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class="vector-toc-list"> <li id="toc-Covariance_and_contravariance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Covariance_and_contravariance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Covariance and contravariance</span> </div> </a> <ul id="toc-Covariance_and_contravariance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Opposite_functor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Opposite_functor"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Opposite functor</span> </div> </a> <ul id="toc-Opposite_functor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bifunctors_and_multifunctors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bifunctors_and_multifunctors"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Bifunctors and multifunctors</span> </div> </a> <ul id="toc-Bifunctors_and_multifunctors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_other_categorical_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_other_categorical_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relation to other categorical concepts</span> </div> </a> <ul id="toc-Relation_to_other_categorical_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_implementations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computer_implementations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Computer implementations</span> </div> </a> <ul id="toc-Computer_implementations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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 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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Functor</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="دال (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="دال (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Functor" title="Functor – Catalan" lang="ca" hreflang="ca" data-title="Functor" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Funktor" title="Funktor – Czech" lang="cs" hreflang="cs" data-title="Funktor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Funktor_(Mathematik)" title="Funktor (Mathematik) – German" lang="de" hreflang="de" data-title="Funktor (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%B1%CF%81%CF%84%CE%B7%CF%84%CE%AE%CF%82" title="Συναρτητής – Greek" lang="el" hreflang="el" data-title="Συναρτητής" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funtor" title="Funtor – Spanish" lang="es" hreflang="es" data-title="Funtor" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9%DA%AF%D9%88%D9%86" title="تابعگون – Persian" lang="fa" hreflang="fa" data-title="تابعگون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Foncteur" title="Foncteur – French" lang="fr" hreflang="fr" data-title="Foncteur" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Functor" title="Functor – Galician" lang="gl" hreflang="gl" data-title="Functor" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A8%EC%9E%90_(%EC%88%98%ED%95%99)" title="함자 (수학) – Korean" lang="ko" hreflang="ko" data-title="함자 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungtor" title="Fungtor – Indonesian" lang="id" hreflang="id" data-title="Fungtor" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funtore_(matematica)" title="Funtore (matematica) – Italian" lang="it" hreflang="it" data-title="Funtore (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%98%D7%95%D7%A8" title="פונקטור – Hebrew" lang="he" hreflang="he" data-title="פונקטור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functor" title="Functor – Dutch" lang="nl" hreflang="nl" data-title="Functor" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%A2%E6%89%8B" title="関手 – Japanese" lang="ja" hreflang="ja" data-title="関手" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Funktor" title="Funktor – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Funktor" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A9%B0%E0%A8%95%E0%A8%9F%E0%A8%B0" title="ਫੰਕਟਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਫੰਕਟਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funktor_(teoria_kategorii)" title="Funktor (teoria kategorii) – Polish" lang="pl" hreflang="pl" data-title="Funktor (teoria kategorii)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Functor" title="Functor – Portuguese" lang="pt" hreflang="pt" data-title="Functor" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Functor" title="Functor – Romanian" lang="ro" hreflang="ro" data-title="Functor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%82%D0%BE%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функтор (математика) – Russian" lang="ru" hreflang="ru" data-title="Функтор (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Funktor" title="Funktor – Slovenian" lang="sl" hreflang="sl" data-title="Funktor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi badge-Q70894304 mw-list-item" title=""><a href="https://fi.wikipedia.org/wiki/Funktori" title="Funktori – Finnish" lang="fi" hreflang="fi" data-title="Funktori" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Funktor" title="Funktor – Swedish" lang="sv" hreflang="sv" data-title="Funktor" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%82%D0%BE%D1%80" title="Функтор – Ukrainian" lang="uk" hreflang="uk" data-title="Функтор" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mapping between categories</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 the mathematical concept. For other uses, see <a href="/wiki/Functor_(disambiguation)" class="mw-disambig" title="Functor (disambiguation)">Functor (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Functoriality" redirects here. For the Langlands functoriality conjecture in number theory, see <a href="/wiki/Langlands_program#Functoriality" title="Langlands program">Langlands program § Functoriality</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, specifically <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a <b>functor</b> is a <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> between <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a>. Functors were first considered in <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, where algebraic objects (such as the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a>) are associated to <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>, and maps between these algebraic objects are associated to <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which <a href="/wiki/Category_theory" title="Category theory">category theory</a> is applied. </p><p>The words <i>category</i> and <i>functor</i> were borrowed by mathematicians from the philosophers <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> and <a href="/wiki/Rudolf_Carnap" title="Rudolf Carnap">Rudolf Carnap</a>, respectively.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The latter used <i>functor</i> in a <a href="/wiki/Linguistics" title="Linguistics">linguistic</a> context;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> see <a href="/wiki/Function_word" title="Function word">function word</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Functor&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">November 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Commutative_diagram_for_morphism.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/220px-Commutative_diagram_for_morphism.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/330px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/440px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption>A category with objects X, Y, Z and morphisms f, g, g ∘ f</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Commutative_diagram_of_a_functor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Commutative_diagram_of_a_functor.svg/220px-Commutative_diagram_of_a_functor.svg.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Commutative_diagram_of_a_functor.svg/330px-Commutative_diagram_of_a_functor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Commutative_diagram_of_a_functor.svg/440px-Commutative_diagram_of_a_functor.svg.png 2x" data-file-width="260" data-file-height="173" /></a><figcaption>Functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> must preserve the composition of morphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span></figcaption></figure> <p>Let <i>C</i> and <i>D</i> be <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a>. A <b>functor</b> <i>F</i> from <i>C</i> to <i>D</i> is a mapping that<sup id="cite_ref-FOOTNOTEJacobson2009p._19,_def._1.2_3-0" class="reference"><a href="#cite_note-FOOTNOTEJacobson2009p._19,_def._1.2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ul><li>associates each <a href="/wiki/Mathematical_object" title="Mathematical object">object</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <i>C</i> to an object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00816772e8dff4e6733c478ec77fab0382264a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}"></span> in <i>D</i>,</li> <li>associates each <a href="/wiki/Morphism" title="Morphism">morphism</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> in <i>C</i> to a morphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)\colon F(X)\to F(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)\colon F(X)\to F(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6706e8089dac8a824056309cb3cdb49919f543cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.33ex; height:2.843ex;" alt="{\displaystyle F(f)\colon F(X)\to F(Y)}"></span> in <i>D</i> such that the following two conditions hold: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34316157c8076290d01ee976aaa01293edc5ecd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:16.689ex; height:3.176ex;" alt="{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}"></span> for every object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <i>C</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(g\circ f)=F(g)\circ F(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(g\circ f)=F(g)\circ F(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5386b2f5cd99fb03896b82b2a5efeec6ca0edb30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.927ex; height:2.843ex;" alt="{\displaystyle F(g\circ f)=F(g)\circ F(f)}"></span> for all morphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e805a664676aac8f884c1cfd043a6ab3a615c0c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.067ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y\,\!}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon Y\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6268a0bd7f3016093edf824725f1ffcba89f8064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.218ex; height:2.509ex;" alt="{\displaystyle g\colon Y\to Z}"></span> in <i>C</i>.</li></ul></li></ul> <p>That is, functors must preserve <a href="/wiki/Morphism#Definition" title="Morphism">identity morphisms</a> and <a href="/wiki/Function_composition" title="Function composition">composition</a> of morphisms. </p> <div class="mw-heading mw-heading3"><h3 id="Covariance_and_contravariance">Covariance and contravariance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=2" title="Edit section: Covariance and contravariance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Covariance_and_contravariance_(computer_science)" title="Covariance and contravariance (computer science)">Covariance and contravariance (computer science)</a></div> <p>There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a <b>contravariant functor</b> <i>F</i> from <i>C</i> to <i>D</i> as a mapping that </p> <ul><li>associates each object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <i>C</i> with an object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00816772e8dff4e6733c478ec77fab0382264a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle F(X)}"></span> in <i>D</i>,</li> <li>associates each morphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> in <i>C</i> with a morphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)\colon F(Y)\to F(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)\colon F(Y)\to F(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2cf37ecba1180285acae23de1a8c1b02558f09d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.33ex; height:2.843ex;" alt="{\displaystyle F(f)\colon F(Y)\to F(X)}"></span> in <i>D</i> such that the following two conditions hold: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34316157c8076290d01ee976aaa01293edc5ecd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.387ex; width:16.689ex; height:3.176ex;" alt="{\displaystyle F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!}"></span> for every object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <i>C</i>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(g\circ f)=F(f)\circ F(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(g\circ f)=F(f)\circ F(g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dde85dd5484b51b860ba87bb3af755ed25e1878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.927ex; height:2.843ex;" alt="{\displaystyle F(g\circ f)=F(f)\circ F(g)}"></span> for all morphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon Y\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6268a0bd7f3016093edf824725f1ffcba89f8064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.218ex; height:2.509ex;" alt="{\displaystyle g\colon Y\to Z}"></span> in <i>C</i>.</li></ul></li></ul> <p>Variance of functor (composite)<sup id="cite_ref-FOOTNOTESimmons2011Exercise_3.1.4_4-0" class="reference"><a href="#cite_note-FOOTNOTESimmons2011Exercise_3.1.4-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The composite of two functors of the same variance: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2834bd5d4cd23cd8394c2a4ff8edc4743b1cd531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:36.254ex; height:2.176ex;" alt="{\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Contravariant} \circ \mathrm {Contravariant} \to \mathrm {Covariant} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Contravariant} \circ \mathrm {Contravariant} \to \mathrm {Covariant} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a42af41c8d5f64700900f9f8a72b4e9bb93ac12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:44.797ex; height:2.176ex;" alt="{\displaystyle \mathrm {Contravariant} \circ \mathrm {Contravariant} \to \mathrm {Covariant} }"></span></li></ul></li> <li>The composite of two functors of opposite variance: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Covariant} \circ \mathrm {Contravariant} \to \mathrm {Contravariant} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Covariant} \circ \mathrm {Contravariant} \to \mathrm {Contravariant} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30b5b711c2a7e3eb1b9d9335397ed3bde993ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:44.797ex; height:2.176ex;" alt="{\displaystyle \mathrm {Covariant} \circ \mathrm {Contravariant} \to \mathrm {Contravariant} }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Contravariant} \circ \mathrm {Covariant} \to \mathrm {Contravariant} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Contravariant} \circ \mathrm {Covariant} \to \mathrm {Contravariant} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58eeb51342216eac6cce638966e344d841a5e3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:44.797ex; height:2.176ex;" alt="{\displaystyle \mathrm {Contravariant} \circ \mathrm {Covariant} \to \mathrm {Contravariant} }"></span></li></ul></li></ul> <p>Note that contravariant functors reverse the direction of composition. </p><p>Ordinary functors are also called <b>covariant functors</b> in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a <i>covariant</i> functor on the <a href="/wiki/Opposite_category" title="Opposite category">opposite category</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a797ad9c8efdf926168c8b6272d0e72420ac8eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.766ex; height:2.343ex;" alt="{\displaystyle C^{\mathrm {op} }}"></span>.<sup id="cite_ref-FOOTNOTEJacobson200919–20_5-0" class="reference"><a href="#cite_note-FOOTNOTEJacobson200919–20-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Some authors prefer to write all expressions covariantly. That is, instead of saying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon C\to D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon C\to D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8299d81a3d96658bf9240b799a248872f36a3772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.079ex; height:2.176ex;" alt="{\displaystyle F\colon C\to D}"></span> is a contravariant functor, they simply write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon C^{\mathrm {op} }\to D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo stretchy="false">→</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon C^{\mathrm {op} }\to D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792272bf8adf357234d5f73095e71dc94d6facce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.079ex; height:2.343ex;" alt="{\displaystyle F\colon C^{\mathrm {op} }\to D}"></span> (or sometimes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon C\to D^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon C\to D^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee86ee1f456c4406fbf4136f37c5eb8339ef0c05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.048ex; height:2.343ex;" alt="{\displaystyle F\colon C\to D^{\mathrm {op} }}"></span>) and call it a functor. </p><p>Contravariant functors are also occasionally called <i>cofunctors</i>.<sup id="cite_ref-Popescu1979_6-0" class="reference"><a href="#cite_note-Popescu1979-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>There is a convention which refers to "vectors"—i.e., <a href="/wiki/Vector_field" title="Vector field">vector fields</a>, elements of the space of sections <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (TM)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (TM)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a0815a341f6a1b87b432c1ba9756425361a729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.341ex; height:2.843ex;" alt="{\displaystyle \Gamma (TM)}"></span> of a <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea000afb5769206ddd5fd43f458430d04422ddeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}"></span>—as "contravariant" and to "covectors"—i.e., <a href="/wiki/1-forms" class="mw-redirect" title="1-forms">1-forms</a>, elements of the space of sections <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1e59ba0ea46a7776c2a132fbaa938725998621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.479ex; height:2.843ex;" alt="{\displaystyle \Gamma {\mathord {\left(T^{*}M\right)}}}"></span> of a <a href="/wiki/Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e9815aa22bca801ff9618269fbbb247575ae86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.216ex; height:2.343ex;" alt="{\displaystyle T^{*}M}"></span>—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in <a href="/wiki/Einstein_summation" class="mw-redirect" title="Einstein summation">expressions</a> such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b72284681bda9de7c6e358d1923caa4d12c6fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.062ex; height:4.009ex;" alt="{\displaystyle {x'}^{\,i}=\Lambda _{j}^{i}x^{j}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64fc2a39bda2e406d64984980805e98581813d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.478ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} '={\boldsymbol {\Lambda }}\mathbf {x} }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d973a904ba4063ad5c26b9134257efb56ca0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.222ex; height:3.509ex;" alt="{\displaystyle \omega '_{i}=\Lambda _{i}^{j}\omega _{j}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38996df33ca2cdee509e4ffe5aa11220323378a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.992ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}{\boldsymbol {\Lambda }}^{\textsf {T}}.}"></span> In this formalism it is observed that the coordinate transformation symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{i}^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda _{i}^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2d946da81939e5de95887d2c40b73369bd4c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.523ex; height:3.509ex;" alt="{\displaystyle \Lambda _{i}^{j}}"></span> (representing the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c406305b3c3cdb54b614b036b53851be6c0c8d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.225ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\Lambda }}^{\textsf {T}}}"></span>) acts on the "covector coordinates" "in the same way" as on the basis vectors: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab0e1b02d085eca84c5767c46b13d21348b0dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.781ex; height:3.509ex;" alt="{\displaystyle \mathbf {e} _{i}=\Lambda _{i}^{j}\mathbf {e} _{j}}"></span>—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ea9e4bc1ed5e8e89d67736d8360b0507b6b653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.781ex; height:3.676ex;" alt="{\displaystyle \mathbf {e} ^{i}=\Lambda _{j}^{i}\mathbf {e} ^{j}}"></span>). This terminology is contrary to the one used in category theory because it is the covectors that have <i>pullbacks</i> in general and are thus <i>contravariant</i>, whereas vectors in general are <i>covariant</i> since they can be <i>pushed forward</i>. See also <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Opposite_functor">Opposite functor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=3" title="Edit section: Opposite functor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon C\to D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon C\to D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8299d81a3d96658bf9240b799a248872f36a3772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.079ex; height:2.176ex;" alt="{\displaystyle F\colon C\to D}"></span> induces the <b>opposite functor</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>:</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884608502ab900086456f21b5290b23a74424fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.089ex; height:2.343ex;" alt="{\displaystyle F^{\mathrm {op} }\colon C^{\mathrm {op} }\to D^{\mathrm {op} }}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a797ad9c8efdf926168c8b6272d0e72420ac8eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.766ex; height:2.343ex;" alt="{\displaystyle C^{\mathrm {op} }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6905ced650c74643aa7b850b3a0a2b2dc364ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.892ex; height:2.343ex;" alt="{\displaystyle D^{\mathrm {op} }}"></span> are the <a href="/wiki/Opposite_category" title="Opposite category">opposite categories</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> By definition, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bea620b787f248a511a2d25dd74e697ff3b9d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.783ex; height:2.343ex;" alt="{\displaystyle F^{\mathrm {op} }}"></span> maps objects and morphisms in the identical way as does <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a797ad9c8efdf926168c8b6272d0e72420ac8eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.766ex; height:2.343ex;" alt="{\displaystyle C^{\mathrm {op} }}"></span> does not coincide with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> as a category, and similarly for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bea620b787f248a511a2d25dd74e697ff3b9d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.783ex; height:2.343ex;" alt="{\displaystyle F^{\mathrm {op} }}"></span> is distinguished from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. For example, when composing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon C_{0}\to C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon C_{0}\to C_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59088ea30015022315090fd875cd546d32d72104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.821ex; height:2.509ex;" alt="{\displaystyle F\colon C_{0}\to C_{1}}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>:</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msubsup> <mo stretchy="false">→</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74fd8063bbbf3331efa9a8d0eb58b3987b97614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.957ex; height:3.176ex;" alt="{\displaystyle G\colon C_{1}^{\mathrm {op} }\to C_{2}}"></span>, one should use either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\circ F^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>∘</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\circ F^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c21742201040133a79ea2aae223e62c35bf0abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.804ex; height:2.343ex;" alt="{\displaystyle G\circ F^{\mathrm {op} }}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{\mathrm {op} }\circ F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>∘</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{\mathrm {op} }\circ F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ac537f23f31e836f14cc33289fab331540e041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.73ex; height:2.343ex;" alt="{\displaystyle G^{\mathrm {op} }\circ F}"></span>. Note that, following the property of <a href="/wiki/Opposite_category" title="Opposite category">opposite category</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27b482a3cb4ac1a4e1157e7e1685a3942cd4302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.4ex; height:3.009ex;" alt="{\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Bifunctors_and_multifunctors">Bifunctors and multifunctors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=4" title="Edit section: Bifunctors and multifunctors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>bifunctor</b> (also known as a <b>binary functor</b>) is a functor whose domain is a <a href="/wiki/Product_category" title="Product category">product category</a>. For example, the <a href="/wiki/Hom_functor" title="Hom functor">Hom functor</a> is of the type <span class="nowrap"><i>C<sup>op</sup></i> × <i>C</i> → <b>Set</b></span>. It can be seen as a functor in <i>two</i> arguments; it is contravariant in one argument, covariant in the other. </p><p>A <b>multifunctor</b> is a generalization of the functor concept to <i>n</i> variables. So, for example, a bifunctor is a multifunctor with <span class="nowrap"><i>n</i> = 2</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=5" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two important consequences of the functor <a href="/wiki/Axiom" title="Axiom">axioms</a> are: </p> <ul><li><i>F</i> transforms each <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagram</a> in <i>C</i> into a commutative diagram in <i>D</i>;</li> <li>if <i>f</i> is an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> in <i>C</i>, then <i>F</i>(<i>f</i>) is an isomorphism in <i>D</i>.</li></ul> <p>One can compose functors, i.e. if <i>F</i> is a functor from <i>A</i> to <i>B</i> and <i>G</i> is a functor from <i>B</i> to <i>C</i> then one can form the composite functor <span class="nowrap"><i>G</i> ∘ <i>F</i></span> from <i>A</i> to <i>C</i>. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the <a href="/wiki/Category_of_small_categories" title="Category of small categories">category of small categories</a>. </p><p>A small category with a single object is the same thing as a <a href="/wiki/Monoid" title="Monoid">monoid</a>: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Diagram_(category_theory)" title="Diagram (category theory)">Diagram</a></dt> <dd>For categories <i>C</i> and <i>J</i>, a diagram of type <i>J</i> in <i>C</i> is a covariant functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\colon J\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>:</mo> <mi>J</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\colon J\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b66b5bc7b5641e22c64c150a56f89a52394a9f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.81ex; height:2.176ex;" alt="{\displaystyle D\colon J\to C}"></span>.</dd> <dt><a href="/wiki/Presheaf_(category_theory)" title="Presheaf (category theory)">(Category theoretical) presheaf</a></dt> <dd>For categories <i>C</i> and <i>J</i>, a <i>J</i>-presheaf on <i>C</i> is a contravariant functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\colon C\to J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\colon C\to J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c2ed16f7a54e053be3673e9f05d47bcdf2e759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.81ex; height:2.176ex;" alt="{\displaystyle D\colon C\to J}"></span>.<div class="paragraphbreak" style="margin-top:0.5em"></div>In the special case when J is <b>Set</b>, the category of sets and functions, <i>D</i> is called a <a href="/wiki/Presheaf_(category_theory)" title="Presheaf (category theory)">presheaf</a> on <i>C</i>.</dd> <dt>Presheaves (over a topological space)</dt> <dd>If <i>X</i> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, then the <a href="/wiki/Open_set" title="Open set">open sets</a> in <i>X</i> form a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered set</a> Open(<i>X</i>) under inclusion. Like every partially ordered set, Open(<i>X</i>) forms a small category by adding a single arrow <span class="nowrap"><i>U</i> → <i>V</i></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7720f1387a2e51d58daf1fb9e9b1b730430b8466" 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 U\subseteq V}"></span>. Contravariant functors on Open(<i>X</i>) are called <i><a href="/wiki/Presheaf" class="mw-redirect" title="Presheaf">presheaves</a></i> on <i>X</i>. For instance, by assigning to every open set <i>U</i> the <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> of real-valued continuous functions on <i>U</i>, one obtains a presheaf of algebras on <i>X</i>.</dd> <dt>Constant functor</dt> <dd>The functor <span class="nowrap"><i>C</i> → <i>D</i></span> which maps every object of <i>C</i> to a fixed object <i>X</i> in <i>D</i> and every morphism in <i>C</i> to the identity morphism on <i>X</i>. Such a functor is called a <i>constant</i> or <i>selection</i> functor.</dd> <dt></dt><dt id="endofunctor"><dfn>Endofunctor</dfn></dt> <dd>A functor that maps a category to that same category; e.g., <a href="/wiki/Polynomial_functor" title="Polynomial functor">polynomial functor</a>.</dd> <dt></dt><dt id="identity_functor"><dfn>Identity functor</dfn></dt> <dd>In category <i>C</i>, written 1<sub><i>C</i></sub> or id<sub><i>C</i></sub>, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.</dd> <dt>Diagonal functor</dt> <dd>The <a href="/wiki/Diagonal_functor" title="Diagonal functor">diagonal functor</a> is defined as the functor from <i>D</i> to the functor category <i>D</i><sup><i>C</i></sup> which sends each object in <i>D</i> to the constant functor at that object.</dd> <dt>Limit functor</dt> <dd>For a fixed <a href="/wiki/Index_category" class="mw-redirect" title="Index category">index category</a> <i>J</i>, if every functor <span class="nowrap"><i>J</i> → <i>C</i></span> has a <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">limit</a> (for instance if <i>C</i> is complete), then the limit functor <span class="nowrap"><i>C</i><sup><i>J</i></sup> → <i>C</i></span> assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the <a href="/wiki/Adjoint_functors" title="Adjoint functors">right-adjoint</a> to the <a href="/wiki/Diagonal_functor" title="Diagonal functor">diagonal functor</a> and invoking the <a href="/wiki/Freyd_adjoint_functor_theorem" class="mw-redirect" title="Freyd adjoint functor theorem">Freyd adjoint functor theorem</a>. This requires a suitable version of the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).</dd> <dt>Power sets functor</dt> <dd>The power set functor <span class="nowrap"><i>P</i> : <b>Set</b> → <b>Set</b></span> maps each set to its <a href="/wiki/Power_set" title="Power set">power set</a> and each function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> to the map which sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\in {\mathcal {P}}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\in {\mathcal {P}}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c42f685bd8ac92506fe554a96b692149e95e4cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.116ex; height:2.843ex;" alt="{\displaystyle U\in {\mathcal {P}}(X)}"></span> to its image <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(U)\in {\mathcal {P}}(Y)}"> <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>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(U)\in {\mathcal {P}}(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863be268f444f631e374d4aebda8300d913d1070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.997ex; height:2.843ex;" alt="{\displaystyle f(U)\in {\mathcal {P}}(Y)}"></span>. One can also consider the <b>contravariant power set functor</b> which sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 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\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> to the map which sends <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subseteq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd1a13c51d9c5c652f6ad2bf6b9f63bee6632e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.659ex; height:2.343ex;" alt="{\displaystyle V\subseteq Y}"></span> to its <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)\subseteq X.}"> <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 stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>X</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)\subseteq X.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee2c50437979a0250f7743005c49a94836c41a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.975ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V)\subseteq X.}"></span><div class="paragraphbreak" style="margin-top:0.5em"></div> For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ee0b41e4e86efb22c4c62463176f000fe045bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.762ex; height:2.843ex;" alt="{\displaystyle X=\{0,1\}}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X)={\mathcal {P}}(X)=\{\{\},\{0\},\{1\},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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(X)={\mathcal {P}}(X)=\{\{\},\{0\},\{1\},X\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68171c79a8cbfec744866c6076e6e8291f374aae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.926ex; height:2.843ex;" alt="{\displaystyle F(X)={\mathcal {P}}(X)=\{\{\},\{0\},\{1\},X\}}"></span>. Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=\{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=\{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936947e384745c162e0604333eba6b71abd916ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.674ex; height:2.843ex;" alt="{\displaystyle f(0)=\{\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe2467152b32433e236a9d66a22fb7d774c0ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.329ex; height:2.843ex;" alt="{\displaystyle f(1)=X}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b338d0157928ab4d2fbcaedb56bff674b9c8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle F(f)}"></span> is the function which sends any subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> to its image <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/562e763c2291125cfb06f14882bc9f9aba8a7d7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:2.843ex;" alt="{\displaystyle f(U)}"></span>, which in this case means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}\mapsto f(\{\})=\{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}\mapsto f(\{\})=\{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0170db6235e6cddedaeb87fbc3dfd6b0c6be10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.775ex; height:2.843ex;" alt="{\displaystyle \{\}\mapsto f(\{\})=\{\}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mapsto }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↦</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mapsto }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc09de045e7d82eef9fe078e7e7606576640c11b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \mapsto }"></span> denotes the mapping under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b338d0157928ab4d2fbcaedb56bff674b9c8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle F(f)}"></span>, so this could also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F(f))(\{\})=\{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F(f))(\{\})=\{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb1d9853bada41f6b401015a2c4399a19e7c164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.843ex;" alt="{\displaystyle (F(f))(\{\})=\{\}}"></span>. For the other values,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}\mapsto f(\{0\})=\{f(0)\}=\{\{\}\},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}\mapsto f(\{0\})=\{f(0)\}=\{\{\}\},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e854d775a070db047ae77451201335786c316b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.713ex; height:2.843ex;" alt="{\displaystyle \{0\}\mapsto f(\{0\})=\{f(0)\}=\{\{\}\},\ }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}\mapsto f(\{1\})=\{f(1)\}=\{X\},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1\}\mapsto f(\{1\})=\{f(1)\}=\{X\},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe1e1dc189aa506f8846082bca20e00a7f53bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.368ex; height:2.843ex;" alt="{\displaystyle \{1\}\mapsto f(\{1\})=\{f(1)\}=\{X\},\ }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}\mapsto f(\{0,1\})=\{f(0),f(1)\}=\{\{\},X\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1\}\mapsto f(\{0,1\})=\{f(0),f(1)\}=\{\{\},X\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034c02a9c9ea098cb2c4636d6f93ade9cf4e5c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.437ex; height:2.843ex;" alt="{\displaystyle \{0,1\}\mapsto f(\{0,1\})=\{f(0),f(1)\}=\{\{\},X\}.}"></span> Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\{0,1\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\{0,1\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c2b44bb4a0232a26941558adca35c4e185ab55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.772ex; height:2.843ex;" alt="{\displaystyle f(\{0,1\})}"></span> consequently generates the <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. Also note that although the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in this example mapped to the power set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, that need not be the case in general.</dd> <dt><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Dual_vector_space"></span><span class="vanchor-text">Dual vector space</span></span></dt> <dd>The map which assigns to every <a href="/wiki/Vector_space" title="Vector space">vector space</a> its <a href="/wiki/Dual_space" title="Dual space">dual space</a> and to every <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear map</a> its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> to itself.</dd> <dt>Fundamental group</dt> <dd>Consider the category of <a href="/wiki/Pointed_topological_space" class="mw-redirect" title="Pointed topological space">pointed topological spaces</a>, i.e. topological spaces with distinguished points. The objects are pairs <span class="nowrap">(<i>X</i>, <i>x</i><sub>0</sub>)</span>, where <i>X</i> is a topological space and <i>x</i><sub>0</sub> is a point in <i>X</i>. A morphism from <span class="nowrap">(<i>X</i>, <i>x</i><sub>0</sub>)</span> to <span class="nowrap">(<i>Y</i>, <i>y</i><sub>0</sub>)</span> is given by a <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> map <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> with <span class="nowrap"><i>f</i>(<i>x</i><sub>0</sub>) = <i>y</i><sub>0</sub></span>.<div class="paragraphbreak" style="margin-top:0.5em"></div> To every topological space <i>X</i> with distinguished point <i>x</i><sub>0</sub>, one can define the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> based at <i>x</i><sub>0</sub>, denoted <span class="nowrap">π<sub>1</sub>(<i>X</i>, <i>x</i><sub>0</sub>)</span>. This is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Homotopy" title="Homotopy">homotopy</a> classes of loops based at <i>x</i><sub>0</sub>, with the group operation of concatenation. If <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> is a morphism of <a href="/wiki/Pointed_space" title="Pointed space">pointed spaces</a>, then every loop in <i>X</i> with base point <i>x</i><sub>0</sub> can be composed with <i>f</i> to yield a loop in <i>Y</i> with base point <i>y</i><sub>0</sub>. This operation is compatible with the homotopy <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> and the composition of loops, and we get a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from <span class="nowrap">π(<i>X</i>, <i>x</i><sub>0</sub>)</span> to <span class="nowrap">π(<i>Y</i>, <i>y</i><sub>0</sub>)</span>. We thus obtain a functor from the category of pointed topological spaces to the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>.<div class="paragraphbreak" style="margin-top:0.5em"></div> In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the <b>fundamental <a href="/wiki/Groupoid" title="Groupoid">groupoid</a></b> instead of the fundamental group, and this construction is functorial.</dd> <dt>Algebra of continuous functions</dt> <dd>A contravariant functor from the category of <a href="/wiki/Topology" title="Topology">topological spaces</a> (with continuous maps as morphisms) to the category of real <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebras</a> is given by assigning to every topological space <i>X</i> the algebra C(<i>X</i>) of all real-valued continuous functions on that space. Every continuous map <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> induces an <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> <span class="nowrap">C(<i>f</i>) : C(<i>Y</i>) → C(<i>X</i>)</span> by the rule <span class="nowrap">C(<i>f</i>)(<i>φ</i>) = <i>φ</i> ∘ <i>f</i></span> for every <i>φ</i> in C(<i>Y</i>).</dd> <dt>Tangent and cotangent bundles</dt> <dd>The map which sends every <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> to its <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> and every <a href="/wiki/Smooth_map" class="mw-redirect" title="Smooth map">smooth map</a> to its <a href="/wiki/Derivative" title="Derivative">derivative</a> is a covariant functor from the category of differentiable manifolds to the category of <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a>. <div class="paragraphbreak" style="margin-top:0.5em"></div>Doing this constructions pointwise gives the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a>, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, <a href="/wiki/Cotangent_space" title="Cotangent space">cotangent space</a> is a contravariant functor, essentially the composition of the tangent space with the <a href="#Dual_vector_space">dual space</a> above.</dd> <dt>Group actions/representations</dt> <dd>Every <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> can be considered as a category with a single object whose morphisms are the elements of <i>G</i>. A functor from <i>G</i> to <b>Set</b> is then nothing but a <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of <i>G</i> on a particular set, i.e. a <i>G</i>-set. Likewise, a functor from <i>G</i> to the <a href="/wiki/Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">category of vector spaces</a>, <b>Vect</b><sub><i>K</i></sub>, is a <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representation</a> of <i>G</i>. In general, a functor <span class="nowrap"><i>G</i> → <i>C</i></span> can be considered as an "action" of <i>G</i> on an object in the category <i>C</i>. If <i>C</i> is a group, then this action is a group homomorphism.</dd> <dt>Lie algebras</dt> <dd>Assigning to every real (complex) <a href="/wiki/Lie_group" title="Lie group">Lie group</a> its real (complex) <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> defines a functor.</dd> <dt>Tensor products</dt> <dd>If <i>C</i> denotes the category of vector spaces over a fixed field, with <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear maps</a> as morphisms, then the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\otimes W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊗</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\otimes W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8e48e05a95d9c68f80f49e3d509ba9de064c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.063ex; height:2.343ex;" alt="{\displaystyle V\otimes W}"></span> defines a functor <span class="nowrap"><i>C</i> × <i>C</i> → <i>C</i></span> which is covariant in both arguments.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></dd> <dt>Forgetful functors</dt> <dd>The functor <span class="nowrap"><i>U</i> : <b>Grp</b> → <b>Set</b></span> which maps a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> to its underlying set and a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> to its underlying function of sets is a functor.<sup id="cite_ref-FOOTNOTEJacobson2009p._20,_ex._2_9-0" class="reference"><a href="#cite_note-FOOTNOTEJacobson2009p._20,_ex._2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Functors like these, which "forget" some structure, are termed <i><a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functors</a></i>. Another example is the functor <span class="nowrap"><b>Rng</b> → <b>Ab</b></span> which maps a <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a> to its underlying additive <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>. Morphisms in <b>Rng</b> (<a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphisms</a>) become morphisms in <b>Ab</b> (abelian group homomorphisms).</dd> <dt>Free functors</dt> <dd>Going in the opposite direction of forgetful functors are free functors. The free functor <span class="nowrap"><i>F</i> : <b>Set</b> → <b>Grp</b></span> sends every set <i>X</i> to the <a href="/wiki/Free_group" title="Free group">free group</a> generated by <i>X</i>. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See <a href="/wiki/Free_object" title="Free object">free object</a>.</dd> <dt>Homomorphism groups</dt> <dd>To every pair <i>A</i>, <i>B</i> of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">abelian groups</a> one can assign the abelian group Hom(<i>A</i>, <i>B</i>) consisting of all <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphisms</a> from <i>A</i> to <i>B</i>. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor <span class="nowrap"><b>Ab</b><sup>op</sup> × <b>Ab</b> → <b>Ab</b></span> (where <b>Ab</b> denotes the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a> with group homomorphisms). If <span class="nowrap"><i>f</i> : <i>A</i><sub>1</sub> → <i>A</i><sub>2</sub></span> and <span class="nowrap"><i>g</i> : <i>B</i><sub>1</sub> → <i>B</i><sub>2</sub></span> are morphisms in <b>Ab</b>, then the group homomorphism <span class="nowrap">Hom(<i>f</i>, <i>g</i>)</span>: <span class="nowrap">Hom(<i>A</i><sub>2</sub>, <i>B</i><sub>1</sub>) → Hom(<i>A</i><sub>1</sub>, <i>B</i><sub>2</sub>)</span> is given by <span class="nowrap"><i>φ</i> ↦ <i>g</i> ∘ <i>φ</i> ∘ <i>f</i></span>. See <a href="/wiki/Hom_functor" title="Hom functor">Hom functor</a>.</dd> <dt>Representable functors</dt> <dd>We can generalize the previous example to any category <i>C</i>. To every pair <i>X</i>, <i>Y</i> of objects in <i>C</i> one can assign the set <span class="nowrap">Hom(<i>X</i>, <i>Y</i>)</span> of morphisms from <i>X</i> to <i>Y</i>. This defines a functor to <b>Set</b> which is contravariant in the first argument and covariant in the second, i.e. it is a functor <span class="nowrap"><i>C</i><sup>op</sup> × <i>C</i> → <b>Set</b></span>. If <span class="nowrap"><i>f</i> : <i>X</i><sub>1</sub> → <i>X</i><sub>2</sub></span> and <span class="nowrap"><i>g</i> : <i>Y</i><sub>1</sub> → <i>Y</i><sub>2</sub></span> are morphisms in <i>C</i>, then the map <span class="nowrap">Hom(<i>f</i>, <i>g</i>) : Hom(<i>X</i><sub>2</sub>, <i>Y</i><sub>1</sub>) → Hom(<i>X</i><sub>1</sub>, <i>Y</i><sub>2</sub>)</span> is given by <span class="nowrap"><i>φ</i> ↦ <i>g</i> ∘ <i>φ</i> ∘ <i>f</i></span>.<div class="paragraphbreak" style="margin-top:0.5em"></div> Functors like these are called <a href="/wiki/Representable_functor" title="Representable functor">representable functors</a>. An important goal in many settings is to determine whether a given functor is representable.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_categorical_concepts">Relation to other categorical concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=7" title="Edit section: Relation to other categorical concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>C</i> and <i>D</i> be categories. The collection of all functors from <i>C</i> to <i>D</i> forms the objects of a category: the <a href="/wiki/Functor_category" title="Functor category">functor category</a>. Morphisms in this category are <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformations</a> between functors. </p><p>Functors are often defined by <a href="/wiki/Universal_property" title="Universal property">universal properties</a>; examples are the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>, the <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> and <a href="/wiki/Direct_product" title="Direct product">direct product</a> of groups or vector spaces, construction of free groups and modules, <a href="/wiki/Direct_limit" title="Direct limit">direct</a> and <a href="/wiki/Inverse_limit" title="Inverse limit">inverse</a> limits. The concepts of <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">limit and colimit</a> generalize several of the above. </p><p>Universal constructions often give rise to pairs of <a href="/wiki/Adjoint_functors" title="Adjoint functors">adjoint functors</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Computer_implementations">Computer implementations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=8" title="Edit section: Computer implementations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Functor_(functional_programming)" title="Functor (functional programming)">Functor (functional programming)</a></div> <p>Functors sometimes appear in <a href="/wiki/Functional_programming" title="Functional programming">functional programming</a>. For instance, the programming language <a href="/wiki/Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> has a <a href="/wiki/Type_class" title="Type class">class</a> <code>Functor</code> where <a href="/wiki/Map_(higher-order_function)#Generalization" title="Map (higher-order function)"><code>fmap</code></a> is a <a href="/wiki/Polytypic_function" class="mw-redirect" title="Polytypic function">polytypic function</a> used to map <a href="/wiki/Function_(computer_programming)" title="Function (computer programming)">functions</a> (<i>morphisms</i> on <i>Hask</i>, the category of Haskell types)<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> between existing types to functions between some new types.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 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.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"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFMac_Lane1971" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1971), <i>Categories for the Working Mathematician</i>, New York: Springer-Verlag, p. 30, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-90035-1" title="Special:BookSources/978-3-540-90035-1"><bdi>978-3-540-90035-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories+for+the+Working+Mathematician&rft.place=New+York&rft.pages=30&rft.pub=Springer-Verlag&rft.date=1971&rft.isbn=978-3-540-90035-1&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Rudolf_Carnap" title="Rudolf Carnap">Carnap, Rudolf</a> (1937). <i>The Logical Syntax of Language</i>, Routledge & Kegan, pp. 13–14.</span> </li> <li id="cite_note-FOOTNOTEJacobson2009p._19,_def._1.2-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJacobson2009p._19,_def._1.2_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 19, def. 1.2.</span> </li> <li id="cite_note-FOOTNOTESimmons2011Exercise_3.1.4-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESimmons2011Exercise_3.1.4_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSimmons2011">Simmons (2011)</a>, Exercise 3.1.4.</span> </li> <li id="cite_note-FOOTNOTEJacobson200919–20-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJacobson200919–20_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson (2009)</a>, pp. 19–20.</span> </li> <li id="cite_note-Popescu1979-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Popescu1979_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopescuPopescu1979" class="citation book cs1">Popescu, Nicolae; Popescu, Liliana (1979). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YnHwCAAAQBAJ&q=cofunctor+covariant&pg=PA12"><i>Theory of categories</i></a>. Dordrecht: Springer. p. 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789400995505" title="Special:BookSources/9789400995505"><bdi>9789400995505</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">23 April</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+categories&rft.place=Dordrecht&rft.pages=12&rft.pub=Springer&rft.date=1979&rft.isbn=9789400995505&rft.aulast=Popescu&rft.aufirst=Nicolae&rft.au=Popescu%2C+Liliana&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYnHwCAAAQBAJ%26q%3Dcofunctor%2Bcovariant%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneMoerdijk1992" 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> (1992), <i>Sheaves in geometry and logic: a first introduction to topos theory</i>, Springer, <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></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.pub=Springer&rft.date=1992&rft.isbn=978-0-387-97710-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Moerdijk%2C+Ieke&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkelGubareniGubareniKirichenko2004" class="citation cs2"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>; <a href="/w/index.php?title=Nadezhda_Mikha%C4%ADlovna&action=edit&redlink=1" class="new" title="Nadezhda Mikhaĭlovna (page does not exist)">Gubareni, Nadezhda Mikhaĭlovna</a>; <a href="/w/index.php?title=Nadiya_Gubareni&action=edit&redlink=1" class="new" title="Nadiya Gubareni (page does not exist)">Gubareni, Nadiya</a>; <a href="/w/index.php?title=Vladimir_V._Kirichenko&action=edit&redlink=1" class="new" title="Vladimir V. Kirichenko (page does not exist)">Kirichenko, Vladimir V.</a> (2004), <i>Algebras, rings and modules</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-2690-4" title="Special:BookSources/978-1-4020-2690-4"><bdi>978-1-4020-2690-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebras%2C+rings+and+modules&rft.pub=Springer&rft.date=2004&rft.isbn=978-1-4020-2690-4&rft.aulast=Hazewinkel&rft.aufirst=Michiel&rft.au=Gubareni%2C+Nadezhda+Mikha%C4%ADlovna&rft.au=Gubareni%2C+Nadiya&rft.au=Kirichenko%2C+Vladimir+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEJacobson2009p._20,_ex._2-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJacobson2009p._20,_ex._2_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 20, ex. 2.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">It's not entirely clear that Haskell datatypes truly form a category. See <a rel="nofollow" class="external free" href="https://wiki.haskell.org/Hask">https://wiki.haskell.org/Hask</a> for more details.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">See <a rel="nofollow" class="external free" href="https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell">https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell</a> for more information.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson2009" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009), <i>Basic algebra</i>, vol. 2 (2nd ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47187-7" title="Special:BookSources/978-0-486-47187-7"><bdi>978-0-486-47187-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover&rft.date=2009&rft.isbn=978-0-486-47187-7&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimmons2011" class="citation cs2">Simmons, Harold (2011), "Functors and natural transformations", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VOCQUC_uiWgC&pg=PA74"><i>An Introduction to Category Theory</i></a>, pp. <span class="nowrap">72–</span>107, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511863226.004">10.1017/CBO9780511863226.004</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-01087-1" title="Special:BookSources/978-1-107-01087-1"><bdi>978-1-107-01087-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Functors+and+natural+transformations&rft.btitle=An+Introduction+to+Category+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E72-%3C%2Fspan%3E107&rft.date=2011&rft_id=info%3Adoi%2F10.1017%2FCBO9780511863226.004&rft.isbn=978-1-107-01087-1&rft.aulast=Simmons&rft.aufirst=Harold&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVOCQUC_uiWgC%26pg%3DPA74&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Functor&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><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" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/functor" class="extiw" title="wiktionary:functor">functor</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Functor">"Functor"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Functor&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DFunctor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></li> <li>see <a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/functor">functor</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a> and the variations discussed and linked to there.</li> <li><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>, <a rel="nofollow" class="external text" href="http://ncatlab.org/nlab">CatLab</a>, a wiki project dedicated to the exposition of categorical mathematics</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHillman2001" class="citation web cs1">Hillman, Chris (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/19970503051012/http://www.math.washington.edu:80/~hillman/PUB/categorical.ps">"A Categorical Primer"</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.3264">10.1.1.24.3264</a></span>. Archived from <a rel="nofollow" class="external text" href="http://www.math.washington.edu:80/~hillman/PUB/categorical.ps">the original</a> on 1997-05-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Categorical+Primer&rft.date=2001&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.24.3264%23id-name%3DCiteSeerX&rft.aulast=Hillman&rft.aufirst=Chris&rft_id=http%3A%2F%2Fwww.math.washington.edu%3A80%2F~hillman%2FPUB%2Fcategorical.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFunctor" class="Z3988"></span></li> <li>J. Adamek, H. Herrlich, G. Stecker, <a rel="nofollow" class="external text" href="http://katmat.math.uni-bremen.de/acc/acc.pdf">Abstract and Concrete Categories-The Joy of Cats</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150421081851/http://katmat.math.uni-bremen.de/acc/acc.pdf">Archived</a> 2015-04-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/category-theory/">Category Theory</a>" — by Jean-Pierre Marquis. Extensive bibliography.</li> <li><a rel="nofollow" class="external text" href="http://www.mta.ca/~cat-dist/">List of academic conferences on category theory</a></li> <li>Baez, John, 1996,"<a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week73.html">The Tale of <i>n</i>-categories.</a>" An informal introduction to higher order categories.</li> <li><a rel="nofollow" class="external text" href="http://wildcatsformma.wordpress.com/">WildCats</a> is a <a href="/wiki/Category_theory" title="Category theory">category theory</a> package for <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>. Manipulation and visualization of objects, <a href="/wiki/Morphism" title="Morphism">morphisms</a>, categories, functors, <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformations</a>, <a href="/wiki/Universal_properties" class="mw-redirect" title="Universal properties">universal properties</a>.</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/user/TheCatsters">The catsters</a>, a YouTube channel about category theory.</li> <li><a rel="nofollow" class="external text" href="http://categorieslogicphysics.wikidot.com/events">Video archive</a> of recorded talks relevant to categories, logic and the foundations of physics.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080916162345/http://www.j-paine.org/cgi-bin/webcats/webcats.php">Interactive Web page</a> which generates examples of categorical constructions in the category of finite sets.</li></ul> <div class="navbox-styles"><style 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style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#e5e5ff;"><div id="Key_concepts427" style="font-size:114%;margin:0 4em">Key concepts</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Category</a> <ul><li><a href="/wiki/Abelian_category" title="Abelian category">Abelian</a></li> <li><a href="/wiki/Additive_category" title="Additive category">Additive</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a></li> <li><a href="/wiki/Pre-abelian_category" title="Pre-abelian category">Pre-abelian</a></li> <li><a href="/wiki/Preadditive_category" title="Preadditive category">Preadditive</a></li> <li><a href="/wiki/Bicategory" title="Bicategory">Bicategory</a></li></ul></li> <li><a href="/wiki/Adjoint_functors" title="Adjoint functors">Adjoint functors</a></li> <li><a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">CCC</a></li> <li><a href="/wiki/Commutative_diagram" title="Commutative diagram">Commutative diagram</a></li> <li><a href="/wiki/End_(category_theory)" title="End (category theory)">End</a></li> <li><a href="/wiki/Exponential_object" title="Exponential object">Exponential</a></li> <li><a class="mw-selflink selflink">Functor</a></li> <li><a href="/wiki/Kan_extension" title="Kan extension">Kan extension</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Natural_transformation" title="Natural transformation">Natural transformation</a></li> <li><a href="/wiki/Universal_property" title="Universal property">Universal property</a></li> <li><a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Universal_construction" class="mw-redirect" title="Universal construction">Universal constructions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Initial_and_terminal_objects" title="Initial and terminal objects">Terminal objects</a></li> <li><a href="/wiki/Product_(category_theory)" title="Product (category theory)">Products</a></li> <li><a href="/wiki/Equaliser_(mathematics)" title="Equaliser (mathematics)">Equalizers</a> <ul><li><a href="/wiki/Kernel_(category_theory)" title="Kernel (category theory)">Kernels</a></li></ul></li> <li><a href="/wiki/Pullback_(category_theory)" title="Pullback (category theory)">Pullbacks</a></li> <li><a href="/wiki/Inverse_limit" title="Inverse limit">Inverse limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Colimit" class="mw-redirect" title="Colimit">Colimits</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Initial_and_terminal_objects" title="Initial and terminal objects">Initial objects</a></li> <li><a href="/wiki/Coproduct" title="Coproduct">Coproducts</a></li> <li><a href="/wiki/Coequalizer" title="Coequalizer">Coequalizers</a> <ul><li><a href="/wiki/Cokernel" title="Cokernel">Cokernels and quotients</a></li></ul></li> <li><a href="/wiki/Pushout_(category_theory)" title="Pushout (category theory)">Pushout</a></li> <li><a href="/wiki/Direct_limit" title="Direct limit">Direct limit</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebraic_category" class="mw-redirect" title="Algebraic category">Algebraic categories</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_of_sets" title="Category of sets">Sets</a></li> <li><a href="/wiki/Category_of_relations" title="Category of relations">Relations</a></li> <li><a href="/wiki/Category_of_magmas" class="mw-redirect" title="Category of magmas">Magmas</a></li> <li><a href="/wiki/Category_of_groups" title="Category of groups">Groups</a></li> <li><a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">Abelian groups</a></li> <li><a href="/wiki/Category_of_rings" title="Category of rings">Rings</a> (<a href="/wiki/Category_of_rings#Category_of_fields" title="Category of rings">Fields</a>)</li> <li><a href="/wiki/Category_of_modules" title="Category of modules">Modules</a> (<a href="/wiki/Category_of_modules#Example:_the_category_of_vector_spaces" title="Category of modules">Vector spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions on categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Free_category" title="Free category">Free category</a></li> <li><a href="/wiki/Functor_category" title="Functor category">Functor category</a></li> <li><a href="/wiki/Kleisli_category" title="Kleisli category">Kleisli category</a></li> <li><a href="/wiki/Opposite_category" title="Opposite category">Opposite category</a></li> <li><a href="/wiki/Quotient_category" title="Quotient category">Quotient category</a></li> <li><a href="/wiki/Product_category" title="Product category">Product category</a></li> <li><a href="/wiki/Comma_category" title="Comma category">Comma category</a></li> <li><a href="/wiki/Subcategory" title="Subcategory">Subcategory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><span class="skin-invert" typeof="mw:File"><a href="/wiki/Commutative_diagram" title="Commutative diagram"><img alt="A simple triangular commutative diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/60px-Commutative_diagram_for_morphism.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/90px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/120px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#e5e5ff;"><div id="Higher_category_theory427" style="font-size:114%;margin:0 4em"><a href="/wiki/Higher_category_theory" title="Higher category theory">Higher category theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li class="mw-empty-elt"></li></ul></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Categorification" title="Categorification">Categorification</a></li> <li><a href="/wiki/Enriched_category" title="Enriched category">Enriched category</a></li> <li><a href="/wiki/Higher-dimensional_algebra" title="Higher-dimensional algebra">Higher-dimensional algebra</a></li> <li><a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a></li> <li><a href="/wiki/Model_category" title="Model category">Model category</a></li> <li><a href="/wiki/Simplex_category" title="Simplex category">Simplex category</a></li> <li><a href="/wiki/String_diagram" title="String diagram">String diagram</a></li> <li><a href="/wiki/Topos" title="Topos">Topos</a></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">n-categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Weak_n-category" title="Weak n-category">Weak <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bicategory" title="Bicategory">Bicategory</a> (<a href="/wiki/Pseudo-functor" title="Pseudo-functor">pseudofunctor</a>)</li> <li><a href="/wiki/Tricategory" title="Tricategory">Tricategory</a></li> <li><a href="/wiki/Tetracategory" title="Tetracategory">Tetracategory</a></li> <li><a href="/wiki/Quasi-category" title="Quasi-category">Kan complex</a></li> <li><a href="/wiki/%E2%88%9E-groupoid" title="∞-groupoid">∞-groupoid</a></li> <li><a href="/wiki/%E2%88%9E-topos" title="∞-topos">∞-topos</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strict_n-category" class="mw-redirect" title="Strict n-category">Strict <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strict_2-category" title="Strict 2-category">2-category</a> (<a href="/wiki/2-functor" title="2-functor">2-functor</a>)</li> <li><a href="/wiki/3-category" class="mw-redirect" title="3-category">3-category</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Categorification" title="Categorification">Categorified</a> concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2-group" title="2-group">2-group</a></li> <li><a href="/wiki/2-ring" title="2-ring">2-ring</a></li> <li><a href="/wiki/En-ring" title="En-ring"><i>E<sub>n</sub></i>-ring</a></li> <li>(<a href="/wiki/Traced_monoidal_category" title="Traced monoidal category">Traced</a>)(<a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">Symmetric</a>) <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a></li> <li><a href="/wiki/N-group_(category_theory)" title="N-group (category theory)">n-group</a></li> <li><a href="/wiki/N-monoid" title="N-monoid">n-monoid</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Category_theory" title="Category:Category theory">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Outline_of_category_theory" title="Outline of category theory">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Glossary_of_category_theory" title="Glossary of category theory">Glossary</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functor_types17" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functors" title="Template:Functors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functors" title="Template talk:Functors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functors" title="Special:EditPage/Template:Functors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functor_types17" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Functor</a> types</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_functor" class="mw-redirect" title="Additive functor">Additive</a></li> <li><a href="/wiki/Adjoint_functors" title="Adjoint functors">Adjoint</a></li> <li><a href="/wiki/Conservative_functor" title="Conservative functor">Conservative</a></li> <li><a href="/wiki/Derived_functor" title="Derived functor">Derived</a></li> <li><a href="/wiki/Diagonal_functor" title="Diagonal functor">Diagonal</a></li> <li><a href="/wiki/Enriched_functor" class="mw-redirect" title="Enriched functor">Enriched</a></li> <li><a href="/wiki/Essentially_surjective_functor" title="Essentially surjective functor">Essentially surjective</a></li> <li><a href="/wiki/Exact_functor" title="Exact functor">Exact</a></li> <li><a href="/wiki/Forgetful_functor" title="Forgetful functor">Forgetful</a></li> <li><a href="/wiki/Full_and_faithful_functors" title="Full and faithful functors">Full and faithful</a></li> <li><a href="/wiki/Logical_functor" class="mw-redirect" title="Logical functor">Logical</a></li> <li><a href="/wiki/Monoidal_functor" title="Monoidal functor">Monoidal</a></li> <li><a href="/wiki/Representable_functor" title="Representable functor">Representable</a></li> <li><a href="/wiki/Smooth_functor" title="Smooth functor">Smooth</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Function330" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functions_navbox" title="Template:Functions navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Functions_navbox&action=edit&redlink=1" class="new" title="Template talk:Functions navbox (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functions_navbox" title="Special:EditPage/Template:Functions navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Function330" style="font-size:114%;margin:0 4em"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">History</a></li> <li><a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function"><span class="texhtml">X → 𝔹</span></a></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair"><span class="texhtml">𝔹 → X</span></a></li> <li><a href="/wiki/Boolean_function" title="Boolean function"><span class="texhtml">𝔹ⁿ → X</span></a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function"><span class="texhtml">X → ℤ</span></a></li> <li><a href="/wiki/Sequence" title="Sequence"><span class="texhtml">ℤ → X</span></a></li> <li><a href="/wiki/Real-valued_function" title="Real-valued function"><span class="texhtml">X → ℝ</span></a></li> <li><a href="/wiki/Function_of_a_real_variable" title="Function of a real variable"><span class="texhtml">ℝ → X</span></a></li> <li><a href="/wiki/Function_of_several_real_variables" title="Function of several real variables"><span class="texhtml">ℝⁿ → X</span></a></li> <li><a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function"><span class="texhtml">X → ℂ</span></a></li> <li><a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable"><span class="texhtml">ℂ → X</span></a></li> <li><a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables"><span class="texhtml">ℂⁿ → X</span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constant_function" title="Constant function">Constant</a></li> <li><a href="/wiki/Identity_function" title="Identity function">Identity</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Rational_function" title="Rational function">Rational</a></li> <li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic</a></li> <li><a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable</a></li> <li><a href="/wiki/Injective_function" title="Injective function">Injective</a></li> <li><a href="/wiki/Surjective_function" title="Surjective function">Surjective</a></li> <li><a href="/wiki/Bijection" title="Bijection">Bijective</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li> <li><a href="/wiki/Function_composition" title="Function composition">Composition</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">λ</a></li> <li><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td 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