">Properties</h2> <h3 id="SpacesOfEquivalences">Hom-spaces of equivalences</h3> <div class="num_theorem" id="DKTheorem"> <h6 id="theorem_2">Theorem</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> and <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> an object, the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Aut_W(X) \subset \mathbb{R}Hom(X,X) </annotation></semantics></math></div> <p>has the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the component on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>W</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C_W)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of weak equivalences:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Aut</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>W</mi></msub><msub><mo stretchy="false">)</mo> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} Aut_W(X) \simeq N(C_W)_X \,. </annotation></semantics></math></div> <p>The equivalence is given by a finite sequence of <a class="existingWikiWord" href="/nlab/show/zig-zags">zig-zags</a> and is natural with respect to <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> again to a fibrant cofibrant object.</p> </div> <p>This is <a href="#DK84">Dwyer-Kan 84, 2.3, 2.4</a>.</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>For <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> a model category, the simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>W</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C_W)</annotation></semantics></math> is a model for the <a class="existingWikiWord" href="/nlab/show/core">core</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> determined by <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> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>That core, like every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> is equivalent to the disjoint union over its connected components of the deloopings of the automorphism <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 of any representatives in each connected component.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-category">hom-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a>, <a class="existingWikiWord" href="/nlab/show/cocycle+space">cocycle space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+mapping+complex">simplicial mapping complex</a></p> </li> </ul> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/homology">homology</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^n,-]</annotation></semantics></math></td><td style="text-align: left;"><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><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,A]</annotation></semantics></math></td><td style="text-align: left;"><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><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coend">coend</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+tensor+product">derived tensor product</a> <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><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>For some original references by <a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a> and <a class="existingWikiWord" href="/nlab/show/Dan+Kan">Dan Kan</a> see <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>. For instance</p> <ul> <li id="DK84"><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Kan">Dan Kan</a>, <em>A classification theorem for diagrams of simplicial sets</em>, Topology 23 (1984), 139-155.</li> </ul> <p>On the derived <a class="existingWikiWord" href="/nlab/show/function+complexes">function complexes</a> in a <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+simplicial+presheaves">projective model structure on simplicial presheaves</a>:</p> <ul> <li><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 for diagrams of simplicial sets</em>, Indagationes Mathematicae (Proceedings) <strong>86</strong> 2 (1983) 139-147 [<a href="https://doi.org/10.1016/1385-7258(83)90051-3">doi:10.1016/1385-7258(83)90051-3</a>, <a href="https://core.ac.uk/download/pdf/82652265.pdf">pdf</a>]</li> </ul> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 1.2.2 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em>, Annals of Mathematics Studies 170, Princeton University Press 2009 (<a href="https://press.princeton.edu/titles/8957.html">pup:8957</a>, <a href="https://www.math.ias.edu/~lurie/papers/HTT.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dan+Dugger">Dan Dugger</a>, <a class="existingWikiWord" href="/nlab/show/David+Spivak">David Spivak</a>, <em>Mapping spaces in quasi-categories</em>, Algebraic & Geometric Topology <strong>11</strong> (2011) 263–325 [<a href="http://arxiv.org/abs/0911.0469">arXiv:0911.0469</a>, <a href="http://dx.doi.org/10.2140/agt.2011.11.263">doi:10.2140/agt.2011.11.263</a>]</p> </li> </ul> <p>The theory of <em>framings</em> is due to</p> <ul> <li id="DHK"><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/Dan+Kan">Dan Kan</a>, <em>Model categories and general abstract homotopy theory</em>, (1997) (<a href="http://www.mimuw.edu.pl/~jacho/literatura/ModelCategory/DHK_ModelCateogories1.pdf">pdf</a>)</li> </ul> <p>and in parallel section 5 of</p> <ul> <li id="Hovey"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Model categories</em> (<a href="http://math.unice.fr/~brunov/SecretPassage/Hovey-Model%20Categories.ps">ps</a>)</li> </ul> <p>and in sections 16, 17 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <em>Model categories and their localization</em> .</li> </ul> <p>A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Clark+Barwick">Clark Barwick</a>, <em>On (enriched) left Bousfield localization of model categories</em> (<a href="http://arxiv.org/abs/0708.2067">arXiv</a>)</li> </ul> <p>Discussion of derived hom spaces for <a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a> is in</p> <ul> <li id="Cisinski10"><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <em>Invariance de la K-théorie par equivalences dérivées</em>, J. K-theory, 6 (2010), 505–546.</li> </ul> <p>and section 3.6.2 of</p> <ul> <li id="NSS12"><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">Principal ∞-bundles – Presentations</a></em> (<a href="http://arxiv.org/abs/1207.0249">arXiv:1207.0249</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 7, 2024 at 14:11:10. See the <a href="/nlab/history/%28infinity%2C1%29-categorical+hom-space" 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/%28infinity%2C1%29-categorical+hom-space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11510/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/%28infinity%2C1%29-categorical+hom-space/53" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/%28infinity%2C1%29-categorical+hom-space" accesskey="S" class="navlink" id="history" rel="nofollow">History (53 revisions)</a> <a href="/nlab/show/%28infinity%2C1%29-categorical+hom-space/cite" style="color: black">Cite</a> <a href="/nlab/print/%28infinity%2C1%29-categorical+hom-space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%28infinity%2C1%29-categorical+hom-space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>