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> sigma-frame 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> sigma-frame </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/13995/#Item_3" 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="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#category_of_frames'>Category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frames</a></li> <li><a href='#see_also'>See also</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frame</strong> is a <a class="existingWikiWord" href="/nlab/show/sigma-complete+lattice"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-complete lattice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mo>≤</mo><mo>,</mo><mo>⊥</mo><mo>,</mo><mo>∨</mo><mo>,</mo><mo>⊤</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L, \leq, \bot, \vee, \top, \wedge, \Vee)</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">a \in L</annotation></semantics></math> and sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">s:\mathbb{N} \to L</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∧</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mrow><mi>n</mi><mo>:</mo><mi>ℕ</mi></mrow></munder><mi>a</mi><mo>∧</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \wedge \Vee_{n:\mathbb{N}} s(n) = \Vee_{n:\mathbb{N}} a \wedge s(n)</annotation></semantics></math></div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mo>⊥</mo></msub></mrow><annotation encoding="application/x-tex">1_\bot</annotation></semantics></math>, is the initial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frame.</p> </li> <li> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a>, the <a class="existingWikiWord" href="/nlab/show/Rosolini+dominance">Rosolini dominance</a></p> </li> <li> <p>The <span class="newWikiWord">intrinsic topology<a href="/nlab/new/intrinsic+topology">?</a></span> on a set <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>, which is the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma^A</annotation></semantics></math></p> </li> <li> <p>Malcev topologies for set <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>, which are a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/subobject">subframe</a> of the intrinsic topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma^A</annotation></semantics></math>.</p> </li> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/frame">frame</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frame. In particular, the <a class="existingWikiWord" href="/nlab/show/poset+of+truth+values">poset of truth values</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frame.</p> </li> </ul> <h2 id="category_of_frames">Category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frames</h2> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mi mathvariant="normal">Frm</mi></mrow><annotation encoding="application/x-tex">\sigma\mathrm{Frm}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frames and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frame <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a>, that is lattice <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> that preserve <a class="existingWikiWord" href="/nlab/show/countable">countable</a> joins. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mi mathvariant="normal">Frm</mi></mrow><annotation encoding="application/x-tex">\sigma\mathrm{Frm}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/replete+subcategory">replete</a> <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Pos">Pos</a> and <a class="existingWikiWord" href="/nlab/show/DistLat">DistLat</a>, and it is an <a class="existingWikiWord" href="/nlab/show/algebraic+category">algebraic category</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mi mathvariant="normal">Frm</mi></mrow><annotation encoding="application/x-tex">\sigma\mathrm{Frm}</annotation></semantics></math> is the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mi mathvariant="normal">Loc</mi></mrow><annotation encoding="application/x-tex">\sigma\mathrm{Loc}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sigma-locale"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-locales</a>; this is an example of the duality between <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>.</p> <h2 id="see_also">See also</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-complete+lattice">sigma-complete lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-complete+Heyting+algebra">sigma-complete Heyting algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-complete+Boolean+algebra">sigma-complete Boolean algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/countably+prime+filter">countably prime filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dedekind+cut">Dedekind cut</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-pretopos">sigma-pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/streak">streak</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-topological+space">sigma-topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-locale">sigma-locale</a></p> </li> <li> <p><span class="newWikiWord">sigma-coframe<a href="/nlab/new/sigma-coframe">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/first+countable+space">first countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limited+principle+of+omniscience">limited principle of omniscience</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-frame+of+propositions">sigma-frame of propositions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a>, <a class="existingWikiWord" href="/nlab/show/DistLat">DistLat</a>, <a class="existingWikiWord" href="/nlab/show/Frm">Frm</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p>Francesco Ciraulo, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locales in Formal Topology</em>, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) (<a href="https://doi.org/10.46298/lmcs-18%281%3A7%292022">doi:10.46298/lmcs-18%281%3A7%292022</a>, <a href="https://arxiv.org/abs/1801.09644">arXiv:1801.09644</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alex+Simpson">Alex Simpson</a>, <em>Measure, randomness and sublocales</em>, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. (<a href="https://doi.org/10.1016/j.apal.2011.12.014">doi:10.1016/j.apal.2011.12.014</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <em>Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem</em> (<a href="https://arxiv.org/abs/2307.07830">arXiv:2307.07830</a>)</p> </li> <li> <p>Martin E. Bidlingmaier, Florian Faissole, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Synthetic topology in Homotopy Type Theory for probabilistic programming</em>. Mathematical Structures in Computer Science, 2021;31(10):1301-1329. [<a href="https://doi.org/10.1017/S0960129521000165">doi:10.1017/S0960129521000165</a>, <a href="https://arxiv.org/abs/1912.07339">arXiv:1912.07339</a>]</p> </li> <li> <p>Univalent Foundations Project, <a class="existingWikiWord" href="/nlab/show/HoTT+book">Homotopy Type Theory – Univalent Foundations of Mathematics</a> (2013)</p> </li> <li> <p><span class="newWikiWord">Raquel Bernardes<a href="/nlab/new/Raquel+Bernardes">?</a></span>, <em>Lebesgue integration on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locales: simple functions</em>, (<a href="https://arxiv.org/abs/2408.13911">arXiv:2408.13911</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 20, 2025 at 18:52:01. See the <a href="/nlab/history/sigma-frame" 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/sigma-frame" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/13995/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/sigma-frame/22" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/sigma-frame" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/sigma-frame" accesskey="S" class="navlink" id="history" rel="nofollow">History (22 revisions)</a> <a href="/nlab/show/sigma-frame/cite" style="color: black">Cite</a> <a href="/nlab/print/sigma-frame" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/sigma-frame" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>