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> complete Segal 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> complete Segal 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/3677/#Item_16" 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="internal_categories">Internal <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</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+object+in+an+%28%E2%88%9E%2C1%29-category">category object in an (∞,1)-category</a></strong>, <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">groupoid object</a></p> <h2 id="in_a_1category">In a 1-category</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/congruence">congruence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a>, <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a></p> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+functor">internal functor</a>, <a class="existingWikiWord" href="/nlab/show/internal+profunctor">internal profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+diagram">internal diagram</a>, <a class="existingWikiWord" href="/nlab/show/internal+presheaf">internal presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-fold+category">n-fold category</a>, <a class="existingWikiWord" href="/nlab/show/cat-n-group">cat-n-group</a></p> </li> </ul> <h2 id="in_a_">In a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-congruence">2-congruence</a></li> </ul> <h2 id="in_">In <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-Segal+space">semi-Segal space</a>, <a class="existingWikiWord" href="/nlab/show/Segal+space">Segal space</a>, <a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+Segal+space">reduced Segal space</a></p> </li> </ul> <h2 id="in__2">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><mo stretchy="false">(</mo><mi>Cat</mi><mo stretchy="false">(</mo><mi>⋯</mi><mi>Cat</mi><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cat(Cat(\cdots Cat(\infty Grpd)))</annotation></semantics></math></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Segal+space">higher Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">Theta space</a></p> </li> </ul> <h2 id="in__3">In <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a></h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-congruence">(n,r)-congruence</a></li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a></li> </ul> <h2 id="internal_category">Internal <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>-category</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/n-category+object+in+an+%28%E2%88%9E%2C1%29-category">n-category object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta_n+space">Theta_n space</a></p> </li> </ul> <h2 id="operadic_case">Operadic case</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">model structure for dendroidal complete Segal spaces</a></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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#CompleteSegalSpaces'>Complete Segal spaces</a></li> <li><a href='#CompleteSegalSpaceObjects'>Complete Segal space objects</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#CharacterizationOfCompleteness'>Characterization of Completeness</a></li> <li><a href='#ModelCategoryStructure'>Model category structure</a></li> <li><a href='#RelationToSimplicialLocalization'>Relation to simplicial localization</a></li> <li><a href='#model_categories_for_presheaves'>Model categories for presheaves</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#OrdinaryCategoriesAsCompleteSegalSpaces'>Ordinary categories as complete Segal spaces</a></li> <ul> <li><a href='#preliminaries'>Preliminaries</a></li> <li><a href='#the_construction'>The construction</a></li> <li><a href='#PropertiesOfTheInclusionOfCategories'>Properties of the inclusion</a></li> </ul> <li><a href='#RelativeCategoryAsCompleteSegalSpace'>Relative and Model categories as complete Segal spaces</a></li> <li><a href='#QuasiCategoriesAsCompleteSegal'>Quasi-categories as complete Segal spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesGroupoidalVersion'>Groupoidal version</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>complete Segal space</em> is a model for an <a class="existingWikiWord" href="/nlab/show/internal+category+in+an+%28%E2%88%9E%2C1%29-category">internal category in an (∞,1)-category</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, with the latter <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>/<a class="existingWikiWord" href="/nlab/show/Top">Top</a>. So complete Segal spaces present <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>. They are also called <em>Rezk categories</em> after <a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>.</p> <p>More in detail, a complete Segal space <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</p> <ul> <li> <p>for each <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> a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math>, thought of as the <em>space of composable sequences of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-morphisms and their composites</em>;</p> </li> <li> <p>forming a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> <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 <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> (a <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a>);</p> </li> </ul> <p>such that</p> <ol> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation well defined up to <a class="existingWikiWord" href="/nlab/show/coherence">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>: exibited by the <a class="existingWikiWord" href="/nlab/show/Segal+maps">Segal maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1 </annotation></semantics></math></div> <p>(into the iterated <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of 1-morphisms over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of objects) being <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a></p> <p>(so far this defines a <em><a class="existingWikiWord" href="/nlab/show/Segal+space">Segal space</a></em>);</p> </li> <li> <p>the notion of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> in <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 compatible with that in the ambient <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (“completeness”): the sub-simplicial object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Core(X_\bullet)</annotation></semantics></math> on the invertible morphisms in each degree is homotopy constant: it has all face and degeneracy maps being <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>.</p> <p>(this says that if a morphism is an equivalence under the explicit composition operation then it is already a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> ).</p> </li> </ol> <h2 id="Definition">Definition</h2> <p>We first discuss</p> <ul> <li><a href="#CompleteSegalSpaces">Complete Segal spaces</a></li> </ul> <p>as such, and then the more general notion of</p> <ul> <li><a href="#CompleteSegalSpaceObjects">Complete Segal space objects</a></li> </ul> <p>internal to a suitable model category/<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 <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">\mathcal{C}</annotation></semantics></math> – this reduces to the previous notion for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C} = sSet_{Quillen}</annotation></semantics></math>.</p> <h3 id="CompleteSegalSpaces">Complete Segal spaces</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/Segal+space">Segal space</a></strong> is a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> X \in [\Delta^{op}, sSet] </annotation></semantics></math></div> <p>such that</p> <ul> <li> <p>it 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><msub><mi>sSet</mi> <mi>Quillen</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, sSet_{Quillen}]_{Reedy}</annotation></semantics></math>;</p> </li> <li> <p>it is a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to the <a class="existingWikiWord" href="/nlab/show/spine">spine</a> inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Sp</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{Sp[n] \hookrightarrow \Delta[n]\}_{n \in \mathbb{N}}</annotation></semantics></math>;</p> <p>equivalently: for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Segal+map">Segal map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> X_n \to X_1 \times_{X_0} \cdots \times_{X_0} X_1 </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+complexes">Kan complexes</a>, in fact).</p> </li> </ul> </div> <p>(<a href="#Rezk">Rezk, 4.1</a>).</p> <div class="num_defn" id="HomotopyCategory"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a Segal space, its <strong><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are the vertices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</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>Obj</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> Obj(X) = (X_0)_0 </annotation></semantics></math></div> <p>and 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>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x,y \in Obj(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a> is the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ho(X)(x,y) := \pi_0 \Big(\{x\} \times_{X_0} X_1 \times_{X_0} \{y\}\Big) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composition</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ho</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Ho</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Ho</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Ho_X(x,y) \times Ho_X(y,z) \to Ho_X(x,z) </annotation></semantics></math></div> <p>is the (uniquely defined) action of the <a class="existingWikiWord" href="/nlab/show/infinity-anafunctor">infinity-anafunctor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub><munderover><mo>←</mo><mo>≃</mo><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></munderover><msub><mi>X</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow></mover><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> X_1 \times_{X_0} X_1 \underoverset{\simeq}{(d_2, d_0)}{\leftarrow} X_2 \stackrel{d_1}{\to} X_1 </annotation></semantics></math></div> <p>on these connected components.</p> </div> <p>(<a href="#Rezk">Rezk, 5.3</a>)</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a Segal space, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>hoequ</mi></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> X_{hoequ} \hookrightarrow X_1 </annotation></semantics></math></div> <p>for the inclusion of the connected components of those vertices that become <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> in the homotopy category, def. <a class="maruku-ref" href="#HomotopyCategory"></a>.</p> </div> <p>(<a href="#Rezk">Rezk, 5.7</a>)</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A Segal space <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 called a <strong>complete Segal space</strong> if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>0</mn></msub><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mi>hoequ</mi></msub></mrow><annotation encoding="application/x-tex"> s_0 : X_0 \to X_{hoequ} </annotation></semantics></math></div> <p>is a weak equivalence.</p> </div> <p>(<a href="#Rezk">Rezk, 6.</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This condition is equivalent to <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> being a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mover><mo>→</mo><mo>≃</mo></mover><mn>1</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">N(\{0 \stackrel{\simeq}{\to} 1\}) \to *</annotation></semantics></math>. This is discussed <a href="#CharacterizationOfCompleteness">below</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The completeness condition may also be thought of as <em><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a></em>. See there for more.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>There is a Segal completion functor given in (<a href="#Rezk">Rezk, 14.</a>).</p> </div> <h3 id="CompleteSegalSpaceObjects">Complete Segal space objects</h3> <p>(…)</p> <h2 id="properties">Properties</h2> <h3 id="CharacterizationOfCompleteness">Characterization of Completeness</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>A Segal space <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 complete Segal space precisely if it is a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mover><mo>→</mo><mo>≃</mo></mover><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">N(0 \stackrel{\simeq}{\to} 1) \to *</annotation></semantics></math>, hence precisely if with respect to the canonical <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> <a class="existingWikiWord" href="/nlab/show/hom+objects">hom objects</a> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mover><mo>→</mo><mo>≃</mo></mover><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X_0 \simeq [\Delta^{op}, sSet](*, X) \to [\Delta^{op}, sSet](N(0 \stackrel{\simeq}{\to} 1), X) </annotation></semantics></math></div> <p>is a weak equivalence.</p> </div> <p>(<a href="#Rezk">Rezk, theorem 6.2</a>)</p> <h3 id="ModelCategoryStructure">Model category structure</h3> <p>The category <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>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta^{op}, sSet]</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> (<a class="existingWikiWord" href="/nlab/show/bisimplicial+sets">bisimplicial sets</a>) supports a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure whose fibrant objects are precisely the complete Segal spaces: the <em><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a></em>. This presents the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a>.</p> <h3 id="RelationToSimplicialLocalization">Relation to simplicial localization</h3> <p>Given <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">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒲</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{W} \subset Mor(\mathcal{C})</annotation></semantics></math> or more generally a “<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>”, then there is canonically a complete Segal space associated with it by the “relative nerve” construction, def. <a class="maruku-ref" href="#RelativeNerve"></a> followed by fibrant replacement, and hence by the <a href="#ModelCategoryStructure">model structure</a> this determines 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><msub><mi>N</mi> <mi>Rezk</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_{Rezk}(\mathcal{C},\mathcal{W})</annotation></semantics></math> .</p> <p>On the other hand classical <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>-theory provides several ways (e.g. <a class="existingWikiWord" href="/nlab/show/hammock+localization">hammock localization</a>) to turn <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">(\mathcal{C}, \mathcal{W})</annotation></semantics></math> into 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><mi>𝒞</mi><mo stretchy="false">[</mo><mi>𝒲</mi><msup><mo stretchy="false">]</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}[\mathcal{W}]^{-1}</annotation></semantics></math> which universally turns the elements 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">\mathcal{W}</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>These constructions are compatible in that there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>Rezk</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒲</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>𝒲</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_{Rezk}(\mathcal{C},\mathcal{W}) \simeq \mathcal{C}[\mathcal{W}^{-1}] \,. </annotation></semantics></math></div></div> <p>For <a class="existingWikiWord" href="/nlab/show/simplicial+model+categories">simplicial model categories</a> this is (<a href="#Rezk">Rezk, theorem 8.3</a>. For general <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> this is (<a href="#Bergner07">Bergner 07, theorem 6.2</a>). For the fully general case this follows from results by <a class="existingWikiWord" href="/nlab/show/Clark+Barwick">Clark Barwick</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a> and <a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a> as pointed out by <a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a> <a href="http://mathoverflow.net/a/93139/381">here on MathOverflow</a>.</p> <p>In fact, the construction <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><mo>↦</mo><msub><mi>N</mi> <mi>Rezk</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\mathcal{W})\mapsto N_{Rezk}(\mathcal{C},\mathcal{W})</annotation></semantics></math> admits a direct generalization in the case where <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">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/quasicategory">quasicategory</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>Rezk</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒲</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_{Rezk}(\mathcal{C},\mathcal{W})</annotation></semantics></math> still presents the localization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">[</mo><msup><mi>𝒲</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}[\mathcal{W}^{-1}]</annotation></semantics></math>. This is (<a href="#MazelGee19">Mazel-Gee19, Theorem 3.8</a>). A generalization can be found in (<a href="#Arakawa23">Arakawa23, Theorem 1.7</a>).</p> <h3 id="model_categories_for_presheaves">Model categories for presheaves</h3> <p>There is a notion of <a class="existingWikiWord" href="/nlab/show/right%2Fleft+fibration+of+complete+Segal+spaces">right/left fibration of complete Segal spaces</a> analogous to <span class="newWikiWord">right/left Kan fibrations<a href="/nlab/new/right%2Fleft+Kan+fibrations">?</a></span> for <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>.</p> <h2 id="Examples">Examples</h2> <p>We discuss some examples. For more and more basic examples see also at <em><a href="Segal+space#Examples">Segal space – Examples</a></em>.</p> <h3 id="OrdinaryCategoriesAsCompleteSegalSpaces">Ordinary categories as complete Segal spaces</h3> <p>We discuss how an ordinary <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> is naturally regarded as a complete Segal space. (<a href="#Rezk">Rezk, 3.5</a>)</p> <h4 id="preliminaries">Preliminaries</h4> <p>We need the following basic ingredients.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><msup><mi>Cat</mi> <mi>op</mi></msup><mo>×</mo><mi>Cat</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(-)^{(-)} : Cat^{op} \times Cat \to Cat</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, sending two categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>A</mi></msup><mo>=</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X^A = Func(A,X)</annotation></semantics></math>.</p> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> we have a canonical functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>↪</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> \Delta \hookrightarrow Cat </annotation></semantics></math></div> <p>including the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> into <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> by regarding the <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[n]</annotation></semantics></math> as the category generated from <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> consecutive morphisms.</p> <p>The <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> itself is then then functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><mi>Cat</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> N : Cat \to sSet </annotation></semantics></math></div> <p>to <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> sending a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mi>k</mi><mo>↦</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N(C) : k \mapsto C^{\Delta[k]} \,. </annotation></semantics></math></div> <p>Its restriction along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo>↪</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Grpd \hookrightarrow Cat</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> lands in <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KanCplx</mi><mo>↪</mo></mrow><annotation encoding="application/x-tex">KanCplx \hookrightarrow </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/core">core</a> operation is the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo>:</mo><mi>Cat</mi><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> Core : Cat \to Grpd </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to the inclusion of <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> into <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>. It sends a category to the groupoid obtained by discarding all non-invertible morphisms.</p> <h4 id="the_construction">The construction</h4> <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/small+category">small category</a>. Define</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>C</mi></mstyle><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{C} \in [\Delta^{op}, sSet] </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mi>k</mi></msub><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Core</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{C}_k := N(Core(C^{\Delta[k]})) \,. </annotation></semantics></math></div> <p>In degree 0 this is the the <a class="existingWikiWord" href="/nlab/show/core">core</a> 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> itself. In degree 1 it is the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{C}_1</annotation></semantics></math> underlying the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> 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>.</p> <p>One sees that the <a class="existingWikiWord" href="/nlab/show/source">source</a> and <a class="existingWikiWord" href="/nlab/show/target">target</a> functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>:</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">s, t : C^{\Delta[1]} \to C</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isofibrations">isofibrations</a> and hence their image under <a class="existingWikiWord" href="/nlab/show/core">core</a> and <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> are <a class="existingWikiWord" href="/nlab/show/Kan+fibrations">Kan fibrations</a>. Therefore it follows that the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> (see there) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>0</mn></msub></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>0</mn></msub></mrow></msub><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1</annotation></semantics></math> is given already be the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in the <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>. Using this, it is immediate that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> the functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Core</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mi>Core</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Core(C^{\Delta[k]}) \to Core(C^{\Delta[1]}) \times_{Core(C)} \cdots \times_{Core(C)} Core(C^{\Delta[1]}) </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, and so in particular</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mi>k</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>0</mn></msub></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>0</mn></msub></mrow></msub><msub><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \mathbf{C}_k \to \mathbf{C}_1 \times_{\mathbf{C}_0} \cdots \times_{\mathbf{C}_0} \mathbf{C}_1 </annotation></semantics></math></div> <p>is an equivalence.</p> <p>It is clear that the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation in the complete Segal space defined this way “is” the composition 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>. In particular the morphisms that are invertible under this composition are precisely those that are already invertible 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>. Therefore we have the core simplicial object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Core</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo><mo>:</mo><mi>k</mi><mo>↦</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><msup><mo stretchy="false">)</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Core(\mathbf{C}) : k \mapsto N(Core(C)^{\Delta[k]}) = N(Core(C))^{\Delta[k]} \,, </annotation></semantics></math></div> <p>where, note, now we <em>first</em> take the core 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 then form morphism categories.</p> <p>This simplicial Kan complex has in each positive degree a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> for the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>Core</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(Core(C))</annotation></semantics></math>.</p> <p>Notably (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[k]</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalent</a> to the point) it follows that indeed all the face and degeneracy maps are weak homotopy equivalences.</p> <p>So for every 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>, the simplicial object <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> constructed as above is a complete Segal space. This construction extends to a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi><mo>→</mo><mi>completeSegalSpace</mi></mrow><annotation encoding="application/x-tex">Cat \to completeSegalSpace</annotation></semantics></math> and this is homotopy full and faithful.</p> <h4 id="PropertiesOfTheInclusionOfCategories">Properties of the inclusion</h4> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo>:</mo><mi>Cat</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Sing_J : Cat \to [\Delta^{op}, sSet] </annotation></semantics></math></div> <p>for the functor just defined</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> two categories, there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>×</mo><mi>D</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sing_J(C \times D) \simeq Sing_J(C) \times Sing_J(D) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>C</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>Sing</mi> <mi>J</mi></msub><mi>D</mi><msup><mo stretchy="false">)</mo> <mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mi>C</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sing_J(D^C) \simeq (Sing_J D)^{Sing_J C} \,. </annotation></semantics></math></div> <p>A functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f : C \to D</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing_J(f)</annotation></semantics></math> is an equivalence 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>sSet</mi><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, sSet]_{Reedy}</annotation></semantics></math> (hence is degreewise a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> of Kan complexes).</p> </div> <p>This appears as (<a href="#Rezk">Rezk, theorem 3.7</a>).</p> <h3 id="RelativeCategoryAsCompleteSegalSpace">Relative and Model categories as complete Segal spaces</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 category with a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Mor(C)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">weak equivalences</a>. 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> could be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> or (much) more generally a “<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>”. Then the <a href="#OrdinaryCategoriesAsCompleteSegalSpaces">above</a> construction has the following evident variant.</p> <div class="num_prop" id="RelativeNerve"> <h6 id="definition_6">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">N(C,W) \in [\Delta^{op}, sSet]</annotation></semantics></math> be given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>:</mo><mi>n</mi><mo>↦</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>Core</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> N(C,W) : n \mapsto N(Core_W(C^{\Delta[n]})) \,, </annotation></semantics></math></div> <p>where now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Core</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Core_W(-)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> on those <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> whose components are <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The typical <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> is not a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> with respect to the base choice of <a class="existingWikiWord" href="/nlab/show/universe">universe</a>. In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C,W)</annotation></semantics></math> will be a “<a class="existingWikiWord" href="/nlab/show/large+set">large</a>” bisimplicial set. In other words, one needs to employ some <a class="existingWikiWord" href="/nlab/show/universe+enlargement">universe enlargement</a> to interpret this definition.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>If <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 model category, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Core</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Core_W(C^{\Delta[n]})</annotation></semantics></math> is the subcategory of weak equivalences in any of the standard <a class="existingWikiWord" href="/nlab/show/model+structures+on+functors">model structures on functors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C^{\Delta[n]}</annotation></semantics></math>. By a <a href="/nlab/show/%28infinity%2C1%29-categorical+hom-space#SpacesOfEquivalences">classical fact</a> discssed at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></em>, its <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> is a model for the <a class="existingWikiWord" href="/nlab/show/core">core</a> of the corresponding <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> </div> <p>The bisimplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C,W)</annotation></semantics></math> is not, in general, a complete Segal space. It does, however, represent the same <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> as the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> 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> at <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>; see <a href="http://mathoverflow.net/questions/92916/does-the-classification-diagram-localize-a-category-with-weak-equivalences/93139">this MO question</a>.</p> <p>We can, of course, always reflect <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C,W)</annotation></semantics></math> into a complete Segal space by passing to a <a class="existingWikiWord" href="/nlab/show/fibrant+replacement">fibrant replacement</a> in the <a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a>. But something better is true here: it suffices to make a <em><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy</a> fibrant</em> replacement (which does not change the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the simplicial sets <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>Core</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(Core_W(C^{\Delta[n]}))</annotation></semantics></math>, but only “arranges them more nicely”).</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy fibrant replacement</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(C,W)</annotation></semantics></math> is a complete Segal space.</p> </div> <p>This is (<a href="#Rezk">Rezk, theorem 8.3</a>).</p> <h3 id="QuasiCategoriesAsCompleteSegal">Quasi-categories as complete Segal spaces</h3> <p>The formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>↦</mo><mi>k</mi><mo>↦</mo><mi>Core</mi><mo stretchy="false">(</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} \mapsto k \mapsto Core(\mathcal{C}^{\Delta[k]})</annotation></semantics></math> also defines a relative functor from quasi-categories to complete segal spaces, which has a one-sided inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><msub><mi>X</mi> <mrow><mo>•</mo><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X \mapsto X_{\bullet,0}</annotation></semantics></math>. This is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> appearing in <a href="#JoyalTierney07">Joyal & Tierney (2007), proposition 4.10</a>.</p> <p>However, to get a right Quillen functor, we need to use a different model of Core.</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>J</mi></msub><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> \Delta_J : \Delta \to sSet </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of the <a class="existingWikiWord" href="/nlab/show/codiscrete+groupoid">codiscrete groupoid</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>J</mi></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mover><mo>→</mo><mo>≃</mo></mover><mi>⋯</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_J[n] = N(0 \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\to} n) \,. </annotation></semantics></math></div> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Sing_J : sSet \to [\Delta^{op}, sSet] </annotation></semantics></math></div> <p>for the functor given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><msub><mi>Δ</mi> <mi>J</mi></msub><mo stretchy="false">[</mo><mo>•</mo><mo stretchy="false">]</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sing_J(X)_n = Hom_{sSet}(\Delta[n] \times \Delta_J[\bullet], X) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_3">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>sSet</mi></mrow><annotation encoding="application/x-tex">X \in sSet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>/<a class="existingWikiWord" href="/nlab/show/inner+Kan+complex">inner Kan complex</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing_J(X)</annotation></semantics></math> is a complete Segal space.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">model structure for dendroidal complete Segal spaces</a></em> the section <em><a href="model+structure+for+dendroidal+complete+Segal+spaces#QuasiOperadsToDendroidalCompleteSegal">Quasi-operads to dendroidal complete Segal spaces</a></em></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-Segal+space">semi-Segal space</a>, <a class="existingWikiWord" href="/nlab/show/Segal+space">Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">model structure for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+complete+Segal+space">higher complete Segal space</a>, <a class="existingWikiWord" href="/nlab/show/dendroidal+complete+Segal+space">dendroidal complete Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rezk+completion">Rezk completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/table+-+models+for+%28infinity%2C1%29-operads">table - models for (infinity,1)-operads</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Complete Segal spaces were originally defined in</p> <ul> <li id="Rezk"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>A model for the homotopy theory of homotopy theory</em> , Trans. Amer. Math. Soc., 353(3), 973-1007 (<a href="https://arxiv.org/abs/math/9811037">arXiv:math/9811037</a>)</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a> is discussed in</p> <ul> <li id="JoyalTierney07"><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>Quasi-categories vs. Segal spaces</em>, in <em>Categories in Algebra, Geometry and Mathematical Physics</em>, Contemporary Mathematics <strong>431</strong> (2007) [<a href="http://arxiv.org/abs/math/0607820">arXiv:math/0607820</a>, <a href="https://doi.org/10.1090/conm/431">doi:10.1090/conm/431</a>]</li> </ul> <p>Further discussion of the relation to <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> is in</p> <ul> <li id="Bergner07"> <p><a class="existingWikiWord" href="/nlab/show/Julia+Bergner">Julia Bergner</a>, <em>Complete Segal spaces arising from simplicial categories</em> (<a href="http://arxiv.org/abs/0704.1624">arXiv:0704.1624</a>)</p> </li> <li id="MazelGee19"> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Mazel-Gee">Aaron Mazel-Gee</a>, <em>The universality of the Rezk nerve</em>, Algebr. Geom. Topol., 19(7), 2019, 3217–3260 (<a href="https://arxiv.org/abs/1510.03150">arXiv:math/1510.03150</a>, <a href="https://doi.org/10.2140/agt.2019.19.3217">doi:10.2140/agt.2019.19.3217</a>)</p> </li> <li id="Arakawa23"> <p><a class="existingWikiWord" href="/nlab/show/Kensuke+Arakawa">Kensuke Arakawa</a>, <em>Classification Diagrams of Marked Simplicial Sets</em> (<a href="https://arxiv.org/abs/2311.01101">arXiv:math/2311.01101</a>).</p> </li> </ul> <p>A survey of the definition and its relation to equivalent definitions is in section 4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Julia+Bergner">Julia Bergner</a>, <em>A survey 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</em> (<a href="http://arxiv.org/abs/math.AT/0610239">arXiv</a>).</li> </ul> <p>See also pages 29 to 31 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a></em></li> </ul> <p>For literature on the variants and refinements see at <em><a class="existingWikiWord" href="/nlab/show/Theta+space">Theta space</a></em> and <em><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></em>.</p> <p>Related MathOverflow discussion includes</p> <ul> <li><a href="http://mathoverflow.net/q/92916/381">Does the classification diagram localize a category with weak equivalences?</a></li> </ul> <h3 id="ReferencesGroupoidalVersion">Groupoidal version</h3> <p>The groupoidal version of complete Segal spaces (that modelling just <a class="existingWikiWord" href="/nlab/show/groupoid+objects+in+an+%28%E2%88%9E%2C1%29-category">groupoid objects in an (∞,1)-category</a> instead of general <a class="existingWikiWord" href="/nlab/show/category+objects+in+an+%28%E2%88%9E%2C1%29-category">category objects in an (∞,1)-category</a>) is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Julia+Bergner">Julia Bergner</a>, <em>Adding inverses to diagrams encoding algebraic structures</em>, Homology, Homotopy Appl. 10(2), 2008, 149-174 (<a href="http://arxiv.org/abs/math/0610291">arXiv:math/0610291</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Julia+Bergner">Julia Bergner</a>, <em>Adding inverses to diagrams II: Invertible homotopy theories are spaces</em>, Homology, Homotopy and Applications, vol. 10(1) 2008 (<a href="http://www.math.ucr.edu/~jbergner/Groupoid.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 30, 2024 at 00:45:05. See the <a href="/nlab/history/complete+Segal+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/complete+Segal+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3677/#Item_16">Discuss</a><span class="backintime"><a href="/nlab/revision/complete+Segal+space/51" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/complete+Segal+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/complete+Segal+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (51 revisions)</a> <a href="/nlab/show/complete+Segal+space/cite" style="color: black">Cite</a> <a href="/nlab/print/complete+Segal+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/complete+Segal+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>