CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> category of V-enriched categories in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> category of V-enriched categories </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/4451/#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>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="categories_of_categories">Categories of categories</h4> <div class="hide"><div> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1,r+1)</annotation></semantics></math>-categories of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rel">Rel</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28Cat%29">Ho(Cat)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PrCat">PrCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/LexCat">LexCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/VCat">VCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CatAdj">CatAdj</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Prof">Prof</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Operad">Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ModCat">ModCat</a>, <a class="existingWikiWord" href="/nlab/show/CombModCat">CombModCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </li> </ul> </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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#possible_contexts'>Possible Contexts</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#extra_structure'>Extra structure</a></li> <ul> <li><a href='#MonoidalStructure'>Monoidal structure</a></li> <li><a href='#Involutions'>Involutions</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <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">\mathcal{V}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> and <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> <a class="existingWikiWord" href="/nlab/show/cosmos+for+enrichment">cosmos for enrichment</a> there is a <a class="existingWikiWord" href="/nlab/show/2-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>2</mn> </mrow> <annotation encoding="application/x-tex">2</annotation> </semantics> </math>-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\mathcal{V} Cat</annotation></semantics></math> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a>; and</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphism"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>2</mn> </mrow> <annotation encoding="application/x-tex">2</annotation> </semantics> </math>-morphisms</a> 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+natural+transformations">enriched natural transformations</a>.</p> </li> </ul> <p>Sometimes one also considers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\mathcal{V} Cat</annotation></semantics></math> as a mere <a class="existingWikiWord" href="/nlab/show/category">category</a> by dropping the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-morphisms (and using enriched <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a>).</p> <h2 id="possible_contexts">Possible Contexts</h2> <ul> <li> <p><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">\mathcal{V}</annotation></semantics></math> can be a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with underlying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math></p> </li> <li> <p><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">\mathcal{V}</annotation></semantics></math> can be a <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a> with underlying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math></p> </li> <li> <p><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">\mathcal{V}</annotation></semantics></math> can be a <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> with underlying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math></p> </li> <li> <p><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">\mathcal{V}</annotation></semantics></math> can be a <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a> with underlying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math></p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mo>=</mo><mo stretchy="false">(</mo></mrow><annotation encoding="application/x-tex">\mathcal{V} = (</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Set">Set</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>,</mo><mo>×</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">, \times)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">\mathcal{V}Cat \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category of <a class="existingWikiWord" href="/nlab/show/locally+small+categories">locally small categories</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mo>=</mo><mo stretchy="false">(</mo></mrow><annotation encoding="application/x-tex">\mathcal{V} = (</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>,</mo><mo>×</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">, \times)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">\mathcal{V}Cat \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/2Cat">Str2Cat</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category of <a class="existingWikiWord" href="/nlab/show/strict+2-categories">strict 2-categories</a>.</p> </li> </ul> <h2 id="extra_structure">Extra structure</h2> <h3 id="MonoidalStructure">Monoidal structure</h3> <p>If <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">\mathcal{V}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}Cat</annotation></semantics></math> becomes itself a (<a class="existingWikiWord" href="/nlab/show/very+large+category">very large</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> given by forming <a class="existingWikiWord" href="/nlab/show/enriched+product+categories">enriched product categories</a>.</p> <p>If <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">\mathcal{V}</annotation></semantics></math> is in addition <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> and <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}Cat</annotation></semantics></math> becomes itself a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> given by forming <a class="existingWikiWord" href="/nlab/show/enriched+functor+categories">enriched functor categories</a>.</p> <p>[<a href="#Kelly82">Kelly (1982), §2.3</a>]</p> <h3 id="Involutions">Involutions</h3> <ul> <li>If <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">\mathcal{V}</annotation></semantics></math> is a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> structure, then <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">\mathcal{V}</annotation></semantics></math>Cat has a <a class="existingWikiWord" href="/nlab/show/unit+enriched+category">unit object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">\mathcal{I}</annotation></semantics></math>, and a designated lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">]</mo> <mi>op</mi></msup><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mo>−</mo></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">[\mathcal{I},-]^{op}\stackrel{{}^-_0L}{\Rightarrow}[[\mathcal{I},-],\mathcal{V}_0]\colon\mathcal{V}</annotation></semantics></math>Cat<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math>Cat, where the former is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor flipping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-morphisms, and the latter is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor flipping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-morphisms (c.f. <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a>). For the sake of simplicity, we note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{I},-]</annotation></semantics></math> is simply the forgetful <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor from <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">\mathcal{V}</annotation></semantics></math>Cat to Cat, and hence abbreviate it as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(-)_0</annotation></semantics></math>. Then the above lax natural transformation is given by the following data:</li> </ul> <ol> <li> <p>For every object (i.e. <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">\mathcal{V}</annotation></semantics></math>-enriched category) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> 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">\mathcal{V}</annotation></semantics></math>Cat, we have to give a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup><mover><mo>→</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><msub><mi>𝒜</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}_0^{op}\stackrel{{}^{\mathcal{A}}_0L}{\rightarrow}[\mathcal{A}_0,\mathcal{V}_0]</annotation></semantics></math>. By the cartesian closed structure of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-category Cat, we define these to be the hom-functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup><mo>×</mo><mi>𝒜</mi><mover><mo>→</mo><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_0^{op}\times\mathcal{A}\stackrel{\mathcal{A}(-,-)}{\rightarrow}\mathcal{V}_0</annotation></semantics></math> which are defined in terms of the monoidal structure <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">\mathcal{V}</annotation></semantics></math> and 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">\mathcal{V}</annotation></semantics></math>-enrichment data of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(B,C)\stackrel{\mathcal{A}(f,g)}{\rightarrow}\mathcal{A}(A,D)</annotation></semantics></math> to be the composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>l</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><msup><mi>r</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><mi>I</mi><mo>⊗</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>I</mi><mover><mo>→</mo><mrow><mi>f</mi><mo>⊗</mo><mi>id</mi><mo>⊗</mo><mi>g</mi></mrow></mover><mi>𝒜</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mo>∘</mo> <mi>𝒜</mi></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></mover><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}{\to}\mathcal{A}(A,D)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-morphism (i.e. <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">\mathcal{V}</annotation></semantics></math>-enriched functor}) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mover><mo>→</mo><mi>F</mi></mover><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}</annotation></semantics></math>, we have to give a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>F</mi></msubsup><mi>L</mi></mrow><annotation encoding="application/x-tex">{}^F_0L</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup></mrow></mover></mtd> <mtd><msubsup><mi>ℬ</mi> <mn>0</mn> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi><mo stretchy="false">↓</mo></mtd> <mtd><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>F</mi></msubsup><mi>L</mi></mrow></mover></mtd> <mtd><mo stretchy="false">↓</mo><msup><mrow></mrow> <mi>ℬ</mi></msup><mi>L</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>𝒜</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>ℬ</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{A}_0^{op}&\stackrel{F_0^{op}}{\rightarrow}&\mathcal{B}_0^{op}\\ {}_0^{\mathcal{A}}L\downarrow&\stackrel{{}^F_0L}{\Rightarrow}&\downarrow{}^{\mathcal{B}}L\\ [\mathcal{A}_0,\mathcal{V}_0]&\stackrel{[F_0,\mathcal{V}_0]}{\leftarrow}&[\mathcal{B}_0,\mathcal{V}_0] } </annotation></semantics></math></div> <p>Since we have defined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi></mrow><annotation encoding="application/x-tex">{}_0^{\mathcal{A}}L</annotation></semantics></math> to be the hom-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(-,-)</annotation></semantics></math>, to give a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>F</mi></msubsup><mi>L</mi></mrow><annotation encoding="application/x-tex">{}^F_0L</annotation></semantics></math> is to give a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⇒</mo><mi>ℬ</mi><mo stretchy="false">(</mo><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup><mo>−</mo><mo>,</mo><msub><mi>F</mi> <mn>0</mn></msub><mo>−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup><mo>×</mo><mi>𝒜</mi><mo>→</mo><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}(-,-)\Rightarrow\mathcal{B}(F_0^{op}-,F_0-)\colon\mathcal{A}_0^{op}\times\mathcal{A}\to\mathcal{V}_0</annotation></semantics></math>. We thus define the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ob</mi><mo stretchy="false">(</mo><msub><mi>𝒜</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ob(\mathcal{A}_0)</annotation></semantics></math>-indexed family of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>FL</mi></msubsup><msub><mo></mo><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow></mover><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(A,B)\stackrel{{}_0^FL_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math> to be simply the family of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>F</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow></mover><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(A,B)\stackrel{F_{A,B}}{\rightarrow}\mathcal{B}(F_0A,F_0B)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math> defining 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">\mathcal{V}</annotation></semantics></math>-enriched functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mover><mo>→</mo><mi>F</mi></mover><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}</annotation></semantics></math>.</p> </li> <li> <p>The lax naturality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mo>−</mo></msubsup><mi>L</mi></mrow><annotation encoding="application/x-tex">{}^-_0L</annotation></semantics></math> says that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-morphism (i.e. a <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">\mathcal{V}</annotation></semantics></math>-enriched natural transformation) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mover><mo>⇒</mo><mi>α</mi></mover><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">F\stackrel{\alpha}{\Rightarrow} G\colon\mathcal{A}\to\mathcal{B}</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">\mathcal{V}</annotation></semantics></math>Cat, the natural transformations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>F</mi></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">{}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ F_0^{op}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>G</mi></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">{}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}_0^{\mathcal{B}}L\circ G_0</annotation></semantics></math> have to satisfy a compatibility condition with the natural transformations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>G</mi> <mn>0</mn> <mi>op</mi></msubsup><mover><mo>⇒</mo><mrow><msubsup><mi>α</mi> <mn>0</mn> <mi>op</mi></msubsup></mrow></mover><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">G_0^{op}\stackrel{\alpha_0^{op}}{\Rightarrow}F_0^{op}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mover><mo>⇒</mo><mrow><mo stretchy="false">[</mo><msub><mi>α</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[F_0,\mathcal{V}_0]\stackrel{[\alpha_0,\mathcal{V}_0]}{\Rightarrow}[G_0,\mathcal{V}_0]</annotation></semantics></math>. Explicitly, the condition is that the composite natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>F</mi></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup><mover><mo>⇒</mo><mrow><mo stretchy="false">[</mo><msub><mi>α</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>.</mo><mo stretchy="false">(</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msub><mi>F</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mover><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup><mo lspace="verythinmathspace">:</mo><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup><mo>→</mo><mo stretchy="false">[</mo><msub><mi>𝒜</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">{}_0^{\mathcal{A}}L\stackrel{{}^F_0L}{\Rightarrow}[F_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\stackrel{[\alpha_0,\mathcal{V}_0].({}^{\mathcal{B}}_0L\circ F_0)}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]</annotation></semantics></math> is the same as the composite natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>𝒜</mi></msubsup><mi>L</mi><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mi>G</mi></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msubsup><mi>G</mi> <mn>0</mn> <mi>op</mi></msubsup><mover><mo>⇒</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo stretchy="false">)</mo><mo>.</mo><msubsup><mi>α</mi> <mn>0</mn> <mi>op</mi></msubsup></mrow></mover><mo stretchy="false">[</mo><msub><mi>G</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>0</mn></msub><mo stretchy="false">]</mo><mo>∘</mo><msubsup><mrow></mrow> <mn>0</mn> <mi>ℬ</mi></msubsup><mi>L</mi><mo>∘</mo><msubsup><mi>F</mi> <mn>0</mn> <mi>op</mi></msubsup><mo lspace="verythinmathspace">:</mo><msubsup><mi>𝒜</mi> <mn>0</mn> <mi>op</mi></msubsup><mo>→</mo><mo stretchy="false">[</mo><msub><mi>𝒜</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">{}_0^{\mathcal{A}}L\stackrel{{}^G_0L}{\Rightarrow}[G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L\circ G_0^{op}\stackrel{([G_0,\mathcal{V}_0]\circ{}^{\mathcal{B}}_0L).\alpha_0^{op}}{\Rightarrow}[G_0,V_0]\circ{}^{\mathcal{B}}_0L\circ F_0^{op}\colon\mathcal{A}_0^{op}\to[\mathcal{A}_0,\mathcal{V}_0]</annotation></semantics></math>. Unraveling the condition leaves us with the requirement that for every pair of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒜</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_0</annotation></semantics></math> the following diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math> must commute:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>F</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>G</mi> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>B</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><msub><mi>G</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>ℬ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mi>A</mi></msub><mo>,</mo><msub><mi>G</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>ℬ</mi><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>0</mn></msub><mi>A</mi><mo>,</mo><msub><mi>G</mi> <mn>0</mn></msub><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{A}(A,B)&\stackrel{F_{A,B}}{\rightarrow}&\mathcal{B}(F_0A,F_0B)\\ G_{A,B}\downarrow&&\downarrow\mathcal{B}(F_0A,(\alpha_0)_{B})\\ \mathcal{B}(G_0A,G_0B)&\stackrel{\mathcal{B}((\alpha_0)_A,G_0B)}{\rightarrow}&\mathcal{B}(F_0A,G_0B) } </annotation></semantics></math></div> <p>But a <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">\mathcal{V}</annotation></semantics></math>-enriched natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is by definition a collection of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\alpha_0</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℬ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{B}_0</annotation></semantics></math> such that the above diagram commutes.</p> </li> </ol> <ul> <li>Supposing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}_0</annotation></semantics></math> was a self-enriched category, i.e. isomorphic to the underlying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒱</mi> <mn>0</mn> <mi>e</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{V}^e_0</annotation></semantics></math> of a <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">\mathcal{V}</annotation></semantics></math>-enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒱</mi> <mi>e</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{V}^e</annotation></semantics></math>, then it is natural to require that the above lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mover><mo>⇒</mo><mrow><msubsup><mrow></mrow> <mn>0</mn> <mo>−</mo></msubsup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{I},-]\stackrel{{}^-_0L}{\Rightarrow}[(-)_0,\mathcal{V}_0]</annotation></semantics></math> is in fact the whiskering of a lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mover><mo>⇒</mo><mrow><msup><mrow></mrow> <mo>−</mo></msup><mi>L</mi></mrow></mover><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>𝒱</mi> <mi>e</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{I},-]\stackrel{{}^-L}{\Rightarrow}[-,\mathcal{V}^e]</annotation></semantics></math> with the forgetful <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo><mo stretchy="false">[</mo><mi>ℐ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(-)_0=[\mathcal{I},-]</annotation></semantics></math>. Such a lax natural transformation should give us most (if not all) of the closed structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒱</mi> <mn>0</mn> <mi>e</mi></msubsup><mo>≅</mo><msub><mi>𝒱</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{V}^e_0\cong\mathcal{V}_0</annotation></semantics></math>…</li> </ul> <p>…</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1,r+1)</annotation></semantics></math>-categories of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rel">Rel</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28Cat%29">Ho(Cat)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PrCat">PrCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/LexCat">LexCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/VCat">VCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CatAdj">CatAdj</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Prof">Prof</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Operad">Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ModCat">ModCat</a>, <a class="existingWikiWord" href="/nlab/show/CombModCat">CombModCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </li> </ul> </div> <h2 id="references">References</h2> <ul> <li id="Kelly82"><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, Chapter 1 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>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/category">category</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on March 10, 2024 at 11:23:10. See the <a href="/nlab/history/category+of+V-enriched+categories" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/category+of+V-enriched+categories" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4451/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/category+of+V-enriched+categories/10" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/category+of+V-enriched+categories" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/category+of+V-enriched+categories" accesskey="S" class="navlink" id="history" rel="nofollow">History (10 revisions)</a> <a href="/nlab/show/category+of+V-enriched+categories/cite" style="color: black">Cite</a> <a href="/nlab/print/category+of+V-enriched+categories" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/category+of+V-enriched+categories" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>