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> closed monoidal structure on presheaves 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> closed monoidal structure on presheaves </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/8649/#Item_2" 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="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#definition_in_terms_of_homs_of_direct_images'>Definition in terms of homs of direct images</a></li> <li><a href='#relation_of_the_two_definitions'>Relation of the two definitions</a></li> <li><a href='#presheaves_over_a_monoidal_category'>Presheaves over a monoidal category</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>As every <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, a <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> is <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal</a>.</p> <h2 id="definition">Definition</h2> <div class="num_prop" id="CartesianClosedMonoidalnessOfCategoriesOfPresheaves"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closure</a> of <a class="existingWikiWord" href="/nlab/show/categories+of+presheaves">categories of presheaves</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}^{op}, Set]</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a>.</p> <p>This is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>, whose <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> is given objectwise in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]</annotation></semantics></math>, their <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X} \times \mathbf{Y}</annotation></semantics></math> exists and is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><msub><mi>c</mi> <mn>1</mn></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↑</mo> <mpadded width="0"><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>c</mi> <mn>2</mn></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) } </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a>, whose <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> is given for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]</annotation></semantics></math> by</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><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><msub><mi>c</mi> <mn>1</mn></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↑</mo> <mpadded width="0"><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>c</mi> <mn>2</mn></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> [\mathbf{X}, \mathbf{Y}] \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{Y} ) \\ {}^{ \mathllap{ f } }\big\downarrow && \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{Y} ) } } \\ c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{Y} ) } </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><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">Hom_{[\mathcal{C}^{op}, Set]}(-,-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> on the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a>.</p> </li> </ol> </div> <p>(e.g. <a href="#MacLaneMoerdijk">MacLane-Moerdijk, section I.6, pages 46-47</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>The first statement is a special case of the general fact that <a class="existingWikiWord" href="/nlab/show/limits+of+presheaves+are+computed+objectwise">limits of presheaves are computed objectwise</a>.</p> <p>For the second statement, first assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbf{X}, \mathbf{Y}]</annotation></semantics></math> does exist. Then by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">hom-adjunction isomorphism</a> we have for any other presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Z}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of the form</p> <div class="maruku-equation" id="eq:InternalHomIsoInPresheaves"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo>,</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) \;\simeq\; Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y}) \,. </annotation></semantics></math></div> <p>This holds in particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Z} = y(c)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> (i.e. in the <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>) and so the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> implies that if it exists, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbf{X}, \mathbf{Y}]</annotation></semantics></math> must have the claimed form:</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><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [\mathbf{X}, \mathbf{Y}](c) & \simeq Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] ) \\ & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} ) \,. \end{aligned} </annotation></semantics></math></div> <p>Hence it remains to show that this formula does make <a class="maruku-eqref" href="#eq:InternalHomIsoInPresheaves">(1)</a> hold generally.</p> <p>For this we use the equivalent definition of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> in terms of the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> providing a system of <a class="existingWikiWord" href="/nlab/show/universal+arrows">universal arrows</a>.</p> <p>Define a would-be <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a>, which here is called an <em><a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a></em> morphism, by</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><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>ev</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ev</mi> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>ϕ</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{X} \times [\mathbf{X} , \mathbf{Y}] &\overset{ev}{\longrightarrow}& \mathbf{Y} \\ \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y}) &\overset{ev_c}{\longrightarrow}& \mathbf{Y}(c) \\ (x, \phi) &\mapsto& \phi_c( id_c, x ) } </annotation></semantics></math></div> <p>Then it remains to show that for every morphism of presheaves of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>A</mi></mstyle><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mi>f</mi><mphantom><mi>A</mi></mphantom></mrow></mover><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y} </annotation></semantics></math> there is a <em>unique</em> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo>⟶</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}]</annotation></semantics></math> such that</p> <div class="maruku-equation" id="eq:UniversalArrowConditionForEvaluationMapInPresheaves"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>A</mi></mstyle></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mover><mi>f</mi><mo>˜</mo></mover></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>×</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mpadded width="0"><mi>f</mi></mpadded></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>ev</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{X} \times \mathbf{A} && \overset{ \mathbf{X} \times \widetilde f }{\longrightarrow} && \mathbf{X} \times [\mathbf{X}, \mathbf{Y}] \\ & {}_{\mathllap{ \mathrlap{f} }}\searrow && \swarrow_{ \mathrlap{ ev } } \\ && \mathbf{Y} } </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutativity</a> of this diagram means in components at <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 \mathcal{C}</annotation></semantics></math> that, that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in \mathbf{X}(c)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \in \mathbf{A}(c)</annotation></semantics></math> we have</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><msub><mi>ev</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>f</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned} </annotation></semantics></math></div> <p>Hence this fixes the component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\widetilde f_c(a)_c</annotation></semantics></math> when its first argument is the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">id_c</annotation></semantics></math>. But let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">g \;\colon\; d \to c</annotation></semantics></math> be any morphism and chase <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(id_c, x )</annotation></semantics></math> through the naturality diagram for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde f_c(a)</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><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mpadded width="0"><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>d</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>AAAA</mi></mphantom><mrow><mtable><mtr><mtd><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>c</mi></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><msup><mi>g</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>g</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c) &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}& \mathbf{Y}(c) \\ {}^{\mathllap{ g^\ast }}\big\downarrow && \big\downarrow^{\mathrlap{ g^\ast }} \\ Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d) &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}& \mathbf{Y}(d) } \phantom{AAAA} \array{ \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \} \\ \big\downarrow && \big\downarrow \\ \{ (g, g^\ast(x)) \} &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \} } </annotation></semantics></math></div> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">(\widetilde f_c(a))_d</annotation></semantics></math> is fixed to be given by</p> <div class="maruku-equation" id="eq:ComponentFormulaForEvaluationMapInPresheaves"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo>,</mo><msup><mi>g</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) ) </annotation></semantics></math></div> <p>at least on those pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,x')</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x'</annotation></semantics></math> is in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">g^\ast</annotation></semantics></math>.</p> <p>But, finally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">(\widetilde f_c(a))_d</annotation></semantics></math> is also natural 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>c</mi></msub></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mpadded width="0"><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mover><mi>f</mi><mo>˜</mo></mover> <mi>d</mi></msub></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant="bold"><mi>Y</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{A}(c) &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c) \\ {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}} \\ \mathbf{A}(d) &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d) } </annotation></semantics></math></div> <p>which implies that <a class="maruku-eqref" href="#eq:ComponentFormulaForEvaluationMapInPresheaves">(3)</a> must hold generally. Hence naturality implies that <a class="maruku-eqref" href="#eq:UniversalArrowConditionForEvaluationMapInPresheaves">(2)</a> indeed has a unique solution.</p> </div> <h2 id="definition_in_terms_of_homs_of_direct_images">Definition in terms of homs of direct images</h2> <p>Often another, equivalent, expression is used to express the internal hom of presheaves:</p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/site">pre-site</a> with underlying <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><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math>. Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/site">site</a> that just means that we have a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math> on which we consider <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>S</mi> <mi>X</mi> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F \in PSh(S_X) := [S_X^{op}, Set]</annotation></semantics></math>, but that it suggests that</p> <ul> <li> <p>to each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in PSh(X)</annotation></semantics></math> and in particular to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>S</mi> <mi>X</mi></msub><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in S_X \hookrightarrow PSh(X)</annotation></semantics></math> there is naturally associated the pre-site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> with underlying category the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>U</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_U = (Y/Y(U))</annotation></semantics></math>;</p> </li> <li> <p>that the canonical <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">forgetful</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow> <mi>t</mi></msubsup><mo>:</mo><msub><mi>S</mi> <mi>U</mi></msub><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">j^t_{U \to X} : S_U \to S_X</annotation></semantics></math>, which can be thought of as a <a class="existingWikiWord" href="/nlab/show/site">morphism of pre-sites</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">j_{U \to X} : X \to U</annotation></semantics></math> induces the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow></msub><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j_{U \to X})_* : PSh(X) \to PSh(U)</annotation></semantics></math> which we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>↦</mo><mi>F</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">F \mapsto F|_U</annotation></semantics></math>.</p> </li> </ul> <p>Then in these terms the above <strong>internal hom</strong> for presheaves</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hom : PSh(X)^{op} \times PSh(X) \to PSh(X) </annotation></semantics></math></div> <p>is expressed for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F,G \in PSh(X)</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>,</mo><mi>G</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hom(F,G) = U \mapsto Hom_{PSh(U)}(F|_U, G|_U) \,. </annotation></semantics></math></div> <h2 id="relation_of_the_two_definitions">Relation of the two definitions</h2> <p>To see the equivalence of the two definitions, notice</p> <ul> <li>that by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">S_U</annotation></semantics></math> is simply the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>U</mi></msub><mo>=</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">/</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">S_U = S_X/U</annotation></semantics></math>;</li> <li>by the general properties of <a class="existingWikiWord" href="/nlab/show/functors+and+comma+categories">functors and comma categories</a> there is an equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S_X/U) \simeq PSh(S_X)/y(U)</annotation></semantics></math>;</li> <li>which identifies the 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><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>U</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)|_U : PSh(S_X) \to PSh(S_U)</annotation></semantics></math> with the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((-)\times y(U) \stackrel{p_2}{\to} y(U)) : PSh(S_X) \to PSh(S_X)/y(U)</annotation></semantics></math>;</li> <li>and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo>,</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{PSh(S_X)/y(U)}(y(U) \times F, y(U) \times G) \simeq Hom_{PSh(S_X)}(y(U) \times F, G)</annotation></semantics></math>.</li> </ul> <h2 id="presheaves_over_a_monoidal_category">Presheaves over a monoidal category</h2> <p>It is worth noting that in the case where <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> is itself 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>X</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \otimes, I)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Psh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Psh(X)</annotation></semantics></math> is equipped with another (bi)closed monoidal structure given by the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> product and its componentwise right adjoints. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be two presheaves over <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>. Their tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⋆</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F \star G</annotation></semantics></math> can be defined by the following <a class="existingWikiWord" href="/nlab/show/coend">coend</a> formula:</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>⋆</mo><mi>G</mi><mo>=</mo><mi>U</mi><mo>↦</mo><msup><mo>∫</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow></msup><msub><mi>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F\star G = U \mapsto \int^{U_1,U_2\in X} Hom_X(U, U_1\otimes U_2) \times F(U_1) \times G(U_2)</annotation></semantics></math></div> <p>Then we can define two right adjoints</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>⋆</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊣</mo><mi>F</mi><mo>\</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mspace width="2em"></mspace><mo>−</mo><mo>⋆</mo><mi>G</mi><mo>⊣</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F\star - \dashv F \backslash - \qquad -\star G \dashv - / G </annotation></semantics></math></div> <p>by the following <a class="existingWikiWord" href="/nlab/show/end">end</a> formulas:</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>\</mo><mi>H</mi><mo>=</mo><mi>V</mi><mo>↦</mo><msub><mo>∫</mo> <mrow><mi>U</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>H</mi><mo stretchy="false">(</mo><mi>U</mi><mo>⊗</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \backslash H = V \mapsto \int_{U\in X} F(U) \to H(U\otimes V)</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>H</mi><mo stretchy="false">/</mo><mi>G</mi><mo>=</mo><mi>U</mi><mo>↦</mo><msub><mo>∫</mo> <mrow><mi>V</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>G</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>H</mi><mo stretchy="false">(</mo><mi>U</mi><mo>⊗</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H / G = U \mapsto \int_{V\in X} G(V) \to H(U\otimes V)</annotation></semantics></math></div> <p>In the case where the monoidal structure on <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> is cartesian, the induced closed monoidal structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Psh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Psh(X)</annotation></semantics></math> coincides with the cartesian closed structure described in the previous sections.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li>The cartesian closure restricts from presheaves to <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a> (e.g. <a href="#MacLaneMoerdijk">MacLane-Moerdijk, section III.6, p. 136-138</a>)</li> </ul> <h2 id="references">References</h2> <p>The first definition is discussed for instance in section I.6 of</p> <ul> <li id="MacLaneMoerdijk"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></li> </ul> <p>The second definition is discussed for instance in section 17.1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2020 at 16:04:50. See the <a href="/nlab/history/closed+monoidal+structure+on+presheaves" 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/closed+monoidal+structure+on+presheaves" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8649/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/closed+monoidal+structure+on+presheaves/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/closed+monoidal+structure+on+presheaves" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/closed+monoidal+structure+on+presheaves" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/closed+monoidal+structure+on+presheaves/cite" style="color: black">Cite</a> <a href="/nlab/print/closed+monoidal+structure+on+presheaves" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/closed+monoidal+structure+on+presheaves" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>