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> point (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <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> point (changes) </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/diff/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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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"> <p class="show_diff"> Showing changes from revision #15 to #16: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <h1 id='points'>Points</h1> <div class='maruku_toc'><ul><li><a href='#the_abstract_point'>The abstract point</a></li><li><a href='#concrete_points'>Concrete points</a></li><li><a href='#related_concepts'>Related concepts</a><ul><li><a href='#as_homotopy_types'>As homotopy types</a></li><ins class='diffins'><li><a href='#as_directed_homotopy_types'>As directed homotopy types</a></li></ins><li><a href='#as_geometric_objects'>As geometric objects</a></li></ul></li></ul></div> <h2 id='the_abstract_point'>The abstract point</h2> <p>The <strong>point</strong> is what is shown here:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_89b9927518379c656cf780203f9506ff292221e0_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math></div> <p>As a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, we can interpret this as a category with a single <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> and a single <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> (the <a class='existingWikiWord' href='/nlab/show/diff/identity+morphism'>identity morphism</a> on the object, which is not shown in the picture since it is automatic). This can be generalised recursively to <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher categories</a>; as an <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/n-category'>category</a>, the point consists of a single object <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> whose endomorphism <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category is the point (now understood as an <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category). Of course, to make this work, the point must be a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal</a> <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category at each stage, but it is (in a unique way). In the limit, the point can even be understood as an <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>category</a>, with a unique <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>-morphism for each <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>j \geq 0</annotation></semantics></math> (each of which is an identity for <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo>></mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>j \gt 0</annotation></semantics></math>).</p> <p>In the other direction, the point is a <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a> <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> with a unique element <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math>. It can also be seen as a <a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth value</a> that is <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>. It can even be understood as the <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-2)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/%28-2%29-category'>category</a>.</p> <p>As a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, the <em><a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></em> is the usual point from <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a>: that which has no part. In more modern language, we might say that it has no <em>structure</em> —except that something exists. (So it is <em>not</em> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty</a>!) This is consistent with the preceding paragraphs using the interpretation of a topological space as an <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>groupoid</a>. (But up to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, any <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a> qualifies as a point.)</p> <p>In all of the above, the point can be seen as a <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in an appropriate category (or <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-category). However, you can also see it as a <a class='existingWikiWord' href='/nlab/show/diff/zero+object'>null object</a> in a category of <a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed objects</a>. (Of course, it's always true that a terminal object <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> becomes a null object in <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>/</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>1/C</annotation></semantics></math>, but the dual argument also holds, so the question is which is the primary picture.)</p> <p>But perhaps the point is best seen as the unique object in itself:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_89b9927518379c656cf780203f9506ff292221e0_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo><mo>=</mo><mo stretchy='false'>{</mo><mo>•</mo><mo stretchy='false'>}</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'> \bullet = \{\bullet\} ,</annotation></semantics></math></div> <p>an equation that makes sense as a definition in the theory of ill-founded <a class='existingWikiWord' href='/nlab/show/diff/pure+set'>pure sets</a>. Another possible definition (this time well-founded) in pure set theory is that the point is <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>∅</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\emptyset\}</annotation></semantics></math>, but this doesn't capture the picture that we get from higher category theory: the <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-1)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/%28-1%29-category'>category</a> (<a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth value</a>) of the <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-2)</annotation></semantics></math>-category (the point) is <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a> (which is also the point), the <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/0-category'>category</a> (set) of the true truth value is <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a> (which is also the point), the <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/1-category'>category</a> (category) of the singleton (and <em>all</em> of its <a class='existingWikiWord' href='/nlab/show/diff/endofunction'>endofunctions</a>!) is the <a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a> (which is also the point), and so on. That is:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_89b9927518379c656cf780203f9506ff292221e0_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo><mo>∈</mo><mo>•</mo><mo>∈</mo><mo>•</mo><mo>∈</mo><mo>•</mo><mo>∈</mo><mi>⋯</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \bullet \in \bullet \in \bullet \in \bullet \in \cdots .</annotation></semantics></math></div> <h2 id='concrete_points'>Concrete points</h2> <p>The term ‘point’ is often used for a <a class='existingWikiWord' href='/nlab/show/diff/global+element'>global element</a>; that is the meaning, for example, in the sense of a <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+topos'>point of a topos</a> or a <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+locale'>point of a locale</a>. The connection is that a global element of <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a map from the point to <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. So one may describe the point above as the <em>abstract</em> point, while a global element is a <em>concrete</em> point.</p> <h2 id='related_concepts'>Related concepts</h2> <h3 id='as_homotopy_types'>As homotopy types</h3> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/homotopy+level'>homotopy level</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/n-truncated+object+of+an+%28infinity%2C1%29-category'>n-truncation</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos+theory'>higher topos theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a></th></tr></thead><tbody><tr><td style='text-align: left;'>h-level 0</td><td style='text-align: left;'>(-2)-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28-2%29-groupoid'>(-2)-groupoid</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>/<a class='existingWikiWord' href='/nlab/show/diff/unit+type'>unit type</a>/<a class='existingWikiWord' href='/nlab/show/diff/contractible+type'>contractible type</a></td></tr> <tr><td style='text-align: left;'>h-level 1</td><td style='text-align: left;'>(-1)-truncated</td><td style='text-align: left;'>contractible-if-<a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>inhabited</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28-1%29-groupoid'>(-1)-groupoid</a>/<a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth value</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/posite'>(0,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/ideal'>ideal</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>mere proposition</a>/<a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>h-proposition</a></td></tr> <tr><td style='text-align: left;'>h-level 2</td><td style='text-align: left;'>0-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+0-type'>homotopy 0-type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/0-groupoid'>0-groupoid</a>/<a class='existingWikiWord' href='/nlab/show/diff/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/h-set'>h-set</a></td></tr> <tr><td style='text-align: left;'>h-level 3</td><td style='text-align: left;'>1-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+1-type'>homotopy 1-type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/1-groupoid'>1-groupoid</a>/<a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-sheaf'>(2,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/h-groupoid'>h-groupoid</a></td></tr> <tr><td style='text-align: left;'>h-level 4</td><td style='text-align: left;'>2-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+2-type'>homotopy 2-type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/2-groupoid'>2-groupoid</a></td><td style='text-align: left;'>(3,1)-sheaf/2-stack</td><td style='text-align: left;'>h-2-groupoid</td></tr> <tr><td style='text-align: left;'>h-level 5</td><td style='text-align: left;'>3-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+3-type'>homotopy 3-type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/3-groupoid'>3-groupoid</a></td><td style='text-align: left;'>(4,1)-sheaf/3-stack</td><td style='text-align: left;'>h-3-groupoid</td></tr> <tr><td style='text-align: left;'>h-level <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n+2</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-truncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+n-type'>homotopy n-type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/n-groupoid'>n-groupoid</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28n%2C1%29-sheaf'>(n+1,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/%28n%2C1%29-sheaf'>n-stack</a></td><td style='text-align: left;'>h-<math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-groupoid</td></tr> <tr><td style='text-align: left;'>h-level <math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math></td><td style='text-align: left;'>untruncated</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a></td><td style='text-align: left;'>h-<math class='maruku-mathml' display='inline' id='mathml_89b9927518379c656cf780203f9506ff292221e0_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid</td></tr> </tbody></table> <ins class='diffins'><h3 id='as_directed_homotopy_types'>As directed homotopy types</h3></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/terminal+category'>terminal category</a></li> </ul></ins><ins class='diffins'> </ins><h3 id='as_geometric_objects'>As geometric objects</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+point'>geometric point</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+element'>generalized element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimally+thickened+point'>infinitesimally thickened point</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/superpoint'>superpoint</a></p> </li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 12, 2018 at 16:03:35. See the <a href="/nlab/history/point" 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/point" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/point/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/point" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/point" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/point/cite" style="color: black">Cite</a> <a href="/nlab/print/point" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/point" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>