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> restriction and extension of sheaves 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> restriction and extension of sheaves </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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <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='#remarks'>Remarks</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Recall that for <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> <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> (with underlying 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>) with values in a 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">A</annotation></semantics></math> that admits small <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and small colimits (so in particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">A =</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><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>S</mi> <mi>X</mi> <mi>op</mi></msubsup><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">PSh(X, A) = [S_X^{op}, A]</annotation></semantics></math>, every functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mi>t</mi></msup><mo>:</mo><msub><mi>S</mi> <mi>Y</mi></msub><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">f^t : S_Y \to S_X</annotation></semantics></math> induces three functors of presheaf catgeories:</p> <table><thead><tr><th>Notation</th><th></th><th>Definition</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mi>t</mi></msup><msub><mo stretchy="false">)</mo> <mo>*</mo></msub><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^t)_* : PSh(X,A) \to PSh(Y,A)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">direct image</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mi>t</mi></msup><msup><mo stretchy="false">)</mo> <mo>†</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^t)^\dagger : PSh(Y,A) \to PSh(X,A)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">left adjoint to direct image</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mi>t</mi></msup><msup><mo stretchy="false">)</mo> <mo>‡</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^t)^\ddagger : PSh(Y,A) \to PSh(X,A)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">right adjoint to direct image</td></tr> </tbody></table> <p>Recall moreover that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/site">morphism of sites</a>, the left adjoint to <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> followed by <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{(-)}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> map of sheaves:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^{-1} : Sh(Y,A) \to Sh(X,A) \,. </annotation></semantics></math></div> <p>Now, if the morphism of sites <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> happens to be restriction to a sub-site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f : X \to U</annotation></semantics></math> with <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>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in PSh(X,A)</annotation></semantics></math> with <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> carrying the induced <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a>, then</p> <ul> <li> <p>the direct image is called <strong>restriction</strong> of sheaves;</p> </li> <li> <p>the right adjoint takes sheaves to sheaves and is called <strong>extension</strong> of sheaves.</p> </li> </ul> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/site">site</a> <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> 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> and given a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <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> with the induced sub-site <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> corresponding to the <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">forgetful</a> functor <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><mo stretchy="false">(</mo><msub><mi>Y</mi> <mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">j^t_{U \to X} : (Y_{S_X}/U) \to S_X</annotation></semantics></math> from 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><msub><mi>Y</mi> <mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_U = (Y_{S_X}/U) \to S_X</annotation></semantics></math> underlying the <a class="existingWikiWord" href="/nlab/show/site">site</a> <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> (as discussed at <a class="existingWikiWord" href="/nlab/show/site">site</a>) the right <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow> <mo>‡</mo></msubsup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>U</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"> j^{\ddagger}_{U \to X} : PSh(U) \to PSh(X) </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> or, in this case, <strong>restriction</strong> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><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>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (j_{U \to X})_* : Sh(X) \to Sh(U) </annotation></semantics></math></div> <p>whose action may suggestively be denoted</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><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>F</mi><mo>↦</mo><mi>F</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex"> (j_{U \to X})_* : F \mapsto F|_U </annotation></semantics></math></div> <p>happens to take sheaves to sheaves (when <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> is equipped with the canonical induced topology as described at <a class="existingWikiWord" href="/nlab/show/site">site</a>):</p> <p>one calls</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow> <mo>‡</mo></msubsup><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X) </annotation></semantics></math></div> <p>the <strong>extension</strong> of sheaves on <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> to sheaves 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>.</p> <p>To summarize notation and terminology:</p> <table><thead><tr><th>Terminology</th><th></th><th>Notation</th><th></th><th>Definition</th></tr></thead><tbody><tr><td style="text-align: left;">morphism of sites</td><td style="text-align: left;"></td><td style="text-align: left;"><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></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">underlying functor</td><td style="text-align: left;"></td><td style="text-align: left;"><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><mo stretchy="false">(</mo><msub><mi>Y</mi> <mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow></msub><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">j^t_{U \to X} : (Y_{S_X}/U) \to S_X </annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">sheaf restriction</td><td style="text-align: left;"></td><td style="text-align: left;"><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>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j_{U \to X})_* : Sh(X) \to Sh(U)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">direct image</td></tr> <tr><td style="text-align: left;">sheaf extension</td><td style="text-align: left;"></td><td style="text-align: left;"><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> <mo>‡</mo></msubsup><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">right adjoint to direct image</td></tr> <tr><td style="text-align: left;">sheaf inverse image</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><msup><mi>j</mi> <mi>t</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow> <mo>†</mo></msubsup></mrow><mo>¯</mo></mover><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{(j^t)^{\dagger}_{U \to X}} : Sh(U) \to Sh(X)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">left adjoint to direct image followed by sheafification</td></tr> </tbody></table> <h2 id="remarks">Remarks</h2> <p>Notice the difference to the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> operation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>j</mi> <mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> j^{-1}_{U \to X} : Sh(U) \to Sh(X) \,. </annotation></semantics></math></div> <h2 id="References">References</h2> <p>For instance section 17.6 of</p> <ul> <li> <p>Kashiwara, Schapira, <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stacks+Project">Stacks Project</a>, <a href="http://stacks.math.columbia.edu/tag/00VC">Tag 00VC</a> and <a href="http://stacks.math.columbia.edu/tag/00XF">Tag 00XF</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 6, 2020 at 06:53:31. See the <a href="/nlab/history/restriction+and+extension+of+sheaves" 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/restriction+and+extension+of+sheaves" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/restriction+and+extension+of+sheaves/11" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/restriction+and+extension+of+sheaves" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/restriction+and+extension+of+sheaves" accesskey="S" class="navlink" id="history" rel="nofollow">History (11 revisions)</a> <a href="/nlab/show/restriction+and+extension+of+sheaves/cite" style="color: black">Cite</a> <a href="/nlab/print/restriction+and+extension+of+sheaves" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/restriction+and+extension+of+sheaves" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>