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> weighted limit 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> weighted limit </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/11870/#Item_10" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <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="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </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></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> <ul> <li><a href='#weighted_limits_for_'>Weighted limits for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V = Set</annotation></semantics></math></a></li> </ul> <li><a href='#motivation_from_enriched_category_theory'>Motivation from enriched category theory</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#homotopy_limits'>Homotopy limits</a></li> <li><a href='#homotopy_pullback'>Homotopy pullback</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#presenting_homotopy_limits'>Presenting homotopy limits</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>weighted limit</em> (also called <em>indexed limit</em> or <em>mean cotensor product</em> in older texts) is naturally understood from the point of view on <a class="existingWikiWord" href="/nlab/show/limits">limits</a> as described at <em><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></em>:</p> <p>Weighted limits make sense and are considered in the general context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>, but restrict attention to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">V=</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> for the moment, in order to motivate the concept.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> which indexes <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> over which we want to consider limits and eventually weighted limits. Notice that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F \colon K \to Set </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-valued <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of <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 canonically identified simply with the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a href="cone#ConesOverADiagram">cones</a> with tip the <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">pt = \{\bullet\}</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>lim</mi><mi>F</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Δ</mi><mi>pt</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> lim F \;=\; [K,Set](\Delta pt, F) \,. </annotation></semantics></math></div> <p>This means, more generally, that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> F \,\colon\, K \to C </annotation></semantics></math></div> <p>a functor with values in an arbitrary <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">C</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/object">object</a>-wise limit of the functor <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> under the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mover><mo>⟶</mo><mi>F</mi></mover><mi>C</mi><mover><mo>⟶</mo><mi>Y</mi></mover><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> C\big(-,F(-)\big) \,\colon\, K \overset{F}{\longrightarrow} C \overset{Y}{\longrightarrow} Set^{C^{op}} </annotation></semantics></math></div> <p>can be expressed by the right hand side of:</p> <div class="maruku-equation" id="eq:ConicalLimitInWeightedLimitForm"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></munder></munder><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>Δ</mi><mi>pt</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{ \underset{k \in K}{\longleftarrow} }{lim} C\big(-,F(k)\big) \;=\; [K,Set]\Big( \Delta pt ,\, C\big(-,F(-)\big) \Big) \,. </annotation></semantics></math></div> <p>(This is the limit over the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">C\big(-,F(-)\big) \,\colon\, K \to Set^{C^{op}}</annotation></semantics></math> which, if <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>, defines the desired limit of <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>, see <a href="representable+functor#ExampleLimits">this example</a> at <em><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></em>).</p> <p>The <strong>idea</strong> of weighted limits is to:</p> <ol> <li> <p>allow in the formula <a class="maruku-eqref" href="#eq:ConicalLimitInWeightedLimitForm">(1)</a> the particular functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mi>pt</mi></mrow><annotation encoding="application/x-tex">\Delta pt</annotation></semantics></math> to be replaced by any other functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">W \,\colon\, K \to Set</annotation></semantics></math>;</p> </li> <li> <p>to generalize everything straightforwardly from the <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> context to arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched contexts (see below).</p> </li> </ol> <p>The idea is that the weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">W \colon K \to V</annotation></semantics></math> encodes the way in which one generalizes the concept of a <a href="cone#ConesOverADiagram">cones</a> over a diagram <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> (that is, something with just a tip from which morphisms are emanating down to <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>) to a more intricate structure over the diagram <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>. For instance in the application to <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> discussed below, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> being <a class="existingWikiWord" href="/nlab/show/SimpSet">SimpSet</a>, the weight is such that it ensures that not only <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are emanating from the tip, but that any triangle formed by these is filled by a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-cell</a>, every tetrahedron by a <a class="existingWikiWord" href="/nlab/show/3-morphism">3-cell</a>, etc.</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>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/closed+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. All categories in the following are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched categories</a>, all functors are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a>.</p> <p>A <strong>weighted limit</strong> over a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> F \,\colon\, K \to C </annotation></semantics></math></div> <p>with respect to a <em>weight</em> or <em>indexing type</em> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>K</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> W \,\colon\, K \to V </annotation></semantics></math></div> <p>is, if it exists, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>lim</mi> <mi>W</mi></msup><mi>F</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">lim^W F \in C</annotation></semantics></math> which <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a> the functor (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in 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><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>W</mi><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [K,V]\Big(W, C\big(c,F(-)\big)\Big) \;\colon\; C^{op} \to V \,, </annotation></semantics></math></div> <p>i.e. such that for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> there is an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>c</mi><mo>,</mo><msup><mi>lim</mi> <mi>W</mi></msup><mi>F</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>W</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> C\big(c, lim^W F\big) \simeq [K,V]\Big(W(-), C\big(c,F(-)\big)\Big) </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[K,V]</annotation></semantics></math> denotes the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a>, as usual.)</p> <p>In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C = V</annotation></semantics></math> itself, then we get the direct formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>lim</mi> <mi>W</mi></msup><mi>F</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> lim^W F \;\simeq\; [K,V](W,F) \,. </annotation></semantics></math></div> <p>This follows from the above by the <a class="existingWikiWord" href="/nlab/show/end">end</a> manipulation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>V</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>V</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>V</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><msub><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo>:</mo><mi>V</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [K,V](W(-),C(c,F(-))) &:= \int_{k \in K} V(W(k),V(c,F(k))) \\ & \simeq \int_{k \in K} V(c,V(W(k),F(k)) \\ & \simeq V(c, \int_{k \in K} V(W(k),F(k)) \\ & =: V(c, [K,V](W,F)) \,. \end{aligned} </annotation></semantics></math></div> <h3 id="weighted_limits_for_">Weighted limits for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V = Set</annotation></semantics></math></h3> <p>Let us spell out what a weighted limit looks like in ordinary category theory, to give intuition for the difference between weighted limits and ordinary limits.</p> <p>Given a weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">W : K \to Set</annotation></semantics></math> and a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : K \to C</annotation></semantics></math>, a weighted limit comprises an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> together with a projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>k</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>:</mo><mi>L</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{k, w} : L \to F(k)</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w \in W(k)</annotation></semantics></math> such that the following diagram commutes for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k, k \in K</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w \in W(k)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>:</mo><mi>k</mi><mo>→</mo><mi>k</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\kappa : k \to k'</annotation></semantics></math>: <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="138.781pt" height="95.198pt" viewBox="0 0 138.781 95.198" version="1.2"> <defs> <g> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-1"> <path style="stroke:none;" d="M 5.4375 -8.96875 C 5.5625 -9.546875 5.609375 -9.6875 6.921875 -9.6875 C 7.3125 -9.6875 7.421875 -9.6875 7.421875 -9.96875 C 7.421875 -10.109375 7.265625 -10.109375 7.203125 -10.109375 C 6.90625 -10.109375 6.5625 -10.09375 6.265625 -10.09375 L 4.28125 -10.09375 C 4 -10.09375 3.671875 -10.109375 3.390625 -10.109375 C 3.28125 -10.109375 3.109375 -10.109375 3.109375 -9.828125 C 3.109375 -9.6875 3.25 -9.6875 3.46875 -9.6875 C 4.375 -9.6875 4.375 -9.5625 4.375 -9.40625 C 4.375 -9.375 4.375 -9.28125 4.3125 -9.0625 L 2.3125 -1.09375 C 2.171875 -0.578125 2.140625 -0.421875 1.109375 -0.421875 C 0.828125 -0.421875 0.6875 -0.421875 0.6875 -0.15625 C 0.6875 0 0.765625 0 1.0625 0 L 7.703125 0 C 8.03125 0 8.046875 -0.015625 8.140625 -0.28125 L 9.28125 -3.4375 C 9.3125 -3.515625 9.34375 -3.59375 9.34375 -3.640625 C 9.34375 -3.734375 9.265625 -3.796875 9.203125 -3.796875 C 9.1875 -3.796875 9.109375 -3.796875 9.078125 -3.734375 C 9.046875 -3.71875 9.046875 -3.6875 8.9375 -3.40625 C 8.453125 -2.109375 7.796875 -0.421875 5.28125 -0.421875 L 3.859375 -0.421875 C 3.65625 -0.421875 3.625 -0.421875 3.546875 -0.4375 C 3.375 -0.453125 3.359375 -0.484375 3.359375 -0.609375 C 3.359375 -0.71875 3.390625 -0.796875 3.421875 -0.9375 Z M 5.4375 -8.96875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-2"> <path style="stroke:none;" d="M 4.40625 -4.828125 L 5.828125 -4.828125 C 6.953125 -4.828125 7.03125 -4.578125 7.03125 -4.140625 C 7.03125 -3.953125 7 -3.75 6.9375 -3.421875 C 6.90625 -3.359375 6.890625 -3.28125 6.890625 -3.265625 C 6.890625 -3.15625 6.953125 -3.09375 7.046875 -3.09375 C 7.171875 -3.09375 7.1875 -3.15625 7.25 -3.390625 L 8.109375 -6.84375 C 8.109375 -6.90625 8.0625 -6.984375 7.953125 -6.984375 C 7.828125 -6.984375 7.8125 -6.9375 7.75 -6.6875 C 7.4375 -5.5625 7.140625 -5.265625 5.84375 -5.265625 L 4.5 -5.265625 L 5.46875 -9.09375 C 5.59375 -9.609375 5.625 -9.65625 6.234375 -9.65625 L 8.21875 -9.65625 C 10.078125 -9.65625 10.34375 -9.109375 10.34375 -8.0625 C 10.34375 -7.96875 10.34375 -7.640625 10.296875 -7.265625 C 10.28125 -7.203125 10.25 -7 10.25 -6.953125 C 10.25 -6.828125 10.328125 -6.78125 10.40625 -6.78125 C 10.515625 -6.78125 10.578125 -6.84375 10.609375 -7.109375 L 10.921875 -9.703125 C 10.921875 -9.75 10.953125 -9.890625 10.953125 -9.921875 C 10.953125 -10.09375 10.8125 -10.09375 10.546875 -10.09375 L 3.53125 -10.09375 C 3.25 -10.09375 3.09375 -10.09375 3.09375 -9.828125 C 3.09375 -9.65625 3.203125 -9.65625 3.453125 -9.65625 C 4.375 -9.65625 4.375 -9.546875 4.375 -9.390625 C 4.375 -9.3125 4.359375 -9.25 4.3125 -9.09375 L 2.3125 -1.09375 C 2.171875 -0.578125 2.140625 -0.421875 1.109375 -0.421875 C 0.828125 -0.421875 0.6875 -0.421875 0.6875 -0.15625 C 0.6875 0 0.8125 0 0.90625 0 C 1.1875 0 1.484375 -0.03125 1.765625 -0.03125 L 3.6875 -0.03125 C 4.015625 -0.03125 4.375 0 4.703125 0 C 4.828125 0 5 0 5 -0.265625 C 5 -0.421875 4.921875 -0.421875 4.59375 -0.421875 C 3.421875 -0.421875 3.390625 -0.53125 3.390625 -0.75 C 3.390625 -0.828125 3.421875 -0.953125 3.453125 -1.046875 Z M 4.40625 -4.828125 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-3"> <path style="stroke:none;" d="M 4.15625 -9.90625 C 4.171875 -9.96875 4.203125 -10.0625 4.203125 -10.125 C 4.203125 -10.28125 4.0625 -10.28125 4.03125 -10.28125 C 4.015625 -10.28125 3.484375 -10.234375 3.21875 -10.203125 C 2.96875 -10.1875 2.734375 -10.15625 2.46875 -10.140625 C 2.125 -10.109375 2.015625 -10.109375 2.015625 -9.828125 C 2.015625 -9.6875 2.15625 -9.6875 2.3125 -9.6875 C 3.0625 -9.6875 3.0625 -9.546875 3.0625 -9.40625 C 3.0625 -9.34375 3.0625 -9.3125 2.984375 -9.046875 L 0.875 -0.578125 C 0.8125 -0.359375 0.8125 -0.328125 0.8125 -0.234375 C 0.8125 0.09375 1.0625 0.140625 1.21875 0.140625 C 1.625 0.140625 1.71875 -0.171875 1.84375 -0.640625 L 2.53125 -3.40625 C 3.59375 -3.28125 4.234375 -2.84375 4.234375 -2.140625 C 4.234375 -2.046875 4.234375 -1.984375 4.1875 -1.765625 C 4.125 -1.546875 4.125 -1.359375 4.125 -1.28125 C 4.125 -0.421875 4.703125 0.140625 5.453125 0.140625 C 6.125 0.140625 6.484375 -0.46875 6.609375 -0.6875 C 6.921875 -1.234375 7.109375 -2.0625 7.109375 -2.125 C 7.109375 -2.1875 7.046875 -2.25 6.96875 -2.25 C 6.828125 -2.25 6.8125 -2.1875 6.75 -1.953125 C 6.546875 -1.1875 6.234375 -0.140625 5.484375 -0.140625 C 5.1875 -0.140625 4.984375 -0.296875 4.984375 -0.859375 C 4.984375 -1.140625 5.046875 -1.46875 5.109375 -1.6875 C 5.171875 -1.953125 5.171875 -1.96875 5.171875 -2.140625 C 5.171875 -3.015625 4.390625 -3.515625 3.015625 -3.6875 C 3.5625 -4.015625 4.09375 -4.59375 4.296875 -4.8125 C 5.140625 -5.765625 5.71875 -6.234375 6.40625 -6.234375 C 6.734375 -6.234375 6.828125 -6.140625 6.9375 -6.0625 C 6.390625 -6 6.171875 -5.609375 6.171875 -5.3125 C 6.171875 -4.96875 6.453125 -4.84375 6.671875 -4.84375 C 7.0625 -4.84375 7.421875 -5.1875 7.421875 -5.65625 C 7.421875 -6.09375 7.078125 -6.53125 6.40625 -6.53125 C 5.59375 -6.53125 4.9375 -5.953125 3.875 -4.765625 C 3.734375 -4.59375 3.1875 -4.03125 2.640625 -3.828125 Z M 4.15625 -9.90625 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-1"> <path style="stroke:none;" d="M 4.8125 3.59375 C 4.8125 3.5625 4.8125 3.53125 4.5625 3.28125 C 3.078125 1.78125 2.25 -0.671875 2.25 -3.6875 C 2.25 -6.5625 2.953125 -9.03125 4.671875 -10.78125 C 4.8125 -10.921875 4.8125 -10.953125 4.8125 -10.984375 C 4.8125 -11.078125 4.734375 -11.109375 4.6875 -11.109375 C 4.484375 -11.109375 3.28125 -10.046875 2.546875 -8.59375 C 1.796875 -7.09375 1.453125 -5.515625 1.453125 -3.6875 C 1.453125 -2.375 1.65625 -0.609375 2.421875 0.984375 C 3.296875 2.75 4.515625 3.71875 4.6875 3.71875 C 4.734375 3.71875 4.8125 3.6875 4.8125 3.59375 Z M 4.8125 3.59375 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-2"> <path style="stroke:none;" d="M 4.171875 -3.6875 C 4.171875 -4.8125 4.03125 -6.65625 3.203125 -8.375 C 2.328125 -10.140625 1.109375 -11.109375 0.953125 -11.109375 C 0.890625 -11.109375 0.8125 -11.078125 0.8125 -10.984375 C 0.8125 -10.953125 0.8125 -10.921875 1.0625 -10.671875 C 2.546875 -9.171875 3.375 -6.71875 3.375 -3.703125 C 3.375 -0.828125 2.6875 1.640625 0.96875 3.390625 C 0.8125 3.53125 0.8125 3.5625 0.8125 3.59375 C 0.8125 3.6875 0.890625 3.71875 0.953125 3.71875 C 1.140625 3.71875 2.359375 2.65625 3.078125 1.203125 C 3.84375 -0.3125 4.171875 -1.90625 4.171875 -3.6875 Z M 4.171875 -3.6875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph2-1"> <path style="stroke:none;" d="M 2.609375 -4.6875 C 2.671875 -4.8125 2.703125 -4.875 2.703125 -4.984375 C 2.703125 -5.296875 2.40625 -5.515625 2.140625 -5.515625 C 1.734375 -5.515625 1.625 -5.171875 1.59375 -5.03125 L 0.328125 -0.78125 C 0.296875 -0.65625 0.296875 -0.625 0.296875 -0.625 C 0.296875 -0.53125 0.359375 -0.515625 0.453125 -0.484375 C 0.625 -0.40625 0.65625 -0.40625 0.671875 -0.40625 C 0.703125 -0.40625 0.765625 -0.40625 0.828125 -0.578125 Z M 2.609375 -4.6875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-1"> <path style="stroke:none;" d="M 3.125 -3.203125 L 4.140625 -3.203125 C 4.953125 -3.203125 5 -3.046875 5 -2.75 C 5 -2.6875 5 -2.59375 4.921875 -2.328125 C 4.921875 -2.296875 4.90625 -2.21875 4.90625 -2.1875 C 4.90625 -2.171875 4.90625 -2.046875 5.0625 -2.046875 C 5.171875 -2.046875 5.203125 -2.140625 5.234375 -2.265625 L 5.75 -4.390625 C 5.765625 -4.40625 5.796875 -4.53125 5.796875 -4.546875 C 5.796875 -4.578125 5.765625 -4.6875 5.640625 -4.6875 C 5.515625 -4.6875 5.5 -4.59375 5.46875 -4.46875 C 5.265625 -3.703125 5.046875 -3.53125 4.171875 -3.53125 L 3.203125 -3.53125 L 3.828125 -6.015625 C 3.921875 -6.375 3.9375 -6.390625 4.328125 -6.390625 L 5.78125 -6.390625 C 6.96875 -6.390625 7.296875 -6.140625 7.296875 -5.296875 C 7.296875 -5.078125 7.25 -4.828125 7.25 -4.640625 C 7.25 -4.515625 7.3125 -4.46875 7.390625 -4.46875 C 7.53125 -4.46875 7.546875 -4.5625 7.5625 -4.734375 L 7.734375 -6.40625 C 7.75 -6.453125 7.75 -6.53125 7.75 -6.578125 C 7.75 -6.71875 7.640625 -6.71875 7.453125 -6.71875 L 2.390625 -6.71875 C 2.21875 -6.71875 2.078125 -6.71875 2.078125 -6.53125 C 2.078125 -6.390625 2.203125 -6.390625 2.375 -6.390625 C 2.4375 -6.390625 2.578125 -6.390625 2.75 -6.375 C 2.953125 -6.34375 2.984375 -6.328125 2.984375 -6.234375 C 2.984375 -6.1875 2.96875 -6.140625 2.9375 -6.03125 L 1.625 -0.78125 C 1.546875 -0.40625 1.515625 -0.328125 0.796875 -0.328125 C 0.609375 -0.328125 0.484375 -0.328125 0.484375 -0.140625 C 0.484375 -0.09375 0.515625 0 0.640625 0 C 0.84375 0 1.09375 -0.03125 1.296875 -0.03125 L 2.671875 -0.03125 C 2.859375 -0.015625 3.203125 0 3.390625 0 C 3.46875 0 3.609375 0 3.609375 -0.1875 C 3.609375 -0.328125 3.484375 -0.328125 3.28125 -0.328125 C 3.078125 -0.328125 2.984375 -0.328125 2.765625 -0.34375 C 2.5 -0.375 2.46875 -0.40625 2.46875 -0.515625 C 2.46875 -0.53125 2.46875 -0.59375 2.515625 -0.734375 Z M 3.125 -3.203125 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-2"> <path style="stroke:none;" d="M 2.390625 -2.515625 C 2.71875 -2.703125 3.15625 -3.046875 3.390625 -3.234375 C 3.984375 -3.703125 4.25 -3.796875 4.421875 -3.875 C 4.40625 -3.84375 4.390625 -3.765625 4.390625 -3.71875 C 4.390625 -3.515625 4.53125 -3.34375 4.78125 -3.34375 C 5.0625 -3.34375 5.328125 -3.578125 5.328125 -3.890625 C 5.328125 -4.09375 5.203125 -4.25 4.921875 -4.25 C 4.28125 -4.25 3.65625 -3.765625 3.203125 -3.40625 C 2.609375 -2.953125 2.3125 -2.734375 1.890625 -2.59375 C 2.015625 -3.0625 2.03125 -3.140625 2.078125 -3.3125 C 2.140625 -3.5625 2.25 -4 2.25 -4.046875 C 2.25 -4.25 2.09375 -4.359375 1.921875 -4.359375 C 1.84375 -4.359375 1.625 -4.3125 1.515625 -4.078125 C 1.5 -4.015625 0.5625 -0.3125 0.5625 -0.203125 C 0.5625 0.015625 0.75 0.09375 0.875 0.09375 C 1.21875 0.09375 1.296875 -0.203125 1.359375 -0.390625 L 1.8125 -2.234375 C 2.640625 -2.21875 3.59375 -2.09375 3.59375 -1.4375 C 3.59375 -1.34375 3.578125 -1.234375 3.5625 -1.171875 C 3.546875 -1.09375 3.53125 -0.9375 3.53125 -0.90625 C 3.53125 -0.3125 3.953125 0.09375 4.53125 0.09375 C 4.921875 0.09375 5.1875 -0.171875 5.328125 -0.390625 C 5.5625 -0.765625 5.734375 -1.359375 5.734375 -1.40625 C 5.734375 -1.515625 5.65625 -1.546875 5.578125 -1.546875 C 5.453125 -1.546875 5.4375 -1.484375 5.40625 -1.296875 C 5.265625 -0.84375 5.015625 -0.171875 4.546875 -0.171875 C 4.296875 -0.171875 4.234375 -0.390625 4.234375 -0.65625 C 4.234375 -0.796875 4.25 -0.90625 4.296875 -1.140625 C 4.328125 -1.25 4.34375 -1.34375 4.34375 -1.46875 C 4.34375 -2.328125 3.25 -2.5 2.390625 -2.515625 Z M 2.390625 -2.515625 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-3"> <path style="stroke:none;" d="M 2.796875 -3.609375 L 3.9375 -3.609375 C 3.734375 -2.703125 3.5625 -1.96875 3.5625 -1.25 C 3.5625 -1.171875 3.5625 -0.75 3.65625 -0.390625 C 3.796875 0.015625 3.890625 0.09375 4.0625 0.09375 C 4.28125 0.09375 4.515625 -0.09375 4.515625 -0.328125 C 4.515625 -0.390625 4.5 -0.421875 4.46875 -0.5 C 4.25 -0.953125 4.109375 -1.4375 4.109375 -2.234375 C 4.109375 -2.46875 4.109375 -2.890625 4.25 -3.609375 L 5.453125 -3.609375 C 5.609375 -3.609375 5.71875 -3.609375 5.8125 -3.6875 C 5.9375 -3.796875 5.96875 -3.921875 5.96875 -3.984375 C 5.96875 -4.25 5.71875 -4.25 5.546875 -4.25 L 1.984375 -4.25 C 1.78125 -4.25 1.40625 -4.25 0.921875 -3.78125 C 0.5625 -3.421875 0.28125 -2.96875 0.28125 -2.90625 C 0.28125 -2.8125 0.359375 -2.78125 0.4375 -2.78125 C 0.53125 -2.78125 0.546875 -2.8125 0.609375 -2.890625 C 1.09375 -3.609375 1.671875 -3.609375 1.90625 -3.609375 L 2.484375 -3.609375 C 2.1875 -2.5625 1.671875 -1.359375 1.296875 -0.640625 C 1.234375 -0.5 1.140625 -0.296875 1.140625 -0.203125 C 1.140625 0 1.296875 0.09375 1.46875 0.09375 C 1.84375 0.09375 1.9375 -0.28125 2.140625 -1.09375 Z M 2.796875 -3.609375 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph4-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph4-1"> <path style="stroke:none;" d="M 3.28125 2.46875 C 3.375 2.46875 3.484375 2.46875 3.484375 2.34375 C 3.484375 2.3125 3.46875 2.296875 3.34375 2.171875 C 2 0.90625 1.65625 -0.9375 1.65625 -2.46875 C 1.65625 -5.3125 2.828125 -6.640625 3.34375 -7.09375 C 3.46875 -7.234375 3.484375 -7.234375 3.484375 -7.28125 C 3.484375 -7.34375 3.453125 -7.40625 3.34375 -7.40625 C 3.1875 -7.40625 2.703125 -6.90625 2.609375 -6.8125 C 1.296875 -5.4375 1.015625 -3.65625 1.015625 -2.46875 C 1.015625 -0.25 1.953125 1.515625 3.28125 2.46875 Z M 3.28125 2.46875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph4-2"> <path style="stroke:none;" d="M 3.046875 -2.46875 C 3.046875 -3.40625 2.890625 -4.53125 2.28125 -5.703125 C 1.796875 -6.609375 0.90625 -7.40625 0.71875 -7.40625 C 0.625 -7.40625 0.59375 -7.34375 0.59375 -7.28125 C 0.59375 -7.25 0.59375 -7.234375 0.71875 -7.109375 C 2.09375 -5.796875 2.40625 -3.984375 2.40625 -2.46875 C 2.40625 0.359375 1.234375 1.703125 0.734375 2.15625 C 0.609375 2.296875 0.59375 2.296875 0.59375 2.34375 C 0.59375 2.40625 0.625 2.46875 0.71875 2.46875 C 0.875 2.46875 1.375 1.96875 1.453125 1.875 C 2.78125 0.5 3.046875 -1.28125 3.046875 -2.46875 Z M 3.046875 -2.46875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-1"> <path style="stroke:none;" d="M 2.453125 -4.921875 C 2.46875 -4.953125 2.484375 -4.984375 2.484375 -5.03125 C 2.484375 -5.140625 2.328125 -5.140625 2.25 -5.125 L 1.421875 -5.0625 C 1.28125 -5.0625 1.1875 -5.046875 1.1875 -4.875 C 1.1875 -4.765625 1.28125 -4.765625 1.40625 -4.765625 C 1.625 -4.765625 1.671875 -4.734375 1.765625 -4.703125 C 1.765625 -4.625 1.765625 -4.609375 1.734375 -4.484375 L 0.703125 -0.34375 C 0.671875 -0.234375 0.671875 -0.1875 0.671875 -0.171875 C 0.671875 0 0.828125 0.078125 0.9375 0.078125 C 1 0.078125 1.140625 0.046875 1.234375 -0.09375 C 1.28125 -0.1875 1.578125 -1.421875 1.640625 -1.65625 C 2.140625 -1.625 2.671875 -1.5 2.671875 -1.0625 C 2.671875 -1.03125 2.671875 -1 2.640625 -0.90625 C 2.625 -0.796875 2.625 -0.75 2.625 -0.703125 C 2.625 -0.203125 3.03125 0.078125 3.453125 0.078125 C 4.171875 0.078125 4.4375 -1 4.4375 -1.0625 C 4.4375 -1.078125 4.421875 -1.171875 4.296875 -1.171875 C 4.1875 -1.171875 4.171875 -1.125 4.140625 -1 C 4.0625 -0.734375 3.859375 -0.171875 3.484375 -0.171875 C 3.203125 -0.171875 3.203125 -0.484375 3.203125 -0.5625 C 3.203125 -0.6875 3.203125 -0.703125 3.234375 -0.84375 C 3.234375 -0.859375 3.28125 -1 3.28125 -1.09375 C 3.28125 -1.65625 2.625 -1.84375 2.0625 -1.890625 C 2.234375 -2 2.390625 -2.125 2.6875 -2.375 C 3.0625 -2.703125 3.4375 -3.015625 3.84375 -3.015625 C 3.953125 -3.015625 4.03125 -3 4.109375 -2.921875 C 3.84375 -2.875 3.734375 -2.671875 3.734375 -2.515625 C 3.734375 -2.28125 3.921875 -2.21875 4.03125 -2.21875 C 4.1875 -2.21875 4.453125 -2.34375 4.453125 -2.703125 C 4.453125 -3.03125 4.21875 -3.265625 3.859375 -3.265625 C 3.390625 -3.265625 2.953125 -2.90625 2.609375 -2.609375 C 2.234375 -2.28125 2 -2.09375 1.71875 -1.984375 Z M 2.453125 -4.921875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-2"> <path style="stroke:none;" d="M 1.703125 -0.109375 C 1.703125 0.59375 1.328125 1 1.140625 1.171875 C 1.078125 1.25 1.046875 1.265625 1.046875 1.296875 C 1.046875 1.375 1.125 1.4375 1.171875 1.4375 C 1.28125 1.4375 1.953125 0.84375 1.953125 -0.0625 C 1.953125 -0.53125 1.75 -0.890625 1.40625 -0.890625 C 1.109375 -0.890625 0.953125 -0.671875 0.953125 -0.453125 C 0.953125 -0.234375 1.109375 0 1.40625 0 C 1.515625 0 1.625 -0.03125 1.703125 -0.109375 Z M 1.703125 -0.109375 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-3"> <path style="stroke:none;" d="M 2.25 -2.328125 C 2.296875 -2.453125 2.3125 -2.5 2.3125 -2.625 C 2.3125 -3.03125 1.9375 -3.265625 1.5625 -3.265625 C 0.8125 -3.265625 0.453125 -2.28125 0.453125 -2.125 C 0.453125 -2.09375 0.484375 -2.015625 0.578125 -2.015625 C 0.6875 -2.015625 0.71875 -2.0625 0.734375 -2.140625 C 0.9375 -2.84375 1.328125 -3.015625 1.546875 -3.015625 C 1.6875 -3.015625 1.734375 -2.921875 1.734375 -2.75 C 1.734375 -2.59375 1.640625 -2.34375 1.5625 -2.140625 C 1.265625 -1.40625 1.21875 -1.125 1.21875 -0.9375 C 1.21875 -0.21875 1.796875 0.078125 2.484375 0.078125 C 2.625 0.078125 3.03125 0.078125 3.359375 -0.4375 L 3.421875 -0.359375 C 3.5 -0.25 3.75 0.078125 4.453125 0.078125 C 5.8125 0.078125 6.15625 -2.359375 6.15625 -2.65625 C 6.15625 -3.1875 5.859375 -3.28125 5.765625 -3.28125 C 5.546875 -3.28125 5.34375 -3.0625 5.34375 -2.859375 C 5.34375 -2.71875 5.4375 -2.65625 5.5 -2.609375 C 5.6875 -2.46875 5.765625 -2.28125 5.765625 -2.046875 C 5.765625 -1.875 5.421875 -0.171875 4.484375 -0.171875 C 3.84375 -0.171875 3.84375 -0.6875 3.84375 -0.828125 C 3.84375 -1.015625 3.90625 -1.296875 3.96875 -1.53125 L 4.15625 -2.28125 L 4.265625 -2.6875 C 4.28125 -2.78125 4.3125 -2.921875 4.3125 -2.953125 C 4.3125 -3.03125 4.25 -3.1875 4.046875 -3.1875 C 3.78125 -3.1875 3.71875 -2.96875 3.65625 -2.703125 C 3.59375 -2.453125 3.328125 -1.421875 3.296875 -1.28125 C 3.28125 -1.171875 3.265625 -1.078125 3.265625 -0.921875 C 3.265625 -0.859375 3.265625 -0.828125 3.265625 -0.78125 C 3.046875 -0.328125 2.828125 -0.171875 2.515625 -0.171875 C 2 -0.171875 1.8125 -0.46875 1.8125 -0.859375 C 1.8125 -1.140625 1.921875 -1.515625 2.140625 -2.078125 Z M 2.25 -2.328125 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-4"> <path style="stroke:none;" d="M 8.15625 -4.171875 C 8.46875 -4.65625 8.734375 -4.734375 9 -4.765625 C 9.09375 -4.765625 9.1875 -4.78125 9.1875 -4.953125 C 9.1875 -5 9.140625 -5.0625 9.078125 -5.0625 C 8.921875 -5.0625 8.671875 -5.03125 8.484375 -5.03125 C 8.296875 -5.03125 7.828125 -5.0625 7.640625 -5.0625 C 7.59375 -5.0625 7.484375 -5.0625 7.484375 -4.875 C 7.484375 -4.765625 7.578125 -4.765625 7.640625 -4.765625 C 7.828125 -4.75 8.03125 -4.6875 8.03125 -4.515625 C 8.03125 -4.421875 7.96875 -4.3125 7.921875 -4.25 L 5.78125 -0.828125 L 5.3125 -4.578125 C 5.3125 -4.625 5.34375 -4.75 5.84375 -4.765625 C 5.921875 -4.765625 6.03125 -4.765625 6.03125 -4.953125 C 6.03125 -5 5.984375 -5.0625 5.90625 -5.0625 C 5.765625 -5.0625 5.59375 -5.03125 5.453125 -5.03125 C 5.296875 -5.03125 5.09375 -5.03125 4.9375 -5.03125 C 4.796875 -5.03125 4.671875 -5.03125 4.53125 -5.03125 C 4.40625 -5.03125 4.25 -5.0625 4.125 -5.0625 C 4.078125 -5.0625 3.96875 -5.0625 3.96875 -4.875 C 3.96875 -4.765625 4.0625 -4.765625 4.15625 -4.765625 C 4.28125 -4.765625 4.5 -4.765625 4.546875 -4.6875 C 4.5625 -4.640625 4.609375 -4.265625 4.609375 -4.21875 C 4.609375 -4.203125 4.609375 -4.1875 4.5625 -4.109375 L 2.515625 -0.828125 L 2.03125 -4.578125 C 2.078125 -4.65625 2.125 -4.765625 2.5625 -4.765625 C 2.65625 -4.765625 2.765625 -4.765625 2.765625 -4.953125 C 2.765625 -4.984375 2.75 -5.0625 2.640625 -5.0625 C 2.5 -5.0625 2.328125 -5.03125 2.1875 -5.03125 C 2.03125 -5.03125 1.828125 -5.03125 1.671875 -5.03125 C 1.53125 -5.03125 1.40625 -5.03125 1.265625 -5.03125 C 1.140625 -5.03125 0.984375 -5.0625 0.859375 -5.0625 C 0.703125 -5.0625 0.6875 -4.90625 0.6875 -4.859375 C 0.71875 -4.765625 0.78125 -4.765625 0.90625 -4.765625 C 1.28125 -4.765625 1.28125 -4.71875 1.296875 -4.53125 L 1.859375 -0.046875 C 1.890625 0.09375 1.90625 0.15625 2.046875 0.15625 C 2.171875 0.15625 2.234375 0.09375 2.296875 -0.015625 L 4.671875 -3.8125 L 5.140625 -0.046875 C 5.15625 0.09375 5.171875 0.15625 5.3125 0.15625 C 5.4375 0.15625 5.484375 0.109375 5.546875 0 Z M 8.15625 -4.171875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph6-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph6-1"> <path style="stroke:none;" d="M 2.328125 -3.484375 C 2.375 -3.59375 2.390625 -3.671875 2.390625 -3.71875 C 2.390625 -3.953125 2.1875 -4.140625 1.9375 -4.140625 C 1.640625 -4.140625 1.5625 -3.875 1.53125 -3.78125 L 0.484375 -0.609375 C 0.46875 -0.5625 0.453125 -0.5 0.453125 -0.46875 C 0.453125 -0.34375 0.75 -0.28125 0.75 -0.28125 C 0.828125 -0.28125 0.859375 -0.359375 0.875 -0.421875 Z M 2.328125 -3.484375 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-1"> <path style="stroke:none;" d="M 2.953125 1.671875 C 2.140625 1.046875 1.46875 -0.125 1.46875 -1.84375 C 1.46875 -3.5625 2.125 -4.75 2.953125 -5.375 C 2.953125 -5.390625 2.96875 -5.40625 2.96875 -5.453125 C 2.96875 -5.5 2.9375 -5.546875 2.84375 -5.546875 C 2.71875 -5.546875 0.9375 -4.390625 0.9375 -1.84375 C 0.9375 0.6875 2.703125 1.84375 2.84375 1.84375 C 2.9375 1.84375 2.96875 1.796875 2.96875 1.75 C 2.96875 1.703125 2.953125 1.6875 2.953125 1.671875 Z M 2.953125 1.671875 "></path> </symbol> <symbol overflow="visible" id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-2"> <path style="stroke:none;" d="M 2.625 -1.84375 C 2.625 -4.390625 0.84375 -5.546875 0.703125 -5.546875 C 0.609375 -5.546875 0.59375 -5.484375 0.59375 -5.453125 C 0.59375 -5.40625 0.59375 -5.40625 0.734375 -5.28125 C 1.40625 -4.703125 2.09375 -3.609375 2.09375 -1.84375 C 2.09375 -0.28125 1.546875 0.890625 0.703125 1.59375 C 0.59375 1.703125 0.59375 1.703125 0.59375 1.75 C 0.59375 1.78125 0.609375 1.84375 0.703125 1.84375 C 0.84375 1.84375 2.625 0.6875 2.625 -1.84375 Z M 2.625 -1.84375 "></path> </symbol> </g> <clipPath id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-clip1"> <path d="M 19 51 L 30 51 L 30 58 L 19 58 Z M 19 51 "></path> </clipPath> </defs> <g id="d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-1" x="13.647113" y="17.818884"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-2" x="93.731439" y="17.818884"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-1" x="105.140324" y="17.818884"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-3" x="110.782789" y="17.818884"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-2" x="118.827516" y="17.818884"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-2" x="91.999033" y="85.2746"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-1" x="103.407919" y="85.2746"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph0-3" x="109.050383" y="85.2746"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph2-1" x="117.09511" y="79.896161"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph1-2" x="120.559673" y="85.2746"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 39.50661 20.820579 L 39.50661 -20.343626 " transform="matrix(0.991646,0,0,-0.991646,69.924997,47.841953)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.484702 2.870784 C -2.031655 1.149347 -1.019255 0.333945 0.00101451 -0.000869795 C -1.019241 -0.335725 -2.031609 -1.147228 -2.488526 -2.868684 " transform="matrix(-0.0000198329,0.991626,0.991626,0.0000198329,109.102425,68.2529)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-1" x="112.588576" y="50.312143"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph4-1" x="120.689083" y="50.312143"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-2" x="124.770945" y="50.312143"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph4-2" x="130.819984" y="50.312143"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -40.111659 34.012821 L 16.844632 34.012821 " transform="matrix(0.991646,0,0,-0.991646,69.924997,47.841953)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486318 2.868228 C -2.033315 1.146816 -1.020951 0.335349 -0.000709329 0.000520753 C -1.020951 -0.334308 -2.033315 -1.145774 -2.486318 -2.867187 " transform="matrix(0.991646,0,0,-0.991646,86.867891,14.113798)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-3" x="47.962025" y="7.444285"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-1" x="53.989992" y="9.187103"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-2" x="59.042583" y="9.187103"></use> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-3" x="61.855259" y="9.187103"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -40.111659 25.330916 L 21.433752 -20.536645 " transform="matrix(0.991646,0,0,-0.991646,69.924997,47.841953)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487365 2.868515 C -2.031714 1.149552 -1.021785 0.334952 -0.000934173 0.000134123 C -1.018321 -0.335541 -2.034799 -1.147096 -2.486002 -2.868801 " transform="matrix(0.795092,0.592528,0.592528,-0.795092,91.367851,68.348316)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph3-3" x="3.661239" y="53.347571"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-1" x="9.689206" y="56.187397"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph6-1" x="14.740402" y="54.060316"></use> </g> <g clip-path="url(#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-clip1)" clip-rule="nonzero"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-2" x="18.067374" y="56.187397"></use> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-4" x="20.88005" y="56.187397"></use> </g> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-1" x="30.609215" y="56.187397"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-1" x="34.176661" y="56.187397"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-2" x="39.227857" y="56.187397"></use> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-1" x="42.795568" y="56.187397"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph5-3" x="46.362748" y="56.187397"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#d4MjUOFVQ6SxGdgg2s-n8jWTQYs=-glyph7-2" x="53.180313" y="56.187397"></use> </g> </g> </svg> This is required to be universal in the sense that given every such diagram as above with domain <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>, there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">C \to L</annotation></semantics></math> making the diagrams commute.</p> <p>It is clear that when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is the constant functor sending everything to a singleton set, this recovers the usual notion of limit for <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> <h2 id="motivation_from_enriched_category_theory">Motivation from enriched category theory</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</p> <p>Imagine you’re tasked to write down the definition of <em><a class="existingWikiWord" href="/nlab/show/limit">limit</a></em> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. You would start saying there is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F \colon K \to C</annotation></semantics></math> and a limit is a <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over it, i.e. it’s the universal choice of an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> together with an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo lspace="verythinmathspace">:</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_k \colon c \to F(k)</annotation></semantics></math> for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>Here’s where you stop and ask yourself: what is ‘an arrow’ in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>? <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has no <em><a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a></em> — it has <em><a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a></em> — hence what’s ‘an element’ of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(c, F(k))</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>?</p> <p>There are two ways to specify an element of an object <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> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>I</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, I, \otimes)</annotation></semantics></math>:</p> <ol> <li>Give an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I \to X</annotation></semantics></math> (think of sets, where elements of <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> are indeed the same thing as arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\{*\} \to X</annotation></semantics></math>. These are called <a class="existingWikiWord" href="/nlab/show/global+elements">global elements</a> of <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>, and are more often than not a misbehaved notion of element, since often <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is ‘too big’ to thoroughly probe <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> (on the other hand, notice the <a class="existingWikiWord" href="/nlab/show/enriched+category#passage_between_ordinary_categories_and_enriched_categories">underlying category of an enriched category</a> is defined by taking global elements of the hom-objects)</li> <li>Give <em>any</em> arrow into <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>. These are called <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a>, and the existence of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> assures us they completely capture the categorical structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</li> </ol> <p>Hence you now say: a cone over <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 a choice of a <em>generalized element</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(c, F(k))</annotation></semantics></math>, for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. This means specifying an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>k</mi></msub><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W_k \to C(c, F(k))</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. It’s now quite natural to ask for the functoriality of this choice in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, hence we end up defining a ‘generalized cone’ over <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> as an element</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><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [K, V](W(-), C(c, F(-))) </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is simply a uniform way to specify the sides of a cone. A confirmation that this is indeed the right definition of limit in the enriched settings come from the fact that conical completeness (a <a class="existingWikiWord" href="/nlab/show/conical+limit">conical limit</a> is one where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>=</mo><mi>Δ</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">W = \Delta I</annotation></semantics></math>, hence we pick only global element) is an inadequate notion, see for example Section 3.9 in <a href="#Kelly">Kelly’s book</a> (aptly named <em>The inadequacy of conical limits</em>).</p> <h2 id="examples">Examples</h2> <h3 id="homotopy_limits">Homotopy limits</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> some category of higher structures, the <em>local</em> definition of <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> over a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : K \to C</annotation></semantics></math> replaces the ordinary notion of <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over <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> by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.</p> <p>One can convince oneself that for the choice of <a class="existingWikiWord" href="/nlab/show/SimpSet">SimpSet</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> this is realized in terms of the weighted limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>lim</mi> <mi>W</mi></msup><mi>F</mi></mrow><annotation encoding="application/x-tex">lim^W F</annotation></semantics></math> with the weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> taken to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><mi>K</mi><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Simp</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex"> W : K \to \Simp\Set </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><mi>k</mi><mo>↦</mo><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">/</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> W : k \mapsto N(K/k) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">/</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">K/k</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">/</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(K/k)</annotation></semantics></math> denotes its <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>.</p> <p>This leads to the classical definition of homotopy limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Simp</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">\Simp\Set</annotation></semantics></math>-enriched categories due to</p> <ul> <li>A.K. Bousfield and D.M. Kan, <em>Homotopy limits, completions, and localizations</em></li> </ul> <p>See for instance also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marc+Cordier">Jean-Marc Cordier</a> and <a class="existingWikiWord" href="/nlab/show/Timothy+Porter">Timothy Porter</a>, <em>Homotopy Coherent Category Theory</em>, Trans. Amer. Math. Soc. 349 (1997) 1-54, (<a href="http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicola+Gambino">Nicola Gambino</a>, <em>Weighted limits in simplicial homotopy theory</em> (<a href="http://www.crm.cat/Publications/08/Pr790.pdf">pdf</a> or <a href="http://www.math.unipa.it/%7Engambino/Research/Papers/weighted.pdf">pdf</a>)</p> </li> </ul> <p>In some nice cases the weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">/</mo><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(K/-)</annotation></semantics></math> can be replaced by a simpler weight; an example is discussed at <a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a>.</p> <h3 id="homotopy_pullback">Homotopy pullback</h3> <p>For instance in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mo stretchy="false">{</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo>←</mo><mi>s</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">K = \{r \to t \leftarrow s\}</annotation></semantics></math> is the shape of <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagrams we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>r</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> W(r) = \{r\} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>s</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> W(s) = \{s\} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo>←</mo><mi>s</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> W(t) = N( \{r \to t \leftarrow s\} ) </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">{</mo><mi>r</mi><mo stretchy="false">}</mo><mo>→</mo><mo stretchy="false">{</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo>←</mo><mi>s</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}</annotation></semantics></math> injects the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo>←</mo><mi>s</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{r \to t \leftarrow s\}</annotation></semantics></math> and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>s</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(s \to t)</annotation></semantics></math>.</p> <p>This implies that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : K \to C</annotation></semantics></math> a pullback diagram in the <a class="existingWikiWord" href="/nlab/show/SimpSet">SimpSet</a>-enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-weighted <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over <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> with tip some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math>, i.e. a natural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⇒</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> W \Rightarrow C(c, F(-)) </annotation></semantics></math></div> <p>is</p> <ul> <li> <p>over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> a “morphism” from the tip <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(r)</annotation></semantics></math> (i.e. a vertex in the Hom-simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(c,F(r))</annotation></semantics></math>);</p> </li> <li> <p>similarly over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>;</p> </li> <li> <p>over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> three “morphisms” from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(t)</annotation></semantics></math> together with 2-cells between them (i.e. a 2-<a class="existingWikiWord" href="/nlab/show/horn">horn</a> in the Hom-simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(c,F(t))</annotation></semantics></math>)</p> </li> <li> <p>such that the two outer morphisms over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> are identified with the morphisms over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>, respectively, postcomposed with the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(r \to t)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(s \to t)</annotation></semantics></math>, respectively.</p> </li> </ul> <p>So in total such a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-weighted cone looks like</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo>⇒</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇐</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>r</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo>→</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &&& c \\ & \swarrow &\Rightarrow& \downarrow &\Leftarrow& \searrow \\ F(r) && \stackrel{F(r \to t)}{\to} & F(t) & \stackrel{F(s \to t)}{\leftarrow} && F(s) } </annotation></semantics></math></div> <p>as one would expect for a “homotopy cone”.</p> <h1 id="related_pages">Related pages</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/saturated+class+of+limits">saturated class of limits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+colimit">weighted colimit</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The notion of weighted limits was introduced (under the name “<em>mean cotensor product</em>”) in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, p. 10 of: <em>Une notion de <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-limite</em>, in <em>Colloque sur l’algèbre des catégories. Amiens 1973. Résumés des conférences</em>, Cahiers de topologie et géométrie différentielle <strong>14</strong> 2 (1973) 153-223 [<a href="http://www.numdam.org/item/CTGDC_1973__14_2_153_0">numdam:CTGDC_1973__14_2_153_0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Maxwell+Kelly">Gregory Maxwell Kelly</a>, <em>A notion of limit for enriched categories</em>, Bulletin of the Australian Mathematical Society <strong>12</strong> 1 (1975) 49-72 [<a href="https://doi.org/10.1017/S0004972700023637">doi:10.1017/S0004972700023637</a>]</p> </li> </ul> <p>and, independently, (under the name “<em>Hom (formel)</em>”) by:</p> <ul> <li>C. Auderset, <em>Adjonction et monade au niveau des 2-categories</em>, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. (<a href="http://www.numdam.org/item/CTGDC_1974__15_1_3_0/">numdam</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Kelly"> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a href="http://www.emis.de/journals/TAC/reprints/articles/10/tr10.pdf#page=37">section 3.1, p. 37</a> in: <em>Basic concepts of enriched category theory</em>, London Math. Soc. Lec. Note Series <strong>64</strong>, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories, <strong>10</strong> (2005) 1-136 [<a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029">ISBN:9780521287029</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html">tac:tr10</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, §6.6 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Vol. 2: <em>Categories and Structures</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a>]</p> </li> </ul> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>Weighted limits and colimits</em> (2008) (<a href="http://www.math.jhu.edu/~eriehl/weighted.pdf">pdf</a>)</li> </ul> <p>is given an account of lectures by <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a> on the subject. The definition appears there as <a href="http://www.math.jhu.edu/~eriehl/weighted.pdf#page=4">definition 3.1, p. 4</a> (in a form a bit more general than the one above).</p> <h3 id="presenting_homotopy_limits">Presenting homotopy limits</h3> <p>On weighted limits as presentations of <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a>:</p> <ul> <li id="Hirschhorn02"><a class="existingWikiWord" href="/nlab/show/Philip+Hirschhorn">Philip Hirschhorn</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories+and+Their+Localizations">Model Categories and Their Localizations</a></em>, AMS Math. Survey and Monographs <strong>99</strong> (2002) [<a href="https://bookstore.ams.org/surv-99-s/">ISBN:978-0-8218-4917-0</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf">pdf</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>]</li> </ul> <p>To compare with the above discussion notice that</p> <ul> <li> <p>The functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">/</mo><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> W \;\coloneqq\; N(K/-) </annotation></semantics></math></div> <p>is discussed there in definition 14.7.8 on p. 269.</p> </li> <li> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched hom-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[K,V]</annotation></semantics></math> which on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S,T</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/end">end</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow></msub><mi>V</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>T</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[K,V](S,T) = \int_{k \in K} V(S(k), T(k))</annotation></semantics></math> appears as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>hom</mi> <mi>K</mi></msup><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom^K(S,T)</annotation></semantics></math> in definition 18.3.1 (see bottom of the page).</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> set to <a class="existingWikiWord" href="/nlab/show/SimpSet">SimpSet</a> the above definition of homotopy limit appears in example 18.3.6 (2).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, §6.6 and Chapter 7 in: <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [<a href="https://doi.org/10.1017/CBO9781107261457">doi:10.1017/CBO9781107261457</a>, <a href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">pdf</a>]</p> </li> </ul> <p>Discussion of weighted <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-limit"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-limits</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martina+Rovelli">Martina Rovelli</a>, <em>Weighted limits in an (∞,1)-category</em> (2019) [<a href="https://arxiv.org/abs/1902.00805">arxiv:1902.00805</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 10, 2023 at 15:21:57. See the <a href="/nlab/history/weighted+limit" 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/weighted+limit" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11870/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/weighted+limit/46" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/weighted+limit" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/weighted+limit" accesskey="S" class="navlink" id="history" rel="nofollow">History (46 revisions)</a> <a href="/nlab/show/weighted+limit/cite" style="color: black">Cite</a> <a href="/nlab/print/weighted+limit" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/weighted+limit" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>