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)-category 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"> <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> nLab (infinity,1)-category </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <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/643/#Item_30" 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> </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="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> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> Background <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> </li> </ul> Basic concepts <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a> </li> <li> <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 </li> <li> <a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a> </li> <li> <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> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> Universal constructions <ul> <li> <a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a> </li> </ul> Local presentation <ul> <li> <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a> </li> </ul> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a> </li> </ul> Extra stuff, structure, properties <ul> <li> <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> </li> </ul> Models <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derivator">derivator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <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> 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>… 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>, … see also <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> Introductions <ul> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a> </li> </ul> Definitions <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <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> Paths and cylinders <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a> </li> <li> <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> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <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> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> Basic facts <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> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> </ul> </div></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> <h4 id="topos_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>-topos theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> <h2 id="sidebar_background">Background</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a> </li> <li> <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> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> (2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a> <ul> <li> <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>/<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">of a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups </li> <li> <a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a> </li> </ul> </li> </ul> <div> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </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='#definitions'>Definitions</a></li> <ul> <li><a href='#quasicategories'>Quasi-categories</a></li> <li><a href='#__and_simplicially_enriched_categories'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Kan</annotation></semantics></math>- and simplicially enriched categories</a></li> <li><a href='#homotopical_categories'>Homotopical categories</a></li> <li><a href='#model_categories'>Model categories</a></li> <li><a href='#segal_categories_and_complete_segal_spaces'>Segal categories and complete Segal spaces</a></li> <li><a href='#categories'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#category_theory_2'><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</a></li> <li><a href='#the_collection_of_all_categories'>The collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</a></li> <li><a href='#model_category_presentations'>Model category presentations</a></li> </ul> <li><a href='#a_modelindependent_approach'>A model-independent approach</a></li> <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='#LectureNotes'>Surveys and lecture notes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> According to the general pattern on <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is a (weak) <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a> in which all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-morphisms for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a>. This is the joint generalization of the notion of <a class="existingWikiWord" href="/nlab/show/category">category</a> and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>. More precisely, this is the notion of <a class="existingWikiWord" href="/nlab/show/category">category</a> up to <a class="existingWikiWord" href="/nlab/show/coherence">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>: an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is equivalently <ul> <li> an <a class="existingWikiWord" href="/nlab/show/internal+category+in+an+%28%E2%88%9E%2C1%29-category">internal category</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>/basic <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (as such usually modeled as a <a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a>). </li> <li> a category <a class="existingWikiWord" href="/nlab/show/enriched+%28infinity%2C1%29-category">homotopy enriched</a> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (as such usually modeled as a <a class="existingWikiWord" href="/nlab/show/Segal+category">Segal category</a>). </li> </ul> Among all <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-categories</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 are special in that they are the simplest structures that at the same time: <ul> <li> admit a <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher version</a> of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> (<a class="existingWikiWord" href="/nlab/show/limit">limit</a>s, <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>s, <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a>, etc, <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a>, etc.) : <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">(infinity,1)-category theory</a> </li> <li> and know everything about higher <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a>. </li> </ul> Notably for understanding the collections of all <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-categories</a> for arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>, which in general is an <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><mn>1</mn><mo>,</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1,r+1)</annotation></semantics></math>-category, the knowledge of the underlying <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,1)</annotation></semantics></math>- (and hence <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 already captures much of the information of interest: it allows to decide if two given <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 are equivalent and allows to obtain new <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 from existing ones by universal constructions. The collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories forms the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a>. <h2 id="definitions">Definitions</h2> There are a number of different ways to make the idea of an <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 precise, including <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicially</a> <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, <a class="existingWikiWord" href="/nlab/show/topologically+enriched+categories">topologically enriched categories</a>, <a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a>, <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity+categories">categories</a> (most of which can be done either simplicially or topologically). Additionally, any notion of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a> can be specialized to a notion 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>-category by simply requiring all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells for <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\gt 1</annotation></semantics></math> to be invertible. Unlike the case for general notions 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>-category, almost all the definitions 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>-category are known to form <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> that are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a>. See also <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> for a summary of the state of the art about definitions 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>-category and comparisons between them. <h3 id="quasicategories">Quasi-categories</h3> We start with the definition 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>-category” that was promoted by <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> as a good model for the theory. This goes back to Boardman-Vogt in the 1970s and was further developed, by <a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a> and <a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a> in the early 1980s. This is a <a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> which conceives an <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 as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> with extra <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">property</a>. It is a straightforward generalization of the definition of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> as a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, and, in fact, one alternative term used early on was ‘weak Kan complex’; see below. Recall that a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> is a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> in which every <a class="existingWikiWord" href="/nlab/show/horn">horn</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^k[n]</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \leq k \leq n</annotation></semantics></math> has a filler. This condition may be read in words as: every collection of adjacent <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>-cells has a composite <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>-cell, even if the orientations of the cells don’t match. This implicitly encodes the invertibility of every cell: if the orientation does not match, we can invert the cell and then compose. From this perspective one observes, by looking closely at the combinatorics, that the invertibility of the 1-cells in the simplicial set is enforced particularly by the condition that the <a class="existingWikiWord" href="/nlab/show/horn">outer horns</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mn>0</mn></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^0[n]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>n</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^n[n]</annotation></semantics></math> have fillers. Therefore in a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> in which only the inner horns <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda^k[n]</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">0 \lt k \lt n</annotation></semantics></math> have fillers all cells are required to have a kind of inverse, except the 1-cells. (They may have inverses, too, but are not required to). This is evidently a realization of the idea of an <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n = \infty</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r = 1</annotation></semantics></math>. Such a simplicial set with fillers for all inner horns <ul> <li> Boardman and Vogt called a weak Kan complex ; </li> <li> <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> called a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a> called an <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. </li> </ul> Here we follow Joyal and say <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> when we mean concretely the simplicial sets with extra property. We use the more general term “<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” for this or any of its equivalent models, discussed below, in order to distinguish from the term <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a> or <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a> that is more traditionally understood to generically mean an <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 with no conditions on invertibility (in terms of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a>: an <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>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,\infty)</annotation></semantics></math>-category). With <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> being just <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> with extra property, there are evident and simple definitions of <ul> <li> the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors">quasi-category of (∞,1)-functors</a> between two <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</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> 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>; </li> <li> the quasi-category of all <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories">quasi-category of all quasi-categories</a>. </li> </ul> Similarly, <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> and <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a> have shown that all other constructions in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> have good generalizations to quasi-categories, which usually have conceptually simple formulations: see <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a> for more. <h3 id="__and_simplicially_enriched_categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>-, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Kan</annotation></semantics></math>- and simplicially enriched categories</h3> Despite the conceptual simplicity of <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a>, for computations and in particular for obtaining examples, it is often useful to pass to a slightly different model. Recall that we said at the beginning that an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is supposed to be like an <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> which is enriched over the category of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>. This turns out to make sense literally if one takes care to remember that <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 themselves form a higher category. As discussed at <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> there is a Quillen equivalence of the <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> of <ul> <li> the <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">standard model structure</a> on the <a class="existingWikiWord" href="/nlab/show/nice+category+of+spaces">nice category</a> of compactly generated weakly Hausdorff <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">standard model structure</a> on the category of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a>. </li> </ul> In fact, this is also equivalent to <ul> <li>the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">standard model structure</a> on the category <a class="existingWikiWord" href="/nlab/show/SimpSet">SSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>.</li> </ul> If we take the notion of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> to be the most manifest incarnation of the idea “<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>”, then under these equivalences one may think of <ul> <li> a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> as representing the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> which is obtained from it by “freely throwing in the missing inverses” of cells (technically: as representing its fibrant replacement); </li> <li> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> as representing 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>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X)</annotation></semantics></math>, whose <ul> <li> 0-cells are the points of <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>; </li> <li> 1-cells are the paths in <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>; </li> <li> 2-cells are the triangles in <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>; </li> <li> etc. </li> </ul> </li> </ul> With this interpretation understood (i.e. with these model structures understood), <a class="existingWikiWord" href="/nlab/show/SimpSet">SSet</a>-enriched categories do model <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. For more see <ul> <li><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation between quasi-categories and simplicial categories</a></li> </ul> <h3 id="homotopical_categories">Homotopical categories</h3> A <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> is 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> equipped with a class <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> of <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">weak equivalences</a>. Every homotopical category <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> has a quasi-localisation <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><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C[W(-1)]</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>. 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>W</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C[W(-1)]</annotation></semantics></math> is obtained from the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</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> by freely gluing a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> inverse to each <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> 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>, and then, by adding simplices to turn it into a quasi-category (this last step is called a fibrant completion). The <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-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><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C[W(-1)]</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/Dwyer-Kan+localisation">Dwyer-Kan localisation</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> with respect to <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>, via the equivalence between <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> and <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicial categories</a> mentioned above. Conversely, every quasi-category is equivalent to the quasi-localisation of a homotopical category. This gives a representation of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories in terms of <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical categories</a>. It follows that many aspects of the theory 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 can be expressed in terms of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. When the homotopical category (C,W) is obtained from a Quillen model structure (by forgetting the cofibrations and the fibrations) the quasi-category C[W^(-1)] has finite limits and colimits. Conversely, I conjecture that every quasi-category with finite limits and colimits is equivalent to the quasi-localisation of a model category. In fact, every locally presentable quasi-category is a quasi-localisation of a combinatorial model by a result of Lurie. More can be said: the underlying category can taken to be a category of presheaves by a result of Daniel Dugger. http://arxiv.org/abs/math/0007070 <h3 id="model_categories">Model categories</h3> A specific notion of <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> is that of a <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><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 obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite <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>-<a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and <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>-<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s. The <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-categories</a> are precisely those presented this way by <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model categeories</a>. At the very beginning, a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> was understood as a “model for the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of topological spaces,” or more precisely <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>: some <a class="existingWikiWord" href="/nlab/show/category">category</a> with extra <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure and properties</a> which allows one to perform all operations familiar of the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. As mentioned above, from the point of view of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>, <a class="existingWikiWord" href="/nlab/show/Top">Top</a> may naturally be regarded an as <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> and is in fact the archetypical example, analogous to how <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is the archetypical example of an ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a>. This indicates that, more generally, a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> should actually be a means to model (i.e. encode) in 1-categorical terms an <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, and of course this is true since indeed any <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> presents an <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 via Dwyer-Kan simplicial localization. In the case of a model category, however, or at least a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>, this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category has a different, simpler construction. <ul> <li> 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><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math> is, in particular, a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a>. </li> <li> the full <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{A}^\circ</annotation></semantics></math> on the fibrant-cofibrant objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math> happens to be <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent 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><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mathbf{A}^\circ)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{A}^\circ</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> presented by <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>. </li> </ul> Up to equivalence, this gives the same <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 as the Dwyer-Kan hammock localization. With the relation between <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> and <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> via <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> understood, we shall here often not distinguish between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{A}^\circ</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>A</mi></mstyle> <mo>∘</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mathbf{A}^\circ)</annotation></semantics></math> as 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 <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presented</a> by a <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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. <h3 id="segal_categories_and_complete_segal_spaces">Segal categories and complete Segal spaces</h3> Other models 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 are <ul> <li> <a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a>. </li> </ul> Segal categories can be thought of as categories which are weakly enriched in topological spaces/simplicial sets/Kan complexes, where the definition of “weak” makes use of the notion of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> or <a class="existingWikiWord" href="/nlab/show/SimpSet">SSet</a>. Complete Segal spaces are like <a class="existingWikiWord" href="/nlab/show/internal+categories+in+an+%28%E2%88%9E%2C1%29-category">internal categories in an (∞,1)-category</a>. This construction principle in particular lends itself to iteration and hence to an inductive definition of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> via <a class="existingWikiWord" href="/nlab/show/Segal+n-categories">Segal n-categories</a> and <a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+spaces">n-fold complete Segal spaces</a>. <h3 id="categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</h3> An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity+category">category</a> can also be thought of as a category “weakly enriched” in spaces (i.e. <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), except that in contrast to the Segal approaches the “weakness” is specified <a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraically</a> and parametrized by an <a class="existingWikiWord" href="/nlab/show/operad">operad</a>. This approach can be generalized to the <a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble</a> definition 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>-category or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-category. <h2 id="properties">Properties</h2> <h3 id="category_theory_2"><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</h3> A crucial point about the notion 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>-category is that it supports all the standard constructions and theorems of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, if only the consistent replacements are made (<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> becomes <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>, etc.). See <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>. <h3 id="the_collection_of_all_categories">The collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</h3> The collection of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories forms an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> called <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a>. Often it is useful to regard that as a (large) <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 itself, by discarding the non-invertible <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>. <h3 id="model_category_presentations">Model category presentations</h3> There is a wealth of different presentations 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. See <a class="existingWikiWord" href="/nlab/show/table+-+models+for+%28%E2%88%9E%2C1%29-categories">table - models for (∞,1)-categories</a>. <h2 id="a_modelindependent_approach">A model-independent approach</h2> In practice, it can be useful to be able to treat all “presentations 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” on the same equal footing (e.g. relative categories and topologically-enriched categories). While truly model-independent foundations 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>-category theory do not (yet) exist, this can be accomplished within any model 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, which we proceed to describe. As quasicategories are by far the most well-developed, we use them as an ambient framework. We also take care to make as few choices (even “contractible” ones) as possible. However, we do not explicitly mention set-theoretic issues, though these are easily handled using Grothendieck universes. <ol> <li> Consider the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Kan</mi></mrow><annotation encoding="application/x-tex">Kan</annotation></semantics></math>-enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>QCat</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{QCat}</annotation></semantics></math> of quasicategories; for quasicategories <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>, the Kan complex of morphisms between them is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>hom</mi><mo>̲</mo></munder> <munder><mi>QCat</mi><mo>̲</mo></munder></msub><mo>=</mo><mi>ι</mi><mo stretchy="false">(</mo><msub><munder><mi>hom</mi><mo>̲</mo></munder> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{hom}_{\underline{QCat}} = \iota(\underline{hom}_{sSet}(C,D))</annotation></semantics></math>, the largest Kan complex contained in their internal hom simplicial set. </li> <li> Define a relative quasicategory to be a quasicategory equipped with a wide sub-quasicategory of “weak equivalences” containing all equivalences. For relative quasicategories <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><msub><mi>W</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,W_C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><msub><mi>W</mi> <mi>D</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D,W_D)</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>hom</mi><mo>̲</mo></munder> <munder><mi>RelQCat</mi><mo>̲</mo></munder></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><msub><mi>W</mi> <mi>C</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><msub><mi>W</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><msub><munder><mi>hom</mi><mo>̲</mo></munder> <munder><mi>QCat</mi><mo>̲</mo></munder></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{hom}_{\underline{RelQCat}}((C,W_C),(D,W_D)) \subset \underline{hom}_{\underline{QCat}}(C,D)</annotation></semantics></math> for the sub-Kan complex consisting of those maps which take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">W_C</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">W_D</annotation></semantics></math>. Note that using this definition, this is actually the inclusion of a disjoint union of connected components among Kan complexes (in the strictest possible sense). </li> <li> There is an evident inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo>:</mo><munder><mi>QCat</mi><mo>̲</mo></munder><mo>→</mo><munder><mi>RelQCat</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">min : \underline{QCat} \to \underline{RelQCat}</annotation></semantics></math>, which takes a quasicategory <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 the relative quasicategory <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><msup><mi>C</mi> <mo>≃</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,C^\simeq)</annotation></semantics></math>. </li> <li> Although a Quillen equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub><mo>⇄</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_1 \rightleftarrows M_2</annotation></semantics></math> between model categories determines an equivalence of homotopy categories, note that neither adjoint functor need preserve weak equivalences. On the other hand, the restrictions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>M</mi> <mn>1</mn> <mi>c</mi></msubsup><mo>↪</mo><msub><mi>M</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_1^c \hookrightarrow M_1 \rightarrow M_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub><mo>←</mo><msub><mi>M</mi> <mn>2</mn></msub><mo>↩</mo><msubsup><mi>M</mi> <mn>2</mn> <mi>f</mi></msubsup></mrow><annotation encoding="application/x-tex">M_1 \leftarrow M_2 \hookleftarrow M_2^f</annotation></semantics></math> (to the cofibrant objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math> and the fibrant objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math>) do preserve weak equivalences, and these determine a hexagonal diagram of weak equivalences between relative categories (in the Barwick–Kan model structure), as in <a href="http://nyjm.albany.edu/j/2016/22-4.html">MazelGee16, Figure 1</a>. </li> <li> Using the previous observation, expand the diagram in the introduction of <a href="http://arxiv.org/abs/1112.0040">BarwickSchommerPries</a> (relating a great many Quillen equivalent model categories presenting “the homotopy theory 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”) into a diagram of weak equivalences between relative categories. As relative categories are particular examples of relative quasicategories, this defines a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>K</mi><mo>→</mo><munder><mi>RelQCat</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">F : K \to \underline{RelQCat}</annotation></semantics></math> among fibrant objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Cat</mi> <mi>sSet</mi></msub><msub><mo stretchy="false">)</mo> <mi>Bergner</mi></msub></mrow><annotation encoding="application/x-tex">(Cat_{sSet})_{Bergner}</annotation></semantics></math>. </li> <li> Now, apply the right Quillen equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mi>hc</mi></msup><mo>:</mo><mo stretchy="false">(</mo><msub><mi>Cat</mi> <mi>sSet</mi></msub><msub><mo stretchy="false">)</mo> <mi>Bergner</mi></msub><mo>→</mo><msub><mi>sSet</mi> <mi>Joyal</mi></msub></mrow><annotation encoding="application/x-tex">N^{hc} : (Cat_{sSet})_{Bergner} \to sSet_{Joyal}</annotation></semantics></math> (the homotopy-coherent nerve) to this cospan <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>QCat</mi><mo>̲</mo></munder><mover><mo>→</mo><mi>min</mi></mover><munder><mi>RelQCat</mi><mo>̲</mo></munder><mo>←</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\underline{QCat} \xrightarrow{min} \underline{RelQCat} \leftarrow K</annotation></semantics></math>. </li> <li> The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><mi>min</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^{hc}(min)</annotation></semantics></math> of quasicategories admits a contractible Kan complex worth of quasicategorical left adjoints, any of which presents the localization of relative quasicategories. Choose one, and denote this quasicategorical adjunction by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>RelQCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>⇄</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>QCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>:</mo><mi>min</mi></mrow><annotation encoding="application/x-tex">L : N^{hc} (\underline{RelQCat}) \rightleftarrows N^{hc} ( \underline{QCat}) : min</annotation></semantics></math>. </li> <li> It follows from the main theorem of <a href="http://arxiv.org/abs/math/0409598">Toen</a> that the composite map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>RelQCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo><mo>→</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>QCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \circ N^{hc}(F) : N(K) \cong N^{hc}(K) \to N^{hc}(\underline{RelQCat}) \to N^{hc}(\underline{QCat})</annotation></semantics></math> is “essentially contractible” in the quasicategorical sense. More precisely, for any cofibration into an acyclic object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo>′</mo><mo>≈</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">i : N(K) \to K' \approx pt</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Joyal</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Joyal}</annotation></semantics></math>, there exists a contractible Kan complex worth of extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \circ N^{hc}(F)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>. </li> <li> Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>The</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cats</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>QCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(The(\infty,1)Cats) \subset N^{hc}(\underline{QCat})</annotation></semantics></math> to be the maximal sub-Kan complex generated by the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \circ N^{hc}(F)</annotation></semantics></math>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>∈</mo><mo stretchy="false">(</mo><mi>The</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cats</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cat_{(\infty,1)} \in (The(\infty,1)Cats)</annotation></semantics></math> for any vertex, and propose that to work “model-independently” is to work within <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Cat_{(\infty,1)}</annotation></semantics></math>. </li> </ol> This sequence of maneuvers balances twin aims. On the one hand, Toen’s theorem asserts that after choosing a basepoint, this Kan complex is a model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(\mathbb{Z}/2)</annotation></semantics></math>. Thus, any sort of object which might be considered as “a presentation of an <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” canonically determines an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Cat_{(\infty,1)}</annotation></semantics></math> (where “canonical” must still be taken in the quasicategorical sense). On the other hand, it is completely independent of which vertex of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>The</mi><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cats</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(The(\infty,1)Cats)</annotation></semantics></math> we choose. <a href="http://ncatlab.org/nlab/files/relcats-modelcats-qcats-inftycats.jpg">This diagram</a>, taking place in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mi>hc</mi></msup><mo stretchy="false">(</mo><munder><mi>QCat</mi><mo>̲</mo></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N^{hc}(\underline{QCat})</annotation></semantics></math>, elaborates on certain salient aspects of the passage from models 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 to a model-independent approach. (For a small amount of explanation of this diagram, see <a href="https://nforum.ncatlab.org/discussion/7029/a-diagram-relating-different-models-of-inftycategories/#Item_0">here</a>.) <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/table+-+models+for+%28%E2%88%9E%2C1%29-categories">table - models for (∞,1)-categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/0-category">0-category</a>, <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/3-category">3-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-category">(n,1)-category</a> </li> <li> (∞,1)-category, <a class="existingWikiWord" href="/nlab/show/internal+%28%E2%88%9E%2C1%29-category">internal (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+%28%E2%88%9E%2C1%29-category">locally cartesian closed (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiadditive+%28%E2%88%9E%2C1%29-category">semiadditive (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/additive+%28%E2%88%9E%2C1%29-category">additive (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/compactly+generated+%28%E2%88%9E%2C1%29-category">compactly generated (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/disjunctive+%28%E2%88%9E%2C1%29-category">disjunctive (∞,1)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28%E2%88%9E%2C1%29-category">model (∞,1)-category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> For several years <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> – who was one of the first to promote the idea that for studying <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> it is good to first study <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 in terms of <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> – has been preparing a textbook on the subject. This still doesn’t quite exist, but an extensive write-up of lecture notes does: <ul> <li id="Joyal08"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/The+Theory+of+Quasi-Categories+and+its+Applications">The Theory of Quasi-Categories and its Applications</a>, lectures at: <a href="https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html">Advanced Course on Simplicial Methods in Higher Categories</a>, Quadern 45 2, Centre de Recerca Matemàtica, Barcelona 2008 ( <a class="existingWikiWord" href="/nlab/files/JoyalTheoryOfQuasiCategories.pdf" title="pdf">pdf</a>)</li> </ul> Further notes (where the term “<a class="existingWikiWord" href="/nlab/show/logos">logos</a>” is used instead of <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>): <ul> <li id="Joyal08"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, Notes on Logoi, 2008 (<a href="http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalOnLogoi2008.pdf" title="pdf">pdf</a>)</li> </ul> Meanwhile <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory 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 in terms of the models <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> and <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> is <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a> .</li> </ul> An brief exposition from the point of view of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, What is… an <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?, Notices of the AMS, September 2008 (<a href="http://www.ams.org/notices/200808/tx080800949p.pdf">pdf</a>)</li> </ul> A useful comparison of the four <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/Segal+categories">Segal categories</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a>. </li> </ul> is in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Julie+Bergner">Julie Bergner</a>, 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, In: <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> (eds.), <a class="existingWikiWord" href="/nlab/show/Towards+Higher+Categories">Towards Higher Categories</a>, The IMA Volumes in Mathematics and its Applications, vol 152, Springer 2007 (<a href="http://arxiv.org/abs/math/0610239">arXiv:math/0610239</a>, <a href="https://doi.org/10.1007/978-1-4419-1524-5_2">doi:10.1007/978-1-4419-1524-5_2</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Julia+Bergner">Julia Bergner</a>, Equivalence of models for equivariant <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, Glasgow Mathematical Journal, Volume 59, Issue 1 (2016) (<a href="https://arxiv.org/abs/1408.0038">arXiv:1408.0038</a>, <a href="https://doi.org/10.1017/S0017089516000136">doi:10.1017/S0017089516000136</a>) </li> </ul> More discussion of the other two models can be found at <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a></li> </ul> and in the references listed at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a>. The relation between <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> and <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> was discussed in detail in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Dan+Dugger">Dan Dugger</a>, <a class="existingWikiWord" href="/nlab/show/David+Spivak">David Spivak</a>, Rigidification of quasi-categories (<a href="http://arxiv.org/abs/0910.0814">arXiv:0910.0814</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Dan+Dugger">Dan Dugger</a>, <a class="existingWikiWord" href="/nlab/show/David+Spivak">David Spivak</a>, Mapping spaces in quasi-categories, Algebraic & Geometric Topology 11 (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>) </li> </ul> The 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 by <a class="existingWikiWord" href="/nlab/show/homotopical+categories">homotopical categories</a> and <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> is discussed in <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Kan">Daniel Kan</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <a class="existingWikiWord" href="/nlab/show/Homotopy+Limit+Functors+on+Model+Categories+and+Homotopical+Categories">Homotopy Limit Functors on Model Categories and Homotopical Categories</a> , volume 113 of Mathematical Surveys and Monographs</li> </ul> A model by <a class="existingWikiWord" href="/nlab/show/stratified+spaces">stratified spaces</a> is in <ul> <li id="AyalaFrancisRozenblyum15"><a class="existingWikiWord" href="/nlab/show/David+Ayala">David Ayala</a>, <a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Rozenblyum">Nick Rozenblyum</a>, A stratified homotopy hypothesis (<a href="http://arxiv.org/abs/1502.01713">arXiv:1502.01713</a>)</li> </ul> A more model-independent abstract formulation is discussed in <ul> <li id="RiehlVerity16"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, Infinity category theory from scratch, Higher Structures 4 1 (2020) [<a href="https://arxiv.org/abs/1608.05314">arXiv:1608.05314</a>, <a href="http://www.math.jhu.edu/~eriehl/scratch.pdf">pdf</a>, <a href="https://www.epfl.ch/labs/hessbellwald-lab/seminar/ytm2015/">lectures</a>] </li> <li id="RiehlVerity22"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, <a class="existingWikiWord" href="/nlab/show/Elements+of+%E2%88%9E-Category+Theory">Elements of ∞-Category Theory</a>, Cambridge studies in advanced mathematics 194, Cambridge University Press (2022) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1017/9781108936880">doi:10.1017/9781108936880</a>, ISBN:978-1-108-83798-9, <a href="https://emilyriehl.github.io/files/elements.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> For discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> see <a class="existingWikiWord" href="/nlab/show/internal+category+in+homotopy+type+theory">internal category in homotopy type theory</a> and see <ul> <li id="RiehlShulman17"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, A type theory for synthetic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories (<a href="https://arxiv.org/abs/1705.07442">arXiv:1705.07442</a>) </li> <li id="Riehl18"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories, talk at <a href="http://www.math.ias.edu/vvmc2018">Vladimir Voevodsky Memorial Conference 2018</a> (<a href="http://www.math.jhu.edu/~eriehl/Voevodsky.pdf">pdf</a>) </li> </ul> <h3 id="LectureNotes">Surveys and lecture notes</h3> An introduction to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> through <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: <ul> <li>Omar Antolín Camarena, A whirlwind tour of the world 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, 2013 (<a href="http://arxiv.org/abs/1303.4669">arXiv:1303.4669</a>)</li> </ul> Elementary exposition with an eye towards <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>: <ul> <li id="Riehl22"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <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 theory for undergraduates, <a href="Center+for+Quantum+and+Topological+Systems#RiehlDec2022">talk</a> at <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Dec. 2022) [<a href="Center+for+Quantum+and+Topological+Systems#RiehlDec2022">web</a>, video: <a href="https://www.youtube.com/watch?v=7g2rkiFsbXo">YT</a>] </li> <li id="Riehl23"> <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a href="https://www.ams.org/journals/notices/202305/noti2692/noti2692.html?adat=May%202023&trk=2692&galt=feature&cat=feature&pdfissue=202305&pdffile=rnoti-p727.pdf">Could <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 theory be taught to undergraduates?</a>, Notices of the AMS (May 2023) [<a href="https://www.ams.org/journals/notices/202305/rnoti-p727.pdf">published pdf</a>, <a href="https://arxiv.org/abs/2302.07855">arxiv:2302.07855</a>] </li> </ul> A foundational set of lecture notes: <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Categories+and+Homotopical+Algebra">Higher Categories and Homotopical Algebra</a>, Cambridge University Press 2019 (<a href="https://doi.org/10.1017/9781108588737">doi:10.1017/9781108588737</a>, <a href="http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf">pdf</a>)</li> </ul> A survey with an eye towards <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Moritz+Groth">Moritz Groth</a>, A short course on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories (<a href="https://arxiv.org/pdf/1007.2925">pdf</a>)</li> </ul> A survey on various notions of <a class="existingWikiWord" href="/nlab/show/homotopical+categories">homotopical categories</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, Homotopical categories: from model categories to (∞,1)-categories, in: <a class="existingWikiWord" href="/nlab/show/Andrew+J.+Blumberg">Andrew J. Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/Teena+Gerhardt">Teena Gerhardt</a>, <a class="existingWikiWord" href="/nlab/show/Michael+A.+Hill">Michael A. Hill</a> (eds,) <a class="existingWikiWord" href="/nlab/show/Stable+categories+and+structured+ring+spectra">Stable categories and structured ring spectra</a>, MSRI Book Series, Cambridge University Press (<a href="https://arxiv.org/abs/1904.00886">arXiv:1904.00886</a>)</li> </ul> Also: <ul> <li><a class="existingWikiWord" href="/nlab/show/Markus+Land">Markus Land</a>, Introduction to Infinity-Categories, Birkhäuser 2021 (<a href="https://link.springer.com/book/10.1007/978-3-030-61524-6">doi:10.1007/978-3-030-61524-6</a>)</li> </ul> Lecture notes: <ul> <li><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></li> </ul> See also <ul> <li><a class="existingWikiWord" href="/nlab/show/Zhen+Lin+Low">Zhen Lin Low</a>, <a class="existingWikiWord" href="/nlab/show/Notes+on+homotopical+algebra">Notes on homotopical algebra</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on March 17, 2025 at 23:50:30. See the <a href="/nlab/history/%28infinity%2C1%29-category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/%28infinity%2C1%29-category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/643/#Item_30">Discuss</a><a href="/nlab/revision/%28infinity%2C1%29-category/103" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/%28infinity%2C1%29-category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/%28infinity%2C1%29-category" accesskey="S" class="navlink" id="history" rel="nofollow">History (103 revisions)</a> <a href="/nlab/show/%28infinity%2C1%29-category/cite" style="color: black">Cite</a> <a href="/nlab/print/%28infinity%2C1%29-category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/%28infinity%2C1%29-category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>