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> category of classes 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> category of classes </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/12048/#Item_18" 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></title></head> <body> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#relations_and_maps'>Relations and maps</a></li> <li><a href='#diagrams'>Diagrams</a></li> <li><a href='#limits_and_colimits_as_adjoint_functors'>Limits and colimits as adjoint functors</a></li> <li><a href='#limits'>Limits</a></li> <li><a href='#colimits'>Colimits</a></li> <li><a href='#other_categorical_properties'>Other categorical properties</a></li> <li><a href='#category_with_class_structure'>Category with class structure</a></li> <li><a href='#related_notions'>Related notions</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> has <a class="existingWikiWord" href="/nlab/show/classes">classes</a> as objects and maps of classes as morphisms.</p> <p>In the <a class="existingWikiWord" href="/nlab/show/Zermelo-Fraenkel+set+theory">Zermelo-Fraenkel set theory</a>, a <a class="existingWikiWord" href="/nlab/show/class">class</a> is a <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> with a designated <a class="existingWikiWord" href="/nlab/show/free+variable">free variable</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>. We interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x)</annotation></semantics></math> as saying that <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> belongs to the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>. We also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x\in P</annotation></semantics></math>, while understanding that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is not a set, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is.</p> <p>For example, any <a class="existingWikiWord" href="/nlab/show/set">set</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> is a <a class="existingWikiWord" href="/nlab/show/class">class</a>, whose proposition is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a\in A</annotation></semantics></math> and the designated <a class="existingWikiWord" href="/nlab/show/free+variable">free variable</a> is <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> <p>The proposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a=a</annotation></semantics></math> defines the <a class="existingWikiWord" href="/nlab/show/class">class</a> of all sets, which does not arise via the above construction from any <a class="existingWikiWord" href="/nlab/show/set">set</a>.</p> <h2 id="relations_and_maps">Relations and maps</h2> <p>Given two classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, we can form their <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math> as the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x)</annotation></semantics></math> means <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x=(a,b)</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a\in A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b\in B</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> denotes the usual <a class="existingWikiWord" href="/nlab/show/ordered+pair">ordered pair</a> constructed in <a class="existingWikiWord" href="/nlab/show/Zermelo-Fraenkel+set+theory">Zermelo-Fraenkel set theory</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(a,b)=\{\{a\},\{a,b\}\}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mi>∅</mi><mo>,</mo><mi>a</mi><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><mi>b</mi><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(a,b)=\{\{\emptyset,a\},\{b\}\}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mo stretchy="false">{</mo><mi>∅</mi><mo stretchy="false">}</mo><mo>,</mo><mi>a</mi><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><mi>b</mi><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(a,b)=\{\{\{\emptyset\},a\},\{b\}\}</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x)</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(x)</annotation></semantics></math>, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is a subclass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/relation">relation</a> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a subclass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/map">map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a relation <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> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> such that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a\in A</annotation></semantics></math> there is a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b\in B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(a,b)</annotation></semantics></math>. In this case we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math> for this unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f(a)=b</annotation></semantics></math> means <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(a,b)</annotation></semantics></math>.</p> <p><a class="existingWikiWord" href="/nlab/show/maps">Maps</a> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> can be composed in the usual manner, which produces a <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>Here a category is understood in the sense of a <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a> with two sorts (objects and morphisms), but at no point we attempt to consider all <a class="existingWikiWord" href="/nlab/show/classes">classes</a> as a single unified whole.</p> <p>A <a class="existingWikiWord" href="/nlab/show/family">family</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> indexed by a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/map">map</a> of classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/class">class</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">{</mo><mi>i</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f^*\{i\}</annotation></semantics></math>. Such families can be pulled back along maps of classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">J\to I</annotation></semantics></math> and pushed forward along maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I\to J</annotation></semantics></math>.</p> <p>We can now define <a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> as having a <a class="existingWikiWord" href="/nlab/show/class">class</a> of objects, a <a class="existingWikiWord" href="/nlab/show/class">class</a> of morphisms, together with <a class="existingWikiWord" href="/nlab/show/composition">composition</a> and <a class="existingWikiWord" href="/nlab/show/identities">identities</a> satisfying the usual axioms. The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> considered above is not a <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> in this sense.</p> <p>We do not require <a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> to be <a class="existingWikiWord" href="/nlab/show/locally+small">locally small</a>.</p> <h2 id="diagrams">Diagrams</h2> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/large+category">large category</a>. An <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>-indexed <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of classes is defined as follows. First, we have an <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>-indexed family of classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math>. Secondly, we have a transition map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>t</mi><mo>∈</mo><mi>T</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>dom</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>→</mo><mi>T</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">tr\colon\{(t,h)\mid t\in T, h\in Mor(I), f(t)=dom(h)\}\to T,</annotation></semantics></math></div> <p>which is a map of classes such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>tr</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>codom</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(tr(t,h))=codom(h)</annotation></semantics></math>. (The <a class="existingWikiWord" href="/nlab/show/domain">domain</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tr</mi></mrow><annotation encoding="application/x-tex">tr</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/class">class</a> because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/sets">sets</a>.) Finally, the transition map satisfies the usual axioms expected from a functor.</p> <p>For any <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> <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> and a class <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> we can define the <a class="existingWikiWord" href="/nlab/show/constant+diagram">constant diagram</a> indexed by <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> with value <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>. We take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mi>I</mi><mo>×</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T=I\times C</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/projection+map">projection map</a>. The transition map sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">(t,h)\mapsto t</annotation></semantics></math>.</p> <h2 id="limits_and_colimits_as_adjoint_functors">Limits and colimits as adjoint functors</h2> <p>For any <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> <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> we have a <a class="existingWikiWord" href="/nlab/show/constant+diagram">constant diagram</a> functor that sends a <a class="existingWikiWord" href="/nlab/show/class">class</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> to the <a class="existingWikiWord" href="/nlab/show/constant+diagram">constant diagram</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>We can talk about left and right <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> to this functor in the sense of a <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a> with two sorts (objects and morphisms), augmented with symbols for the functor, its adjoint, together with <a class="existingWikiWord" href="/nlab/show/unit">unit</a> and <a class="existingWikiWord" href="/nlab/show/counit">counit</a> maps that satisfy the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a>.</p> <h2 id="limits">Limits</h2> <p>The category of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> admits all <a class="existingWikiWord" href="/nlab/show/small+limits">small limits</a>.</p> <p>First, it admits <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>: the equalizer of <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 lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f,g\colon A\to B</annotation></semantics></math> is the subclass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of <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> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>∈</mo><mi>A</mi><mo>∧</mo><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(e)=(e\in A \wedge f(e)=g(e))</annotation></semantics></math>.</p> <p>Secondly, it admits <a class="existingWikiWord" href="/nlab/show/small+products">small products</a>: the <a class="existingWikiWord" href="/nlab/show/small+product">small product</a> of an <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>-indexed family of classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is an arbitrary <a class="existingWikiWord" href="/nlab/show/set">set</a> (considered as a <a class="existingWikiWord" href="/nlab/show/class">class</a> when used with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>) can be constructed as the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">p\in P</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <span class="newWikiWord">map of sets<a href="/nlab/new/map+of+sets">?</a></span> whose <a class="existingWikiWord" href="/nlab/show/domain">domain</a> is <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> and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">p(i)\in T</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">f(p(i))=i</annotation></semantics></math>. (Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is indeed a class.)</p> <p>Finally, it admits all <a class="existingWikiWord" href="/nlab/show/small+limits">small limits</a> because the usual reduction of <a class="existingWikiWord" href="/nlab/show/small+limits">small limits</a> to <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> of <a class="existingWikiWord" href="/nlab/show/small+products">small products</a> continues to work provided that we adhere to the above convention on the definition of families of classes.</p> <h2 id="colimits">Colimits</h2> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of classes admits all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> indexed by arbitrary <a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> <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>, i.e., large colimits.</p> <p>First, the standard reduction of <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>-indexed <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> to a <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of a pair of arrows between <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(I)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(I)</annotation></semantics></math> still works in this context since class-indexed families of classes can be pulled back along source and target maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(I)\to Ob(I)</annotation></semantics></math>.</p> <p>Secondly, class-indexed <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> can be computed simply by taking the total class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of the corresponding class-indexed family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math> of classes.</p> <p>Thirdly, <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> exist by <a class="existingWikiWord" href="/nlab/show/Scott%27s+trick">Scott's trick</a>. Observe that given a pair of arrows <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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g:X\to Y</annotation></semantics></math> between classes, we can define an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> by saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>~</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">y~y'</annotation></semantics></math> if there is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h:[0,n]\to Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">h(0)=y</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h(n)=y'</annotation></semantics></math> and for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i\in[0,n)</annotation></semantics></math> there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(i)=f(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(i+1)=g(x)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(i)=g(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(i+1)=f(x)</annotation></semantics></math>. The quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> by this equivalence relation exists by <a class="existingWikiWord" href="/nlab/show/Scott%27s+trick">Scott's trick</a> and is precisely the desired coequalizer.</p> <h2 id="other_categorical_properties">Other categorical properties</h2> <p>The category of classes is a <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a>: the obvious notion of <a class="existingWikiWord" href="/nlab/show/image+factorization">image factorization</a> is stable under <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>. It is also a <a class="existingWikiWord" href="/nlab/show/Barr-exact+category">Barr-exact category</a>: every <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <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> on a class <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 induced by the <a class="existingWikiWord" href="/nlab/show/quotient+map">quotient map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>C</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">C\to C/R</annotation></semantics></math>.</p> <p>It is also an <a class="existingWikiWord" href="/nlab/show/infinitary+extensive+category">infinitary extensive category</a>, where “infinitary” means “class-indexed”. Indeed, <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> of <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> injections along arbitrary maps of classes exist and class-indexed <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> are <a class="existingWikiWord" href="/nlab/show/disjoint+coproducts">disjoint</a> and stable under <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> <p>It is also <a class="existingWikiWord" href="/nlab/show/well-pointed+category">well-pointed</a>: for every two maps between classes <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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g:X\rightarrow Y</annotation></semantics></math> and every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = g(y)</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f = g</annotation></semantics></math>, and the category of classes is not the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>.</p> <p>It likewise has all objects corresponding to <a class="existingWikiWord" href="/nlab/show/large+cardinal">large cardinal</a>s, most notably a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. Otherwise, the category of finite sets <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> is vacuously a category of classes, as the notions of ‘finitary’ and ‘class-indexed’/‘infinitary’ coincide.</p> <p>As such, the category of classes is a well-pointed infinitary <a class="existingWikiWord" href="/nlab/show/Heyting+category">Heyting</a> or <a class="existingWikiWord" href="/nlab/show/Boolean+category">Boolean</a> <a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a>, depending upon the external logic used, with a natural numbers object and other large cardinals, and where “infinitary” is used in the rather strong sense of “class-indexed”.</p> <p>The category of classes is not <a class="existingWikiWord" href="/nlab/show/cartesian+closed">cartesian closed</a> or <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed">locally cartesian closed</a> and does not have <a class="existingWikiWord" href="/nlab/show/power+objects">power objects</a>. Indeed, the class of all sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> does not have a <a class="existingWikiWord" href="/nlab/show/power+object">power object</a>, or, equivalently, there is no <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(S,\{0,1\})</annotation></semantics></math>.</p> <h2 id="category_with_class_structure">Category with class structure</h2> <p>The category of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> is a primordial example of a <a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a>. Its open maps are precisely those maps of classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f\colon T\to I</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">{</mo><mi>i</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f^*\{i\}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/set">set</a> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math>. Small maps coincide with open maps. The powerclass of a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the class of all sets <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> such that <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> is a subclass 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>. The universal class is the class of all sets.</p> <h2 id="related_notions">Related notions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/class">class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+of+classes">type of classes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+category">large category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 22, 2023 at 14:50:18. See the <a href="/nlab/history/category+of+classes" 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/category+of+classes" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12048/#Item_18">Discuss</a><span class="backintime"><a href="/nlab/revision/category+of+classes/11" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/category+of+classes" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/category+of+classes" accesskey="S" class="navlink" id="history" rel="nofollow">History (11 revisions)</a> <a href="/nlab/show/category+of+classes/cite" style="color: black">Cite</a> <a href="/nlab/print/category+of+classes" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/category+of+classes" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>