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id="toc-Presheaves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sheaves"> <div class="vector-toc-text"> 1.2 Sheaves </div> </a> <ul id="toc-Sheaves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_examples"> <div class="vector-toc-text"> 1.3 Further examples </div> </a> <ul id="toc-Further_examples-sublist" class="vector-toc-list"> <li id="toc-Sheaf_of_sections_of_a_continuous_map" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sheaf_of_sections_of_a_continuous_map"> <div class="vector-toc-text"> 1.3.1 Sheaf of sections of a continuous map </div> </a> <ul id="toc-Sheaf_of_sections_of_a_continuous_map-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaves_on_manifolds" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sheaves_on_manifolds"> <div class="vector-toc-text"> 1.3.2 Sheaves on manifolds </div> </a> <ul id="toc-Sheaves_on_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Presheaves_that_are_not_sheaves" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Presheaves_that_are_not_sheaves"> <div class="vector-toc-text"> 1.3.3 Presheaves that are not sheaves </div> </a> <ul id="toc-Presheaves_that_are_not_sheaves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Motivating_sheaves_from_complex_analytic_spaces_and_algebraic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Motivating_sheaves_from_complex_analytic_spaces_and_algebraic_geometry"> <div class="vector-toc-text"> 1.4 Motivating sheaves from complex analytic spaces and algebraic geometry </div> </a> <ul id="toc-Motivating_sheaves_from_complex_analytic_spaces_and_algebraic_geometry-sublist" class="vector-toc-list"> <li id="toc-Technical_challenges_with_complex_manifolds" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Technical_challenges_with_complex_manifolds"> <div class="vector-toc-text"> 1.4.1 Technical challenges with complex manifolds </div> </a> <ul id="toc-Technical_challenges_with_complex_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tracking_submanifolds_with_sheaves" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tracking_submanifolds_with_sheaves"> <div class="vector-toc-text"> 1.4.2 Tracking submanifolds with sheaves </div> </a> <ul id="toc-Tracking_submanifolds_with_sheaves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Operations_with_sheaves" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Operations_with_sheaves"> <div class="vector-toc-text"> 2 Operations with sheaves </div> </a> <button aria-controls="toc-Operations_with_sheaves-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Operations with sheaves subsection </button> <ul id="toc-Operations_with_sheaves-sublist" class="vector-toc-list"> <li id="toc-Morphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Morphisms"> <div class="vector-toc-text"> 2.1 Morphisms </div> </a> <ul id="toc-Morphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stalks_of_a_sheaf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stalks_of_a_sheaf"> <div class="vector-toc-text"> 2.2 Stalks of a sheaf </div> </a> <ul id="toc-Stalks_of_a_sheaf-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Turning_a_presheaf_into_a_sheaf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turning_a_presheaf_into_a_sheaf"> <div class="vector-toc-text"> 2.3 Turning a presheaf into a sheaf </div> </a> <ul id="toc-Turning_a_presheaf_into_a_sheaf-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subsheaves,_quotient_sheaves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subsheaves,_quotient_sheaves"> <div class="vector-toc-text"> 2.4 Subsheaves, quotient sheaves </div> </a> <ul id="toc-Subsheaves,_quotient_sheaves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_functoriality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_functoriality"> <div class="vector-toc-text"> 2.5 Basic functoriality </div> </a> <ul id="toc-Basic_functoriality-sublist" class="vector-toc-list"> <li id="toc-Direct_image" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Direct_image"> <div class="vector-toc-text"> 2.5.1 Direct image </div> </a> <ul id="toc-Direct_image-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_image" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inverse_image"> <div class="vector-toc-text"> 2.5.2 Inverse image </div> </a> <ul id="toc-Inverse_image-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_by_zero" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Extension_by_zero"> <div class="vector-toc-text"> 2.5.3 Extension by zero </div> </a> <ul id="toc-Extension_by_zero-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Complements" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complements"> <div class="vector-toc-text"> 3 Complements </div> </a> <button aria-controls="toc-Complements-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Complements subsection </button> <ul id="toc-Complements-sublist" class="vector-toc-list"> <li id="toc-Sheaves_in_more_general_categories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sheaves_in_more_general_categories"> <div class="vector-toc-text"> 3.1 Sheaves in more general categories </div> </a> <ul id="toc-Sheaves_in_more_general_categories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ringed_spaces_and_sheaves_of_modules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ringed_spaces_and_sheaves_of_modules"> <div class="vector-toc-text"> 3.2 Ringed spaces and sheaves of modules </div> </a> <ul id="toc-Ringed_spaces_and_sheaves_of_modules-sublist" class="vector-toc-list"> <li id="toc-Finiteness_conditions_for_sheaves_of_modules" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finiteness_conditions_for_sheaves_of_modules"> <div class="vector-toc-text"> 3.2.1 Finiteness conditions for sheaves of modules </div> </a> <ul id="toc-Finiteness_conditions_for_sheaves_of_modules-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_étalé_space_of_a_sheaf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_étalé_space_of_a_sheaf"> <div class="vector-toc-text"> 3.3 The étalé space of a sheaf </div> </a> <ul id="toc-The_étalé_space_of_a_sheaf-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sheaf_cohomology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sheaf_cohomology"> <div class="vector-toc-text"> 4 Sheaf cohomology </div> </a> <button aria-controls="toc-Sheaf_cohomology-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Sheaf cohomology subsection </button> <ul id="toc-Sheaf_cohomology-sublist" class="vector-toc-list"> <li id="toc-Computing_sheaf_cohomology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computing_sheaf_cohomology"> <div class="vector-toc-text"> 4.1 Computing sheaf cohomology </div> </a> <ul id="toc-Computing_sheaf_cohomology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derived_categories_of_sheaves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derived_categories_of_sheaves"> <div class="vector-toc-text"> 4.2 Derived categories of sheaves </div> </a> <ul id="toc-Derived_categories_of_sheaves-sublist" class="vector-toc-list"> <li id="toc-Derived_categories_of_coherent_sheaves_and_the_Grothendieck_group" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Derived_categories_of_coherent_sheaves_and_the_Grothendieck_group"> <div class="vector-toc-text"> 4.2.1 Derived categories of coherent sheaves and the Grothendieck group </div> </a> <ul id="toc-Derived_categories_of_coherent_sheaves_and_the_Grothendieck_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Sites_and_topoi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sites_and_topoi"> <div class="vector-toc-text"> 5 Sites and topoi </div> </a> <ul id="toc-Sites_and_topoi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 6 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div 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(matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Garbe_(Mathematik)" title="Garbe (Mathematik) – German" lang="de" hreflang="de" data-title="Garbe (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Haces_(herramienta_matem%C3%A1tica)" title="Haces (herramienta matemática) – Spanish" lang="es" hreflang="es" data-title="Haces (herramienta matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Garbo_(matematiko)" title="Garbo (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Garbo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Sorta" title="Sorta – Basque" lang="eu" hreflang="eu" data-title="Sorta" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%A7%D9%81%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="بافه (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="بافه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Faisceau_(math%C3%A9matiques)" title="Faisceau (mathématiques) – French" lang="fr" hreflang="fr" data-title="Faisceau (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B8%B5_(%EC%88%98%ED%95%99)" title="층 (수학) – Korean" lang="ko" hreflang="ko" data-title="층 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fascio_(teoria_delle_categorie)" title="Fascio (teoria delle categorie) – Italian" lang="it" hreflang="it" data-title="Fascio (teoria delle categorie)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%95%D7%9E%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" 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locally defined data attached to the open sets of a topological space</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about sheaves on <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. For sheaves on a site, see <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> and <a href="/wiki/Topos" title="Topos">Topos</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></div> <div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/sheaf" class="extiw" title="wiktionary:sheaf">sheaf</a> in Wiktionary, the free dictionary.</div></div> </div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a sheaf (<abbr title="plural form">pl.</abbr>: sheaves) is a tool for systematically tracking data (such as <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>, <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>) attached to the <a href="/wiki/Open_set" title="Open set">open sets</a> of a <a href="/wiki/Topological_space" title="Topological space">topological space</a> and defined locally with regard to them. For example, for each open set, the data could be the ring of <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets <a href="/wiki/Cover_(topology)" title="Cover (topology)">covering</a> the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract <a href="/wiki/Mathematical_object" title="Mathematical object">objects</a>. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as <a href="/wiki/Sheaf_of_rings" class="mw-redirect" title="Sheaf of rings">sheaves of rings</a>, for example, depending on the type of data assigned to the open sets. There are also <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">maps</a> (or <a href="/wiki/Morphism" title="Morphism">morphisms</a>) from one sheaf to another; sheaves (of a specific type, such as sheaves of <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a>) with their morphisms on a fixed topological space form a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. On the other hand, to each <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous map</a> there is associated both a <a href="/wiki/Direct_image_functor" title="Direct image functor">direct image functor</a>, taking sheaves and their morphisms on the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> to sheaves and morphisms on the <a href="/wiki/Codomain" title="Codomain">codomain</a>, and an <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image functor</a> operating in the opposite direction. These <a href="/wiki/Functor" title="Functor">functors</a>, and certain variants of them, are essential parts of sheaf theory. Due to their general nature and versatility, sheaves have several applications in topology and especially in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic</a> and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. First, geometric structures such as that of a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> or a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as <a href="/wiki/Vector_bundles" class="mw-redirect" title="Vector bundles">vector bundles</a> or <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a> are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">cohomology theory</a>, which encompasses also the "usual" topological cohomology theories such as <a href="/wiki/Singular_cohomology" class="mw-redirect" title="Singular cohomology">singular cohomology</a>. Especially in algebraic geometry and the theory of <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of <a href="/wiki/D-module" title="D-module">D-modules</a>, which provide applications to the theory of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. In addition, generalisations of sheaves to more general settings than topological spaces, such as <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a>, have provided applications to <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and to <a href="/wiki/Number_theory" title="Number theory">number theory</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions_and_examples">Definitions and examples</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=1" title="Edit section: Definitions and examples">edit</a>]</div> In many mathematical branches, several structures defined on a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (e.g., a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>) can be naturally localised or restricted to <a href="/wiki/Open_set" title="Open set">open</a> <a href="/wiki/Subset" title="Subset">subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}">: typical examples include <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a>-valued or <a href="/wiki/Complex_number" title="Complex number">complex</a>-valued functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-times <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> (real-valued or complex-valued) functions, <a href="/wiki/Bounded_function" title="Bounded function">bounded</a> real-valued functions, <a href="/wiki/Vector_field" title="Vector field">vector fields</a>, and <a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of any <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> on the space. The ability to restrict data to smaller open subsets gives rise to the concept of presheaves. Roughly speaking, sheaves are then those presheaves, where local data can be glued to global data. <div class="mw-heading mw-heading3"><h3 id="Presheaves">Presheaves</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=2" title="Edit section: Presheaves">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Presheaf_(category_theory)" title="Presheaf (category theory)">Presheaf (category theory)</a></div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be a topological space. A presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> of sets on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> consists of the following data: <ul><li>For each open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}">, there exists a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}">. This set is also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (U,{\mathcal {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (U,{\mathcal {F}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665ac6fb625e737c956780dbb9a0a54ac8bdee01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.005ex; height:2.843ex;" alt="{\displaystyle \Gamma (U,{\mathcal {F}})}">. The elements in this set are called the sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. The sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> are called the global sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">.</li> <li>For each inclusion of open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.668ex; height:2.343ex;" alt="{\displaystyle V\subseteq U}">, a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {res} _{V}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {res} _{V}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a215957f297c268b402191642915df5cf524e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.046ex; height:3.176ex;" alt="{\displaystyle \operatorname {res} _{V}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)}">. In view of many of the examples below, the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{res}}_{V}^{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{res}}_{V}^{U}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/226109b755dcd16b6821f7588f40404156ae2a4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.356ex; height:3.176ex;" alt="{\displaystyle {\text{res}}_{V}^{U}}"> are called restriction morphisms. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247691a5af59cf1fdc88db4a8581d0e91c2175d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.45ex; height:2.843ex;" alt="{\displaystyle s\in {\mathcal {F}}(U)}">, then its restriction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{res}}_{V}^{U}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{res}}_{V}^{U}(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eddeef57e1566a40ad70ab75c8a720151cc8264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.256ex; height:3.176ex;" alt="{\displaystyle {\text{res}}_{V}^{U}(s)}"> is often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s|_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s|_{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7327140ba5fae591dbfa18cf3fb7b89f6ce6b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.233ex; height:3.009ex;" alt="{\displaystyle s|_{V}}"> by analogy with restriction of functions.</li></ul> The restriction morphisms are required to satisfy two additional (<a href="/wiki/Functorial" class="mw-redirect" title="Functorial">functorial</a>) properties: <ul><li>For every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, the restriction morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {res} _{U}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {res} _{U}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af3f7fc29e9ddef00833b97d8f19de158adff26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.038ex; height:3.176ex;" alt="{\displaystyle \operatorname {res} _{U}^{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {F}}(U)}"> is the identity morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}">.</li> <li>If we have three open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subseteq V\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊆</mo> <mi>V</mi> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\subseteq V\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7945770f01cf7cc73bd3bd966e634c8041e0384b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.202ex; height:2.343ex;" alt="{\displaystyle W\subseteq V\subseteq U}">, then the <a href="/wiki/Function_composition" title="Function composition">composite</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{res}}_{W}^{V}\circ {\text{res}}_{V}^{U}={\text{res}}_{W}^{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msubsup> <mo>∘</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mtext>res</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{res}}_{W}^{V}\circ {\text{res}}_{V}^{U}={\text{res}}_{W}^{U}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a21fea4186936808a2af2309840419ac15845bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.278ex; height:3.176ex;" alt="{\displaystyle {\text{res}}_{W}^{V}\circ {\text{res}}_{V}^{U}={\text{res}}_{W}^{U}}"></li></ul> Informally, the second axiom says it does not matter whether we restrict to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> in one step or restrict first to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">, then to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}">. A concise functorial reformulation of this definition is given further below. Many examples of presheaves come from different classes of functions: to any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">, one can assign the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{0}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d96dc6e268e1c1f7a2d5e60c543e3304b09025f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:3.176ex;" alt="{\displaystyle C^{0}(U)}"> of continuous real-valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. The restriction maps are then just given by restricting a continuous function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> to a smaller open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.668ex; height:2.343ex;" alt="{\displaystyle V\subseteq U}">, which again is a continuous function. The two presheaf axioms are immediately checked, thereby giving an example of a presheaf. This can be extended to a presheaf of holomorphic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(-)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9856289e9ab9da4985aa0c81c77579ee41b120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}(-)}"> and a presheaf of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(-)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f29d4d992871ccedc84bf2f70c9b237e4399bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.291ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(-)}">. Another common class of examples is assigning to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> the set of constant real-valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. This presheaf is called the constant presheaf associated to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> and is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbb {R} }}^{\text{psh}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>psh</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbb {R} }}^{\text{psh}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8830539f90f788cb704fe41d85af294b86861572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.513ex; margin-bottom: -0.825ex; width:4.388ex; height:3.676ex;" alt="{\displaystyle {\underline {\mathbb {R} }}^{\text{psh}}}">. <div class="mw-heading mw-heading3"><h3 id="Sheaves">Sheaves</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=3" title="Edit section: Sheaves">edit</a>]</div> Given a presheaf, a natural question to ask is to what extent its sections over an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> are specified by their restrictions to open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. A sheaf is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions. Axiomatically, a sheaf is a presheaf that satisfies both of the following axioms: <ol><li>(Locality) Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open set, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U_{i}\}_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"> is an open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31870b4d162e8f53e74758ca53837f3cd399d2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.268ex; height:2.509ex;" alt="{\displaystyle U_{i}\subseteq U}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t\in {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s,t\in {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d151a3fb8225264bfa097b589dd65c6703d7029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.323ex; height:2.843ex;" alt="{\displaystyle s,t\in {\mathcal {F}}(U)}"> are sections. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s|_{U_{i}}=t|_{U_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi>t</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s|_{U_{i}}=t|_{U_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc03ef690f2c8724ef178aacfb9edb176066805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.282ex; height:3.176ex;" alt="{\displaystyle s|_{U_{i}}=t|_{U_{i}}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0fd4cbdea3f3c94e1dd79dee73d87a150dbf3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.029ex; height:2.009ex;" alt="{\displaystyle s=t}">.</li> <li>(<a href="/wiki/Gluing_axiom" title="Gluing axiom">Gluing</a>) Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open set, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U_{i}\}_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"> is an open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31870b4d162e8f53e74758ca53837f3cd399d2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.268ex; height:2.509ex;" alt="{\displaystyle U_{i}\subseteq U}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{s_{i}\in {\mathcal {F}}(U_{i})\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{s_{i}\in {\mathcal {F}}(U_{i})\}_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4643d2718947e49a2a94edd638e03cd63c52dbb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.903ex; height:2.843ex;" alt="{\displaystyle \{s_{i}\in {\mathcal {F}}(U_{i})\}_{i\in I}}"> is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c83198ef4aca351893f01fc21864a3c15e3f5e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:18.108ex; height:3.509ex;" alt="{\displaystyle s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7596675a16b3250fc6b02be961d315d3ed467af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.509ex;" alt="{\displaystyle i,j\in I}">, then there exists a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247691a5af59cf1fdc88db4a8581d0e91c2175d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.45ex; height:2.843ex;" alt="{\displaystyle s\in {\mathcal {F}}(U)}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s|_{U_{i}}=s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s|_{U_{i}}=s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4f26e44ee831829f9f9b3badf02f0fe03d548e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.706ex; height:3.176ex;" alt="{\displaystyle s|_{U_{i}}=s_{i}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}">.<a href="#cite_note-1">[1]</a></li></ol> In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigcup _{i\in I}U_{i}=U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigcup _{i\in I}U_{i}=U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2721dca07fc64b14fa478d14ef8be3dc01471ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.316ex; height:3.009ex;" alt="{\textstyle \bigcup _{i\in I}U_{i}=U}">. The section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> whose existence is guaranteed by axiom 2 is called the gluing, concatenation, or collation of the sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}">. By axiom 1 it is unique. Sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a350c64508aef872d4e72ee677746ef7a20f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.343ex;" alt="{\displaystyle s_{j}}"> satisfying the agreement precondition of axiom 2 are often called compatible ; thus axioms 1 and 2 together state that any collection of pairwise compatible sections can be uniquely glued together. A separated presheaf, or monopresheaf, is a presheaf satisfying axiom 1.<a href="#cite_note-2">[2]</a> The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}:U_{i}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}:U_{i}\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865df46156b5263bd856d3ce5347d3dd290edcce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.555ex; height:2.509ex;" alt="{\displaystyle f_{i}:U_{i}\to \mathbb {R} }"> which agree on the intersections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}\cap U_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}\cap U_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be592e07a07b4ce92cc30774ae3a8d984eca426e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.467ex; height:2.843ex;" alt="{\displaystyle U_{i}\cap U_{j}}">, there is a unique continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f481901ff501baa824d1eab35eba6d9410ba57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {R} }"> whose restriction equals the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}">. By contrast, the constant presheaf is usually not a sheaf as it fails to satisfy the locality axiom on the empty set (this is explained in more detail at <a href="/wiki/Constant_sheaf" title="Constant sheaf">constant sheaf</a>). Presheaves and sheaves are typically denoted by capital letters, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> being particularly common, presumably for the <a href="/wiki/French_language" title="French language">French</a> word for sheaf, faisceau. Use of calligraphic letters such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> is also common. It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a <a href="/wiki/Basis_(topology)" class="mw-redirect" title="Basis (topology)">basis</a> for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. This observation is used to construct another example which is crucial in algebraic geometry, namely <a href="/wiki/Quasi-coherent_sheaf" class="mw-redirect" title="Quasi-coherent sheaf">quasi-coherent sheaves</a>. Here the topological space in question is the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></a>, whose points are the <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">. The open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}:=\{{\mathfrak {p}}\subseteq R,f\notin {\mathfrak {p}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>⊆</mo> <mi>R</mi> <mo>,</mo> <mi>f</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{f}:=\{{\mathfrak {p}}\subseteq R,f\notin {\mathfrak {p}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b64e833cfb345bd14726cf2c0b6fee589822edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.471ex; height:3.009ex;" alt="{\displaystyle D_{f}:=\{{\mathfrak {p}}\subseteq R,f\notin {\mathfrak {p}}\}}"> form a basis for the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a> on this space. Given an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">-module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, there is a sheaf, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/782dd22843a23d3c71e4dca3f51bdd724819e83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.676ex;" alt="{\displaystyle {\tilde {M}}}"> on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac8125a353f23571cda07bfbd46b21dba38c88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.801ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} R}">, that satisfies <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {M}}(D_{f}):=M[1/f],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>M</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {M}}(D_{f}):=M[1/f],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c95e3ca9b913d8c460360de8ffca80aae4236d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.044ex; height:3.343ex;" alt="{\displaystyle {\tilde {M}}(D_{f}):=M[1/f],}"> the <a href="/wiki/Localization_(commutative_algebra)" title="Localization (commutative algebra)">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">.</dd></dl> There is another characterization of sheaves that is equivalent to the previously discussed. A presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> is a sheaf if and only if for any open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> and any open cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{a}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U_{a}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c846c0b192aa90c0e1b5a6e13afa22903768d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle \{U_{a}\}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}"> is the fibre product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)\cong {\mathcal {F}}(U_{a})\times _{{\mathcal {F}}(U_{a}\cap U_{b})}{\mathcal {F}}(U_{b})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)\cong {\mathcal {F}}(U_{a})\times _{{\mathcal {F}}(U_{a}\cap U_{b})}{\mathcal {F}}(U_{b})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7861bdbebe2bc24ca4645a98868eb28eb630da42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.966ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}(U)\cong {\mathcal {F}}(U_{a})\times _{{\mathcal {F}}(U_{a}\cap U_{b})}{\mathcal {F}}(U_{b})}">. This characterization is useful in construction of sheaves, for example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}},{\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}},{\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2113943dd96ba6d806045d445bfaedf676bed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.353ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}},{\mathcal {G}}}"> are <a href="/wiki/Sheaf_of_abelian_groups" class="mw-redirect" title="Sheaf of abelian groups">abelian sheaves</a>, then the kernel of sheaves morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fba7f07f364f53efaaf999a67db3f374bb8dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.933ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}"> is a sheaf, since projective limits commutes with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limit not necessarily commutes with projective limits. One of the way to fix this is to consider Noetherian topological spaces; every open sets are compact so that the cokernel is a sheaf, since finite projective limits commutes with inductive limits. <div class="mw-heading mw-heading3"><h3 id="Further_examples">Further examples</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=4" title="Edit section: Further examples">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Sheaf_of_sections_of_a_continuous_map">Sheaf of sections of a continuous map</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=5" title="Edit section: Sheaf of sections of a continuous map">edit</a>]</div> Any continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e8fe3d0dc6757c546d482a59b9c5220eaff7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:Y\to X}"> of topological spaces determines a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (Y/X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (Y/X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b06f6dd8c9d21695e91d8e5acf0284e543adf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.178ex; height:2.843ex;" alt="{\displaystyle \Gamma (Y/X)}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> by setting <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (Y/X)(U)=\{s:U\to Y,f\circ s=\operatorname {id} _{U}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>,</mo> <mi>f</mi> <mo>∘</mo> <mi>s</mi> <mo>=</mo> <msub> <mi>id</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (Y/X)(U)=\{s:U\to Y,f\circ s=\operatorname {id} _{U}\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2849fff10f78fb5f2303dab21bd5555cb44f93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.166ex; height:2.843ex;" alt="{\displaystyle \Gamma (Y/X)(U)=\{s:U\to Y,f\circ s=\operatorname {id} _{U}\}.}"></dd></dl> Any such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> is commonly called a <a href="/wiki/Section_(category_theory)" title="Section (category theory)">section</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, and this example is the reason why the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}"> are generally called sections. This construction is especially important when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the projection of a <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundle</a> onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the <a href="/wiki/Trivial_bundle" class="mw-redirect" title="Trivial bundle">trivial bundle</a>. Another example: the sheaf of sections of <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} {\stackrel {\exp }{\longrightarrow }}\mathbb {C} \setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>exp</mi> </mrow> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} {\stackrel {\exp }{\longrightarrow }}\mathbb {C} \setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f57e448a94e886ecb30d42f25ee1df5e76eefdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.844ex; height:3.843ex;" alt="{\displaystyle \mathbb {C} {\stackrel {\exp }{\longrightarrow }}\mathbb {C} \setminus \{0\}}"></dd></dl> is the sheaf which assigns to any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq \mathbb {C} \setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq \mathbb {C} \setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332ab88a8856c9d9d3686c6bbcb9a059a7e467fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.241ex; height:2.843ex;" alt="{\displaystyle U\subseteq \mathbb {C} \setminus \{0\}}"> the set of branches of the <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. Given a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and an abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, the skyscraper sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17297f0fdd502bf3cf9e999b890ffb872869c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.597ex; height:2.509ex;" alt="{\displaystyle S_{x}}"> is defined as follows: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is an open set containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}(U)=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(U)=S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42cee3156371c5ff5c6c65bbb069a0c628f48db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.787ex; height:2.843ex;" alt="{\displaystyle S_{x}(U)=S}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> does not contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}(U)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(U)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05298becd8cd207e4ef18a66467b086af042d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.45ex; height:2.843ex;" alt="{\displaystyle S_{x}(U)=0}">, the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a>. The restriction maps are either the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, if both open sets contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, or the zero map otherwise. <div class="mw-heading mw-heading4"><h4 id="Sheaves_on_manifolds">Sheaves on manifolds</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=6" title="Edit section: Sheaves on manifolds">edit</a>]</div> On an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}">-manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, there are a number of important sheaves, such as the sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">-times continuously differentiable functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{M}^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{M}^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5d2f1d34e55c3b1f1ace59ed95239caf163f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.809ex; height:3.509ex;" alt="{\displaystyle {\mathcal {O}}_{M}^{j}}"> (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\leq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f39ddd5ca00e7daa67a5ff8b5de690f1344aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.294ex; height:2.509ex;" alt="{\displaystyle j\leq k}">). Its sections on some open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> are the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750276e39741e496309284ddd0b837da0b0da3ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.708ex; height:2.676ex;" alt="{\displaystyle C^{j}}">-functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144ad25c662c157f4e365d4abc2c8398de3d6fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.075ex; height:2.176ex;" alt="{\displaystyle U\to \mathbb {R} }">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b954ae5b92d2350b5af4f8d1b0e91d26a01bbe70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.294ex; height:2.509ex;" alt="{\displaystyle j=k}">, this sheaf is called the structure sheaf and is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d784cc240e46457fe2b72cf60e2447b5579af11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.809ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{M}}">. The nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}"> functions also form a sheaf, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb33a0eb556018cd971d155f97600290845215a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.482ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X}^{\times }}">. <a href="/wiki/Differential_form" title="Differential form">Differential forms</a> (of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">) also form a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{M}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{M}^{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89ba9d51b03c5be678b7d77ebfb80bcdffcc500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.637ex; height:3.176ex;" alt="{\displaystyle \Omega _{M}^{p}}">. In all these examples, the restriction morphisms are given by restricting functions or forms. The assignment sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> to the compactly supported functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset. Instead, this forms a <a href="/wiki/Cosheaf" title="Cosheaf">cosheaf</a>, a <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dual</a> concept where the restriction maps go in the opposite direction than with sheaves.<a href="#cite_note-3">[3]</a> However, taking the <a href="/wiki/Dual_vector_space" class="mw-redirect" title="Dual vector space">dual</a> of these vector spaces does give a sheaf, the sheaf of <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. <div class="mw-heading mw-heading4"><h4 id="Presheaves_that_are_not_sheaves">Presheaves that are not sheaves</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=7" title="Edit section: Presheaves that are not sheaves">edit</a>]</div> In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves: <ul><li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> be the <a href="/wiki/Discrete_two-point_space" title="Discrete two-point space">two-point topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x,y\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2611cdc8fecaffa28cb0ea888dbba55f3a31077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.844ex; height:2.843ex;" alt="{\displaystyle \{x,y\}}"> with the discrete topology. Define a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varnothing )=\{\varnothing \},\ F(\{x\})=\mathbb {R} ,\ F(\{y\})=\mathbb {R} ,\ F(\{x,y\})=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\varnothing )=\{\varnothing \},\ F(\{x\})=\mathbb {R} ,\ F(\{y\})=\mathbb {R} ,\ F(\{x,y\})=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f2d763f37d077eab0daeb3f28541e18577448f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.429ex; height:2.843ex;" alt="{\displaystyle F(\varnothing )=\{\varnothing \},\ F(\{x\})=\mathbb {R} ,\ F(\{y\})=\mathbb {R} ,\ F(\{x,y\})=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }">The restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\{x,y\})\to F(\{x\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\{x,y\})\to F(\{x\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/807d5c98a88d2f458751a826c8e073194aebbd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.213ex; height:2.843ex;" alt="{\displaystyle F(\{x,y\})\to F(\{x\})}"> is the projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17007c98c10928212ba41ac10c3c2d494477a2de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.715ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }"> onto its first coordinate, and the restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\{x,y\})\to F(\{y\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\{x,y\})\to F(\{y\})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160a59404dc0723755b0c97c32445b8c9933dd1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.039ex; height:2.843ex;" alt="{\displaystyle F(\{x,y\})\to F(\{y\})}"> is the projection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17007c98c10928212ba41ac10c3c2d494477a2de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.715ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }"> onto its second coordinate. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120eeb8a091b516595765bd08b306f2394e7721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144dbb8cb5d9e202a8d1ead595f39c167238858c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.48ex; height:2.843ex;" alt="{\displaystyle \{y\}}"> determine only two of those numbers. So while we can glue any two sections over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a120eeb8a091b516595765bd08b306f2394e7721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="{\displaystyle \{x\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144dbb8cb5d9e202a8d1ead595f39c167238858c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.48ex; height:2.843ex;" alt="{\displaystyle \{y\}}">, we cannot glue them uniquely.</li> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2e2b6427cd2b517be352b378a1830c1540e3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.757ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {R} }"> be the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462254eb7da92fe91d24532826f4e036cf1eeaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.333ex; height:2.843ex;" alt="{\displaystyle F(U)}"> be the set of <a href="/wiki/Bounded_function" title="Bounded function">bounded</a> continuous functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. This is not a sheaf because it is not always possible to glue. For example, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> be the set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ca1ad0a1244df1140911b00c0437057e3d08e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.524ex; height:2.843ex;" alt="{\displaystyle |x|<i}">. The identity function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f690285952308aa49e3c6aac892df31cad6d1b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.846ex; height:2.843ex;" alt="{\displaystyle f(x)=x}"> is bounded on each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}">. Consequently, we get a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfda82668232cbdc0874ed28ab8b6079420d1ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.89ex; height:2.009ex;" alt="{\displaystyle s_{i}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}">. However, these sections do not glue, because the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is not bounded on the real line. Consequently <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> is a presheaf, but not a sheaf. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> is separated because it is a sub-presheaf of the sheaf of continuous functions.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Motivating_sheaves_from_complex_analytic_spaces_and_algebraic_geometry">Motivating sheaves from complex analytic spaces and algebraic geometry</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=8" title="Edit section: Motivating sheaves from complex analytic spaces and algebraic geometry">edit</a>]</div> One of the historical motivations for sheaves have come from studying <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>,<a href="#cite_note-4">[4]</a> <a href="/wiki/Complex_analytic_geometry" class="mw-redirect" title="Complex analytic geometry">complex analytic geometry</a>,<a href="#cite_note-5">[5]</a> and <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme theory</a> from <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. This is because in all of the previous cases, we consider a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> together with a structure sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"> giving it the structure of a complex manifold, complex analytic space, or scheme. This perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces (see below). <div class="mw-heading mw-heading4"><h4 id="Technical_challenges_with_complex_manifolds">Technical challenges with complex manifolds</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=9" title="Edit section: Technical challenges with complex manifolds">edit</a>]</div> One of the main historical motivations for introducing sheaves was constructing a device which keeps track of <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> on <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>. For example, on a <a href="/wiki/Compact_space" title="Compact space">compact</a> complex manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> (like <a href="/wiki/Complex_projective_space" title="Complex projective space">complex projective space</a> or the <a href="/wiki/Vanishing_locus" class="mw-redirect" title="Vanishing locus">vanishing locus</a> in projective space of a <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomial</a>), the only holomorphic functions<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a57fe120f564b03d50ef328f6a5e09d0e860781e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {C} }"></blockquote>are the constant functions.<a href="#cite_note-stackexchange1-6">[6]</a><a href="#cite_note-7">[7]</a> This means there exist two compact complex manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,X'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,X'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c04b15c46d27512014ffa88b4d34425e878a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.696ex; height:2.843ex;" alt="{\displaystyle X,X'}"> which are not isomorphic, but nevertheless their rings of global holomorphic functions, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2084059d1b597ad91444021e9852e02b2bc7814a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.242ex; height:3.009ex;" alt="{\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')}">, are isomorphic. Contrast this with <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifolds</a> where every manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> can be embedded inside some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">, hence its ring of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b543485314a4426ee2ff956f206cfd802d68d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}"> comes from restricting the smooth functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e0692b99aae72807a3f239a799c0fdadaee711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} ^{n})}">. Another complexity when considering the ring of holomorphic functions on a complex manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is given a small enough open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}">, the holomorphic functions will be isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709003fbcdef8a66fb7d4e58a7f987565eee8010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.324ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})}">. Sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> on arbitrary open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}">. This means as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> becomes more complex topologically, the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c59263a17e08b492445a022e0ddbaa445cae685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.556ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}(U)}"> can be expressed from gluing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}(U_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}(U_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c509bc8e41b2eb7a4db0162aa4a8850fc1136d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.16ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}(U_{i})}">. Note that sometimes this sheaf is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(-)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f945b63886825fadd0e69c5304e437f311dee0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.467ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(-)}"> or just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}">, or even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}"> when we want to emphasize the space the structure sheaf is associated to. <div class="mw-heading mw-heading4"><h4 id="Tracking_submanifolds_with_sheaves">Tracking submanifolds with sheaves</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=10" title="Edit section: Tracking submanifolds with sheaves">edit</a>]</div> Another common example of sheaves can be constructed by considering a complex submanifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\hookrightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">↪</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\hookrightarrow X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2020d885f78fc6f81ceaf1c28234c1e4880cbba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.66ex; height:2.176ex;" alt="{\displaystyle Y\hookrightarrow X}">. There is an associated sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{Y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a953f47b204676743cc08273ab1b1f952b41ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.336ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{Y}}"> which takes an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.861ex; height:2.343ex;" alt="{\displaystyle U\subseteq X}"> and gives the ring of holomorphic functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cap Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∩</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cap Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5237a175ef39017498275e3008b03ce974d3ef9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.138ex; height:2.176ex;" alt="{\displaystyle U\cap Y}">. This kind of formalism was found to be extremely powerful and motivates a lot of <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a> such as <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> since an <a href="/wiki/Intersection_theory" title="Intersection theory">intersection theory</a> <a href="/wiki/Intersection_number" title="Intersection number">can be built using these kinds of sheaves</a> from the Serre intersection formula. <div class="mw-heading mw-heading2"><h2 id="Operations_with_sheaves">Operations with sheaves</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=11" title="Edit section: Operations with sheaves">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Morphisms">Morphisms</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=12" title="Edit section: Morphisms">edit</a>]</div> Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which is simply an assignment of outputs to inputs, morphisms of sheaves are also required to be compatible with the local–global structures of the underlying sheaves. This idea is made precise in the following definition. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}"> be two sheaves of sets (respectively abelian groups, rings, etc.) on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. A <a href="/wiki/Morphism" title="Morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :{\mathcal {F}}\to {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :{\mathcal {F}}\to {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9847b1891aefbe0fe18ec04bcb21fe9b6a78e257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.39ex; height:2.676ex;" alt="{\displaystyle \varphi :{\mathcal {F}}\to {\mathcal {G}}}"> consists of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{U}:{\mathcal {F}}(U)\to {\mathcal {G}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{U}:{\mathcal {F}}(U)\to {\mathcal {G}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b4434d08f48208347f2e95962932c5a58fef1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.067ex; height:2.843ex;" alt="{\displaystyle \varphi _{U}:{\mathcal {F}}(U)\to {\mathcal {G}}(U)}"> of sets (respectively abelian groups, rings, etc.) for each open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> of an open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">, the following diagram is <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}{\mathcal {F}}(U)&\xrightarrow {\quad \varphi _{U}\quad } &{\mathcal {G}}(U)\\r_{V}^{U}{\Biggl \downarrow }&&{\Biggl \downarrow }{r'}_{V}^{U}\\{\mathcal {F}}(V)&{\xrightarrow[{\quad \varphi _{V}\quad }]{}}&{\mathcal {G}}(V)\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mspace width="1em" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mspace width="1em" /> </mpadded> </mover> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">↓</mo> </mrow> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo fence="true" symmetric="true" maxsize="2.470em" minsize="2.470em">↓</mo> </mrow> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="1em" /> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em" /> </munderover> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}{\mathcal {F}}(U)&\xrightarrow {\quad \varphi _{U}\quad } &{\mathcal {G}}(U)\\r_{V}^{U}{\Biggl \downarrow }&&{\Biggl \downarrow }{r'}_{V}^{U}\\{\mathcal {F}}(V)&{\xrightarrow[{\quad \varphi _{V}\quad }]{}}&{\mathcal {G}}(V)\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44efdc087f1535826546c9b33bdfefe954da8c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.939ex; margin-top: -0.405ex; margin-bottom: -0.566ex; width:24.143ex; height:16.009ex;" alt="{\displaystyle {\begin{array}{rcl}{\mathcal {F}}(U)&\xrightarrow {\quad \varphi _{U}\quad } &{\mathcal {G}}(U)\\r_{V}^{U}{\Biggl \downarrow }&&{\Biggl \downarrow }{r'}_{V}^{U}\\{\mathcal {F}}(V)&{\xrightarrow[{\quad \varphi _{V}\quad }]{}}&{\mathcal {G}}(V)\end{array}}}"></dd></dl> For example, taking the derivative gives a morphism of sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\colon {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>:</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\colon {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6903ae1c42a0863da5a287b08800e64d11d41b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.191ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\colon {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.}"> Indeed, given an (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-times continuously differentiable) function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f481901ff501baa824d1eab35eba6d9410ba57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.29ex; height:2.509ex;" alt="{\displaystyle f:U\to \mathbb {R} }"> (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> open), the restriction (to a smaller open subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">) of its derivative equals the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f|_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671304ca54797698a07c26fb1e4ff8194dec3d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.421ex; height:3.009ex;" alt="{\displaystyle f|_{V}}">. With this notion of morphism, sheaves of sets (respectively abelian groups, rings, etc.) on a fixed topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> form a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. The general categorical notions of <a href="/wiki/Monomorphisms" class="mw-redirect" title="Monomorphisms">mono-</a>, <a href="/wiki/Epimorphism" title="Epimorphism">epi-</a> and <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a> can therefore be applied to sheaves. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon {\mathcal {F}}\rightarrow {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon {\mathcal {F}}\rightarrow {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5a7fa9f3c869f9e6a56191cee8aeed96538bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.487ex; height:2.676ex;" alt="{\displaystyle \varphi \colon {\mathcal {F}}\rightarrow {\mathcal {G}}}"> of sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is an isomorphism (respectively monomorphism) if and only if there exists an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{\alpha }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U_{\alpha }\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/681f8e25b4f4a48037221f84af293974138ee1c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.197ex; height:2.843ex;" alt="{\displaystyle \{U_{\alpha }\}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi |_{U_{\alpha }}\colon {\mathcal {F}}(U_{\alpha })\rightarrow {\mathcal {G}}(U_{\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi |_{U_{\alpha }}\colon {\mathcal {F}}(U_{\alpha })\rightarrow {\mathcal {G}}(U_{\alpha })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5f3072a4e24d8e163abf4963134db09c85e47f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.869ex; height:3.176ex;" alt="{\displaystyle \varphi |_{U_{\alpha }}\colon {\mathcal {F}}(U_{\alpha })\rightarrow {\mathcal {G}}(U_{\alpha })}"> are isomorphisms (respectively injective morphisms) of sets (respectively abelian groups, rings, etc.) for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">. These statements give examples of how to work with sheaves using local information, but it's important to note that we cannot check if a morphism of sheaves is an epimorphism in the same manner. Indeed the statement that maps on the level of open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {G}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {G}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841db39bcf970a7041e22b92918209d629047bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.164ex; height:2.843ex;" alt="{\displaystyle \varphi _{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {G}}(U)}"> are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor—or equivalently, to non-triviality of <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a>. <div class="mw-heading mw-heading3"><h3 id="Stalks_of_a_sheaf">Stalks of a sheaf</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=13" title="Edit section: Stalks of a sheaf">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stalk_(sheaf)" title="Stalk (sheaf)">Stalk (sheaf)</a></div> The stalk <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d72460f7036cb595f6158fd0e9e125a595cf2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.844ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{x}}"> of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> captures the properties of a sheaf "around" a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}">, generalizing the <a href="/wiki/Germ_(mathematics)" title="Germ (mathematics)">germs of functions</a>. Here, "around" means that, conceptually speaking, one looks at smaller and smaller <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhoods</a> of the point. Of course, no single neighborhood will be small enough, which requires considering a limit of some sort. More precisely, the stalk is defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{x}=\varinjlim _{U\ni x}{\mathcal {F}}(U),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mo>∋</mo> <mi>x</mi> </mrow> </munder> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{x}=\varinjlim _{U\ni x}{\mathcal {F}}(U),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4abd75396241e6eed743441f2e6311ac20859163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:15.854ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}_{x}=\varinjlim _{U\ni x}{\mathcal {F}}(U),}"></dd></dl> the <a href="/wiki/Direct_limit" title="Direct limit">direct limit</a> being over all open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> containing the given point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. In other words, an element of the stalk is given by a section over some open neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood. The natural morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)\to {\mathcal {F}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)\to {\mathcal {F}}_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f1a7f2302df56aa8435caa95d45910d919fc36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.976ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)\to {\mathcal {F}}_{x}}"> takes a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}"> to its germ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f738a5f93f694ded413a8f8da71081d0fdff4501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.009ex;" alt="{\displaystyle s_{x}}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. This generalises the usual definition of a <a href="/wiki/Germ_(mathematics)" title="Germ (mathematics)">germ</a>. In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. In this sense, a sheaf is determined by its stalks, which are a local data. By contrast, the global information present in a sheaf, i.e., the global sections, i.e., the sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0863c8da8bca35eee88c4898e0439e26d1c3d62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(X)}"> on the whole space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, typically carry less information. For example, for a <a href="/wiki/Compact_space" title="Compact space">compact</a> complex manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, the global sections of the sheaf of holomorphic functions are just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">, since any holomorphic function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d22812d4739fce70b9bd178d5e71bdf51d26785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.272ex; height:2.176ex;" alt="{\displaystyle X\to \mathbb {C} }"></dd></dl> is constant by <a href="/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>.<a href="#cite_note-stackexchange1-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Turning_a_presheaf_into_a_sheaf">Turning a presheaf into a sheaf</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=14" title="Edit section: Turning a presheaf into a sheaf">edit</a>]</div> It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> and produces a new sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5baea99fabc0fde396d69e1886291f903ad7692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle a{\mathcal {F}}}"> called the sheafification or sheaf associated to the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">. For example, the sheafification of the constant presheaf (see above) is called the <a href="/wiki/Constant_sheaf" title="Constant sheaf">constant sheaf</a>. Despite its name, its sections are <a href="/wiki/Locally_constant" class="mw-redirect" title="Locally constant">locally constant</a> functions. The sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5baea99fabc0fde396d69e1886291f903ad7692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle a{\mathcal {F}}}"> can be constructed using the <a href="/wiki/%C3%89tal%C3%A9_space" class="mw-redirect" title="Étalé space">étalé space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">, namely as the sheaf of sections of the map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\to X.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\to X.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49b9848cdf6268e1dda1357d06260156adfdc0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.017ex; height:2.176ex;" alt="{\displaystyle E\to X.}"></dd></dl> Another construction of the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5baea99fabc0fde396d69e1886291f903ad7692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle a{\mathcal {F}}}"> proceeds by means of a functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9ce742a7a5468fc0e78566d0f0a51197144da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle L{\mathcal {F}}}"> is a separated presheaf, and for any separated presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9ce742a7a5468fc0e78566d0f0a51197144da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.509ex; height:2.176ex;" alt="{\displaystyle L{\mathcal {F}}}"> is a sheaf. The associated sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5baea99fabc0fde396d69e1886291f903ad7692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle a{\mathcal {F}}}"> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle LL{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle LL{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42dc4626803814cc5806167fe5132ca50f05a3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.092ex; height:2.176ex;" alt="{\displaystyle LL{\mathcal {F}}}">.<a href="#cite_note-8">[8]</a> The idea that the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5baea99fabc0fde396d69e1886291f903ad7692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.156ex; height:2.176ex;" alt="{\displaystyle a{\mathcal {F}}}"> is the best possible approximation to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> by a sheaf is made precise using the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: there is a natural morphism of presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\colon {\mathcal {F}}\to a{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\colon {\mathcal {F}}\to a{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f00d63b01fbf98bc1bb3ce931e01fdfafbf00dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.533ex; height:2.176ex;" alt="{\displaystyle i\colon {\mathcal {F}}\to a{\mathcal {F}}}"> so that for any sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}"> and any morphism of presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon {\mathcal {F}}\to {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon {\mathcal {F}}\to {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06c9a7d31b517cde6cddd4125b9c1e1c393b841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.246ex; height:2.509ex;" alt="{\displaystyle f\colon {\mathcal {F}}\to {\mathcal {G}}}">, there is a unique morphism of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}\colon a{\mathcal {F}}\rightarrow {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>:</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\colon a{\mathcal {F}}\rightarrow {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab629cccfdcecdf9197d1f9723a6c8a7a42e5b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.896ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}\colon a{\mathcal {F}}\rightarrow {\mathcal {G}}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\tilde {f}}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\tilde {f}}i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3949c575dac9d848545f1f88939b87064d6f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.879ex; height:3.009ex;" alt="{\displaystyle f={\tilde {f}}i}">. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is the left <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functor</a> to the inclusion functor (or <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a>) from the category of sheaves to the category of presheaves, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> is the <a href="/wiki/Adjoint_functor#Unit_and_co-unit" class="mw-redirect" title="Adjoint functor">unit</a> of the adjunction. In this way, the category of sheaves turns into a <a href="/wiki/Giraud_subcategory" title="Giraud subcategory">Giraud subcategory</a> of presheaves. This categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves, but not for kernels, say. <div class="mw-heading mw-heading3"><h3 id="Subsheaves,_quotient_sheaves">Subsheaves, quotient sheaves</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=15" title="Edit section: Subsheaves, quotient sheaves">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is a <a href="/wiki/Subobject" title="Subobject">subsheaf</a> of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> of abelian groups, then the quotient sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> is the sheaf associated to the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto F(U)/K(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto F(U)/K(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad3f3661aaee4a2fe9013000d68fdd907fc32e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.55ex; height:2.843ex;" alt="{\displaystyle U\mapsto F(U)/K(U)}">; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups; <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to K\to F\to Q\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to K\to F\to Q\to 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52b898b2c1d77784ad560f730a3ab545e5034c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.073ex; height:2.509ex;" alt="{\displaystyle 0\to K\to F\to Q\to 0.}"></dd></dl> (this is also called a <a href="/wiki/Sheaf_extension" class="mw-redirect" title="Sheaf extension">sheaf extension</a>.) Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>,</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F,G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/986666e6e8feddc018c06deca748781e42646cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.601ex; height:2.509ex;" alt="{\displaystyle F,G}"> be sheaves of abelian groups. The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (F,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (F,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08579c199665838aaf69b386ceaaea460b52188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.252ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (F,G)}"> of morphisms of sheaves from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> forms an abelian group (by the abelian group structure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">). The sheaf hom of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}">, denoted by, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Hom}}(F,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">o</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Hom}}(F,G)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d13c342e4695f21ccb450a8c821e12dcbdb9c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.542ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Hom}}(F,G)}"></dd></dl> is the sheaf of abelian groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto \operatorname {Hom} (F|_{U},G|_{U})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>,</mo> <mi>G</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto \operatorname {Hom} (F|_{U},G|_{U})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd8b6f3d4f0b062506f158318261b681ed91f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.928ex; height:3.009ex;" alt="{\displaystyle U\mapsto \operatorname {Hom} (F|_{U},G|_{U})}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F|_{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F|_{U}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579265d2b4ac072976349554e498e2a1f60560ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.88ex; height:3.009ex;" alt="{\displaystyle F|_{U}}"> is the sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F|_{U})(V)=F(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F|_{U})(V)=F(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcc496d99e24425a2be942840898320d658d356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.722ex; height:3.009ex;" alt="{\displaystyle (F|_{U})(V)=F(V)}"> (note sheafification is not needed here). The direct sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is the sheaf given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto F(U)\oplus G(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto F(U)\oplus G(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31386ae31250f49d45ad32131f0cc7ea2f6a9947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.988ex; height:2.843ex;" alt="{\displaystyle U\mapsto F(U)\oplus G(U)}">, and the tensor product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> is the sheaf associated to the presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto F(U)\otimes G(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto F(U)\otimes G(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631f7cd4de0e34075ff57622d7ab3ac698684329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.988ex; height:2.843ex;" alt="{\displaystyle U\mapsto F(U)\otimes G(U)}">. All of these operations extend to <a href="/wiki/Sheaf_of_modules" title="Sheaf of modules">sheaves of modules</a> over a <a href="/wiki/Sheaf_of_rings" class="mw-redirect" title="Sheaf of rings">sheaf of rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; the above is the special case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the <a href="/wiki/Constant_sheaf" title="Constant sheaf">constant sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {Z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {Z} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16ec565233e46a4c1f7107e2e86e4b222f4d80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-bottom: -0.827ex; width:1.636ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {Z} }}}">. <div class="mw-heading mw-heading3"><h3 id="Basic_functoriality">Basic functoriality</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=16" title="Edit section: Basic functoriality">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Image_functors_for_sheaves" title="Image functors for sheaves">Image functors for sheaves</a></div> Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> between two topological spaces, pushforward and pullback relate sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> to those on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> and vice versa. <div class="mw-heading mw-heading4"><h4 id="Direct_image">Direct image</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=17" title="Edit section: Direct image">edit</a>]</div> The pushforward (also known as <a href="/wiki/Direct_image_functor" title="Direct image functor">direct image</a>) of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is the sheaf defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{*}{\mathcal {F}})(V)={\mathcal {F}}(f^{-1}(V)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{*}{\mathcal {F}})(V)={\mathcal {F}}(f^{-1}(V)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f975911da4fb9155ffb0a6262341456bd6de97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.257ex; height:3.176ex;" alt="{\displaystyle (f_{*}{\mathcal {F}})(V)={\mathcal {F}}(f^{-1}(V)).}"></dd></dl> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">, so that its preimage is open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> by the continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. This construction recovers the skyscraper sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17297f0fdd502bf3cf9e999b890ffb872869c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.597ex; height:2.509ex;" alt="{\displaystyle S_{x}}"> mentioned above: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}=i_{*}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}=i_{*}(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7954a44e05da9bcd1feb7058637fb9ab1d0b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.861ex; height:2.843ex;" alt="{\displaystyle S_{x}=i_{*}(S)}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:\{x\}\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:\{x\}\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb5a904fd7e5b91e6a70222fca184b1deef7d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.988ex; height:2.843ex;" alt="{\displaystyle i:\{x\}\to X}"> is the inclusion, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is regarded as a sheaf on the <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singleton</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\{*\})=S,S(\emptyset )=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\{*\})=S,S(\emptyset )=\emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9727d71a2932e2bf1ee2160ea7ecb75a821156f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.159ex; height:2.843ex;" alt="{\displaystyle S(\{*\})=S,S(\emptyset )=\emptyset }">. For a map between <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact spaces</a>, the <a href="/wiki/Direct_image_with_compact_support" title="Direct image with compact support">direct image with compact support</a> is a subsheaf of the direct image.<a href="#cite_note-9">[9]</a> By definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{!}{\mathcal {F}})(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{!}{\mathcal {F}})(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c827b696902bc538615f599615c3f667ddc58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.161ex; height:2.843ex;" alt="{\displaystyle (f_{!}{\mathcal {F}})(V)}"> consists of those <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in {\mathcal {F}}(f^{-1}(V))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in {\mathcal {F}}(f^{-1}(V))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c2d19ad8af1f48c18e8441a7ace82a58377729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.917ex; height:3.176ex;" alt="{\displaystyle s\in {\mathcal {F}}(f^{-1}(V))}"> whose <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a> is mapped <a href="/wiki/Proper_map" title="Proper map">properly</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is proper itself, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{!}{\mathcal {F}}=f_{*}{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{!}{\mathcal {F}}=f_{*}{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a5611e68f1242b17087eeadab22262ab08e81df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.974ex; height:2.509ex;" alt="{\displaystyle f_{!}{\mathcal {F}}=f_{*}{\mathcal {F}}}">, but in general they disagree. <div class="mw-heading mw-heading4"><h4 id="Inverse_image">Inverse image</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=18" title="Edit section: Inverse image">edit</a>]</div> The pullback or <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image</a> goes the other way: it produces a sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}{\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}{\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36121bcf46bedbfe1e77c280a27dacc246bca22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.046ex; height:3.009ex;" alt="{\displaystyle f^{-1}{\mathcal {G}}}"> out of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is the inclusion of an open subset, then the inverse image is just a restriction, i.e., it is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{-1}{\mathcal {G}})(U)={\mathcal {G}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f^{-1}{\mathcal {G}})(U)={\mathcal {G}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945fa523d6e8fa20f9d92613479c5a7ddd109933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.53ex; height:3.176ex;" alt="{\displaystyle (f^{-1}{\mathcal {G}})(U)={\mathcal {G}}(U)}"> for an open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. A sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> (on some space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">) is called <a href="/wiki/Locally_constant_sheaf" title="Locally constant sheaf">locally constant</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\bigcup _{i\in I}U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\bigcup _{i\in I}U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/238a86e8ced1924f5c41300d8af6790015d63e9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.434ex; height:5.676ex;" alt="{\displaystyle X=\bigcup _{i\in I}U_{i}}"> by some open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> such that the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> to all these open subsets is constant. On a wide range of topological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, such sheaves are <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to <a href="/wiki/Group_representation" title="Group representation">representations</a> of the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c502a8f1054224532cb495be2f6b5e65f660c2aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(X)}">. For general maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">, the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}{\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}{\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36121bcf46bedbfe1e77c280a27dacc246bca22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.046ex; height:3.009ex;" alt="{\displaystyle f^{-1}{\mathcal {G}}}"> is more involved; it is detailed at <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image functor</a>. The stalk is an essential special case of the pullback in view of a natural identification, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> is as above: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{x}=i^{-1}{\mathcal {G}}(\{x\}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{x}=i^{-1}{\mathcal {G}}(\{x\}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e582217e3587ec573cf1602b74ffe13378e4a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.292ex; height:3.176ex;" alt="{\displaystyle {\mathcal {G}}_{x}=i^{-1}{\mathcal {G}}(\{x\}).}"></dd></dl> More generally, stalks satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1af72c4d6f8eaf13a5458837a71b63e19b6b952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.865ex; height:3.509ex;" alt="{\displaystyle (f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}}">. <div class="mw-heading mw-heading4"><h4 id="Extension_by_zero">Extension by zero</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=19" title="Edit section: Extension by zero">edit</a>]</div> For the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j:U\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j:U\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a7f2f35febd238a03138d51bc4f78c7e460600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.299ex; height:2.509ex;" alt="{\displaystyle j:U\to X}"> of an open subset, the extension by zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{!}{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{!}{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/426214bed61df7a78b162e58e5242d9b605d14bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.601ex; height:2.509ex;" alt="{\displaystyle j_{!}{\mathcal {F}}}"> (pronounced "j lower shriek of F") of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> of abelian groups on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is the sheafification of the presheaf defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\mapsto {\mathcal {F}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\mapsto {\mathcal {F}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8e24d763fd75c20ef76c6be3609adc7bcd0c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.924ex; height:2.843ex;" alt="{\displaystyle V\mapsto {\mathcal {F}}(V)}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.668ex; height:2.343ex;" alt="{\displaystyle V\subseteq U}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\mapsto 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">↦</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\mapsto 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df80731f3e44ad89880db8b009d9b300810b1dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V\mapsto 0}"> otherwise.</dd></dl> For a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, this construction is in a sense complementary to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66dc282423598fe1c56516de137b99ab43d2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:1.857ex; height:2.509ex;" alt="{\displaystyle i_{*}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\setminus U\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>X</mi> <mo class="MJX-variant">∖</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:X\setminus U\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51bb710103f8e467510aa657ac0490bafb1dbb00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.291ex; height:2.843ex;" alt="{\displaystyle i:X\setminus U\to X}"> is the inclusion of the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (j_{!}j^{*}{\mathcal {G}})_{x}={\mathcal {G}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (j_{!}j^{*}{\mathcal {G}})_{x}={\mathcal {G}}_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b01dec1c885f1f5a1cc89137a703b5bfd2bafd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.688ex; height:2.843ex;" alt="{\displaystyle (j_{!}j^{*}{\mathcal {G}})_{x}={\mathcal {G}}_{x}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">, and the stalk is zero otherwise, while</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i_{*}i^{*}{\mathcal {G}})_{x}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i_{*}i^{*}{\mathcal {G}})_{x}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6525e070306c8d95dda02132ae407cf9b059ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.348ex; height:2.843ex;" alt="{\displaystyle (i_{*}i^{*}{\mathcal {G}})_{x}=0}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">, and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6efd33e635b4a1ce8a0bc3e2868681ccd916f189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.556ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{x}}"> otherwise.</dd></dl> More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826569be03f873b81cdc6f12637ef5520c369d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}"> is a locally closed subset, then there exists an open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb4d220a869f247f17f6fdc745826ed3c178dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.356ex; height:2.509ex;" alt="{\displaystyle f:A\to U}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j:U\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j:U\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a7f2f35febd238a03138d51bc4f78c7e460600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:10.299ex; height:2.509ex;" alt="{\displaystyle j:U\to X}"> be the natural inclusions. Then the extension by zero of a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{!}f_{*}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{!}f_{*}F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a43709c8b842cb37a0f7456b973f101840c27e7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.609ex; height:2.509ex;" alt="{\displaystyle j_{!}f_{*}F}">. Due to its nice behavior on stalks, the extension by zero functor is useful for reducing sheaf-theoretic questions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> to ones on the strata of a <a href="/wiki/Stratification_(mathematics)" title="Stratification (mathematics)">stratification</a>, i.e., a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> into smaller, locally closed subsets. <div class="mw-heading mw-heading2"><h2 id="Complements">Complements</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=20" title="Edit section: Complements">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Sheaves_in_more_general_categories">Sheaves in more general categories</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=21" title="Edit section: Sheaves in more general categories">edit</a>]</div> In addition to (pre-)sheaves as introduced above, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}"> is merely a set, it is in many cases important to keep track of additional structure on these sections. For example, the sections of the sheaf of continuous functions naturally form a real <a href="/wiki/Vector_space" title="Vector space">vector space</a>, and restriction is a <a href="/wiki/Linear_map" title="Linear map">linear map</a> between these vector spaces. Presheaves with values in an arbitrary category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> are defined by first considering the category of open sets on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> to be the <a href="/wiki/Posetal_category" title="Posetal category">posetal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e6b588bf3d9ada6cd5342cff5a72d1044cd500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle O(X)}"> whose objects are the open sets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and whose morphisms are inclusions. Then a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">-valued presheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is the same as a <a href="/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e6b588bf3d9ada6cd5342cff5a72d1044cd500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle O(X)}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">. Morphisms in this category of functors, also known as <a href="/wiki/Natural_transformations" class="mw-redirect" title="Natural transformations">natural transformations</a>, are the same as the morphisms defined above, as can be seen by unraveling the definitions. If the target category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> admits all <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">limits</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">-valued presheaf is a sheaf if the following diagram is an <a href="/wiki/Equalizer_(mathematics)" class="mw-redirect" title="Equalizer (mathematics)">equalizer</a> for every open cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf91339c1648ed6bcadc0f8e06fbde97a884bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.038ex; width:12.17ex; height:2.843ex;" alt="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}"> of any open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\rightarrow \prod _{i}F(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j}F(U_{i}\cap U_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">⟶</mo> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mfrac> </mrow> </mfrac> </mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(U)\rightarrow \prod _{i}F(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j}F(U_{i}\cap U_{j}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd870fcdad5ca37e28ae23af83ad94b6c1e91825" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.454ex; height:6.509ex;" alt="{\displaystyle F(U)\rightarrow \prod _{i}F(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j}F(U_{i}\cap U_{j}).}"></dd></dl> Here the first map is the product of the restriction maps <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {res} _{U_{i},U}\colon F(U)\rightarrow F(U_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>U</mi> </mrow> </msub> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {res} _{U_{i},U}\colon F(U)\rightarrow F(U_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f0abc960183c3bf3bef32353f5bd3d1b770158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.476ex; height:3.009ex;" alt="{\displaystyle \operatorname {res} _{U_{i},U}\colon F(U)\rightarrow F(U_{i})}"></dd></dl> and the pair of arrows the products of the two sets of restrictions <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}\cap U_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}\cap U_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fceecbb790b1bc43c1ed56968b0feff067320511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.58ex; height:3.176ex;" alt="{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}\cap U_{j})}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}\cap U_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}\cap U_{j}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa79301c93180bd0908f14ffd52f5149485b8e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.426ex; height:3.176ex;" alt="{\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}\cap U_{j}).}"></dd></dl> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a>, this condition can also be rephrased by requiring that there is an <a href="/wiki/Exact_sequence" title="Exact sequence">exact sequence</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to F(U)\to \prod _{i}F(U_{i})\xrightarrow {\operatorname {res} _{U_{i}\cap U_{j},U_{i}}-\operatorname {res} _{U_{i}\cap U_{j},U_{j}}} \prod _{i,j}F(U_{i}\cap U_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <msub> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi>res</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mpadded> </mover> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to F(U)\to \prod _{i}F(U_{i})\xrightarrow {\operatorname {res} _{U_{i}\cap U_{j},U_{i}}-\operatorname {res} _{U_{i}\cap U_{j},U_{j}}} \prod _{i,j}F(U_{i}\cap U_{j}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ebff2ba75dbebcac49cb521aba82cacb9668b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-top: -0.323ex; width:58.9ex; height:7.343ex;" alt="{\displaystyle 0\to F(U)\to \prod _{i}F(U_{i})\xrightarrow {\operatorname {res} _{U_{i}\cap U_{j},U_{i}}-\operatorname {res} _{U_{i}\cap U_{j},U_{j}}} \prod _{i,j}F(U_{i}\cap U_{j}).}"></dd></dl> A particular case of this sheaf condition occurs for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> being the empty set, and the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> also being empty. In this case, the sheaf condition requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\emptyset )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\emptyset )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61cfe6e8cf075ac9d634194c83fd2a20cd2d7c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.898ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\emptyset )}"> to be the <a href="/wiki/Terminal_object" class="mw-redirect" title="Terminal object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">. <div class="mw-heading mw-heading3"><h3 id="Ringed_spaces_and_sheaves_of_modules">Ringed spaces and sheaves of modules</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=22" title="Edit section: Ringed spaces and sheaves of modules">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Ringed_space" title="Ringed space">Ringed space</a> and <a href="/wiki/Sheaf_of_modules" title="Sheaf of modules">Sheaf of modules</a></div> In several geometrical disciplines, including <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}">. Such a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {O}}_{X})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46716c1fec068ffe9981107a5a215fa1fc9e7d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})}"> is called a <a href="/wiki/Ringed_space" title="Ringed space">ringed space</a>. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X,x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X,x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0617c163a03d1584ad15e192f2d488ffaaef7d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X,x}}"> of the structure sheaf are <a href="/wiki/Local_ring" title="Local ring">local rings</a>, in which case the pair is called a locally ringed space. For example, an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}"> manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a locally ringed space whose structure sheaf consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}">-functions on the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">. The property of being a locally ringed space translates into the fact that such a function, which is nonzero at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, is also non-zero on a sufficiently small open neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. Some authors actually define real (or complex) manifolds to be locally ringed spaces that are locally isomorphic to the pair consisting of an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> (respectively <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}">) together with the sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167fdb0cfb5644c4623b5842e1a9141acd83b534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.887ex; height:2.676ex;" alt="{\displaystyle C^{k}}"> (respectively holomorphic) functions.<a href="#cite_note-10">[10]</a> Similarly, <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a>. Given a ringed space, a sheaf of modules is a sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"> such that on every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153116181b0098f5d7038b1e5212c7f51cce980c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.382ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(U)}"> is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db03ab826a2c84d86dc4a6326cd1b881473b4863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X}(U)}">-module and for every inclusion of open sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.668ex; height:2.343ex;" alt="{\displaystyle V\subseteq U}">, the restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(U)\to {\mathcal {M}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(U)\to {\mathcal {M}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99fe12d77b343d5500dd5e8deda5eb46aff8ad78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.384ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(U)\to {\mathcal {M}}(V)}"> is compatible with the restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(U)\to {\mathcal {O}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(U)\to {\mathcal {O}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5f2963b055062d15f7e35f3a985dede3805385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.502ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(U)\to {\mathcal {O}}(V)}">: the restriction of fs is the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> times that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5569fb1106ea2f1d95c5b42192dd819077f4bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.442ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(U)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153116181b0098f5d7038b1e5212c7f51cce980c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.382ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(U)}">. Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> and <a href="/wiki/Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free sheaves</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}">-modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"> consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are <a href="/wiki/D-module" title="D-module"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">-modules</a>, that is, modules over the sheaf of <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a>. On any topological space, modules over the constant sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {Z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {Z} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16ec565233e46a4c1f7107e2e86e4b222f4d80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-bottom: -0.827ex; width:1.636ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {Z} }}}"> are the same as <a href="/wiki/Abelian_sheaf" class="mw-redirect" title="Abelian sheaf">sheaves of abelian groups</a> in the sense above. There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.676ex;" alt="{\displaystyle f^{*}}"> and it is distinct from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}">. See <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image functor</a>. <div class="mw-heading mw-heading4"><h4 id="Finiteness_conditions_for_sheaves_of_modules">Finiteness conditions for sheaves of modules</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=23" title="Edit section: Finiteness conditions for sheaves of modules">edit</a>]</div> Finiteness conditions for module over <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a> give rise to similar finiteness conditions for sheaves of modules: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"> is called finitely generated (respectively finitely presented) if, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, there exists an open neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> (possibly depending on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">), and a surjective morphism of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7119ecb38996b26f495e078db6a5e26b4bc5f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.166ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}}"> (respectively, in addition a natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">, and an exact sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61796f02cfbc1818a8ed03807075798a7ff1c7a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.221ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}\to 0}">.) Paralleling the notion of a <a href="/wiki/Coherent_module" class="mw-redirect" title="Coherent module">coherent module</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}"> is called a <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaf</a> if it is of finite type and if, for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> and every morphism of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :{\mathcal {O}}_{X}^{n}\to {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :{\mathcal {O}}_{X}^{n}\to {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e930370ea1a8129e6be63c2240bcbb4f94b833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.21ex; height:2.843ex;" alt="{\displaystyle \phi :{\mathcal {O}}_{X}^{n}\to {\mathcal {M}}}"> (not necessarily surjective), the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> is of finite type. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}"> is coherent if it is coherent as a module over itself. Like for modules, coherence is in general a strictly stronger condition than finite presentation. The <a href="/wiki/Oka_coherence_theorem" title="Oka coherence theorem">Oka coherence theorem</a> states that the sheaf of holomorphic functions on a <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a> is coherent. <div class="mw-heading mw-heading3"><h3 id="The_étalé_space_of_a_sheaf">The étalé space of a sheaf</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=24" title="Edit section: The étalé space of a sheaf">edit</a>]</div> In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the étalé space, from the French word étalé <a href="/wiki/Help:IPA/French" title="Help:IPA/French">[etale]</a>, meaning roughly "spread out". If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\in {\text{Sh}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sh</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\in {\text{Sh}}(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71bdf4bf1b8e024bd7029404e33025b28619c0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.956ex; height:2.843ex;" alt="{\displaystyle F\in {\text{Sh}}(X)}"> is a sheaf over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, then the étalé space (sometimes called the étale space) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> is a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> together with a <a href="/wiki/Local_homeomorphism" title="Local homeomorphism">local homeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"> such that the sheaf of sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (\pi ,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (\pi ,-)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418cc7c80017d4b04ee99448cfcd9b88a6c85019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.436ex; height:2.843ex;" alt="{\displaystyle \Gamma (\pi ,-)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is usually very strange, and even if the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> arises from a natural topological situation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> may not have any clear topological interpretation. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> is the sheaf of sections of a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e8fe3d0dc6757c546d482a59b9c5220eaff7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:Y\to X}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7eba308c0d2304559e402392f1883bc841ce8be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.647ex; height:2.176ex;" alt="{\displaystyle E=Y}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is a <a href="/wiki/Local_homeomorphism" title="Local homeomorphism">local homeomorphism</a>. The étalé space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is constructed from the stalks of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. As a set, it is their <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is the obvious map that takes the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> on the stalk of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}">. The topology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is defined as follows. For each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in F(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in F(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da1a1db9aaf5b167e9d1d260a6a5846dffd8dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.264ex; height:2.843ex;" alt="{\displaystyle s\in F(U)}"> and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}">, we get a germ of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [s]_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>s</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [s]_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b007e53586d2bd81d4ce2657174eec3347d05131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.557ex; height:2.843ex;" alt="{\displaystyle [s]_{x}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f738a5f93f694ded413a8f8da71081d0fdff4501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.009ex;" alt="{\displaystyle s_{x}}">. These germs determine points of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">. For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in F(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>∈</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\in F(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da1a1db9aaf5b167e9d1d260a6a5846dffd8dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.264ex; height:2.843ex;" alt="{\displaystyle s\in F(U)}">, the union of these points (for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}">) is declared to be open in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">. Notice that each stalk has the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a> as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor. The construction above determines an <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalence of categories</a> between the category of sheaves of sets on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and the category of étalé spaces over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf. This construction makes all sheaves into <a href="/wiki/Representable_functor" title="Representable functor">representable functors</a> on certain categories of topological spaces. As above, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> be a sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> be its étalé space, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"> be the natural projection. Consider the <a href="/wiki/Overcategory" title="Overcategory">overcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Top}}/X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Top</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Top}}/X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2244568c09ce079a67cd6d0f88628961625e19cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.276ex; height:2.843ex;" alt="{\displaystyle {\text{Top}}/X}"> of topological spaces over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, that is, the category of topological spaces together with fixed continuous maps to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. Every object of this category is a continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e8fe3d0dc6757c546d482a59b9c5220eaff7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:Y\to X}">, and a morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772ca21e4bcba5ccedd53ff7cb37add671628747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle Y\to X}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910c5602ddbc171446f929a662ecf62603d9f325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.274ex; height:2.176ex;" alt="{\displaystyle Z\to X}"> is a continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\to Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0c8a75448c17c05895b2f2ac236d70c9ef16ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.176ex;" alt="{\displaystyle Y\to Z}"> that commutes with the two maps to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. There is a functor<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma :{\text{Top}}/X\to {\text{Sets}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Top</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sets</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma :{\text{Top}}/X\to {\text{Sets}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/800d147eb643bd5c8b01992043e48adcafee5d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.425ex; height:2.843ex;" alt="{\displaystyle \Gamma :{\text{Top}}/X\to {\text{Sets}}}"></blockquote>sending an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:Y\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e8fe3d0dc6757c546d482a59b9c5220eaff7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:Y\to X}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}F(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}F(Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64eff4407d5e937835fec86a56d01ad387ac6446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.977ex; height:3.176ex;" alt="{\displaystyle f^{-1}F(Y)}">. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:U\hookrightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">↪</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:U\hookrightarrow X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189f22305defaeaf378846965df227e88aac0607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.409ex; height:2.176ex;" alt="{\displaystyle i:U\hookrightarrow X}"> is the inclusion of an open subset, then<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (i)=f^{-1}F(U)=F(U)=\Gamma (F,U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (i)=f^{-1}F(U)=F(U)=\Gamma (F,U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5458fe2d7a53833ce45b15327e80adf6510953a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.498ex; height:3.176ex;" alt="{\displaystyle \Gamma (i)=f^{-1}F(U)=F(U)=\Gamma (F,U)}"></blockquote>and for the inclusion of a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:\{x\}\hookrightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">↪</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:\{x\}\hookrightarrow X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2260c0041443f0b4d656817d8a6773c0fedc40bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.281ex; height:2.843ex;" alt="{\displaystyle i:\{x\}\hookrightarrow X}">, then<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (i)=f^{-1}F(\{x\})=F|_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (i)=f^{-1}F(\{x\})=F|_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb21f50b2aea1807127619ba9d5b92b3f57fb95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.68ex; height:3.343ex;" alt="{\displaystyle \Gamma (i)=f^{-1}F(\{x\})=F|_{x}}"></blockquote>is the stalk of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">. There is a natural isomorphism<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{-1}F)(Y)\cong \operatorname {Hom} _{\mathbf {Top} /X}(f,\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> <mi mathvariant="bold">o</mi> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>X</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f^{-1}F)(Y)\cong \operatorname {Hom} _{\mathbf {Top} /X}(f,\pi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec03363e2e571d3d8660e54dddba2019d0cba50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.944ex; height:3.509ex;" alt="{\displaystyle (f^{-1}F)(Y)\cong \operatorname {Hom} _{\mathbf {Top} /X}(f,\pi )}">,</blockquote>which shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :E\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi :E\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf8de3c25d0905f50abaf4f9374daa0a4bd796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.639ex; height:2.176ex;" alt="{\displaystyle \pi :E\to X}"> (for the étalé space) represents the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> is constructed so that the projection map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is a covering map. In algebraic geometry, the natural analog of a covering map is called an <a href="/wiki/%C3%89tale_morphism" title="Étale morphism">étale morphism</a>. Despite its similarity to "étalé", the word étale <a href="/wiki/Help:IPA/French" title="Help:IPA/French">[etal]</a> has a different meaning in French. It is possible to turn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> into a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> into a morphism of schemes in such a way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> retains the same universal property, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> is not in general an étale morphism because it is not quasi-finite. It is, however, <a href="/wiki/Formally_%C3%A9tale" class="mw-redirect" title="Formally étale">formally étale</a>. The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>. <div class="mw-heading mw-heading2"><h2 id="Sheaf_cohomology">Sheaf cohomology</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=25" title="Edit section: Sheaf cohomology">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">Sheaf cohomology</a></div> In contexts where the open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> is fixed, and the sheaf is regarded as a variable, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462254eb7da92fe91d24532826f4e036cf1eeaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.333ex; height:2.843ex;" alt="{\displaystyle F(U)}"> is also often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (U,F).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (U,F).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c01f50852e300bde3d234821ccc0047b5f82f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.466ex; height:2.843ex;" alt="{\displaystyle \Gamma (U,F).}"> As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fba7f07f364f53efaaf999a67db3f374bb8dc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.933ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}\to {\mathcal {G}}}"> is a map with the following property: for any section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in {\mathcal {G}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in {\mathcal {G}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139978a98b40690b0ccb5e17aa5b36ea3db865f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle g\in {\mathcal {G}}(U)}"> there is a covering <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf91339c1648ed6bcadc0f8e06fbde97a884bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.038ex; width:12.17ex; height:2.843ex;" alt="{\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}"> where <blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\bigcup _{i\in I}U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\bigcup _{i\in I}U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7305e8635b8c2d7a8f11b9a7ae1f85857147512" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.237ex; height:5.676ex;" alt="{\displaystyle U=\bigcup _{i\in I}U_{i}}"> </blockquote>of open subsets, such that the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g|_{U_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g|_{U_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11deb088fa2201bd11656f9d6d2470da787061e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.743ex; height:3.176ex;" alt="{\displaystyle g|_{U_{i}}}"> are in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7cd8c81cf3632ba66063c97c5d2d3284ca4452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.123ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U_{i})}">. However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> itself need not be in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.518ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(U)}">. A concrete example of this phenomenon is the exponential map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}{\stackrel {\exp }{\to }}{\mathcal {O}}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>exp</mi> </mrow> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}{\stackrel {\exp }{\to }}{\mathcal {O}}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a28b32c3727e1f062f2fc92e7b909357baa783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.723ex; height:3.343ex;" alt="{\displaystyle {\mathcal {O}}{\stackrel {\exp }{\to }}{\mathcal {O}}^{\times }}"></dd></dl> between the sheaf of <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> (on some open subset in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">, say), admits a <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> locally, i.e., after restricting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> to appropriate open subsets. However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> need not have a logarithm globally. Sheaf cohomology captures this phenomenon. More precisely, for an <a href="/wiki/Exact_sequence" title="Exact sequence">exact sequence</a> of sheaves of abelian groups <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to {\mathcal {F}}_{1}\to {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}\to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {F}}_{1}\to {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}\to 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ce7f51d14aba5d0379830eae91828dff76e50c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.604ex; height:2.509ex;" alt="{\displaystyle 0\to {\mathcal {F}}_{1}\to {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}\to 0,}"></dd></dl> (i.e., an epimorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2111aa983de54c53779ebd7fab4a8e34bb0b0aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.065ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}}"> whose kernel is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d392bc6fb93f473201227731129df52c55d996e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.725ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{1}}">), there is a long exact sequence<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to \Gamma (U,{\mathcal {F}}_{1})\to \Gamma (U,{\mathcal {F}}_{2})\to \Gamma (U,{\mathcal {F}}_{3})\to H^{1}(U,{\mathcal {F}}_{1})\to H^{1}(U,{\mathcal {F}}_{2})\to H^{1}(U,{\mathcal {F}}_{3})\to H^{2}(U,{\mathcal {F}}_{1})\to \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to \Gamma (U,{\mathcal {F}}_{1})\to \Gamma (U,{\mathcal {F}}_{2})\to \Gamma (U,{\mathcal {F}}_{3})\to H^{1}(U,{\mathcal {F}}_{1})\to H^{1}(U,{\mathcal {F}}_{2})\to H^{1}(U,{\mathcal {F}}_{3})\to H^{2}(U,{\mathcal {F}}_{1})\to \dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716752c0d85ef762d936462037667faf743d2b69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.246ex; height:3.176ex;" alt="{\displaystyle 0\to \Gamma (U,{\mathcal {F}}_{1})\to \Gamma (U,{\mathcal {F}}_{2})\to \Gamma (U,{\mathcal {F}}_{3})\to H^{1}(U,{\mathcal {F}}_{1})\to H^{1}(U,{\mathcal {F}}_{2})\to H^{1}(U,{\mathcal {F}}_{3})\to H^{2}(U,{\mathcal {F}}_{1})\to \dots }">By means of this sequence, the first cohomology group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{1}(U,{\mathcal {F}}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{1}(U,{\mathcal {F}}_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75de654f68dfb512e23696c9ea28ad724167091b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.509ex; height:3.176ex;" alt="{\displaystyle H^{1}(U,{\mathcal {F}}_{1})}"> is a measure for the non-surjectivity of the map between sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50876b20378d232fb8274ed5c8e46c51bd56125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.725ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2b04c31347b3db88c716ccff11dfe28b764647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.725ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{3}}">. There are several different ways of constructing sheaf cohomology. <a href="#CITEREFGrothendieck1957">Grothendieck (1957)</a> introduced them by defining sheaf cohomology as the <a href="/wiki/Derived_functor" title="Derived functor">derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">. This method is theoretically satisfactory, but, being based on <a href="/wiki/Injective_resolution" class="mw-redirect" title="Injective resolution">injective resolutions</a>, of little use in concrete computations. <a href="/wiki/Godement_resolution" title="Godement resolution">Godement resolutions</a> are another general, but practically inaccessible approach. <div class="mw-heading mw-heading3"><h3 id="Computing_sheaf_cohomology">Computing sheaf cohomology</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=26" title="Edit section: Computing sheaf cohomology">edit</a>]</div> Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by <a href="/wiki/Soft_sheaf" class="mw-redirect" title="Soft sheaf">soft sheaves</a>, <a href="/wiki/Fine_sheaf" class="mw-redirect" title="Fine sheaf">fine sheaves</a>, and <a href="/wiki/Flabby_sheaf" class="mw-redirect" title="Flabby sheaf">flabby sheaves</a> (also known as flasque sheaves from the French flasque meaning flabby). For example, a <a href="/wiki/Partition_of_unity" title="Partition of unity">partition of unity</a> argument shows that the sheaf of smooth functions on a manifold is soft. The higher cohomology groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}(U,{\mathcal {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{i}(U,{\mathcal {F}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859747e81e331844f463ba039d7f9a493d3f7bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.456ex; height:3.176ex;" alt="{\displaystyle H^{i}(U,{\mathcal {F}})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f49f2878fd68a89c3da37eb537198e887cf0293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>0}"> vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, the <a href="/wiki/De_Rham_complex" class="mw-redirect" title="De Rham complex">de Rham complex</a> is a resolution of the constant sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {R} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad35a3aa0e2f7be12f98f9921494ef4408a7443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.537ex; margin-bottom: -0.801ex; width:2.006ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {R} }}}"> on any smooth manifold, so the sheaf cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {R} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad35a3aa0e2f7be12f98f9921494ef4408a7443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.537ex; margin-bottom: -0.801ex; width:2.006ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {R} }}}"> is equal to its <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a>. A different approach is by <a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing the <a href="/wiki/Coherent_sheaf_cohomology" title="Coherent sheaf cohomology">coherent sheaf cohomology</a> of complex projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}">.<a href="#cite_note-11">[11]</a> It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246d198ccb2f5e5488a7afd13093aab7b005139b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.158ex; height:2.676ex;" alt="{\displaystyle H^{1}}"> but incorrect higher cohomology groups. To get around this, <a href="/wiki/Jean-Louis_Verdier" title="Jean-Louis Verdier">Jean-Louis Verdier</a> developed <a href="/wiki/Hypercover" class="mw-redirect" title="Hypercover">hypercoverings</a>. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of <a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a>'s <a href="/wiki/Mixed_Hodge_structure" title="Mixed Hodge structure">mixed Hodge structures</a>. Many other coherent sheaf cohomology groups are found using an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:X\hookrightarrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">↪</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i:X\hookrightarrow Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f3f0662c620237a4fae4347bc9e48f13e34e47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.4ex; height:2.176ex;" alt="{\displaystyle i:X\hookrightarrow Y}"> of a space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> into a space with known cohomology, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}">, or some <a href="/wiki/Weighted_projective_space" title="Weighted projective space">weighted projective space</a>. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{*}{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{*}{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d550b273df68f3369104b6eb8db481c4faa12294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.293ex; margin-bottom: -0.379ex; width:3.783ex; height:2.509ex;" alt="{\displaystyle i_{*}{\mathcal {F}}}">, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}(Y,i_{*}{\mathcal {F}})\cong H^{i}(X,{\mathcal {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>≅</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{i}(Y,i_{*}{\mathcal {F}})\cong H^{i}(X,{\mathcal {F}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efeb81f72dd29e18e92948c564f45dbcc26abed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.054ex; height:3.176ex;" alt="{\displaystyle H^{i}(Y,i_{*}{\mathcal {F}})\cong H^{i}(X,{\mathcal {F}})}">. For example, computing the <a href="/wiki/Coherent_sheaf_cohomology#Sheaf_cohomology_of_plane-curves" title="Coherent sheaf cohomology">coherent sheaf cohomology of projective plane curves</a> is easily found. One big theorem in this space is the <a href="/wiki/Hodge_structure" title="Hodge structure">Hodge decomposition</a> found using a <a href="/wiki/Leray_spectral_sequence" title="Leray spectral sequence">spectral sequence associated to sheaf cohomology groups</a>, proved by Deligne.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> Essentially, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}">-page with terms<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}^{p,q}=H^{p}(X,\Omega _{X}^{q})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msubsup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}^{p,q}=H^{p}(X,\Omega _{X}^{q})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcf9e07d7e5e126a7c95223d54f6377c37a7035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.461ex; height:3.176ex;" alt="{\displaystyle E_{1}^{p,q}=H^{p}(X,\Omega _{X}^{q})}"></blockquote>the sheaf cohomology of a <a href="/wiki/Smooth_variety" class="mw-redirect" title="Smooth variety">smooth</a> <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, degenerates, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}=E_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}=E_{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c6ae10566d1a068d8ea940db4b7d28f9c7a609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.459ex; height:2.509ex;" alt="{\displaystyle E_{1}=E_{\infty }}">. This gives the canonical Hodge structure on the cohomology groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{k}(X,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{k}(X,\mathbb {C} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324298d64251920a92e34ace4c26b65654317150" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.693ex; height:3.176ex;" alt="{\displaystyle H^{k}(X,\mathbb {C} )}">. It was later found these cohomology groups can be easily explicitly computed using <a href="/wiki/Poincar%C3%A9_residue" title="Poincaré residue">Griffiths residues</a>. See <a href="/wiki/Jacobian_ideal" title="Jacobian ideal">Jacobian ideal</a>. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, <a href="/wiki/Decomposition_theorem" class="mw-redirect" title="Decomposition theorem">the decomposition theorem</a>, paving the path for <a href="/wiki/Mixed_Hodge_module" title="Mixed Hodge module">Mixed Hodge modules</a>. Another clean approach to the computation of some cohomology groups is the <a href="/wiki/Borel%E2%80%93Bott%E2%80%93Weil_theorem" class="mw-redirect" title="Borel–Bott–Weil theorem">Borel–Bott–Weil theorem</a>, which identifies the cohomology groups of some <a href="/wiki/Line_bundle" title="Line bundle">line bundles</a> on <a href="/wiki/Flag_manifold" class="mw-redirect" title="Flag manifold">flag manifolds</a> with <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representations</a> of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and <a href="/wiki/Grassmann_manifold" class="mw-redirect" title="Grassmann manifold">grassmann manifolds</a>. In many cases there is a duality theory for sheaves that generalizes <a href="/wiki/Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a>. See <a href="/wiki/Coherent_duality" title="Coherent duality">Grothendieck duality</a> and <a href="/wiki/Verdier_duality" title="Verdier duality">Verdier duality</a>. <div class="mw-heading mw-heading3"><h3 id="Derived_categories_of_sheaves">Derived categories of sheaves</h3>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=27" title="Edit section: Derived categories of sheaves">edit</a>]</div> The <a href="/wiki/Derived_category" title="Derived category">derived category</a> of the category of sheaves of, say, abelian groups on some space X, denoted here as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054ce5cebeedabbdd1656658239c5da5264c2b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.843ex;" alt="{\displaystyle D(X)}">, is the conceptual haven for sheaf cohomology, by virtue of the following relation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{n}(X,{\mathcal {F}})=\operatorname {Hom} _{D(X)}(\mathbf {Z} ,{\mathcal {F}}[n]).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{n}(X,{\mathcal {F}})=\operatorname {Hom} _{D(X)}(\mathbf {Z} ,{\mathcal {F}}[n]).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0a6c2e271e9f42d5480b26d32498a5f2800b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.023ex; height:3.176ex;" alt="{\displaystyle H^{n}(X,{\mathcal {F}})=\operatorname {Hom} _{D(X)}(\mathbf {Z} ,{\mathcal {F}}[n]).}"></dd></dl> The adjunction between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}">, which is the left adjoint of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903e97c260c5d157bff320b5e38e0e4b42cd20ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{*}}"> (already on the level of sheaves of abelian groups) gives rise to an adjunction <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}:D(Y)\rightleftarrows D(X):Rf_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇄</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>R</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}:D(Y)\rightleftarrows D(X):Rf_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e8f8553403f0a0db37ec9795a8773f0eef8f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.32ex; height:3.176ex;" alt="{\displaystyle f^{-1}:D(Y)\rightleftarrows D(X):Rf_{*}}"> (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}">),</dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rf_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rf_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f756896d20fb9481d5c8a1769fbb8adfe66f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.957ex; height:2.509ex;" alt="{\displaystyle Rf_{*}}"> is the derived functor. This latter functor encompasses the notion of sheaf cohomology since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{n}(X,{\mathcal {F}})=R^{n}f_{*}{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{n}(X,{\mathcal {F}})=R^{n}f_{*}{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e758c16cd63edc249b432d4c45d90a33940776c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.273ex; height:2.843ex;" alt="{\displaystyle H^{n}(X,{\mathcal {F}})=R^{n}f_{*}{\mathcal {F}}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \{*\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to \{*\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4fde8b1413caae276eeea5e9e27ebadfa76a226" class="mwe-math-fallback-image-inline mw-invert skin-invert" 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.sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="background:#ccccff;"><a href="/wiki/Image_functors_for_sheaves" title="Image functors for sheaves">Image functors for sheaves</a></th></tr><tr><td class="sidebar-content"> <a href="/wiki/Direct_image_functor" title="Direct image functor">direct image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903e97c260c5d157bff320b5e38e0e4b42cd20ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{*}}"></a></td> </tr><tr><td class="sidebar-content"> <a href="/wiki/Inverse_image_functor" title="Inverse image functor">inverse image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.676ex;" alt="{\displaystyle f^{*}}"></a></td> </tr><tr><td class="sidebar-content"> <a href="/wiki/Direct_image_with_compact_support" title="Direct image with compact support">direct image with compact support <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b2946320392521c0aa3d0b60d6d873adf0d10d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.829ex; height:2.509ex;" alt="{\displaystyle f_{!}}"></a></td> </tr><tr><td class="sidebar-content"> <a href="/wiki/Exceptional_inverse_image_functor" title="Exceptional inverse image functor">exceptional inverse image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rf^{!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rf^{!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93f1fa319348f7ac8d43f0b86fffbd07ee942a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.774ex; height:3.009ex;" alt="{\displaystyle Rf^{!}}"></a></td> </tr><tr><td class="sidebar-content"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\leftrightarrows f_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">⇆</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}\leftrightarrows f_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107fa29a670ca64bfb82538b319ecab4738cf6d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.182ex; height:2.676ex;" alt="{\displaystyle f^{*}\leftrightarrows f_{*}}"> </td> </tr><tr><td class="sidebar-content"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mo stretchy="false">⇆</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e14c904da2b1178f56d8511c223ea071a5590f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.6ex; height:3.176ex;" alt="{\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}"></td> </tr><tr><td class="sidebar-content"> <a href="/wiki/Base_change_theorems" title="Base change theorems">Base change theorems</a></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Images_of_sheaves" title="Template:Images of sheaves"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Images_of_sheaves" title="Template talk:Images of sheaves"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Images_of_sheaves" title="Special:EditPage/Template:Images of sheaves"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903e97c260c5d157bff320b5e38e0e4b42cd20ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{*}}">, the direct image with compact support <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b2946320392521c0aa3d0b60d6d873adf0d10d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.829ex; height:2.509ex;" alt="{\displaystyle f_{!}}"> can also be derived. By virtue of the following isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rf_{!}{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rf_{!}{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb0d45e2acb99f2491ae16f0b03f83e4000ec6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.519ex; height:2.509ex;" alt="{\displaystyle Rf_{!}{\mathcal {F}}}"> parametrizes the <a href="/wiki/Cohomology_with_compact_support" title="Cohomology with compact support">cohomology with compact support</a> of the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fibers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R^{i}f_{!}{\mathcal {F}})_{y}=H_{c}^{i}(f^{-1}(y),{\mathcal {F}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R^{i}f_{!}{\mathcal {F}})_{y}=H_{c}^{i}(f^{-1}(y),{\mathcal {F}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d71af9a12601a8b27f1dd5c36fc90a2c319a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.214ex; height:3.343ex;" alt="{\displaystyle (R^{i}f_{!}{\mathcal {F}})_{y}=H_{c}^{i}(f^{-1}(y),{\mathcal {F}}).}"><a href="#cite_note-14">[14]</a></dd></dl> This isomorphism is an example of a <a href="/wiki/Base_change_theorems" title="Base change theorems">base change theorem</a>. There is another adjunction <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rf_{!}:D(X)\rightleftarrows D(Y):f^{!}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msub> <mo>:</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇄</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rf_{!}:D(X)\rightleftarrows D(Y):f^{!}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709da09c08e43dfc92cab56dd312f5ab90ca5993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.959ex; height:3.176ex;" alt="{\displaystyle Rf_{!}:D(X)\rightleftarrows D(Y):f^{!}.}"></dd></dl> Unlike all the functors considered above, the twisted (or exceptional) inverse image functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{!}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4233fec5953b2348765f253f6fc1f2f2736ef03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.01ex; height:3.009ex;" alt="{\displaystyle f^{!}}"> is in general only defined on the level of <a href="/wiki/Derived_category" title="Derived category">derived categories</a>, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \{*\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to \{*\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4fde8b1413caae276eeea5e9e27ebadfa76a226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.297ex; height:2.843ex;" alt="{\displaystyle f:X\to \{*\}}"> and X is a smooth <a href="/wiki/Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable manifold</a> of dimension n, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{!}{\underline {\mathbf {R} }}\cong {\underline {\mathbf {R} }}[n].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>!</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>_</mo> </munder> </mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{!}{\underline {\mathbf {R} }}\cong {\underline {\mathbf {R} }}[n].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2324c49616982fdcdebcf4c6f7e4d0c10f59e4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:12.455ex; height:3.676ex;" alt="{\displaystyle f^{!}{\underline {\mathbf {R} }}\cong {\underline {\mathbf {R} }}[n].}"><a href="#cite_note-15">[15]</a></dd></dl> This computation, and the compatibility of the functors with duality (see <a href="/wiki/Verdier_duality" title="Verdier duality">Verdier duality</a>) can be used to obtain a high-brow explanation of <a href="/wiki/Poincar%C3%A9_duality" title="Poincaré duality">Poincaré duality</a>. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as <a href="/wiki/Coherent_duality" title="Coherent duality">coherent duality</a>. <a href="/wiki/Perverse_sheaf" title="Perverse sheaf">Perverse sheaves</a> are certain objects in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054ce5cebeedabbdd1656658239c5da5264c2b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.843ex;" alt="{\displaystyle D(X)}">, i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry of <a href="/wiki/Singularity_(mathematics)" title="Singularity (mathematics)">singularities</a>.<a href="#cite_note-16">[16]</a> <div class="mw-heading mw-heading4"><h4 id="Derived_categories_of_coherent_sheaves_and_the_Grothendieck_group">Derived categories of coherent sheaves and the Grothendieck group</h4>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=28" title="Edit section: Derived categories of coherent sheaves and the Grothendieck group">edit</a>]</div> Another important application of derived categories of sheaves is with the derived category of <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a> on a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{Coh}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>o</mi> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{Coh}(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ce971db10697d3160d7c26e035119bb78ca49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.939ex; height:2.843ex;" alt="{\displaystyle D_{Coh}(X)}">. This was used by Grothendieck in his development of <a href="/wiki/Intersection_theory" title="Intersection theory">intersection theory</a><a href="#cite_note-17">[17]</a> using <a href="/wiki/Derived_categories" class="mw-redirect" title="Derived categories">derived categories</a> and <a href="/wiki/K-theory" title="K-theory">K-theory</a>, that the intersection product of subschemes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1},Y_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{1},Y_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f047ad58c0eaef5603829006972f2abfc46b59d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.844ex; height:2.509ex;" alt="{\displaystyle Y_{1},Y_{2}}"> is represented in <a href="/wiki/Grothendieck_group" title="Grothendieck group">K-theory</a> as<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [Y_{1}]\cdot [Y_{2}]=[{\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{Y_{2}}]\in K({\text{Coh(X)}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msubsup> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∈</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Coh(X)</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [Y_{1}]\cdot [Y_{2}]=[{\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{Y_{2}}]\in K({\text{Coh(X)}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973a10eb4b2de4bbcb338569dbe3a83e24caf3ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.289ex; height:3.509ex;" alt="{\displaystyle [Y_{1}]\cdot [Y_{2}]=[{\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{Y_{2}}]\in K({\text{Coh(X)}})}"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{Y_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{Y_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6e0dcbcb89db49c812315e24bde3796d82f202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.662ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{Y_{i}}}"> are <a href="/wiki/Coherent_sheaves" class="mw-redirect" title="Coherent sheaves">coherent sheaves</a> defined by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}">-modules given by their <a href="/wiki/Structure_sheaf" class="mw-redirect" title="Structure sheaf">structure sheaves</a>. <div class="mw-heading mw-heading2"><h2 id="Sites_and_topoi">Sites and topoi</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=29" title="Edit section: Sites and topoi">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> and <a href="/wiki/Topos" title="Topos">Topos</a></div> <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a>'s <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a> stated that there was a <a href="/wiki/Weil_cohomology_theory" title="Weil cohomology theory">cohomology theory</a> for <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a> that would give an analogue of the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {C} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {C} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becee48d7f45ea2d736cfdb4b0133651a30bab0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.537ex; margin-bottom: -0.801ex; width:1.934ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {C} }}}"> in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a>, and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). <a href="/wiki/Alexandre_Grothendieck" class="mw-redirect" title="Alexandre Grothendieck">Alexandre Grothendieck</a> solved this problem by introducing <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topologies</a>, which axiomatize the notion of covering. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define <a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">étale cohomology</a> and <a href="/wiki/%E2%84%93-adic_cohomology" class="mw-redirect" title="ℓ-adic cohomology">ℓ-adic cohomology</a>, which eventually were used to prove the Weil conjectures. A category with a Grothendieck topology is called a site. A category of sheaves on a site is called a topos or a Grothendieck topos. The notion of a topos was later abstracted by <a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a> and Miles Tierney to define an <a href="/wiki/Elementary_topos" class="mw-redirect" title="Elementary topos">elementary topos</a>, which has connections to <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=30" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>. Please help <a href="/wiki/Special:EditPage/Sheaf_(mathematics)" title="Special:EditPage/Sheaf (mathematics)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed. (January 2023) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> The first origins of sheaf theory are hard to pin down – they may be co-extensive with the idea of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>]. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on <a href="/wiki/Cohomology" title="Cohomology">cohomology</a>. <ul><li>1936 <a href="/wiki/Eduard_%C4%8Cech" title="Eduard Čech">Eduard Čech</a> introduces the <a href="/wiki/Nerve_of_an_open_covering" class="mw-redirect" title="Nerve of an open covering">nerve</a> construction, for associating a <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> to an open covering.</li> <li>1938 <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a> gives a 'modern' definition of cohomology, summarizing the work since <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">J. W. Alexander</a> and <a href="/wiki/Kolmogorov" class="mw-redirect" title="Kolmogorov">Kolmogorov</a> first defined <a href="/wiki/Cochain" class="mw-redirect" title="Cochain">cochains</a>.</li> <li>1943 <a href="/wiki/Norman_Steenrod" title="Norman Steenrod">Norman Steenrod</a> publishes on homology with <a href="/wiki/Local_coefficients" class="mw-redirect" title="Local coefficients">local coefficients</a>.<a href="#cite_note-18">[18]</a></li> <li>1945 <a href="/wiki/Jean_Leray" title="Jean Leray">Jean Leray</a> publishes work carried out as a <a href="/wiki/Prisoner_of_war" title="Prisoner of war">prisoner of war</a>, motivated by proving <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed-point</a> theorems for application to <a href="/wiki/Partial_differential_equation" title="Partial differential equation">PDE</a> theory; it is the start of sheaf theory and <a href="/wiki/Spectral_sequence" title="Spectral sequence">spectral sequences</a>.<a href="#cite_note-Dieudonné123-19">[19]</a></li> <li>1947 <a href="/wiki/Henri_Cartan" title="Henri Cartan">Henri Cartan</a> reproves the <a href="/wiki/De_Rham_theorem" title="De Rham theorem">de Rham theorem</a> by sheaf methods, in correspondence with <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a> (see <a href="/wiki/De_Rham%E2%80%93Weil_theorem" title="De Rham–Weil theorem">De Rham–Weil theorem</a>). Leray gives a sheaf definition in his courses via closed sets (the later carapaces).</li> <li>1948 The Cartan seminar writes up sheaf theory for the first time.</li> <li>1950 The "second edition" sheaf theory from the Cartan seminar: the <a href="/wiki/Sheaf_space" class="mw-redirect" title="Sheaf space">sheaf space</a> (espace étalé) definition is used, with stalkwise structure. <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Supports</a> are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time <a href="/wiki/Kiyoshi_Oka" title="Kiyoshi Oka">Kiyoshi Oka</a> introduces an idea (adjacent to that) of a sheaf of ideals, in <a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables">several complex variables</a>.</li> <li>1951 The Cartan seminar proves <a href="/wiki/Theorems_A_and_B" class="mw-redirect" title="Theorems A and B">theorems A and B</a>, based on Oka's work.</li> <li>1953 The finiteness theorem for <a href="/wiki/Coherent_sheaf_cohomology#Finite-dimensionality" title="Coherent sheaf cohomology">coherent sheaves</a> in the analytic theory is proved by Cartan and <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a>,<a href="#cite_note-20">[20]</a> as is <a href="/wiki/Serre_duality" title="Serre duality">Serre duality</a>.</li> <li>1954 Serre's paper <a href="/wiki/List_of_important_publications_in_mathematics#Faisceaux_Algébriques_Cohérents" title="List of important publications in mathematics">Faisceaux algébriques cohérents</a><a href="#cite_note-21">[21]</a> (published in 1955) introduces sheaves into <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. These ideas are immediately exploited by <a href="/wiki/Friedrich_Hirzebruch" title="Friedrich Hirzebruch">Friedrich Hirzebruch</a>, who writes a major 1956 book on topological methods.</li> <li>1955 <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> in lectures in <a href="/wiki/Kansas" title="Kansas">Kansas</a> defines <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> and presheaf, and by using <a href="/wiki/Injective_resolution" class="mw-redirect" title="Injective resolution">injective resolutions</a> allows direct use of sheaf cohomology on all topological spaces, as <a href="/wiki/Derived_functor" title="Derived functor">derived functors</a>.</li> <li>1956 <a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Oscar Zariski</a>'s report Algebraic sheaf theory<a href="#cite_note-22">[22]</a></li> <li>1957 Grothendieck's <a href="/wiki/Grothendieck%27s_T%C3%B4hoku_paper" title="Grothendieck's Tôhoku paper">Tohoku paper</a><a href="#cite_note-23">[23]</a> rewrites <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>; he proves <a href="/wiki/Coherent_duality" title="Coherent duality">Grothendieck duality</a> (i.e., Serre duality for possibly <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singular</a> algebraic varieties).</li> <li>1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> and general sheaves on them, <a href="/wiki/Local_cohomology" title="Local cohomology">local cohomology</a>, <a href="/wiki/Derived_category" title="Derived category">derived categories</a> (with Verdier), and <a href="/wiki/Grothendieck_topologies" class="mw-redirect" title="Grothendieck topologies">Grothendieck topologies</a>. There emerges also his influential schematic idea of '<a href="/wiki/Grothendieck%27s_six_operations" class="mw-redirect" title="Grothendieck's six operations">six operations</a>' in homological algebra.</li> <li>1958 <a href="/wiki/Roger_Godement" title="Roger Godement">Roger Godement</a>'s book on sheaf theory is published. At around this time <a href="/wiki/Mikio_Sato" title="Mikio Sato">Mikio Sato</a> proposes his <a href="/wiki/Hyperfunction" title="Hyperfunction">hyperfunctions</a>, which will turn out to have sheaf-theoretic nature.</li></ul> At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>. It was later discovered that the logic in categories of sheaves is <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a> (this observation is now often referred to as <a href="/wiki/Kripke%E2%80%93Joyal_semantics" class="mw-redirect" title="Kripke–Joyal semantics">Kripke–Joyal semantics</a>, but probably should be attributed to a number of authors). <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=31" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Coherent_sheaf" title="Coherent sheaf">Coherent sheaf</a></li> <li><a href="/wiki/Gerbe" title="Gerbe">Gerbe</a></li> <li><a href="/wiki/Stack_(mathematics)" title="Stack (mathematics)">Stack (mathematics)</a></li> <li><a href="/wiki/Sheaf_of_spectra" title="Sheaf of spectra">Sheaf of spectra</a></li> <li><a href="/wiki/Perverse_sheaf" title="Perverse sheaf">Perverse sheaf</a></li> <li><a href="/wiki/Presheaf_of_spaces" class="mw-redirect" title="Presheaf of spaces">Presheaf of spaces</a></li> <li><a href="/wiki/Constructible_sheaf" title="Constructible sheaf">Constructible sheaf</a></li> <li><a href="/wiki/De_Rham%27s_theorem" class="mw-redirect" title="De Rham's theorem">De Rham's theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=32" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEisenbudHarris2006" class="citation cs2">Eisenbud, David; Harris, Joe (6 April 2006), The Geometry of Schemes, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>, New York, NY: Springer, pp. 11–18, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22639-2" title="Special:BookSources/978-0-387-22639-2"><bdi>978-0-387-22639-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Schemes&rft.place=New+York%2C+NY&rft.series=GTM&rft.pages=11-18&rft.pub=Springer&rft.date=2006-04-06&rft.isbn=978-0-387-22639-2&rft.aulast=Eisenbud&rft.aufirst=David&rft.au=Harris%2C+Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTennison1975" class="citation cs2">Tennison, B. 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9 (2): 119–221, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2748%2Ftmj%2F1178244839">10.2748/tmj/1178244839</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0040-8735">0040-8735</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0102537">0102537</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Tohoku+Mathematical+Journal&rft.atitle=Sur+quelques+points+d%27alg%C3%A8bre+homologique&rft.volume=9&rft.issue=2&rft.pages=119-221&rft.date=1957&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0102537%23id-name%3DMR&rft.issn=0040-8735&rft_id=info%3Adoi%2F10.2748%2Ftmj%2F1178244839&rft.aulast=Grothendieck&rft.aufirst=Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Sheaf_(mathematics)&action=edit&section=33" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBredon1997" class="citation cs2"><a href="/wiki/Glen_Bredon" title="Glen Bredon">Bredon, Glen E.</a> (1997), Sheaf theory, Graduate Texts in Mathematics, vol. 170 (2nd ed.), <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94905-5" title="Special:BookSources/978-0-387-94905-5"><bdi>978-0-387-94905-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1481706">1481706</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sheaf+theory&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=978-0-387-94905-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1481706%23id-name%3DMR&rft.aulast=Bredon&rft.aufirst=Glen+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (oriented towards conventional topological applications)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_CataldoMigliorini2010" class="citation journal cs1">de Cataldo, Andrea Mark; Migliorini, Luca (2010). <a rel="nofollow" class="external text" href="http://www.ams.org/notices/201005/rtx100500632p.pdf">"What is a perverse sheaf?"</a> (PDF). <a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>. 57 (5): 632–4. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1004.2983">1004.2983</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010arXiv1004.2983D">2010arXiv1004.2983D</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2664042">2664042</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=What+is+a+perverse+sheaf%3F&rft.volume=57&rft.issue=5&rft.pages=632-4&rft.date=2010&rft_id=info%3Aarxiv%2F1004.2983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2664042%23id-name%3DMR&rft_id=info%3Abibcode%2F2010arXiv1004.2983D&rft.aulast=de+Cataldo&rft.aufirst=Andrea+Mark&rft.au=Migliorini%2C+Luca&rft_id=http%3A%2F%2Fwww.ams.org%2Fnotices%2F201005%2Frtx100500632p.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGodement2006" class="citation cs2"><a href="/wiki/Roger_Godement" title="Roger Godement">Godement, Roger</a> (2006) [1973], Topologie algébrique et théorie des faisceaux, Paris: Hermann, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/2705612521" title="Special:BookSources/2705612521"><bdi>2705612521</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0345092">0345092</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topologie+alg%C3%A9brique+et+th%C3%A9orie+des+faisceaux&rft.place=Paris&rft.pub=Hermann&rft.date=2006&rft.isbn=2705612521&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0345092%23id-name%3DMR&rft.aulast=Godement&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieck1957" class="citation cs2"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a> (1957), "Sur quelques points d'algèbre homologique", The Tohoku Mathematical Journal, Second Series, 9 (2): 119–221, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2748%2Ftmj%2F1178244839">10.2748/tmj/1178244839</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0040-8735">0040-8735</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0102537">0102537</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Tohoku+Mathematical+Journal&rft.atitle=Sur+quelques+points+d%27alg%C3%A8bre+homologique&rft.volume=9&rft.issue=2&rft.pages=119-221&rft.date=1957&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0102537%23id-name%3DMR&rft.issn=0040-8735&rft_id=info%3Adoi%2F10.2748%2Ftmj%2F1178244839&rft.aulast=Grothendieck&rft.aufirst=Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHirzebruch1995" class="citation cs2"><a href="/wiki/Friedrich_Hirzebruch" title="Friedrich Hirzebruch">Hirzebruch, Friedrich</a> (1995), Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58663-0" title="Special:BookSources/978-3-540-58663-0"><bdi>978-3-540-58663-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1335917">1335917</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+methods+in+algebraic+geometry&rft.series=Classics+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-3-540-58663-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1335917%23id-name%3DMR&rft.aulast=Hirzebruch&rft.aufirst=Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (updated edition of a classic using enough sheaf theory to show its power)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIversen1986" class="citation cs2">Iversen, Birger (1986), Cohomology of sheaves, Universitext, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-82783-9">10.1007/978-3-642-82783-9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-16389-1" title="Special:BookSources/3-540-16389-1"><bdi>3-540-16389-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0842190">0842190</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cohomology+of+sheaves&rft.series=Universitext&rft.pub=Springer&rft.date=1986&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D842190%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-82783-9&rft.isbn=3-540-16389-1&rft.aulast=Iversen&rft.aufirst=Birger&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKashiwaraSchapira1994" class="citation cs2"><a href="/wiki/Masaki_Kashiwara" title="Masaki Kashiwara">Kashiwara, Masaki</a>; Schapira, Pierre (1994), Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-51861-7" title="Special:BookSources/978-3-540-51861-7"><bdi>978-3-540-51861-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1299726">1299726</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sheaves+on+manifolds&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+%5BFundamental+Principles+of+Mathematical+Sciences%5D&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-3-540-51861-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1299726%23id-name%3DMR&rft.aulast=Kashiwara&rft.aufirst=Masaki&rft.au=Schapira%2C+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (advanced techniques such as the <a href="/wiki/Derived_category" title="Derived category">derived category</a> and <a href="/wiki/Vanishing_cycle" title="Vanishing cycle">vanishing cycles</a> on the most reasonable spaces)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneMoerdijk1994" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Ieke_Moerdijk" title="Ieke Moerdijk">Moerdijk, Ieke</a> (1994), Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97710-2" title="Special:BookSources/978-0-387-97710-2"><bdi>978-0-387-97710-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1300636">1300636</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sheaves+in+Geometry+and+Logic%3A+A+First+Introduction+to+Topos+Theory&rft.series=Universitext&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-0-387-97710-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1300636%23id-name%3DMR&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Moerdijk%2C+Ieke&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (category theory and toposes emphasised)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartinChernZariski1956" class="citation cs2">Martin, William T.; <a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern, Shiing-Shen</a>; <a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1956), "Scientific report on the Second Summer Institute, several complex variables", <a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>, 62 (2): 79–141, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1956-10013-X">10.1090/S0002-9904-1956-10013-X</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0077995">0077995</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Scientific+report+on+the+Second+Summer+Institute%2C+several+complex+variables&rft.volume=62&rft.issue=2&rft.pages=79-141&rft.date=1956&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0077995%23id-name%3DMR&rft.issn=0002-9904&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1956-10013-X&rft.aulast=Martin&rft.aufirst=William+T.&rft.au=Chern%2C+Shiing-Shen&rft.au=Zariski%2C+Oscar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamanan2005" class="citation cs2">Ramanan, S. (2005), Global calculus, Graduate Studies in Mathematics, vol. 65, American Mathematical Society, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fgsm%2F065">10.1090/gsm/065</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-3702-8" title="Special:BookSources/0-8218-3702-8"><bdi>0-8218-3702-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104612">2104612</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Global+calculus&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2104612%23id-name%3DMR&rft_id=info%3Adoi%2F10.1090%2Fgsm%2F065&rft.isbn=0-8218-3702-8&rft.aulast=Ramanan&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeebachSeebachSteen1970" class="citation cs2"><a href="/wiki/J._Arthur_Seebach" class="mw-redirect" title="J. Arthur Seebach">Seebach, J. Arthur</a>; Seebach, Linda A.; <a href="/wiki/Lynn_A._Steen" class="mw-redirect" title="Lynn A. Steen">Steen, Lynn A.</a> (1970), "What is a Sheaf", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, 77 (7): 681–703, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1970.11992563">10.1080/00029890.1970.11992563</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0263073">0263073</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:203043621">203043621</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=What+is+a+Sheaf&rft.volume=77&rft.issue=7&rft.pages=681-703&rft.date=1970&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0263073%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A203043621%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F00029890.1970.11992563&rft.aulast=Seebach&rft.aufirst=J.+Arthur&rft.au=Seebach%2C+Linda+A.&rft.au=Steen%2C+Lynn+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1955" class="citation cs2"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1955), <a rel="nofollow" class="external text" href="http://www.mat.uniroma1.it/people/arbarello/FAC.pdf">"Faisceaux algébriques cohérents"</a> (PDF), <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, Second Series, 61 (2): 197–278, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969915">10.2307/1969915</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969915">1969915</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0068874">0068874</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Faisceaux+alg%C3%A9briques+coh%C3%A9rents&rft.volume=61&rft.issue=2&rft.pages=197-278&rft.date=1955&rft.issn=0003-486X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0068874%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969915%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1969915&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=http%3A%2F%2Fwww.mat.uniroma1.it%2Fpeople%2Farbarello%2FFAC.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwan1964" class="citation cs2"><a href="/wiki/Richard_Swan" title="Richard Swan">Swan, Richard G.</a> (1964), The Theory of Sheaves, Chicago lectures in mathematics (3 ed.), <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780226783291" title="Special:BookSources/9780226783291"><bdi>9780226783291</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Sheaves&rft.series=Chicago+lectures+in+mathematics&rft.edition=3&rft.pub=University+of+Chicago+Press&rft.date=1964&rft.isbn=9780226783291&rft.aulast=Swan&rft.aufirst=Richard+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (concise lecture notes)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTennison1975" class="citation cs2">Tennison, Barry R. (1975), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JOglpKIw6kIC">Sheaf theory</a>, London Mathematical Society Lecture Note Series, vol. 20, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-20784-3" title="Special:BookSources/978-0-521-20784-3"><bdi>978-0-521-20784-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0404390">0404390</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sheaf+theory&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1975&rft.isbn=978-0-521-20784-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0404390%23id-name%3DMR&rft.aulast=Tennison&rft.aufirst=Barry+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJOglpKIw6kIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"> (pedagogic treatment)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosiak2022" class="citation book cs1">Rosiak, Daniel (2022). <a rel="nofollow" class="external text" href="https://direct.mit.edu/books/oa-monograph/5460/Sheaf-Theory-through-Examples">Sheaf theory through examples</a>. Cambridge, Massachusetts. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7551%2Fmitpress%2F12581.001.0001">10.7551/mitpress/12581.001.0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-37042-4" title="Special:BookSources/978-0-262-37042-4"><bdi>978-0-262-37042-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1333708310">1333708310</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:253133215">253133215</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sheaf+theory+through+examples&rft.place=Cambridge%2C+Massachusetts&rft.date=2022&rft_id=info%3Aoclcnum%2F1333708310&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A253133215%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.7551%2Fmitpress%2F12581.001.0001&rft.isbn=978-0-262-37042-4&rft.aulast=Rosiak&rft.aufirst=Daniel&rft_id=https%3A%2F%2Fdirect.mit.edu%2Fbooks%2Foa-monograph%2F5460%2FSheaf-Theory-through-Examples&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+%28mathematics%29" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) (introductory book with open access)</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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