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> 2-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"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> 2-category </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1942/#Item_34" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="2category_theory">2-category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#strict_2categories'>Strict 2-categories</a></li> <li><a href='#Weak'>General 2-categories</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#double_nerve'>Double nerve</a></li> <li><a href='#ModelCategoryStructure'>Model category structure</a></li> <ul> <li><a href='#FreeResolution'>Free resolutions</a></li> </ul> </ul> <li><a href='#closed_structures_on_the_category_of_2categories'>Closed structures on the category of 2-categories</a></li> <ul> <li><a href='#references'>References</a></li> </ul> <li><a href='#2categorical_concepts'>2-categorical concepts</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of a <em>2-category</em> generalizes that of <a class="existingWikiWord" href="/nlab/show/category">category</a>: a 2-category is a <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category</a>, where on top of the objects and morphisms, there are also 2-morphisms. In old texts, strict 2-categories are occasionally called <em>hypercategories</em>.</p> <p>A 2-category consists of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a>;</p> </li> <li> <p>1-<a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between objects;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> between morphisms.</p> </li> </ul> <p>The morphisms can be <a class="existingWikiWord" href="/nlab/show/composition">composed</a> along the objects, while the 2-morphisms can be composed in two different directions: along objects – called <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> – and along morphisms – called <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a>. The composition of morphisms is allowed to be associative only up to <a class="existingWikiWord" href="/nlab/show/coherent">coherent</a> <a class="existingWikiWord" href="/nlab/show/associator">associator</a> 2-morphisms.</p> <p>2-categories are also a <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>: they are like monoidal categories with many objects.</p> <p>2-categories provide the context for discussing</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>s;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a>s.</p> </li> </ul> <p>The concept of 2-category generalizes further in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> to <a class="existingWikiWord" href="/nlab/show/n-categories">n-categories</a>, which have <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>s for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k\le n</annotation></semantics></math>.</p> <p>2-categories form a <a class="existingWikiWord" href="/nlab/show/3-category">3-category</a>, <a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a>.</p> <h2 id="definitions">Definitions</h2> <h3 id="strict_2categories">Strict 2-categories</h3> <p>The easiest definition of 2-category is that it is a category <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over the <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>. Thus it has a collection of objects, and for each pair of objects a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(x,y)</annotation></semantics></math>. The objects of these hom-categories are the morphisms, and the morphisms of these hom-categories are the 2-morphisms. This produces the classical notion of <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>.</p> <h3 id="Weak">General 2-categories</h3> <p>For some purposes, strict 2-categories are too strict: one would like to allow composition of morphisms to be associative and unital only up to coherent invertible 2-morphisms. A direct generalization of the above “enriched” definition produces the classical notion of <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>.</p> <p>One can also obtain notions of 2-category by specialization from the case of higher categories. Specifically, if we fix a meaning of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-category">category</a>, however weak or strict we wish, then we can define a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category</strong> to be 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 such that every 3-morphism is an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>, and all parallel pairs of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-morphisms are equivalent for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">j \geq 3</annotation></semantics></math>. It follows that, up to equivalence, there is no point in mentioning anything beyond <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-morphisms, except whether two given parallel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-morphisms are equivalent. In some models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories, it is possible to make this precise by demanding that all parallel pairs of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>-morphisms are actually <em>equal</em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">j\geq 3</annotation></semantics></math>, producing a simpler notion of 2-category in which we can speak about <a class="existingWikiWord" href="/nlab/show/equality">equality</a> of 2-morphisms instead of equivalence. (This is the case for both strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories and bicategories.)</p> <p>All of the above definitions produce “equivalent” theories of 2-category, although in some cases (such as the fact that every bicategory is equivalent to a strict 2-category) this requires some work to prove. On the nLab, we often use the word “2-category” in the general sense of referring to whatever model one may prefer, but usually one in which composition is weak; a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> is an adequate definition. One should beware, however, that in the literature it is common for “2-category” to refer only to <em>strict</em> 2-categories.</p> <p>A 2-category in which all 1-morphisms and 2-morphisms are invertible is a <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The archetypical 2-category is <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, the 2-category whose</p> <ul> <li> <p>objects are <a class="existingWikiWord" href="/nlab/show/categories">categories</a>;</p> </li> <li> <p>morphisms are <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s;</p> </li> <li> <p>2-morphisms are <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>;</p> <p>horizontal composition of 2-morphisms is the <a class="existingWikiWord" href="/nlab/show/Godement+product">Godement product</a>.</p> </li> </ul> <p>This happens to be a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a>.</p> </li> <li> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> any enriching category (such as a Benabou <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>), there is a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V Cat</annotation></semantics></math> whose</p> <ul> <li>objects are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>;</li> <li>morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched functors; and</li> <li>2-morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-natural transformations.</li> </ul> </li> <li> <p>On the other hand, for any such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> we also have a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Prof">Prof</a> whose</p> <ul> <li>objects are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>;</li> <li>morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a>s; and</li> <li>2-morphisms are natural transformations between these.</li> </ul> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a category with <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, then there is a bicategory <a class="existingWikiWord" href="/nlab/show/Span">Span</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C)</annotation></semantics></math> whose</p> <ul> <li>objects are the objects 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>;</li> <li>morphisms are <a class="existingWikiWord" href="/nlab/show/spans">spans</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>; and</li> <li>2-morphisms are morphisms of spans.</li> </ul> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> may be thought of as a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}C</annotation></semantics></math> (its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>). This has</p> <ul> <li> <p>a single object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math>;</p> </li> <li> <p>morphisms are the objects 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>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(\mathbf{B}C)_1 = C_0</annotation></semantics></math>;</p> </li> <li> <p>2-morphisms are the morphisms 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> : <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mo>=</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(\mathbf{B}C)_2 = C_1</annotation></semantics></math>;</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}C</annotation></semantics></math> is composition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>Conversely, every 2-category with a single object comes from a monoidal category this way, so the concepts are effectively equivalent. (Precisely: the 2-category of <em>pointed</em> 2-categories with a single object is equivalent to that of monoidal categories). For more on this relation see <a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>, <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a>, and <a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a>.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> is a 2-category. For instance</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, the double <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 A</annotation></semantics></math> is the strict 2-category with a single object, a single 1-morphisms, set of 2-moprhisms being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and both horizontal composition as well as <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a> being the product in <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>.</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>, its single <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> is a 2-groupoid with a single object.</p> </li> </ul> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> has a <a class="existingWikiWord" href="/nlab/show/path+2-groupoid">path 2-groupoid</a>.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> has a <strong>homotopy 2-category</strong>, obtained by dividing out all 3-morphisms and higher.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="double_nerve">Double nerve</h3> <p>An ordinary <a class="existingWikiWord" href="/nlab/show/category">category</a> has a <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> which is a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>. For 2-categories one may consider their <a class="existingWikiWord" href="/nlab/show/double+nerve">double nerve</a> which is a <a class="existingWikiWord" href="/nlab/show/bisimplicial+set">bisimplicial set</a>.</p> <p>There is also a 2-nerve. (<a href="#LackPaoli">LackPaoli</a>)</p> <p>(…)</p> <h3 id="ModelCategoryStructure">Model category structure</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on 2-categories – sometimes known as the <a class="existingWikiWord" href="/nlab/show/folk+model+structure">folk model structure</a> – that models the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> underlying <a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a> (<a href="#LackFolkModel">Lack</a>).</p> <p>For strict 2-categories this is the restriction of the corresponding <a class="existingWikiWord" href="/nlab/show/folk+model+structure">folk model structure</a> on <a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a>.</p> <ul> <li> <p>The weak equivalences are the <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a>s that are equivalences of 2-categories.</p> </li> <li> <p>The acyclic fibrations are the <a class="existingWikiWord" href="/nlab/show/k-surjective+functor">k-surjective functor</a>s for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </li> </ul> <h4 id="FreeResolution">Free resolutions</h4> <p><strong>Theorem</strong> A strict 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is cofibrant precisely if the underlying 1-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+category">free category</a>.</p> <p>This is theorem 4.8 in (<a href="#LackStrict">LackStrict</a>). This is a special case of the more general statement that free strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories are given by <a class="existingWikiWord" href="/nlab/show/computad">computad</a>s.</p> <p><strong>Example (free resolution of a 1-category).</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an ordinary category (a 1-category) regarded as a strict 2-category. Then the cofibrant resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">\hat C \stackrel{\simeq}{\to} C</annotation></semantics></math> is the strict 2-category given as follows:</p> <ul> <li> <p>the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> are those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>;</p> </li> <li> <p>the morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> are finite sequences of composable morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, and composition is concatenation of such sequences</p> <p>(hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>C</mi><mo stretchy="false">^</mo></mover><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(\hat C)_1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+category">free category</a> on the <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a> underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>);</p> </li> <li> <p>the 2-morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> are <em>generated</em> from 2-morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></msub></mrow><annotation encoding="application/x-tex">c_{f,g}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>c</mi> <mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></msub></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>f</mi></mrow></munder></mtd> <mtd></mtd> <mtd><mi>z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && y \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{c_{f,g}}& \searrow^{\mathrlap{g}} \\ x && \underset{g \circ_C f }{\to} && z } </annotation></semantics></math></div> <p>and their formal inverses</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo>⇑</mo> <mrow><msubsup><mi>c</mi> <mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>f</mi></mrow></munder></mtd> <mtd></mtd> <mtd><mi>z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && y \\ & {}^{\mathllap{f}}\nearrow &\Uparrow^{c_{f,g}^{-1}}& \searrow^{\mathrlap{g}} \\ x && \underset{g \circ_C f }{\to} && z } </annotation></semantics></math></div> <p>for all composable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in Mor(C)</annotation></semantics></math> with composite (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>!) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">g \circ_C f</annotation></semantics></math>;</p> <p>subject to the <em>relation</em> that for all composable triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g,h \in Mor(C)</annotation></semantics></math> the following equation of 2-morphisms holds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msup><mo>⇘</mo> <mrow><msub><mi>c</mi> <mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>c</mi> <mrow><mi>h</mi><mo>,</mo><mo stretchy="false">(</mo><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mi>h</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>f</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>w</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mrow><msub><mi>c</mi> <mrow><mi>g</mi><mo>,</mo><mi>h</mi></mrow></msub></mrow></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>⇓</mo> <mrow><msub><mi>c</mi> <mrow><mi>f</mi><mo>,</mo><mo stretchy="false">(</mo><mi>g</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>h</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>h</mi><msub><mo>∘</mo> <mi>C</mi></msub><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi></mrow></munder></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>w</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ y &\to& &\stackrel{g}{\to}& &\to& && z \\ \uparrow &\seArrow^{c_{f,g}}& && & \nearrow & && \downarrow \\ {}^{\mathllap{f}}\uparrow && & \nearrow & &&&& \downarrow^{\mathrlap{h}} \\ \uparrow & \nearrow & && &\Downarrow^{c_{h,(g\circ_C f)}}& && \downarrow \\ x &\to& &\underset{h \circ (g \circ_C f)}{\to}& &\to& &\to& w } \;\;\; = \;\;\; \array{ y &\to& &\stackrel{g}{\to}& &\to& && z \\ \uparrow &\searrow& && & & &\swArrow_{c_{g,h}}& \downarrow \\ {}^{\mathllap{f}}\uparrow && & \searrow & &&&& \downarrow^{\mathrlap{h}} \\ \uparrow &\Downarrow_{c_{f,(g \circ_C h)}}& && &\searrow& && \downarrow \\ x &\to& &\underset{( h \circ_C g) \circ f}{\to}& &\to& &\to& w } </annotation></semantics></math></div></li> </ul> <p><strong>Observation</strong> Let <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> be any strict 2-catgeory. Then a <a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C \to D</annotation></semantics></math> is the same as a strict 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\hat C \to D</annotation></semantics></math>.</p> <h2 id="closed_structures_on_the_category_of_2categories">Closed structures on the category of 2-categories</h2> <table><thead><tr><th>Category</th><th>Internal hom</th><th>Monoidal</th><th>Comments</th><th>Reference</th></tr></thead><tbody><tr><td style="text-align: left;">2-categories, strict functors</td><td style="text-align: left;">Strict functors, <a class="existingWikiWord" href="/nlab/show/2-natural+transformations">2-natural transformations</a>, <a class="existingWikiWord" href="/nlab/show/modifications">modifications</a></td><td style="text-align: left;">Yes</td><td style="text-align: left;">Cartesian</td><td style="text-align: left;">Follows from the theory of <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a></td></tr> <tr><td style="text-align: left;">2-categories, strict functors</td><td style="text-align: left;">Strict functors, <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformations">pseudonatural transformations</a>, modifications</td><td style="text-align: left;">Yes</td><td style="text-align: left;">Symmetric</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></td></tr> <tr><td style="text-align: left;">2-categories, strict functors</td><td style="text-align: left;">Strict functors, op/<a class="existingWikiWord" href="/nlab/show/lax+natural+transformations">lax natural transformations</a>, modifications</td><td style="text-align: left;">Yes</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></td></tr> </tbody></table> <h3 id="references">References</h3> <p>Related constructions appear in:</p> <ul> <li><span class="newWikiWord">Branko Nikolić<a href="/nlab/new/Branko+Nikoli%C4%87">?</a></span>, <em>Strictification tensor product of 2-categories</em>, <a href="https://arxiv.org/abs/1810.12213">arXiv:1810.12213</a> (2018).</li> <li>Peter F. Faul, Graham Manuell, and José Siqueira, <em>2-dimensional bifunctor theorems and distributive laws</em>, <a href="https://arxiv.org/abs/2010.07926">arXiv:2010.07926</a> (2020).</li> </ul> <h2 id="2categorical_concepts">2-categorical concepts</h2> <p><strong>constructions</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/opposite+2-category">opposite 2-category</a></li> </ul> <p><strong>extra properties</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/regular+2-category">regular 2-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/exact+2-category">exact 2-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/coherent+2-category">coherent 2-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/extensive+2-category">extensive 2-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-pretopos">2-pretopos</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></li> </ul> <p><strong>types of morphisms</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></li> <li><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></li> <li><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></li> <li><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></li> <li><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></li> </ul> <p><strong>specific versions</strong></p> <ul> <li>globular <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> </ul> <p><strong>limit notions</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/flexible+limit">flexible limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/PIE+limit">PIE limit</a></li> </ul> <p><strong>model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">canonical model structure</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-trivial+model+structure">2-trivial model structure</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><strong>2-category</strong></p> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+2-category">equivalence in a 2-category</a></p> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+2-category">localization of a 2-category</a></p> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/nlab/show/directed+type+theory">directed type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-category">3-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+category">double category</a></p> </li> </ul> <h2 id="References">References</h2> <p>Despite its being frequently attributed to Ehresmann, the notion of <a class="existingWikiWord" href="/nlab/show/strict+2-categories">strict 2-categories</a> is due to:</p> <ul> <li id="Bénabou65"> <p><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, Example (5) of: <em>Catégories relatives</em>, C. R. Acad. Sci. Paris <strong>260</strong> (1965) 3824-3827 [<a href="https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item">gallica</a>]</p> <blockquote> <p>(conceived as <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and called <em>2-categories</em>)</p> </blockquote> </li> <li id="Maranda65"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Maranda">Jean-Marie Maranda</a>, Def. 1 in: <em>Formal categories</em>, Canadian Journal of Mathematics <strong>17</strong> (1965) 758-801 [<a href="https://doi.org/10.4153/CJM-1965-076-0">doi:10.4153/CJM-1965-076-0</a>, <a href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf">pdf</a>]</p> <blockquote> <p>(conceived as <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and called <em>categories of the second type</em>)</p> </blockquote> </li> </ul> <p>both apparently following or inspired by the earlier definition of <em><a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a></em> due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Catégories double et catégories structurées</em>, C.R. Acad. Paris 256 (1963) 1198-1201 [<a class="existingWikiWord" href="/nlab/files/Ehresmann-CategoriesDoubles.pdf" title="pdf">pdf</a>, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246">gallica</a>]</li> </ul> <p>An early definition also appears in the following, where it is mistakenly attributed to <a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>:</p> <ul> <li id="EK65"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a>, <em>Closed Categories</em>, p. 425 in: <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">S. Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/D.+K.+Harrison">D. K. Harrison</a>, <a class="existingWikiWord" href="/nlab/show/S.+MacLane">S. MacLane</a>, <a class="existingWikiWord" href="/nlab/show/H.+R%C3%B6hrl">H. Röhrl</a> (eds.): <em><a class="existingWikiWord" href="/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965">Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) [<a href="https://doi.org/10.1007/978-3-642-99902-4">doi:10.1007/978-3-642-99902-4</a>]</p> <blockquote> <p>(expressed entirely in components, under the name <em>hypercategories</em>)</p> </blockquote> </li> </ul> <p>The fundamental structure of the <a class="existingWikiWord" href="/nlab/show/2-category+of+categories">2-category of categories</a> (namely the <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> and the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>) was first described in:</p> <ul> <li id="Godement58"><a class="existingWikiWord" href="/nlab/show/Roger+Godement">Roger Godement</a>, Appendix (pp. 269) of: <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) [<a href="https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement">webpage</a>, <a class="existingWikiWord" href="/nlab/files/Godement-TopologieAlgebrique.pdf" title="pdf">pdf</a>]</li> </ul> <p>Early discussion of the general notion of bicategories:</p> <ul> <li id="Bénabou1967"><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, <em>Introduction to Bicategories</em>, Lecture Notes in Mathematics <strong>47</strong> Springer (1967) 1-77 [<a href="http://dx.doi.org/10.1007/BFb0074299">doi:10.1007/BFb0074299</a>]</li> </ul> <p>Exposition and review:</p> <ul> <li id="KellyStreet74"> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Review of the elements of 2-categories</em>, Sydney Category Seminar 1972/1973, in <a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a> (ed.) Lecture Notes in Mathematics <strong>420</strong>, Springer (1974) [<a href="https://doi.org/10.1007/BFb0063101">doi:10.1007/BFb0063101</a>]</p> </li> <li id="Street96"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Categorical Structures</em>, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [<a href="https://doi.org/10.1016/S1570-7954(96)80019-2">doi:10.1016/S1570-7954(96)80019-2</a>, <a href="http://maths.mq.edu.au/~street/45.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Street-CategoricalStructures.pdf" title="pdf">pdf</a>, <a href="https://shop.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3">ISBN:978-0-444-82212-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Encyclopedia article on 2-categories and bicategories</em> (<a href="http://www.maths.mq.edu.au/~street/Encyclopedia.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Basic bicategories</em> (<a href="http://arxiv.org/abs/math/9810017">arXiv:9810017</a>)</p> </li> <li id="Lack10"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A 2-categories companion</em>, In: Baez J., May J. (eds.) <em><a class="existingWikiWord" href="/nlab/show/Towards+Higher+Categories">Towards Higher Categories</a></em>. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 (<a href="http://arxiv.org/abs/math.CT/0702535">arXiv:math.CT/0702535</a>, <a href="https://doi.org/10.1007/978-1-4419-1524-5_4">doi:10.1007/978-1-4419-1524-5_4</a>)</p> <blockquote> <p>(including discussion of (<a class="existingWikiWord" href="/nlab/show/strict+2-limits">strict</a>) <a class="existingWikiWord" href="/nlab/show/2-limits">2-limits</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>2-Categories</em>, BRICS Notes Series 1998 (<a href="http://www.brics.dk/NS/98/7/BRICS-NS-98-7.pdf">pdf</a>)</p> </li> </ul> <p>Comprehensive textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ofer+Gabber">Ofer Gabber</a>, <a class="existingWikiWord" href="/nlab/show/Lorenzo+Ramero">Lorenzo Ramero</a>, Chapter 2 of: <em>Foundations for almost ring theory</em> (<a href="https://arxiv.org/abs/math/0409584">arXiv:math/0409584</a>)</p> </li> <li id="JohnsonYau20"> <p><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, <a class="existingWikiWord" href="/nlab/show/Donald+Yau">Donald Yau</a>, <em>2-Dimensional Categories</em>, Oxford University Press (2021) [<a href="http://arxiv.org/abs/2002.06055">arXiv:2002.06055</a>, <a href="https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378">doi:10.1093/oso/9780198871378.001.0001</a>]</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/coherence+theorems">coherence theorems</a>:</p> <ul> <li id="Power89"><a class="existingWikiWord" href="/nlab/show/A.+John+Power">A. John Power</a>, <em>A general coherence result.</em> J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. <a href="http://dx.doi.org/10.1016/0022-4049%2889%2990113-8">doi:10.1016/0022-4049(89)90113-8</a> <a href="http://www.ams.org/mathscinet-getitem?mr=985657">MR0985657</a></li> </ul> <p>Relation between bicategories and Tamsamani weak 2-categories:</p> <ul> <li id="LackPaoli"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <a class="existingWikiWord" href="/nlab/show/Simona+Paoli">Simona Paoli</a>, <em>2-nerves for bicategories</em> (<a href="http://arxiv.org/abs/math/0607271">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simona+Paoli">Simona Paoli</a>, <em>From Tamsamani weak 2-categories to bicategories</em> (<a href="http://www.maths.mq.edu.au/~simonap/Bicategories_Rev_4.pdf">arXiv</a>)</p> </li> </ul> <p>There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on 2-categories – the <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">canonical model structure</a> – that models the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> underlying <a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a>:</p> <ul> <li id="LackStrict"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A Quillen Model Structure for 2-Categories</em>, K-Theory 26: 171–205, 2002. (<a href="http://www.maths.usyd.edu.au/u/stevel/papers/qmc2cat.html">website</a>)</p> </li> <li id="LackFolkModel"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A Quillen Model Structure for Biategories</em>, K-Theory 33: 185-197, 2004. (<a href="http://www.maths.usyd.edu.au/u/stevel/papers/qmcbicat.html">website</a>)</p> </li> </ul> <p>Discussion of weak 2-categories in the style of <a class="existingWikiWord" href="/nlab/show/A-infinity+categories">A-infinity categories</a> is (using <a class="existingWikiWord" href="/nlab/show/dendroidal+sets">dendroidal sets</a> to model the higher <a class="existingWikiWord" href="/nlab/show/operads">operads</a>) in</p> <ul> <li> <p>Andor Lucacs, <em>Dendroidal weak 2-categories</em> (<a href="http://de.arxiv.org/abs/1304.4278">arXiv:1304.4278</a>)</p> </li> <li> <p>Jonathan Chiche, <em>La théorie de l’homotopie des 2-catégories</em>, thesis, <a href="http://arxiv.org/abs/1411.6936">arXiv</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 8, 2024 at 20:44:36. See the <a href="/nlab/history/2-category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/2-category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1942/#Item_34">Discuss</a><span class="backintime"><a href="/nlab/revision/2-category/63" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/2-category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/2-category" accesskey="S" class="navlink" id="history" rel="nofollow">History (63 revisions)</a> <a href="/nlab/show/2-category/cite" style="color: black">Cite</a> <a href="/nlab/print/2-category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/2-category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>