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</a> <ul id="toc-Morphisms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Functors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Functors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Functors</span> </div> </a> <ul id="toc-Functors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natural_transformations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Natural_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Natural transformations</span> </div> </a> <ul id="toc-Natural_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other concepts</span> </div> </a> <button aria-controls="toc-Other_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other concepts subsection</span> </button> <ul id="toc-Other_concepts-sublist" class="vector-toc-list"> <li id="toc-Universal_constructions,_limits,_and_colimits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Universal_constructions,_limits,_and_colimits"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Universal constructions, limits, and colimits</span> </div> </a> <ul id="toc-Universal_constructions,_limits,_and_colimits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_categories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_categories"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Equivalent categories</span> </div> </a> <ul id="toc-Equivalent_categories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_concepts_and_results" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_concepts_and_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Further concepts and results</span> </div> </a> <ul id="toc-Further_concepts_and_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-dimensional_categories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher-dimensional_categories"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Higher-dimensional categories</span> </div> </a> <ul id="toc-Higher-dimensional_categories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Historical_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Historical notes</span> </div> </a> <ul id="toc-Historical_notes-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> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">10</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 class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Category theory</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 49 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-49" 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">49 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/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D9%81%D8%A6%D8%A9" title="نظرية الفئة – Arabic" lang="ar" hreflang="ar" data-title="نظرية الفئة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_categor%C3%ADes" title="Teoría de categoríes – Asturian" lang="ast" hreflang="ast" data-title="Teoría de categoríes" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%9F%E0%A6%BE%E0%A6%97%E0%A6%B0%E0%A6%BF_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="ক্যাটাগরি তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="ক্যাটাগরি তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8%D1%82%D0%B5" title="Теория на категориите – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на категориите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_categories" title="Teoria de categories – Catalan" lang="ca" hreflang="ca" data-title="Teoria de categories" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Категорисен теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Категорисен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_kategori%C3%AD" title="Teorie kategorií – Czech" lang="cs" hreflang="cs" data-title="Teorie kategorií" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Theori_categori" title="Theori categori – Welsh" lang="cy" hreflang="cy" data-title="Theori categori" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kategoriteori" title="Kategoriteori – Danish" lang="da" hreflang="da" data-title="Kategoriteori" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kategorientheorie" title="Kategorientheorie – German" lang="de" hreflang="de" data-title="Kategorientheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kategooriateooria" title="Kategooriateooria – Estonian" lang="et" hreflang="et" data-title="Kategooriateooria" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%BA%CE%B1%CF%84%CE%B7%CE%B3%CE%BF%CF%81%CE%B9%CF%8E%CE%BD" title="Θεωρία κατηγοριών – Greek" lang="el" hreflang="el" data-title="Θεωρία κατηγοριών" 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/Teor%C3%ADa_de_categor%C3%ADas" title="Teoría de categorías – Spanish" lang="es" hreflang="es" data-title="Teoría de categorías" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Teorio_de_kategorioj" title="Teorio de kategorioj – Esperanto" lang="eo" hreflang="eo" data-title="Teorio de kategorioj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kategorien_teoria" title="Kategorien teoria – Basque" lang="eu" hreflang="eu" data-title="Kategorien teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%B1%D8%B3%D8%AA%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه رسته‌ها – Persian" lang="fa" hreflang="fa" data-title="نظریه رسته‌ها" 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/Th%C3%A9orie_des_cat%C3%A9gories" title="Théorie des catégories – French" lang="fr" hreflang="fr" data-title="Théorie des catégories" 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/Teor%C3%ADa_das_categor%C3%ADas" title="Teoría das categorías – Galician" lang="gl" hreflang="gl" data-title="Teoría das categorías" 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/%EB%B2%94%EC%A3%BC%EB%A1%A0" title="범주론 – Korean" lang="ko" hreflang="ko" data-title="범주론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%BF%D5%A5%D5%A3%D5%B8%D6%80%D5%AB%D5%A1%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D5%A5%D5%BD%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Կատեգորիաների տեսություն – Armenian" lang="hy" hreflang="hy" data-title="Կատեգորիաների տեսություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_kategorija" title="Teorija kategorija – Croatian" lang="hr" hreflang="hr" data-title="Teorija kategorija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_kategori" title="Teori kategori – Indonesian" lang="id" hreflang="id" data-title="Teori kategori" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%ADkjafr%C3%A6%C3%B0i" title="Ríkjafræði – Icelandic" lang="is" hreflang="is" data-title="Ríkjafræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_delle_categorie" title="Teoria delle categorie – Italian" lang="it" hreflang="it" data-title="Teoria delle categorie" 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%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%95%D7%AA" title="תורת הקטגוריות – Hebrew" lang="he" hreflang="he" data-title="תורת הקטגוריות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%90%E1%83%A2%E1%83%94%E1%83%92%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90%E1%83%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="კატეგორიათა თეორია – Georgian" lang="ka" hreflang="ka" data-title="კატეგორიათა თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoria_categoriarum" title="Theoria categoriarum – Latin" lang="la" hreflang="la" data-title="Theoria categoriarum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kateg%C3%B3riaelm%C3%A9let" title="Kategóriaelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Kategóriaelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kategori" title="Teori kategori – Malay" lang="ms" hreflang="ms" data-title="Teori kategori" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%90%E1%80%BA%E1%80%90%E1%80%82%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="ကတ်တဂိုရီသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="ကတ်တဂိုရီသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Categorietheorie_(wiskunde)" title="Categorietheorie (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Categorietheorie (wiskunde)" 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/%E5%9C%8F%E8%AB%96" title="圏論 – Japanese" lang="ja" hreflang="ja" data-title="圏論" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_kategorii" title="Teoria kategorii – Polish" lang="pl" hreflang="pl" data-title="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/Teoria_das_categorias" title="Teoria das categorias – Portuguese" lang="pt" hreflang="pt" data-title="Teoria das categorias" 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/Teoria_categoriilor" title="Teoria categoriilor – Romanian" lang="ro" hreflang="ro" data-title="Teoria categoriilor" 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%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B9" title="Теория категорий – Russian" lang="ru" hreflang="ru" data-title="Теория категорий" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Category_theory" title="Category theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Category theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_kategorij" title="Teorija kategorij – Slovenian" lang="sl" hreflang="sl" data-title="Teorija kategorij" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%98%D0%B0" 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.mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Category+theory%22">"Category theory"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Category+theory%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Category+theory%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Category+theory%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Category+theory%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Category+theory%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2024</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-halign-right" 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/200px-Commutative_diagram_for_morphism.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/300px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/400px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a><figcaption>Schematic representation of a category with objects <i>X</i>, <i>Y</i>, <i>Z</i> and morphisms <i>f</i>, <i>g</i>, <span class="nowrap"><i>g</i> ∘ <i>f</i></span>. (The category's three identity morphisms 1<sub><i>X</i></sub>, 1<sub><i>Y</i></sub> and 1<sub><i>Z</i></sub>, if explicitly represented, would appear as three arrows, from the letters <i>X</i>, <i>Y</i>, and <i>Z</i> to themselves, respectively.)</figcaption></figure> <p><b>Category theory</b> is a general theory of <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> and their relations. It was introduced by <a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a> and <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> in the middle of the 20th century in their foundational work on <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>.<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> Category theory is used in almost all areas of mathematics. In particular, many constructions of new <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include <a href="/wiki/Quotient_space_(disambiguation)" class="mw-redirect mw-disambig" title="Quotient space (disambiguation)">quotient spaces</a>, <a href="/wiki/Direct_product" title="Direct product">direct products</a>, completion, and <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">duality</a>. </p><p>Many areas of <a href="/wiki/Computer_science" title="Computer science">computer science</a> also rely on category theory, such as <a href="/wiki/Functional_programming" title="Functional programming">functional programming</a> and <a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">semantics</a>. </p><p>A <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> is formed by two sorts of <a href="/wiki/Mathematical_object" title="Mathematical object">objects</a>: the <a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a> of the category, and the <a href="/wiki/Morphism" title="Morphism">morphisms</a>, which relate two objects called the <i>source</i> and the <i>target</i> of the morphism. Metaphorically, a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one. Morphism composition has similar properties as <a href="/wiki/Function_composition" title="Function composition">function composition</a> (<a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> and existence of an <a href="/wiki/Identity_element" title="Identity element">identity morphism</a> for each object). Morphisms are often some sort of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, but this is not always the case. For example, a <a href="/wiki/Monoid" title="Monoid">monoid</a> may be viewed as a category with a single object, whose morphisms are the elements of the monoid. </p><p>The second fundamental concept of category theory is the concept of a <a href="/wiki/Functor" title="Functor">functor</a>, which plays the role of a morphism between two categories <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 {\mathcal {C}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8f21a3eeba8a3d7e57262d1d056af4d24ae2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{1}}"></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 {\mathcal {C}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e4f90850fb306af575d9a4a37b15b6d9178ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{2}}"></span>: it maps objects 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 {\mathcal {C}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8f21a3eeba8a3d7e57262d1d056af4d24ae2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{1}}"></span> to objects 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 {\mathcal {C}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e4f90850fb306af575d9a4a37b15b6d9178ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{2}}"></span> and morphisms 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 {\mathcal {C}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8f21a3eeba8a3d7e57262d1d056af4d24ae2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{1}}"></span> to morphisms 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 {\mathcal {C}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e4f90850fb306af575d9a4a37b15b6d9178ce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.279ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{2}}"></span> in such a way that sources are mapped to sources, and targets are mapped to targets (or, in the case of a <a href="/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant functor</a>, sources are mapped to targets and <i>vice-versa</i>). A third fundamental concept is a <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformation</a> that may be viewed as a morphism of functors. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Categories,_objects,_and_morphisms"><span id="Categories.2C_objects.2C_and_morphisms"></span>Categories, objects, and morphisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=1" title="Edit section: Categories, objects, and morphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main articles: <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Category (mathematics)</a> and <a href="/wiki/Morphism" title="Morphism">Morphism</a></div> <div class="mw-heading mw-heading3"><h3 id="Categories">Categories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=2" title="Edit section: Categories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>category</i> <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 {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"></span> consists of the following three mathematical entities: </p> <ul><li>A <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</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 {\text{ob}}({\mathcal {C}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ob</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ob}}({\mathcal {C}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0379d04df76227110a5b3490fa6b2360849ecd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle {\text{ob}}({\mathcal {C}})}"></span>, whose elements are called <i>objects</i>;</li> <li>A class <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 {\text{hom}}({\mathcal {C}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{hom}}({\mathcal {C}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919295ea287370e307a6e5370d73b2a006b084ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:2.843ex;" alt="{\displaystyle {\text{hom}}({\mathcal {C}})}"></span>, whose elements are called <a href="/wiki/Morphism" title="Morphism">morphisms</a> or <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">maps</a> or <i>arrows</i>. <br />Each morphism <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}"> <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></b> has a <i>source object </i> <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></b> and <i>target object</i> <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></b>.</li></ul> <dl><dd>The expression <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:a\mapsto b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:a\mapsto b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447954784e9409a043850d10ff6e2467ce084341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.057ex; height:2.509ex;" alt="{\displaystyle f:a\mapsto b}"></span>, would be verbally stated 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}"> <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> is a morphism from <span class="texhtml mvar" style="font-style:italic;">a</span> to <span class="texhtml mvar" style="font-style:italic;">b</span>".</dd> <dd>The expression <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 {\text{hom}}(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{hom}}(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a30bca6292fa64695eaa7c96dbffce85b90e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.461ex; height:2.843ex;" alt="{\displaystyle {\text{hom}}(a,b)}"></span> – alternatively expressed 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 {\text{hom}}_{\mathcal {C}}(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{hom}}_{\mathcal {C}}(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfef171d61c26111481e3d97efd0bb1ea85d78b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.57ex; height:2.843ex;" alt="{\displaystyle {\text{hom}}_{\mathcal {C}}(a,b)}"></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 {\text{mor}}(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>mor</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{mor}}(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0afa8f3916060a401bd73e482d9b2e8dca3d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.081ex; height:2.843ex;" alt="{\displaystyle {\text{mor}}(a,b)}"></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 {\mathcal {C}}(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89418b6b41c85096bfadf1f877d9113642270f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.31ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}(a,b)}"></span> – denotes the <i>hom-class</i> of all morphisms 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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>.</dd></dl> <ul><li>A <a href="/wiki/Binary_operation" title="Binary operation">binary operation</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 \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span>, called <i>composition of morphisms</i>, such that</li></ul> <dl><dd>for any three objects <i><span class="texhtml mvar" style="font-style:italic;">a</span></i>, <i><span class="texhtml mvar" style="font-style:italic;">b</span></i>, and <i><span class="texhtml mvar" style="font-style:italic;">c</span></i>, we have <dl><dd><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 \circ :{\text{hom}}(b,c)\times {\text{hom}}(a,b)\mapsto {\text{hom}}(a,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ :{\text{hom}}(b,c)\times {\text{hom}}(a,b)\mapsto {\text{hom}}(a,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0651882a4ec41fbd9f60d8aed8be15647d5a00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.725ex; height:2.843ex;" alt="{\displaystyle \circ :{\text{hom}}(b,c)\times {\text{hom}}(a,b)\mapsto {\text{hom}}(a,c)}"></span></dd></dl></dd> <dd>The composition 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 f:a\mapsto b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:a\mapsto b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447954784e9409a043850d10ff6e2467ce084341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.057ex; height:2.509ex;" alt="{\displaystyle f:a\mapsto b}"></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:b\mapsto c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:b\mapsto c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab11e2b0ba2311debc6719ea32927f2be9a7ed91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.672ex; height:2.509ex;" alt="{\displaystyle g:b\mapsto c}"></span> is 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 g\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}"></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 gf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae6992a2a836c1ff200f058911a5a15f32de24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle gf}"></span>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> governed by two axioms: <dl><dd>1. <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">Associativity</a>: 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 f:a\mapsto b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:a\mapsto b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447954784e9409a043850d10ff6e2467ce084341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.057ex; height:2.509ex;" alt="{\displaystyle f:a\mapsto b}"></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 g:b\mapsto c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>b</mi> <mo stretchy="false">↦</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:b\mapsto c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab11e2b0ba2311debc6719ea32927f2be9a7ed91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.672ex; height:2.509ex;" alt="{\displaystyle g:b\mapsto 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 h:c\mapsto d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>:</mo> <mi>c</mi> <mo stretchy="false">↦</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h:c\mapsto d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19471f0a5c97e1d7002e6532a4002849f0009d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.113ex; height:2.176ex;" alt="{\displaystyle h:c\mapsto d}"></span> then <dl><dd><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 h\circ (g\circ f)=(h\circ g)\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∘</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\circ (g\circ f)=(h\circ g)\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54241bc77220479e208a59d88e5eca826086ab97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.963ex; height:2.843ex;" alt="{\displaystyle h\circ (g\circ f)=(h\circ g)\circ f}"></span></dd></dl></dd> <dd>2. <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">Identity</a>: For every object <span class="texhtml mvar" style="font-style:italic;">x</span>, there exists 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 1_{x}:x\mapsto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{x}:x\mapsto x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2264f09e92c00f119ae19f9dc0c91bf5e9103196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.546ex; height:2.509ex;" alt="{\displaystyle 1_{x}:x\mapsto x}"></span> (also denoted 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 {\text{id}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>id</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{id}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5ceb6893f5720cdd9d2496f160884497836e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.112ex; height:2.509ex;" alt="{\displaystyle {\text{id}}_{x}}"></span>) called the <i><a href="/wiki/Identity_morphism" class="mw-redirect" title="Identity morphism">identity morphism</a> for <span class="texhtml mvar" style="font-style:italic;">x</span></i>, such that for every 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:a\mapsto b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>a</mi> <mo stretchy="false">↦</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:a\mapsto b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447954784e9409a043850d10ff6e2467ce084341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.057ex; height:2.509ex;" alt="{\displaystyle f:a\mapsto b}"></span>, we have <dl><dd><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_{b}\circ f=f=f\circ 1_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>∘</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{b}\circ f=f=f\circ 1_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d6001a39eb0b0f57476eb8e259831564348b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.787ex; height:2.509ex;" alt="{\displaystyle 1_{b}\circ f=f=f\circ 1_{a}}"></span></dd></dl></dd> <dd>From the axioms, it can be proved that there is exactly one <a href="/wiki/Identity_morphism" class="mw-redirect" title="Identity morphism">identity morphism</a> for every object.</dd></dl></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The category <b><a href="/wiki/Category_of_sets" title="Category of sets">Set</a></b> <ul><li>As the class of objects <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 {\text{ob}}({\text{Set}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ob</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Set</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{ob}}({\text{Set}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b88a2363b0108576c4a81891930720abaeaccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.494ex; height:2.843ex;" alt="{\displaystyle {\text{ob}}({\text{Set}})}"></span>, we choose the class of all sets.</li> <li>As the class 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 {\text{hom}}({\text{Set}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Set</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{hom}}({\text{Set}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe7d1e86e0a269aba4c6f663d35f11b17f92b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.43ex; height:2.843ex;" alt="{\displaystyle {\text{hom}}({\text{Set}})}"></span>, we choose the class of all <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>. Therefore, for two objects <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, i.e. sets, we have <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 {\text{hom}}(A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>hom</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{hom}}(A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14855c352a54e8ef20e1055cd9df1b92aad3fe2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.741ex; height:2.843ex;" alt="{\displaystyle {\text{hom}}(A,B)}"></span> to be the class of all functions <span class="nowrap">⁠<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>⁠</span> such that <span class="nowrap">⁠<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:A\mapsto B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">↦</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\mapsto B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95138b7a1c4e935668228a44c617bfe09e7cce74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\mapsto B}"></span>⁠</span>.</li> <li>The composition of morphisms <span class="nowrap">⁠<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 \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span>⁠</span> is simply the usual <a href="/wiki/Function_composition" title="Function composition">function composition</a>, i.e. for two morphisms <span class="nowrap">⁠<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:A\mapsto B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">↦</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\mapsto B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95138b7a1c4e935668228a44c617bfe09e7cce74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\mapsto B}"></span>⁠</span> and <span class="nowrap">⁠<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:B\mapsto C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">↦</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:B\mapsto C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0cfd5a1ce31169e39aa8b128c9caa704f2bd45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.198ex; height:2.509ex;" alt="{\displaystyle g:B\mapsto C}"></span>⁠</span>, we have <span class="nowrap">⁠<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:A\mapsto C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">↦</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f:A\mapsto C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3cf3b2e37c9f53ff9498266e768edb03292d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.65ex; height:2.509ex;" alt="{\displaystyle g\circ f:A\mapsto C}"></span>⁠</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 (g\circ f)(x)=g(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=g(f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a73f8d834a602ee506ac323b8a36ce17ac2b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.979ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g(f(x))}"></span>, which is obviously associative. Furthermore, for every object <span class="texhtml mvar" style="font-style:italic;">A</span> we have the identity 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 {\text{id}}_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>id</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{id}}_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8010f5ff2b373727f8a483644024e23500c5ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.404ex; height:2.509ex;" alt="{\displaystyle {\text{id}}_{A}}"></span> to be the identity map <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 {\text{id}}_{A}:A\mapsto A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>id</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">↦</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{id}}_{A}:A\mapsto A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78352bd65faddffe72e364a9153e8f2b89a41cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.442ex; height:2.509ex;" alt="{\displaystyle {\text{id}}_{A}:A\mapsto A}"></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 {\text{id}}_{A}(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>id</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{id}}_{A}(x)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4c5bad815677a58214ad3ec048d40820837e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.971ex; height:2.843ex;" alt="{\displaystyle {\text{id}}_{A}(x)=x}"></span> on <span class="texhtml mvar" style="font-style:italic;">A</span></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Morphisms">Morphisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=4" title="Edit section: Morphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Relations among morphisms (such as <span class="nowrap"><i>fg</i> = <i>h</i></span>) are often depicted using <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagrams</a>, with "points" (corners) representing objects and "arrows" representing morphisms. </p><p><a href="/wiki/Morphism" title="Morphism">Morphisms</a> can have any of the following properties. A morphism <span class="nowrap"><i>f</i> : <i>a</i> → <i>b</i></span> is a: </p> <ul><li><a href="/wiki/Monomorphism" title="Monomorphism">monomorphism</a> (or <i>monic</i>) if <span class="nowrap"><i>f</i> ∘ <i>g</i><sub>1</sub> = <i>f</i> ∘ <i>g</i><sub>2</sub></span> implies <span class="nowrap"><i>g</i><sub>1</sub> = <i>g</i><sub>2</sub></span> for all morphisms <span class="nowrap"><i>g</i><sub>1</sub>, <i>g<sub>2</sub></i> : <i>x</i> → <i>a</i></span>.</li> <li><a href="/wiki/Epimorphism" title="Epimorphism">epimorphism</a> (or <i>epic</i>) if <span class="nowrap"><i>g</i><sub>1</sub> ∘ <i>f</i> = <i>g</i><sub>2</sub> ∘ <i>f</i></span> implies <span class="nowrap"><i>g<sub>1</sub></i> = <i>g<sub>2</sub></i></span> for all morphisms <span class="nowrap"><i>g</i><sub>1</sub>, <i>g</i><sub>2</sub> : <i>b</i> → <i>x</i></span>.</li> <li><i>bimorphism</i> if <i>f</i> is both epic and monic.</li> <li><a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> if there exists a morphism <span class="nowrap"><i>g</i> : <i>b</i> → <i>a</i></span> such that <span class="nowrap"><i>f</i> ∘ <i>g</i> = 1<sub><i>b</i></sub> and <i>g</i> ∘ <i>f</i> = 1<sub><i>a</i></sub></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> if <span class="nowrap"><i>a</i> = <i>b</i></span>. end(<i>a</i>) denotes the class of endomorphisms of <i>a</i>.</li> <li><a href="/wiki/Automorphism" title="Automorphism">automorphism</a> if <i>f</i> is both an endomorphism and an isomorphism. aut(<i>a</i>) denotes the class of automorphisms of <i>a</i>.</li> <li><a href="/wiki/Retract_(category_theory)" class="mw-redirect" title="Retract (category theory)">retraction</a> if a right inverse of <i>f</i> exists, i.e. if there exists a morphism <span class="nowrap"><i>g</i> : <i>b</i> → <i>a</i></span> with <span class="nowrap"><i>f</i> ∘ <i>g</i> = 1<sub><i>b</i></sub></span>.</li> <li><a href="/wiki/Section_(category_theory)" title="Section (category theory)">section</a> if a left inverse of <i>f</i> exists, i.e. if there exists a morphism <span class="nowrap"><i>g</i> : <i>b</i> → <i>a</i></span> with <span class="nowrap"><i>g</i> ∘ <i>f</i> = 1<sub><i>a</i></sub></span>.</li></ul> <p>Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent: </p> <ul><li><i>f</i> is a monomorphism and a retraction;</li> <li><i>f</i> is an epimorphism and a section;</li> <li><i>f</i> is an isomorphism.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Functors">Functors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=5" title="Edit section: Functors"><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" title="Functor">Functor</a></div> <p><a href="/wiki/Functor" title="Functor">Functors</a> are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. </p><p>A (<b>covariant</b>) functor <i>F</i> from a category <i>C</i> to a category <i>D</i>, written <span class="nowrap"><i>F</i> : <i>C</i> → <i>D</i></span>, consists of: </p> <ul><li>for each object <i>x</i> in <i>C</i>, an object <i>F</i>(<i>x</i>) in <i>D</i>; and</li> <li>for each morphism <span class="nowrap"><i>f</i> : <i>x</i> → <i>y</i></span> in <i>C</i>, a morphism <span class="nowrap"><i>F</i>(<i>f</i>) : <i>F</i>(<i>x</i>) → <i>F</i>(<i>y</i>)</span> in <i>D</i>,</li></ul> <p>such that the following two properties hold: </p> <ul><li>For every object <i>x</i> in <i>C</i>, <span class="nowrap"><i>F</i>(1<sub><i>x</i></sub>) = 1<sub><i>F</i>(<i>x</i>)</sub></span>;</li> <li>For all morphisms <span class="nowrap"><i>f</i> : <i>x</i> → <i>y</i></span> and <span class="nowrap"><i>g</i> : <i>y</i> → <i>z</i></span>, <span class="nowrap"><i>F</i>(<i>g</i> ∘ <i>f</i>) = <i>F</i>(<i>g</i>) ∘ <i>F</i>(<i>f</i>)</span>.</li></ul> <p>A <b>contravariant</b> functor <span class="nowrap"><i>F</i>: <i>C</i> → <i>D</i></span> is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism <span class="nowrap"><i>f</i> : <i>x</i> → <i>y</i></span> in <i>C</i> must be assigned to a morphism <span class="nowrap"><i>F</i>(<i>f</i>) : <i>F</i>(<i>y</i>) → <i>F</i>(<i>x</i>)</span> in <i>D</i>. In other words, a contravariant functor acts as a covariant functor from the <a href="/wiki/Opposite_category" title="Opposite category">opposite category</a> <i>C</i><sup>op</sup> to <i>D</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Natural_transformations">Natural transformations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=6" title="Edit section: Natural transformations"><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/Natural_transformation" title="Natural transformation">Natural transformation</a></div> <p>A <i>natural transformation</i> is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors. </p><p>If <i>F</i> and <i>G</i> are (covariant) functors between the categories <i>C</i> and <i>D</i>, then a natural transformation <i>η</i> from <i>F</i> to <i>G</i> associates to every object <i>X</i> in <i>C</i> a morphism <span class="nowrap"><i>η</i><sub><i>X</i></sub> : <i>F</i>(<i>X</i>) → <i>G</i>(<i>X</i>)</span> in <i>D</i> such that for every morphism <span class="nowrap"><i>f</i> : <i>X</i> → <i>Y</i></span> in <i>C</i>, we have <span class="nowrap"><i>η</i><sub><i>Y</i></sub> ∘ <i>F</i>(<i>f</i>) = <i>G</i>(<i>f</i>) ∘ <i>η</i><sub><i>X</i></sub></span>; this means that the following diagram is <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative</a>: </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Natural_transformation.svg" class="mw-file-description" title="Commutative diagram defining natural transformations"><img alt="Commutative diagram defining natural transformations" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Natural_transformation.svg/175px-Natural_transformation.svg.png" decoding="async" width="175" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Natural_transformation.svg/263px-Natural_transformation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Natural_transformation.svg/350px-Natural_transformation.svg.png 2x" data-file-width="125" data-file-height="101" /></a><figcaption>Commutative diagram defining natural transformations</figcaption></figure> <p>The two functors <i>F</i> and <i>G</i> are called <i>naturally isomorphic</i> if there exists a natural transformation from <i>F</i> to <i>G</i> such that <i>η</i><sub><i>X</i></sub> is an isomorphism for every object <i>X</i> in <i>C</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Other_concepts">Other concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=7" title="Edit section: Other concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Universal_constructions,_limits,_and_colimits"><span id="Universal_constructions.2C_limits.2C_and_colimits"></span>Universal constructions, limits, and colimits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=8" title="Edit section: Universal constructions, limits, and colimits"><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 articles: <a href="/wiki/Universal_property" title="Universal property">Universal property</a> and <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">Limit (category theory)</a></div> <p>Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies. </p><p>Each category is distinguished by properties that all its objects have in common, such as the <a href="/wiki/Empty_set" title="Empty set">empty set</a> or the <a href="/wiki/Product_topology" title="Product topology">product of two topologies</a>, yet in the definition of a category, objects are considered atomic, i.e., we <i>do not know</i> whether an object <i>A</i> is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find <i><a href="/wiki/Universal_property" title="Universal property">universal properties</a></i> that uniquely determine the objects of interest. </p><p>Numerous important constructions can be described in a purely categorical way if the <i>category limit</i> can be developed and dualized to yield the notion of a <i>colimit</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_categories">Equivalent categories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=9" title="Edit section: Equivalent categories"><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 articles: <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">Equivalence of categories</a> and <a href="/wiki/Isomorphism_of_categories" title="Isomorphism of categories">Isomorphism of categories</a></div> <p>It is a natural question to ask: under which conditions can two categories be considered <i>essentially the same</i>, in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called <i>equivalence of categories</i>, which is given by appropriate functors between two categories. Categorical equivalence has found <a href="/wiki/Equivalence_of_categories#Examples" title="Equivalence of categories">numerous applications</a> in mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Further_concepts_and_results">Further concepts and results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=10" title="Edit section: Further concepts and results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading. </p> <ul><li>The <a href="/wiki/Functor_category" title="Functor category">functor category</a> <i>D</i><sup><i>C</i></sup> has as objects the functors from <i>C</i> to <i>D</i> and as morphisms the natural transformations of such functors. The <a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a> is one of the most famous basic results of category theory; it describes representable functors in functor categories.</li> <li><a href="/wiki/Dual_(category_theory)" title="Dual (category theory)">Duality</a>: Every statement, theorem, or definition in category theory has a <i>dual</i> which is essentially obtained by "reversing all the arrows". If one statement is true in a category <i>C</i> then its dual is true in the dual category <i>C</i><sup>op</sup>. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.</li> <li><a href="/wiki/Adjoint_functors" title="Adjoint functors">Adjoint functors</a>: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Higher-dimensional_categories">Higher-dimensional categories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=11" title="Edit section: Higher-dimensional categories"><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/Higher_category_theory" title="Higher category theory">Higher category theory</a></div> <p>Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of <i>higher-dimensional categories</i>. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes". </p><p>For example, a (strict) <a href="/wiki/2-category" class="mw-redirect" title="2-category">2-category</a> is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is <b>Cat</b>, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformations</a> of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal categories</a>. <a href="/wiki/Bicategory" title="Bicategory">Bicategories</a> are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism. </p><p>This process can be extended for all <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <i>n</i>, and these are called <a href="/wiki/N-category" class="mw-redirect" title="N-category"><i>n</i>-categories</a>. There is even a notion of <i><a href="/wiki/Quasi-category" title="Quasi-category">ω-category</a></i> corresponding to the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> <a href="/wiki/%CE%A9_(ordinal_number)" class="mw-redirect" title="Ω (ordinal number)">ω</a>. </p><p>Higher-dimensional categories are part of the broader mathematical field of <a href="/wiki/Higher-dimensional_algebra" title="Higher-dimensional algebra">higher-dimensional algebra</a>, a concept introduced by <a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Ronald Brown</a>. For a conversational introduction to these ideas, see <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week73.html">John Baez, 'A Tale of <i>n</i>-categories' (1996).</a> </p> <div class="mw-heading mw-heading2"><h2 id="Historical_notes">Historical notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=12" title="Edit section: Historical notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Category_theory" title="Special:EditPage/Category theory">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">November 2015</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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Timeline_of_category_theory_and_related_mathematics" title="Timeline of category theory and related mathematics">Timeline of category theory and related mathematics</a></div> <style data-mw-deduplicate="TemplateStyles:r1023981488">@media all and (max-width:720px){.mw-parser-output .rquote{width:auto!important;float:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote rquote" style="float: right; width: 33%;"><p>It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation [...]</p><div class="templatequotecite">— <cite><a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Eilenberg</a> and <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane</a> (1945) <sup id="cite_ref-Eilenberg-1945_4-0" class="reference"><a href="#cite_note-Eilenberg-1945-4"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p>Whilst specific examples of functors and natural transformations had been given by <a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a> and <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> in a 1942 paper on <a href="/wiki/Group_theory" title="Group theory">group theory</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> these concepts were introduced in a more general sense, together with the additional notion of categories, in a 1945 paper by the same authors<sup id="cite_ref-Eilenberg-1945_4-1" class="reference"><a href="#cite_note-Eilenberg-1945-4"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (who discussed applications of category theory to the field of <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>).<sup id="cite_ref-Marquis-2019_6-0" class="reference"><a href="#cite_note-Marquis-2019-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Their work was an important part of the transition from intuitive and geometric <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> to <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, Eilenberg and Mac Lane later writing that their goal was to understand natural transformations, which first required the definition of functors, then categories. </p><p><a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Stanislaw Ulam</a>, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Who claimed this and where? Please see discussion on talk page. This claim of a claim needs sources. (June 2024)">citation needed</span></a></i>]</sup> Eilenberg was Polish, and studied mathematics in Poland in the 1930s.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Category theory is also, in some sense, a continuation of the work of <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> (one of Mac Lane's teachers) in formalizing abstract processes;<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure (<a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2020)">citation needed</span></a></i>]</sup> Eilenberg and Mac Lane introduced categories for understanding and formalizing the processes (<a href="/wiki/Functor" title="Functor">functors</a>) that relate <a href="/wiki/Topology" title="Topology">topological structures</a> to algebraic structures (<a href="/wiki/Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariants</a>) that characterize them. </p><p>Category theory was originally introduced for the need of <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, and widely extended for the need of modern <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> (<a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a>). Category theory may be viewed as an extension of <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>, as the latter studies <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>, and the former applies to any kind of <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> and studies also the relationships between structures of different nature. For this reason, it is used throughout mathematics. Applications to <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">semantics</a> (<a href="/wiki/Categorical_abstract_machine" title="Categorical abstract machine">categorical abstract machine</a>) came later. </p><p>Certain categories called <a href="/wiki/Topos" title="Topos">topoi</a> (singular <i>topos</i>) can even serve as an alternative to <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> as a foundation of mathematics. A topos can also be considered as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, <a href="/wiki/Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructive mathematics</a>. <a href="/wiki/Topos" title="Topos">Topos theory</a> is a form of abstract <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf theory</a>, with geometric origins, and leads to ideas such as <a href="/wiki/Pointless_topology" title="Pointless topology">pointless topology</a>. </p><p><a href="/wiki/Categorical_logic" title="Categorical logic">Categorical logic</a> is now a well-defined field based on <a href="/wiki/Type_theory" title="Type theory">type theory</a> for <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logics</a>, with applications in <a href="/wiki/Functional_programming" title="Functional programming">functional programming</a> and <a href="/wiki/Domain_theory" title="Domain theory">domain theory</a>, where a <a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">cartesian closed category</a> is taken as a non-syntactic description of a <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some <a href="https://en.wiktionary.org/wiki/abstract" class="extiw" title="wikt:abstract">abstract</a> sense). </p><p>Category theory has been applied in other fields as well, see <a href="/wiki/Applied_category_theory" title="Applied category theory">applied category theory</a>. For example, <a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a> has shown a link between <a href="/wiki/Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a> in <a href="/wiki/Physics" title="Physics">physics</a> and monoidal categories.<sup id="cite_ref-Baez09_9-0" class="reference"><a href="#cite_note-Baez09-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Another application of category theory, more specifically topos theory, has been made in mathematical music theory, see for example the book <i>The Topos of Music, Geometric Logic of Concepts, Theory, and Performance</i> by <a href="/wiki/Guerino_Mazzola" title="Guerino Mazzola">Guerino Mazzola</a>. </p><p>More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of <a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a> and Rosebrugh (2003) and Lawvere and <a href="/wiki/Stephen_Schanuel" title="Stephen Schanuel">Stephen Schanuel</a> (1997) and Mirroslav Yotov (2012). </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=Category_theory&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.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 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href="/wiki/Domain_theory" title="Domain theory">Domain theory</a></li> <li><a href="/wiki/Enriched_category" title="Enriched category">Enriched category theory</a></li> <li><a href="/wiki/Glossary_of_category_theory" title="Glossary of category theory">Glossary of category theory</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Higher_category_theory" title="Higher category theory">Higher category theory</a></li> <li><a href="/wiki/Higher-dimensional_algebra" title="Higher-dimensional algebra">Higher-dimensional algebra</a></li> <li><a href="/wiki/List_of_publications_in_mathematics#Category_theory" class="mw-redirect" title="List of publications in mathematics">Important publications in category theory</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Outline_of_category_theory" title="Outline of category theory">Outline of category theory</a></li> <li><a href="/wiki/Timeline_of_category_theory_and_related_mathematics" title="Timeline of category theory and related mathematics">Timeline of category theory and related mathematics</a></li> <li><a href="/wiki/Applied_category_theory" title="Applied category theory">Applied category theory</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-lower-alpha" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Some authors compose in the opposite order, writing <i>fg</i> or <span class="nowrap"><i>f</i> ∘ <i>g</i></span> for <span class="nowrap"><i>g</i> ∘ <i>f</i></span>. Computer scientists using category theory very commonly write <span class="nowrap"><i>f</i> ; <i>g</i></span> for <span class="nowrap"><i>g</i> ∘ <i>f</i></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">A morphism that is both epic and monic is not necessarily an isomorphism. An elementary counterexample: in the category consisting of two objects <i>A</i> and <i>B</i>, the identity morphisms, and a single morphism <i>f</i> from <i>A</i> to <i>B</i>, <i>f</i> is both epic and monic but is not an isomorphism.</span> </li> </ol></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=Category_theory&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=16" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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="CITEREFMarquis2023" class="citation cs2">Marquis, Jean-Pierre (2023), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2023/entries/category-theory/">"Category Theory"</a>, in Zalta, Edward N.; Nodelman, Uri (eds.), <i>The Stanford Encyclopedia of Philosophy</i> (Fall 2023 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">2024-04-23</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Category+Theory&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Fall+2023&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2023&rft.aulast=Marquis&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Ffall2023%2Fentries%2Fcategory-theory%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></span> </li> <li id="cite_note-Eilenberg-1945-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Eilenberg-1945_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Eilenberg-1945_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEilenbergMac_Lane1945" class="citation journal cs1">Eilenberg, Samuel; Mac Lane, Saunders (1945). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf">"General theory of natural equivalences"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Mathematical Society</i>. <b>58</b>: 247. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1945-0013131-6">10.1090/S0002-9947-1945-0013131-6</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9947">0002-9947</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221010/https://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=General+theory+of+natural+equivalences&rft.volume=58&rft.pages=247&rft.date=1945&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1945-0013131-6&rft.issn=0002-9947&rft.aulast=Eilenberg&rft.aufirst=Samuel&rft.au=Mac+Lane%2C+Saunders&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1945-058-00%2FS0002-9947-1945-0013131-6%2FS0002-9947-1945-0013131-6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEilenbergMac_Lane1942" class="citation journal cs1">Eilenberg, S.; Mac Lane, S. (1942). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968966">"Group Extensions and Homology"</a></span>. <i>Annals of Mathematics</i>. <b>43</b> (4): 757–831. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968966">10.2307/1968966</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968966">1968966</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Group+Extensions+and+Homology&rft.volume=43&rft.issue=4&rft.pages=757-831&rft.date=1942&rft.issn=0003-486X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968966%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1968966&rft.aulast=Eilenberg&rft.aufirst=S.&rft.au=Mac+Lane%2C+S.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968966&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></span> </li> <li id="cite_note-Marquis-2019-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Marquis-2019_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarquis2019" class="citation web cs1">Marquis, Jean-Pierre (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/category-theory/">"Category Theory"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. Department of Philosophy, <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 September</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Category+Theory&rft.date=2019&rft.aulast=Marquis&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fcategory-theory%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Eilenberg/">"Samuel Eilenberg – Biography"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Samuel+Eilenberg+%E2%80%93+Biography&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FEilenberg%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" 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="CITEREFReck2020" class="citation book cs1">Reck, Erich (2020). <i>The Prehistory of Mathematical Structuralism</i> (1st ed.). Oxford University Press. pp. 215–219. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780190641221" title="Special:BookSources/9780190641221"><bdi>9780190641221</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Prehistory+of+Mathematical+Structuralism&rft.pages=215-219&rft.edition=1st&rft.pub=Oxford+University+Press&rft.date=2020&rft.isbn=9780190641221&rft.aulast=Reck&rft.aufirst=Erich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></span> </li> <li id="cite_note-Baez09-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Baez09_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaezStay2010" class="citation book cs1">Baez, J.C.; Stay, M. (2010). "Physics, Topology, Logic and Computation: A Rosetta Stone". <i>New Structures for Physics</i>. Lecture Notes in Physics. Vol. 813. pp. 95–172. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0903.0340">0903.0340</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-12821-9_2">10.1007/978-3-642-12821-9_2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-12820-2" title="Special:BookSources/978-3-642-12820-2"><bdi>978-3-642-12820-2</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115169297">115169297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Physics%2C+Topology%2C+Logic+and+Computation%3A+A+Rosetta+Stone&rft.btitle=New+Structures+for+Physics&rft.series=Lecture+Notes+in+Physics&rft.pages=95-172&rft.date=2010&rft_id=info%3Aarxiv%2F0903.0340&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A115169297%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-642-12821-9_2&rft.isbn=978-3-642-12820-2&rft.aulast=Baez&rft.aufirst=J.C.&rft.au=Stay%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=17" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdámekHerrlichStrecker2004" class="citation book cs1">Adámek, Jiří; <a href="/wiki/Horst_Herrlich" title="Horst Herrlich">Herrlich, Horst</a>; Strecker, George E. (2004). <a rel="nofollow" class="external text" href="http://katmat.math.uni-bremen.de/acc/acc.htm"><i>Abstract and Concrete Categories</i></a>. Heldermann Verlag Berlin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+and+Concrete+Categories&rft.pub=Heldermann+Verlag+Berlin&rft.date=2004&rft.aulast=Ad%C3%A1mek&rft.aufirst=Ji%C5%99%C3%AD&rft.au=Herrlich%2C+Horst&rft.au=Strecker%2C+George+E.&rft_id=http%3A%2F%2Fkatmat.math.uni-bremen.de%2Facc%2Facc.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAwodey2010" class="citation book cs1">Awodey, Steve (2010). <i>Category Theory</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199237180" title="Special:BookSources/978-0199237180"><bdi>978-0199237180</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory&rft.pub=Oxford+University+Press&rft.date=2010&rft.isbn=978-0199237180&rft.aulast=Awodey&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarrWells2012" class="citation cs2"><a href="/wiki/Michael_Barr_(mathematician)" title="Michael Barr (mathematician)">Barr, Michael</a>; <a href="/wiki/Charles_Wells_(mathematician)" title="Charles Wells (mathematician)">Wells, Charles</a> (2012) [1995], <a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html"><i>Category Theory for Computing Science</i></a>, Reprints in Theory and Applications of Categories, vol. 22 (3rd ed.)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory+for+Computing+Science&rft.series=Reprints+in+Theory+and+Applications+of+Categories&rft.edition=3rd&rft.date=2012&rft.aulast=Barr&rft.aufirst=Michael&rft.au=Wells%2C+Charles&rft_id=http%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Freprints%2Farticles%2F22%2Ftr22abs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarrWells2005" class="citation cs2"><a href="/wiki/Michael_Barr_(mathematician)" title="Michael Barr (mathematician)">Barr, Michael</a>; <a href="/wiki/Charles_Wells_(mathematician)" title="Charles Wells (mathematician)">Wells, Charles</a> (2005), <a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html"><i>Toposes, Triples and Theories</i></a>, Reprints in Theory and Applications of Categories, vol. 12, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2178101">2178101</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Toposes%2C+Triples+and+Theories&rft.series=Reprints+in+Theory+and+Applications+of+Categories&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2178101%23id-name%3DMR&rft.aulast=Barr&rft.aufirst=Michael&rft.au=Wells%2C+Charles&rft_id=http%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Freprints%2Farticles%2F12%2Ftr12abs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorceux1994" class="citation book cs1">Borceux, Francis (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YfzImoopB-IC&q=%22Handbook+of+categorical+algebra%22&pg=PP1"><i>Handbook of categorical algebra</i></a>. Encyclopedia of Mathematics and its Applications. Cambridge University Press. pp. 50–52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521441780" title="Special:BookSources/9780521441780"><bdi>9780521441780</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+categorical+algebra&rft.series=Encyclopedia+of+Mathematics+and+its+Applications&rft.pages=50-52&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=9780521441780&rft.aulast=Borceux&rft.aufirst=Francis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYfzImoopB-IC%26q%3D%2522Handbook%2Bof%2Bcategorical%2Balgebra%2522%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreyd2003" class="citation book cs1"><a href="/wiki/Peter_J._Freyd" title="Peter J. Freyd">Freyd, Peter J.</a> (2003) [1964]. <a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html"><i>Abelian Categories</i></a>. Reprints in Theory and Applications of Categories. Vol. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abelian+Categories&rft.series=Reprints+in+Theory+and+Applications+of+Categories&rft.date=2003&rft.aulast=Freyd&rft.aufirst=Peter+J.&rft_id=http%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Freprints%2Farticles%2F3%2Ftr3abs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreydScedrov1990" class="citation book cs1"><a href="/wiki/Peter_J._Freyd" title="Peter J. Freyd">Freyd, Peter J.</a>; Scedrov, Andre (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fCSJRegkKdoC"><i>Categories, allegories</i></a>. North Holland Mathematical Library. Vol. 39. North Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-088701-2" title="Special:BookSources/978-0-08-088701-2"><bdi>978-0-08-088701-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories%2C+allegories&rft.series=North+Holland+Mathematical+Library&rft.pub=North+Holland&rft.date=1990&rft.isbn=978-0-08-088701-2&rft.aulast=Freyd&rft.aufirst=Peter+J.&rft.au=Scedrov%2C+Andre&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfCSJRegkKdoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldblatt2006" class="citation book cs1"><a href="/wiki/Robert_Goldblatt" title="Robert Goldblatt">Goldblatt, Robert</a> (2006) [1979]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AwLc-12-7LMC"><i>Topoi: The Categorial Analysis of Logic</i></a>. Studies in logic and the foundations of mathematics. Vol. 94. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45026-1" title="Special:BookSources/978-0-486-45026-1"><bdi>978-0-486-45026-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topoi%3A+The+Categorial+Analysis+of+Logic&rft.series=Studies+in+logic+and+the+foundations+of+mathematics&rft.pub=Dover&rft.date=2006&rft.isbn=978-0-486-45026-1&rft.aulast=Goldblatt&rft.aufirst=Robert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAwLc-12-7LMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerrlichStrecker2007" class="citation book cs1"><a href="/wiki/Horst_Herrlich" title="Horst Herrlich">Herrlich, Horst</a>; Strecker, George E. (2007). <i>Category Theory</i> (3rd ed.). Heldermann Verlag Berlin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-88538-001-6" title="Special:BookSources/978-3-88538-001-6"><bdi>978-3-88538-001-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory&rft.edition=3rd&rft.pub=Heldermann+Verlag+Berlin&rft.date=2007&rft.isbn=978-3-88538-001-6&rft.aulast=Herrlich&rft.aufirst=Horst&rft.au=Strecker%2C+George+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKashiwaraSchapira2006" class="citation book cs1"><a href="/wiki/Masaki_Kashiwara" title="Masaki Kashiwara">Kashiwara, Masaki</a>; <a href="/wiki/Pierre_Schapira_(mathematician)" title="Pierre Schapira (mathematician)">Schapira, Pierre</a> (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K-SjOw_2gXwC"><i>Categories and Sheaves</i></a>. Grundlehren der Mathematischen Wissenschaften. Vol. 332. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-27949-5" title="Special:BookSources/978-3-540-27949-5"><bdi>978-3-540-27949-5</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+and+Sheaves&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-27949-5&rft.aulast=Kashiwara&rft.aufirst=Masaki&rft.au=Schapira%2C+Pierre&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK-SjOw_2gXwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawvereRosebrugh2003" class="citation book cs1"><a href="/wiki/William_Lawvere" title="William Lawvere">Lawvere, F. William</a>; Rosebrugh, Robert (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/setsformathemati0000lawv"><i>Sets for Mathematics</i></a></span>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-01060-3" title="Special:BookSources/978-0-521-01060-3"><bdi>978-0-521-01060-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets+for+Mathematics&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-01060-3&rft.aulast=Lawvere&rft.aufirst=F.+William&rft.au=Rosebrugh%2C+Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsetsformathemati0000lawv&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawvereSchanuel2009" class="citation book cs1">Lawvere, F. William; <a href="/wiki/Stephen_Schanuel" title="Stephen Schanuel">Schanuel, Stephen Hoel</a> (2009) [1997]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/conceptualmathem00lawv"><i>Conceptual Mathematics: A First Introduction to Categories</i></a></span> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-89485-2" title="Special:BookSources/978-0-521-89485-2"><bdi>978-0-521-89485-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conceptual+Mathematics%3A+A+First+Introduction+to+Categories&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-89485-2&rft.aulast=Lawvere&rft.aufirst=F.+William&rft.au=Schanuel%2C+Stephen+Hoel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconceptualmathem00lawv&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeinster2004" class="citation book cs1 cs1-prop-interwiki-linked-name"><a href="https://de.wikipedia.org/wiki/Tom_Leinster" class="extiw" title="de:Tom Leinster">Leinster, Tom</a> <span class="cs1-format">[in German]</span> (2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html"><i>Higher Operads, Higher Categories</i></a>. London Math. Society Lecture Note Series. Vol. 298. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 448. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004hohc.book.....L">2004hohc.book.....L</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53215-0" title="Special:BookSources/978-0-521-53215-0"><bdi>978-0-521-53215-0</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.maths.gla.ac.uk/~tl/book.html">the original</a> on 2003-10-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-04-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+Operads%2C+Higher+Categories&rft.series=London+Math.+Society+Lecture+Note+Series&rft.pages=448&rft.pub=Cambridge+University+Press&rft.date=2004&rft_id=info%3Abibcode%2F2004hohc.book.....L&rft.isbn=978-0-521-53215-0&rft.aulast=Leinster&rft.aufirst=Tom&rft_id=http%3A%2F%2Fwww.maths.gla.ac.uk%2F~tl%2Fbook.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeinster2014" class="citation book cs1 cs1-prop-interwiki-linked-name"><a href="https://de.wikipedia.org/wiki/Tom_Leinster" class="extiw" title="de:Tom Leinster">Leinster, Tom</a> <span class="cs1-format">[in German]</span> (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Q3vsAwAAQBAJ"><i>Basic Category Theory</i></a>. Cambridge Studies in Advanced Mathematics. Vol. 143. Cambridge University Press. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.09375">1612.09375</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107044241" title="Special:BookSources/9781107044241"><bdi>9781107044241</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+Category+Theory&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=2014&rft_id=info%3Aarxiv%2F1612.09375&rft.isbn=9781107044241&rft.aulast=Leinster&rft.aufirst=Tom&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQ3vsAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLurie2009" class="citation book cs1"><a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Lurie, Jacob</a> (2009). <i>Higher Topos Theory</i>. Annals of Mathematics Studies. Vol. 170. Princeton University Press. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.CT/0608040">math.CT/0608040</a></span>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14049-0" title="Special:BookSources/978-0-691-14049-0"><bdi>978-0-691-14049-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2522659">2522659</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+Topos+Theory&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=2009&rft_id=info%3Aarxiv%2Fmath.CT%2F0608040&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2522659%23id-name%3DMR&rft.isbn=978-0-691-14049-0&rft.aulast=Lurie&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1998" class="citation book cs1"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1998). <a href="/wiki/Categories_for_the_Working_Mathematician" title="Categories for the Working Mathematician"><i>Categories for the Working Mathematician</i></a>. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98403-2" title="Special:BookSources/978-0-387-98403-2"><bdi>978-0-387-98403-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1712872">1712872</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.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-98403-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1712872%23id-name%3DMR&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_LaneBirkhoff1999" class="citation book cs1"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a>; <a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1999) [1967]. <i>Algebra</i> (2nd ed.). Chelsea. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1646-2" title="Special:BookSources/978-0-8218-1646-2"><bdi>978-0-8218-1646-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=2nd&rft.pub=Chelsea&rft.date=1999&rft.isbn=978-0-8218-1646-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartiniEhrigNunes1996" class="citation journal cs1">Martini, A.; Ehrig, H.; Nunes, D. (1996). <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/martini96element.html">"Elements of basic category theory"</a>. <i>Technical Report</i>. <b>96</b> (5).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Technical+Report&rft.atitle=Elements+of+basic+category+theory&rft.volume=96&rft.issue=5&rft.date=1996&rft.aulast=Martini&rft.aufirst=A.&rft.au=Ehrig%2C+H.&rft.au=Nunes%2C+D.&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2Fmartini96element.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMazzola2002" class="citation book cs1"><a href="/wiki/Guerino_Mazzola" title="Guerino Mazzola">Mazzola, Guerino</a> (2002). <i>The Topos of Music, Geometric Logic of Concepts, Theory, and Performance</i>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-5731-3" title="Special:BookSources/978-3-7643-5731-3"><bdi>978-3-7643-5731-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Topos+of+Music%2C+Geometric+Logic+of+Concepts%2C+Theory%2C+and+Performance&rft.pub=Birkh%C3%A4user&rft.date=2002&rft.isbn=978-3-7643-5731-3&rft.aulast=Mazzola&rft.aufirst=Guerino&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedicchioTholen2004" class="citation book cs1">Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). <i>Categorical foundations. Special topics in order, topology, algebra, and sheaf theory</i>. Encyclopedia of Mathematics and Its Applications. Vol. 97. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83414-8" title="Special:BookSources/978-0-521-83414-8"><bdi>978-0-521-83414-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1034.18001">1034.18001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categorical+foundations.+Special+topics+in+order%2C+topology%2C+algebra%2C+and+sheaf+theory&rft.series=Encyclopedia+of+Mathematics+and+Its+Applications&rft.pub=Cambridge+University+Press&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1034.18001%23id-name%3DZbl&rft.isbn=978-0-521-83414-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPierce1991" class="citation book cs1"><a href="/wiki/Benjamin_C._Pierce" title="Benjamin C. Pierce">Pierce, Benjamin C.</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ezdeaHfpYPwC"><i>Basic Category Theory for Computer Scientists</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-66071-6" title="Special:BookSources/978-0-262-66071-6"><bdi>978-0-262-66071-6</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+Category+Theory+for+Computer+Scientists&rft.pub=MIT+Press&rft.date=1991&rft.isbn=978-0-262-66071-6&rft.aulast=Pierce&rft.aufirst=Benjamin+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DezdeaHfpYPwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchalkSimmons2005" class="citation book cs1">Schalk, A.; Simmons, H. (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170321152109/http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf"><i>An introduction to Category Theory in four easy movements</i></a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-03-21<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-12-03</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Category+Theory+in+four+easy+movements&rft.date=2005&rft.aulast=Schalk&rft.aufirst=A.&rft.au=Simmons%2C+H.&rft_id=http%3A%2F%2Fwww.cs.man.ac.uk%2F~hsimmons%2FBOOKS%2FCatTheory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span> Notes for a course offered as part of the MSc. in <a href="/wiki/Mathematical_Logic" class="mw-redirect" title="Mathematical Logic">Mathematical Logic</a>, <a href="/wiki/Manchester_University" class="mw-redirect" title="Manchester University">Manchester University</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimmons2011" class="citation cs2">Simmons, Harold (2011), <i>An Introduction to Category Theory</i>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521283045" title="Special:BookSources/978-0521283045"><bdi>978-0521283045</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Category+Theory&rft.date=2011&rft.isbn=978-0521283045&rft.aulast=Simmons&rft.aufirst=Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimpson2010" class="citation book cs1"><a href="/wiki/Carlos_Simpson" title="Carlos Simpson">Simpson, Carlos</a> (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/arxiv-1001.4071"><i>Homotopy theory of higher categories</i></a>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1001.4071">1001.4071</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010arXiv1001.4071S">2010arXiv1001.4071S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Homotopy+theory+of+higher+categories&rft.date=2010&rft_id=info%3Aarxiv%2F1001.4071&rft_id=info%3Abibcode%2F2010arXiv1001.4071S&rft.aulast=Simpson&rft.aufirst=Carlos&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farxiv-1001.4071&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>, draft of a book.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylor1999" class="citation book cs1">Taylor, Paul (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iSCqyNgzamcC"><i>Practical Foundations of Mathematics</i></a>. Cambridge Studies in Advanced Mathematics. Vol. 59. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-63107-5" title="Special:BookSources/978-0-521-63107-5"><bdi>978-0-521-63107-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Foundations+of+Mathematics&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-63107-5&rft.aulast=Taylor&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DiSCqyNgzamcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuri1996–2001" class="citation web cs1">Turi, Daniele (1996–2001). <a rel="nofollow" class="external text" href="http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf">"Category Theory Lecture Notes"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">11 December</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Category+Theory+Lecture+Notes&rft.date=1996%2F2001&rft.aulast=Turi&rft.aufirst=Daniele&rft_id=http%3A%2F%2Fwww.dcs.ed.ac.uk%2Fhome%2Fdt%2FCT%2Fcategories.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span> Based on <a href="#CITEREFMac_Lane1998">Mac Lane 1998</a>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Category_theory&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarquis2008" class="citation book cs1">Marquis, Jean-Pierre (2008). <i>From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-9384-5" title="Special:BookSources/978-1-4020-9384-5"><bdi>978-1-4020-9384-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+a+Geometrical+Point+of+View%3A+A+Study+of+the+History+and+Philosophy+of+Category+Theory&rft.pub=Springer&rft.date=2008&rft.isbn=978-1-4020-9384-5&rft.aulast=Marquis&rft.aufirst=Jean-Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li></ul> </div> <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=Category_theory&action=edit&section=19" 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"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Category_theory" class="extiw" title="commons:Category:Category theory">Category theory</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Category_theory" class="extiw" title="q:Special:Search/Category theory">Category theory</a></b></i>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><a rel="nofollow" class="external text" href="http://www.tac.mta.ca/tac/">Theory and Application of Categories</a>, an electronic journal of category theory, full text, free, since 1995.</li> <li><a rel="nofollow" class="external text" href="https://cahierstgdc.com/">Cahiers de Topologie et Géométrie Différentielle Catégoriques</a>, an electronic journal of category theory, full text, free, funded in 1957.</li> <li><a rel="nofollow" class="external text" href="http://ncatlab.org/nlab">nLab</a>, a wiki project on mathematics, physics and philosophy with emphasis on the <i>n</i>-categorical point of view.</li> <li><a rel="nofollow" class="external text" href="https://golem.ph.utexas.edu/category/">The n-Category Café</a>, essentially a colloquium on topics in category theory.</li> <li><a rel="nofollow" class="external text" href="http://www.logicmatters.net/categories/">Category Theory</a>, a web page of links to lecture notes and freely available books on category theory.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHillman2001" class="citation cs2">Hillman, Chris (2001), <i>A Categorical Primer</i>, <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></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&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&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span>, a formal introduction to category theory.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamekHerrlichStecker" class="citation web cs1">Adamek, J.; Herrlich, H.; Stecker, G. <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> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060610174819/http://katmat.math.uni-bremen.de/acc/acc.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2006-06-10.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Abstract+and+Concrete+Categories-The+Joy+of+Cats&rft.aulast=Adamek&rft.aufirst=J.&rft.au=Herrlich%2C+H.&rft.au=Stecker%2C+G.&rft_id=http%3A%2F%2Fkatmat.math.uni-bremen.de%2Facc%2Facc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/category-theory/">"Category Theory"</a> entry by Jean-Pierre Marquis in the <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>, with an 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><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaez1996" class="citation web cs1">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>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Tale+of+n-categories&rft.date=1996&rft.aulast=Baez&rft.aufirst=John&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fweek73.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACategory+theory" class="Z3988"></span> — An informal introduction to higher order categories.</li> <li><a rel="nofollow" class="external text" href="http://wildcatsformma.wordpress.com">WildCats</a> is a category theory 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, <a href="/wiki/Functor" title="Functor">functors</a>, <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"><span class="plainlinks">The catsters</span>'s channel</a> on <a href="/wiki/YouTube_user_(identifier)" class="mw-redirect" title="YouTube user (identifier)">YouTube</a>, a channel about category theory.</li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/9categorytheory">Category theory</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>..</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> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150109111227/http://category-theory.mitpress.mit.edu/index.html">Category Theory for the Sciences</a>, an instruction on category theory as a tool throughout the sciences.</li> <li><a rel="nofollow" class="external text" href="https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/">Category Theory for Programmers</a> A book in blog form explaining category theory for computer programmers.</li> <li><a rel="nofollow" class="external text" href="http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf">Introduction to category theory.</a></li></ul> </div> <div 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algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</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/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential 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navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</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/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</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/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</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/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</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/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</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/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <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> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></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="Category_theory" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Category_theory" title="Template:Category theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Category_theory" title="Template talk:Category theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Category_theory" title="Special:EditPage/Template:Category theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Category_theory" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">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"></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_concepts" 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 href="/wiki/Functor" title="Functor">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 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_theory" 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="Computer_science" 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:Computer_science" title="Template:Computer science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computer_science" title="Template talk:Computer science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computer_science" title="Special:EditPage/Template:Computer science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Computer_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Computer_science" title="Computer science">Computer science</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>Note: This template roughly follows the 2012 <a href="/wiki/ACM_Computing_Classification_System" title="ACM Computing Classification System">ACM Computing Classification System</a>.</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_hardware" title="Computer hardware">Hardware</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/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</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/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</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/Network_architecture" title="Network architecture">Network architecture</a></li> <li><a href="/wiki/Network_protocol" class="mw-redirect" title="Network protocol">Network protocol</a></li> <li><a href="/wiki/Networking_hardware" title="Networking hardware">Network components</a></li> <li><a href="/wiki/Network_scheduler" title="Network scheduler">Network scheduler</a></li> <li><a href="/wiki/Network_performance" title="Network performance">Network performance evaluation</a></li> <li><a href="/wiki/Network_service" title="Network service">Network service</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software organization</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/Interpreter_(computing)" title="Interpreter (computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</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/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a href="/wiki/Integrated_development_environment" title="Integrated development environment">Integrated development environment</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration management">Software configuration management</a></li> <li><a href="/wiki/Library_(computing)" title="Library (computing)">Software library</a></li> <li><a href="/wiki/Software_repository" title="Software repository">Software repository</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software_development" title="Software development">Software development</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/Control_variable_(programming)" class="mw-redirect" title="Control variable (programming)">Control variable</a></li> <li><a href="/wiki/Software_development_process" title="Software development process">Software development process</a></li> <li><a href="/wiki/Requirements_analysis" title="Requirements analysis">Requirements analysis</a></li> <li><a href="/wiki/Software_design" title="Software design">Software design</a></li> <li><a href="/wiki/Software_construction" title="Software construction">Software construction</a></li> <li><a href="/wiki/Software_deployment" title="Software deployment">Software deployment</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">Software maintenance</a></li> <li><a href="/wiki/Programming_team" title="Programming team">Programming team</a></li> <li><a href="/wiki/Open-source_software" title="Open-source software">Open-source model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</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/Model_of_computation" title="Model of computation">Model of computation</a> <ul><li><a href="/wiki/Stochastic_computing" title="Stochastic computing">Stochastic</a></li></ul></li> <li><a href="/wiki/Formal_language" title="Formal language">Formal language</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic</a></li> <li><a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">Semantics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algorithm" title="Algorithm">Algorithms</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/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</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/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</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/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</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/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</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/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</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/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</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/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</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/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</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/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed reality</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual reality</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</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/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" title="Enterprise software">Enterprise software</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li> <li><a href="/wiki/Computational_biology" title="Computational biology">Computational biology</a></li> <li><a href="/wiki/Computational_social_science" title="Computational social science">Computational social science</a></li> <li><a href="/wiki/Computational_engineering" title="Computational engineering">Computational engineering</a></li> <li><a href="/wiki/Template:Differentiable_computing" title="Template:Differentiable computing">Differentiable computing</a></li> <li><a href="/wiki/Health_informatics" title="Health informatics">Computational healthcare</a></li> <li><a href="/wiki/Digital_art" title="Digital art">Digital art</a></li> <li><a href="/wiki/Electronic_publishing" title="Electronic publishing">Electronic publishing</a></li> <li><a href="/wiki/Cyberwarfare" title="Cyberwarfare">Cyberwarfare</a></li> <li><a href="/wiki/Electronic_voting" title="Electronic voting">Electronic voting</a></li> <li><a href="/wiki/Video_game" title="Video game">Video games</a></li> <li><a href="/wiki/Word_processor" title="Word processor">Word processing</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Educational_technology" title="Educational technology">Educational technology</a></li> <li><a href="/wiki/Document_management_system" title="Document management system">Document management</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><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:Computer_science" title="Category:Computer science">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Outline"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/10px-Global_thinking.svg.png" decoding="async" width="10" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/15px-Global_thinking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/21px-Global_thinking.svg.png 2x" data-file-width="130" data-file-height="200" /></span></span> <a href="/wiki/Outline_of_computer_science" title="Outline of computer science">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/16px-Symbol_question.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/23px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/31px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Glossaries_of_computers" title="Template:Glossaries of 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href="/wiki/Template_talk:Foundations-footer" title="Template talk:Foundations-footer"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Foundations-footer" title="Special:EditPage/Template:Foundations-footer"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_topics_in_Foundations_of_Mathematics" style="font-size:114%;margin:0 4em">Major topics in <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of Mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="background:#fdf;;width:1%"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a></li> <li><a href="/wiki/Mathematical_induction" title="Mathematical induction">Mathematical induction</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Formal system</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert system</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li></ul></li> <li><a href="/wiki/Mathematical_proof" title="Mathematical proof">Mathematical proof</a></li> <li><a href="/wiki/Model_theory" title="Model theory">Model theory</a></li> <li><a href="/wiki/Mathematical_constructivism" class="mw-redirect" title="Mathematical constructivism">Mathematical constructivism</a></li> <li><a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a></li> <li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">List of mathematical logic topics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#fdf;;width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive set theory</a></li> <li><a href="/wiki/Set_theory#Axiomatic_set_theory" title="Set theory">Axiomatic set theory</a></li> <li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive set theory</a></li> <li><a href="/wiki/Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></li> <li><a href="/wiki/Determinacy" title="Determinacy">Determinacy</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/List_of_set_theory_topics" title="List of set theory topics">List of set theory topics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#fdf;;width:1%"><a href="/wiki/Type_theory" title="Type theory">Type theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom_of_reducibility" title="Axiom of reducibility">Axiom of reducibility</a></li> <li><a href="/wiki/History_of_type_theory#Theory_of_simple_types" title="History of type theory">Simple type theory</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent type theory</a></li> <li><a href="/wiki/Intuitionistic_type_theory" title="Intuitionistic type theory">Intuitionistic type theory</a></li> <li><a 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