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> dependent sum 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> dependent sum </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/4331/#Item_9" 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">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div> <p>bec</p> </div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_the_product'>Relation to the product</a></li> <li><a href='#relation_to_type_theory'>Relation to type theory</a></li> <li><a href='#RelationToSomeLimits'>Relation to some limits</a></li> </ul> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>dependent sum</em> is a <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. It generalizes the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> to the situation where one factor may <em>depend</em> on the other. It is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> functor between <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a>.</p> <p>Beware that the term “sum” in “dependent sum” is really referring to <strong><a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a><em>. The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> notion is that of <em><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></em>.</em></strong></p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f \colon A \to I</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> such that <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> along <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> exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. These then constitute a <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>I</mi></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> f^* \colon \mathcal{C}_{/I} \to \mathcal{C}_{/A} </annotation></semantics></math></div> <p>between the corresponding <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>dependent sum</strong> or <strong>dependent coproduct</strong> along the morphism <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> is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\sum_f \colon \mathcal{C}_{/A} \to \mathcal{C}_{/I}</annotation></semantics></math> of the base change functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mover><munder><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></munder><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder></mrow></mover></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>I</mi></mrow></msub><mspace width="2em"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\sum_f \dashv f^* ) \colon \mathcal{C}_{/A} \stackrel{\overset{\sum_f}{\to}}{\underset{f^*}{\leftarrow}} \mathcal{C}_{/I} \qquad. </annotation></semantics></math></div></div> <p>This is directly seen to be equivalent to the following.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The <strong>dependent sum</strong> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">f \colon A \to I</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></munder><mo>≔</mo><mi>f</mi><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \sum_f \coloneqq f\circ (-) \colon \mathcal{C}_{/A} \to \mathcal{C}_{/I} </annotation></semantics></math></div> <p>given by <a class="existingWikiWord" href="/nlab/show/composition">composition</a> 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>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="relation_to_the_product">Relation to the product</h3> <p>Assume that the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">* \in \mathcal{C}</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/object">object</a> and assume that the terminal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">f \colon X \to *</annotation></semantics></math> admits all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> along it.</p> <p>Notice that a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">A \to *</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> is simply the <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>X</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \times A</annotation></semantics></math>, equipped with its <a class="existingWikiWord" href="/nlab/show/projection">projection</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times A \to X</annotation></semantics></math>. We may regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times A \to X</annotation></semantics></math> as the image of the <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mo>*</mo></mrow></msub><mo>→</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f^* \colon \mathcal{C}_{/*} \to \mathcal{C}_{/X}</annotation></semantics></math>, but this is not quite just the product in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, which instead corresponds to the terminal morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>A</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \times A \to *</annotation></semantics></math>. But we have:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The <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>X</mi><mo>×</mo><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \times A \in \mathcal{C}</annotation></semantics></math> is, if it exists, equivalently the dependent sum of the base change of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">A \to *</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>X</mi></munder><msup><mi>X</mi> <mo>*</mo></msup><mi>A</mi><mo>≃</mo><mi>X</mi><mo>×</mo><mi>A</mi><mo>∈</mo><mi>𝒞</mi><mspace width="1em"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sum_{X} X^* A \simeq X \times A \in \mathcal{C} \quad . </annotation></semantics></math></div></div> <p>Here we write “<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>” also for the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math>.</p> <h3 id="relation_to_type_theory">Relation to type theory</h3> <p>Under the <a class="existingWikiWord" href="/nlab/show/relation+between+category+theory+and+type+theory">relation between category theory and type theory</a> the dependent sum is the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of <a class="existingWikiWord" href="/nlab/show/dependent+sum+types">dependent sum types</a> .</p> <p>Notice that under the identification of <a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>, <a class="existingWikiWord" href="/nlab/show/dependent+sum+types">dependent sum types</a> (or rather their <a class="existingWikiWord" href="/nlab/show/bracket+types">bracket types</a>) correspond to <a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>,</mo><mi>P</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\exists x\colon X, P x</annotation></semantics></math>.</p> <p>The following table shows how the <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> rules for dependent sum types correspond to their <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> given by the dependent sum universal construction.</p> <div> <p>The <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for <a class="existingWikiWord" href="/nlab/show/dependent+pair+types">dependent pair types</a> (aka “dependent sum types” or “<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>-types”):</p> <div style="margin: -30px 0px 20px 10px"> <img src="/nlab/files/DependentPairTypeInference-230215.jpg" width="770px" /> </div></div> <h3 id="RelationToSomeLimits">Relation to some limits</h3> <div class="num_prop" id="AbsoluteDependentSumPreservesFiberProducts"> <h6 id="proposition_2">Proposition</h6> <p>For <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">\mathcal{C}</annotation></semantics></math> a category with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> an object, then dependent sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mo>:</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \underset{X}{\sum}: \mathcal{C}_{/X} \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> and <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflects</a> <a class="existingWikiWord" href="/nlab/show/fiber+products">fiber products</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By <a href="overcategory#LimitsInSliceViaLimitsOfCoconedDiagram">this proposition</a> limits over a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> are computed as limits over the <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> diagram under the cospan in the base category. By <a href="final+functor#CoconeUnderCospan">this proposition</a> this inclusion is a <a class="existingWikiWord" href="/nlab/show/final+functor">final functor</a>, hence preserves limits. Since the dependent sum of the diagram is the restriction along this final functor, the result follows.</p> </div> <div class="num_prop" id="NaturalitySquareOfUnitIsPullback"> <h6 id="proposition_3">Proposition</h6> <p>For <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">\mathcal{C}</annotation></semantics></math> a category with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X\in \mathcal{C}</annotation></semantics></math> any object, the <a class="existingWikiWord" href="/nlab/show/naturality+square">naturality square</a> of the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mo>⊣</mo><msup><mi>X</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\underset{X}{\sum} \dashv X^\ast)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> on any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f \colon A \to B)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/X}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>X</mi> <mo>*</mo></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X^\ast \underset{X}{\sum} A \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{X^\ast \underset{X}{\sum} f}} \\ B &\longrightarrow& X^\ast \underset{X}{\sum} B } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By prop. <a class="maruku-ref" href="#AbsoluteDependentSumPreservesFiberProducts"></a> it suffices to see that the diagram is a pullback in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder></mrow><annotation encoding="application/x-tex">\underset{X}{\sum}</annotation></semantics></math>, where, by <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a>, it becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="2em"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{X}{\sum} A &\stackrel{(A,id)}{\longrightarrow}& X \times \underset{X}{\sum} A \\ \downarrow^{\mathrlap{\underset{X}{\sum}f}} && \downarrow^{\mathrlap{(id, \underset{X}{\sum} f)}} \\ \underset{X}{\sum} B &\stackrel{(B,id)}{\longrightarrow}& X \times \underset{X}{\sum} B } \qquad . </annotation></semantics></math></div></div> <div class="num_prop" id="NaturalitySquareOfCounitIsPullback"> <h6 id="proposition_4">Proposition</h6> <p>For <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">\mathcal{C}</annotation></semantics></math> a category with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X\in \mathcal{C}</annotation></semantics></math> any object, the <a class="existingWikiWord" href="/nlab/show/naturality+square">naturality square</a> of the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><mo>⊣</mo><msup><mi>X</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\underset{X}{\sum} \dashv X^\ast)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> on any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f \colon A \to B)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><msup><mi>X</mi> <mo>*</mo></msup><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><msup><mi>X</mi> <mo>*</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>X</mi></munder><msup><mi>X</mi> <mo>*</mo></msup><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{X}{\sum} X^\ast A &\longrightarrow& A \\ \downarrow^{\mathrlap{\underset{X}{\sum}X^\ast f}} && \downarrow^{\mathrlap{f}} \\ \underset{X}{\sum} X^\ast B &\longrightarrow& B } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a> the diagram is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>×</mo><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="2em"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X\times A & \longrightarrow& A \\ \downarrow^{\mathrlap{(id,f)}} && \downarrow^{\mathrlap{f}} \\ X \times B &\longrightarrow& B } \qquad. </annotation></semantics></math></div></div> <h2 id="in_higher_category_theory">In higher category theory</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an (∞,1)-category. We still want to define the dependent sum as the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>→</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/X} \to C_{/Y}</annotation></semantics></math> given by composing with the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>, but the details are more complex.</p> <p>The <a class="existingWikiWord" href="/nlab/show/codomain+fibration">codomain fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tgt</mi><mo>:</mo><msup><mi>C</mi> <mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">tgt : C^{[1]} \to C</annotation></semantics></math> is a cocartesian fibration classified by a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo>:</mo><mi>X</mi><mo>↦</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L : C \to (\infty,1)Cat : X \mapsto C_{/X}</annotation></semantics></math>.</p> <p>Then for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math>, we can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub><mo>=</mo><mi>L</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma_f = L(f)</annotation></semantics></math> to be the dependent sum.</p> <p>Since the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>z</mi><mo>→</mo><mi>x</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>z</mi><mo>→</mo><mi>y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[z \to x] \to [z \to y]</annotation></semantics></math> induced by composition are cocartesian morphisms, we see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_f</annotation></semantics></math> is indeed given by composition 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>.</p> <p>By proposition 6.1.1.1 and the following remarks of <a href="#Lurie">Lurie</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has pullbacks, then this is also a cartesian fibration, and is classified by a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">R : C^{op} \to (\infty,1)Cat</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub><mo>→</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f^* = R(f) : C_{/Y} \to C_{/X}</annotation></semantics></math> is the functor computing pullbacks along <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>, and we have the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma_f \dashv f^*</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sink">sink</a>, <a class="existingWikiWord" href="/nlab/show/cosink">cosink</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+colimit">internal colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> <ul> <li> <p><strong>dependent sum</strong>, <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/necessity">necessity</a>, <a class="existingWikiWord" href="/nlab/show/possibility">possibility</a>, <a class="existingWikiWord" href="/nlab/show/reader+monad">reader monad</a>, <a class="existingWikiWord" href="/nlab/show/writer+comonad">writer comonad</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Standard textbook accounts include section A1.5.3 of</p> <ul> <li id="Johnstone"><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></li> </ul> <p>and section IV of</p> <ul> <li id="MacLaneMoerdijk"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></li> </ul> <p>Other references:</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>Only fairly indirectly related is the notion of “dependent sums” in</p> <ul> <li>Peter Bonart, <em>Towards a General Theory of Dependent Sums</em> (2024). [<a href="https://arxiv.org/abs/2404.08139">arXiv:2404.08139</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 15, 2024 at 13:28:54. See the <a href="/nlab/history/dependent+sum" 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/dependent+sum" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4331/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/dependent+sum/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/dependent+sum" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/dependent+sum" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/dependent+sum/cite" style="color: black">Cite</a> <a href="/nlab/print/dependent+sum" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/dependent+sum" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>