CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="zh" dir="ltr"> <head> <meta charset="UTF-8"> <title>範疇 (數學) - 维基百科，自由的百科全书</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )zhwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"zh", "wgMonthNames":["","1月","2月","3月","4月","5月","6月","7月","8月","9月","10月","11月","12月"],"wgRequestId":"ca51d7ab-6575-42dd-ba0f-414e3c249987","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"範疇_(數學)","wgTitle":"範疇 (數學)","wgCurRevisionId":85069390,"wgRevisionId":85069390,"wgArticleId":405159,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["自2018年1月缺少注脚的条目","自2024年11月有未列明来源语句的条目","使用ISBN魔术链接的页面","范畴论"],"wgPageViewLanguage":"zh","wgPageContentLanguage":"zh","wgPageContentModel":"wikitext","wgRelevantPageName":"範疇_(數學)","wgRelevantArticleId":405159,"wgUserVariant":"zh","wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true, "wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"zh","pageLanguageDir":"ltr","pageVariantFallbacks":["zh-hans","zh-hant","zh-cn","zh-tw","zh-hk","zh-sg","zh-mo","zh-my"]},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q719395","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.gadget.large-font":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.edit0","ext.gadget.WikiMiniAtlas","ext.gadget.UnihanTooltips","ext.gadget.Difflink","ext.gadget.pseudonamespace-UI","ext.gadget.SpecialWikitext","ext.gadget.switcher","ext.gadget.VariantAlly", "ext.gadget.AdvancedSiteNotices","ext.gadget.hideConversionTab","ext.gadget.internalLinkHelper-altcolor","ext.gadget.noteTA","ext.gadget.NavFrame","ext.gadget.collapsibleTables","ext.gadget.scrollUpButton","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=zh&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.gadget.large-font&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="範疇 (數學) - 维基百科，自由的百科全书"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//zh.m.wikipedia.org/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" type="application/x-wiki" title="编辑本页" href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (zh)"> <link rel="EditURI" type="application/rsd+xml" href="//zh.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://zh.wikipedia.org/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh" href="https://zh.wikipedia.org/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hans" href="https://zh.wikipedia.org/zh-hans/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hans-CN" href="https://zh.wikipedia.org/zh-cn/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hans-MY" href="https://zh.wikipedia.org/zh-my/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hans-SG" href="https://zh.wikipedia.org/zh-sg/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hant" href="https://zh.wikipedia.org/zh-hant/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hant-HK" href="https://zh.wikipedia.org/zh-hk/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hant-MO" href="https://zh.wikipedia.org/zh-mo/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="zh-Hant-TW" href="https://zh.wikipedia.org/zh-tw/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="alternate" hreflang="x-default" href="https://zh.wikipedia.org/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"> <link rel="alternate" type="application/atom+xml" title="Wikipedia的Atom feed" href="/w/index.php?title=Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-範疇_數學 rootpage-範疇_數學 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">跳转到内容</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="主菜单" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">主菜单</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">主菜单</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">隐藏</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 导航 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" title="访问首页[z]" accesskey="z"><span>首页</span></a></li><li id="n-indexpage" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%88%86%E7%B1%BB%E7%B4%A2%E5%BC%95" title="以分类索引搜寻中文维基百科"><span>分类索引</span></a></li><li id="n-Featured_content" class="mw-list-item"><a href="/wiki/Portal:%E7%89%B9%E8%89%B2%E5%85%A7%E5%AE%B9" title="查看中文维基百科的特色内容"><span>特色内容</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:%E6%96%B0%E8%81%9E%E5%8B%95%E6%85%8B" title="提供当前新闻事件的背景资料"><span>新闻动态</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9" title="列出维基百科中的最近修改[r]" accesskey="r"><span>最近更改</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:%E9%9A%8F%E6%9C%BA%E9%A1%B5%E9%9D%A2" title="随机载入一个页面[x]" accesskey="x"><span>随机条目</span></a></li> </ul> </div> </div> <div id="p-help" class="vector-menu mw-portlet mw-portlet-help" > <div class="vector-menu-heading"> 帮助 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:%E7%9B%AE%E5%BD%95" title="寻求帮助"><span>帮助</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%A4%BE%E7%BE%A4%E9%A6%96%E9%A1%B5" title="关于本计划、你可以做什么、应该如何做"><span>维基社群</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/Wikipedia:%E6%96%B9%E9%87%9D%E8%88%87%E6%8C%87%E5%BC%95" title="查看维基百科的方针和指引"><span>方针与指引</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:%E4%BA%92%E5%8A%A9%E5%AE%A2%E6%A0%88" title="参与维基百科社群的讨论"><span>互助客栈</span></a></li><li id="n-Information_desk" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%9F%A5%E8%AF%86%E9%97%AE%E7%AD%94" title="解答任何与维基百科无关的问题的地方"><span>知识问答</span></a></li><li id="n-conversion" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%AD%97%E8%AF%8D%E8%BD%AC%E6%8D%A2" title="提出字词转换请求"><span>字词转换</span></a></li><li id="n-IRC" class="mw-list-item"><a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93"><span>IRC即时聊天</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:%E8%81%94%E7%BB%9C%E6%88%91%E4%BB%AC" title="如何联络维基百科"><span>联络我们</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E" title="查看维基百科的简介"><span>关于维基百科</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="维基百科" src="/static/images/mobile/copyright/wikipedia-wordmark-zh.svg" style="width: 6.5625em; height: 1.375em;"> <img class="mw-logo-tagline" alt="自由的百科全书" src="/static/images/mobile/copyright/wikipedia-tagline-zh.svg" width="103" height="14" style="width: 6.4375em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:%E6%90%9C%E7%B4%A2" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="搜索维基百科[f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>搜索</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="搜索维基百科" aria-label="搜索维基百科" autocapitalize="sentences" title="搜索维基百科[f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:搜索"> </div> <button class="cdx-button cdx-search-input__end-button">搜索</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="个人工具"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="更改页面字体大小、宽度和颜色的外观" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="外观" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">外观</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans" class=""><span>资助维基百科</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29" title="我们推荐您创建账号并登录，但这不是强制性的" class=""><span>创建账号</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29" title="建议你登录，尽管并非必须。[o]" accesskey="o" class=""><span>登录</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="更多选项" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="个人工具" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-定義" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定義"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>定義</span> </div> </a> <button aria-controls="toc-定義-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>开关定義子章节</span> </button> <ul id="toc-定義-sublist" class="vector-toc-list"> <li id="toc-范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>范畴</span> </div> </a> <ul id="toc-范畴-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-小范畴和局部小范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#小范畴和局部小范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>小范畴和局部小范畴</span> </div> </a> <ul id="toc-小范畴和局部小范畴-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-范畴举例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#范畴举例"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>范畴举例</span> </div> </a> <ul id="toc-范畴举例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-态射类型" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#态射类型"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>态射类型</span> </div> </a> <ul id="toc-态射类型-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-特别的范畴" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#特别的范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>特别的范畴</span> </div> </a> <button aria-controls="toc-特别的范畴-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>开关特别的范畴子章节</span> </button> <ul id="toc-特别的范畴-sublist" class="vector-toc-list"> <li id="toc-子范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#子范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>子范畴</span> </div> </a> <ul id="toc-子范畴-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-群胚" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#群胚"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>群胚</span> </div> </a> <ul id="toc-群胚-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-对偶范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#对偶范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>对偶范畴</span> </div> </a> <ul id="toc-对偶范畴-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-积范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#积范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>积范畴</span> </div> </a> <ul id="toc-积范畴-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-逗号范畴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#逗号范畴"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>逗号范畴</span> </div> </a> <ul id="toc-逗号范畴-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-範疇類型" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#範疇類型"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>範疇類型</span> </div> </a> <ul id="toc-範疇類型-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-注释" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#注释"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>注释</span> </div> </a> <ul id="toc-注释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">範疇 (數學)</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="前往另一种语言写成的文章。34种语言可用" > <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-34" 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">34种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Kategorie_(wiskunde)" title="Kategorie (wiskunde) – 南非荷兰语" lang="af" hreflang="af" data-title="Kategorie (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%A6%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="فئة (رياضيات) – 阿拉伯语" lang="ar" hreflang="ar" data-title="فئة (رياضيات)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D1%8D%D0%B3%D0%BE%D1%80%D1%8B%D1%8F" title="Катэгорыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Катэгорыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Категория (математика) – 保加利亚语" lang="bg" hreflang="bg" data-title="Категория (математика)" data-language-autonym="Български" data-language-local-name="保加利亚语" 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/Categoria_(matem%C3%A0tiques)" title="Categoria (matemàtiques) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Categoria (matemàtiques)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kategorie_(matematika)" title="Kategorie (matematika) – 捷克语" lang="cs" hreflang="cs" data-title="Kategorie (matematika)" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kategori_(matematik)" title="Kategori (matematik) – 丹麦语" lang="da" hreflang="da" data-title="Kategori (matematik)" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Category_(mathematics)" title="Category (mathematics) – 英语" lang="en" hreflang="en" data-title="Category (mathematics)" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kategorio_(matematiko)" title="Kategorio (matematiko) – 世界语" lang="eo" hreflang="eo" data-title="Kategorio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Categor%C3%ADa_(matem%C3%A1ticas)" title="Categoría (matemáticas) – 西班牙语" lang="es" hreflang="es" data-title="Categoría (matemáticas)" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kategooria_(matemaatika)" title="Kategooria (matemaatika) – 爱沙尼亚语" lang="et" hreflang="et" data-title="Kategooria (matemaatika)" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kategoria_(matematika)" title="Kategoria (matematika) – 巴斯克语" lang="eu" hreflang="eu" data-title="Kategoria (matematika)" data-language-autonym="Euskara" data-language-local-name="巴斯克语" 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/%D8%B1%D8%B3%D8%AA%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="رسته (ریاضیات) – 波斯语" lang="fa" hreflang="fa" data-title="رسته (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kategoria_(matematiikka)" title="Kategoria (matematiikka) – 芬兰语" lang="fi" hreflang="fi" data-title="Kategoria (matematiikka)" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Categor%C3%ADa_(matem%C3%A1ticas)" title="Categoría (matemáticas) – 加利西亚语" lang="gl" hreflang="gl" data-title="Categoría (matemáticas)" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קטגוריה (מתמטיקה) – 希伯来语" lang="he" hreflang="he" data-title="קטגוריה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kateg%C3%B3ria_(matematika)" title="Kategória (matematika) – 匈牙利语" lang="hu" hreflang="hu" data-title="Kategória (matematika)" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</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%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Կատեգորիա (մաթեմատիկա) – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Կատեգորիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kategori_(matematika)" title="Kategori (matematika) – 印度尼西亚语" lang="id" hreflang="id" data-title="Kategori (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9C%8F_(%E6%95%B0%E5%AD%A6)" title="圏 (数学) – 日语" lang="ja" hreflang="ja" data-title="圏 (数学)" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</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_(%EC%88%98%ED%95%99)" title="범주 (수학) – 韩语" lang="ko" hreflang="ko" data-title="범주 (수학)" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Categorie_(wiskunde)" title="Categorie (wiskunde) – 荷兰语" lang="nl" hreflang="nl" data-title="Categorie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kategori_i_matematikk" title="Kategori i matematikk – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Kategori i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kategori_(matematikk)" title="Kategori (matematikk) – 书面挪威语" lang="nb" hreflang="nb" data-title="Kategori (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kategoria_(matematyka)" title="Kategoria (matematyka) – 波兰语" lang="pl" hreflang="pl" data-title="Kategoria (matematyka)" data-language-autonym="Polski" data-language-local-name="波兰语" 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/Categoria_(teoria_das_categorias)" title="Categoria (teoria das categorias) – 葡萄牙语" lang="pt" hreflang="pt" data-title="Categoria (teoria das categorias)" data-language-autonym="Português" data-language-local-name="葡萄牙语" 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/Categorie_(matematic%C4%83)" title="Categorie (matematică) – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Categorie (matematică)" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Категория (математика) – 俄语" lang="ru" hreflang="ru" data-title="Категория (математика)" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kategorija_(matematika)" title="Kategorija (matematika) – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Kategorija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" 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/Kategorija_(matematika)" title="Kategorija (matematika) – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Kategorija (matematika)" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kategori_(matematik)" title="Kategori (matematik) – 瑞典语" lang="sv" hreflang="sv" data-title="Kategori (matematik)" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kategori_(matematik)" title="Kategori (matematik) – 土耳其语" lang="tr" hreflang="tr" data-title="Kategori (matematik)" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Категорія (математика) – 乌克兰语" lang="uk" hreflang="uk" data-title="Категорія (математика)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" title="範疇 (數學) – 粤语" lang="yue" hreflang="yue" data-title="範疇 (數學)" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q719395#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="工具" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">工具</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&oldid=85069390" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E7%AF%84%E7%96%87_%28%E6%95%B8%E5%AD%B8%29&id=85069390&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25AF%2584%25E7%2596%2587_%28%25E6%2595%25B8%25E5%25AD%25B8%29"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%25AF%2584%25E7%2596%2587_%28%25E6%2595%25B8%25E5%25AD%25B8%29"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E7%AF%84%E7%96%87_%28%E6%95%B8%E5%AD%B8%29&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> 在其他项目中 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q719395" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r83732972">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-night .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-night .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Text_document_with_red_question_mark.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目已<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E4%BE%86%E6%BA%90" class="mw-redirect" title="Wikipedia:列明來源">列出參考文獻</a>，但<b>因為沒有<a href="/wiki/Help:%E8%84%9A%E6%B3%A8" title="Help:脚注">文內引註</a>而使來源仍然不明</b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2018年1月29日</span>)</i></small><span class="hide-when-compact"><br /><small>请加上合适的文內引註来<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit">改善这篇条目</a>。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <p>在<a href="/wiki/%E8%8C%83%E7%95%B4%E8%AE%BA" title="范畴论">范畴论</a>中，<b>范畴</b>这一概念代表一些数学对象及这些对象间的一些关系，以及这些关系之间的关系。利用范畴可以公式化抽象结构并保留结构上的关系，如运算。范畴几乎可以出现于现代数学的任意分支，同时也统合了这些分支的底层理念。对范畴本身的研究就称作<a href="/wiki/%E8%8C%83%E7%95%B4%E8%AE%BA" title="范畴论">范畴论</a>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定義"><span id=".E5.AE.9A.E7.BE.A9"></span>定義</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=1" title="编辑章节：定義"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="范畴"><span id=".E8.8C.83.E7.95.B4"></span>范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=2" title="编辑章节：范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个<b>范畴</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 {\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> 意指资料 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathrm {Ob\ } {\mathcal {C}},\mathrm {Mor\ } {\mathcal {C}};\circ )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>;</mo> <mo>∘</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {Ob\ } {\mathcal {C}},\mathrm {Mor\ } {\mathcal {C}};\circ )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ef6c91aa14d8348d7d9efcf6323be1994afeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.985ex; height:2.843ex;" alt="{\displaystyle (\mathrm {Ob\ } {\mathcal {C}},\mathrm {Mor\ } {\mathcal {C}};\circ )}"></span>，其中： </p> <ul><li>一個由<b>对象</b>（<b>Ob</b>ject）所構成的<a href="/wiki/%E9%A1%9E_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="類 (數學)">類</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 \mathrm {Ob\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> <mtext> </mtext> </mrow> <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 \mathrm {Ob\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb5ea2a1c58abb3edff515a25016b00fe89d4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.92ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ob\ } {\mathcal {C}}}"></span>；</li> <li>物件間的<b><a href="/wiki/%E6%80%81%E5%B0%84" title="态射">态射</a></b>（<b>Mor</b>phism）所構成的類 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <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 \mathrm {Mor\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6980f2e294d88a954a10fe14c3150af9c6830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"></span>。每一個態射 <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\in {\mathrm {Mor\ } }{\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> </mrow> <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 f\in {\mathrm {Mor\ } }{\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e10ce12ab8b462c67368ef8255d724ed44e1abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.144ex; height:2.509ex;" alt="{\displaystyle f\in {\mathrm {Mor\ } }{\mathcal {C}}}"></span> 均蕴含确定的「始对象（<b>Dom</b>ain）」<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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 和「终对象（<b>Cod</b>omain）」<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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>，且 <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,B\in \mathrm {Ob\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> <mtext> </mtext> </mrow> <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 A,B\in \mathrm {Ob\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739a7290763f65808d3f3ca820f289372fdc759f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.302ex; height:2.509ex;" alt="{\displaystyle A,B\in \mathrm {Ob\ } {\mathcal {C}}}"></span></b>。此时记 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to 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\colon A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dec1893560fabff9fa9c17b83b71f7f97996119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.434ex; height:2.509ex;" alt="{\displaystyle f\colon A\to B}"></span>，称 <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> 为从 <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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> 到 <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 的<b>一个</b>态射<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>注释 1<span class="cite-bracket">]</span></a></sup>。所有由 <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 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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></i> 至 <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/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 的态射构成类，记作 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895cbf53566e59cb8b510e61b2e4cb29619b4ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.881ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}"></span>，不致混淆时，也记作 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} \ (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} \ (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022d458db76e0e68c67cf5880f0d8778e63ea801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.772ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} \ (A,B)}"></span>；</li> <li>对任意态射对 <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,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce67314185650d6f0deba39db7dcec9378f4d4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.843ex;" alt="{\displaystyle (A,B)}"></span> 有态射复合 <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> 如下：</li></ul> <p><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 {\begin{aligned}\circ (-,-)\colon \ &\mathrm {Hom} \ (A,B)\times \mathrm {Hom} \ (B,C)&\to &\quad \mathrm {Hom} \ (A,C),\\&(f,g)&\mapsto &\quad f\circ g,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∘</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> <mo>:</mo> <mtext> </mtext> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">→</mo> </mtd> <mtd> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mspace width="1em" /> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\circ (-,-)\colon \ &\mathrm {Hom} \ (A,B)\times \mathrm {Hom} \ (B,C)&\to &\quad \mathrm {Hom} \ (A,C),\\&(f,g)&\mapsto &\quad f\circ g,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3e967788ce3d9b0a9e28039f03d4814a1f5a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.11ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\circ (-,-)\colon \ &\mathrm {Hom} \ (A,B)\times \mathrm {Hom} \ (B,C)&\to &\quad \mathrm {Hom} \ (A,C),\\&(f,g)&\mapsto &\quad f\circ g,\end{aligned}}}"></span> </p><p>其中，<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\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" 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 f\circ g}"></span> 在不致混淆时也记作 <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 fg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" 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 fg}"></span>。 </p><p>此態射複合滿足下列公理： </p> <ul><li>（結合律）对态射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\to 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\colon A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dec1893560fabff9fa9c17b83b71f7f97996119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.434ex; height:2.509ex;" alt="{\displaystyle f\colon A\to 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\colon B\to 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\colon B\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02572d111e174b9ad0bf94d95d47c8e4f7d9649d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.294ex; height:2.509ex;" alt="{\displaystyle g\colon B\to C}"></span> 和 <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\colon C\to 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\colon C\to D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99568af8e0c5cfae564eaa33ac2935749b1c456e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.678ex; height:2.176ex;" alt="{\displaystyle h\colon C\to D}"></span>，有 <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(gh)=(fg)h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(gh)=(fg)h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b798dac2934bd8db6e35ee4cd7331949d6ef50d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.184ex; height:2.843ex;" alt="{\displaystyle f(gh)=(fg)h}"></span>；</li> <li>（幺元）对任意对象 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>，存在一态射 <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}\in \mathrm {Hom} \ (X,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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{X}\in \mathrm {Hom} \ (X,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243d95160366ba0feed70a3a185d0f63ada93118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.861ex; height:2.843ex;" alt="{\displaystyle 1_{X}\in \mathrm {Hom} \ (X,X)}"></span>，使得对任意态射 <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\in \mathrm {Hom} \ (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathrm {Hom} \ (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0053051b3c199ec308f22413ae10523504b65b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.892ex; height:2.843ex;" alt="{\displaystyle f\in \mathrm {Hom} \ (A,B)}"></span>，均满足 <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}f=f=f1_{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> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{B}f=f=f1_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c5f9817a4c0e61d46b81daf7cce1ce9c78495d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.302ex; height:2.509ex;" alt="{\displaystyle 1_{B}f=f=f1_{A}}"></span>。态射 <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649d43631e05d324d6f53dedb5d044f1b80f3e86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.795ex; height:2.509ex;" alt="{\displaystyle 1_{X}}"></span> 称作「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 的单位态射」。</li></ul> <p>根据上述公理可以证明，对每个特定对象而言，单位态射具唯一性。<style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><mark class="template-facttext" title="需要提供文献来源">在这样的等价关系上，部分作者视对象与其单位态射为同一概念。</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求开始于2024年11月22日。">[來源請求]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Functors_Between_Ob_C_and_Mor_C,_Dom,_Cod,_Id.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg/220px-Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg.png" decoding="async" width="220" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg/330px-Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg/440px-Functors_Between_Ob_C_and_Mor_C%2C_Dom%2C_Cod%2C_Id.svg.png 2x" data-file-width="144" data-file-height="70" /></a><figcaption>图 1：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <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 \mathrm {Mor\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6980f2e294d88a954a10fe14c3150af9c6830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ob\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> <mtext> </mtext> </mrow> <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 \mathrm {Ob\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb5ea2a1c58abb3edff515a25016b00fe89d4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.92ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ob\ } {\mathcal {C}}}"></span> 间的映射</figcaption></figure> <p>显然，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <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 \mathrm {Mor\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6980f2e294d88a954a10fe14c3150af9c6830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ob\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> <mtext> </mtext> </mrow> <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 \mathrm {Ob\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb5ea2a1c58abb3edff515a25016b00fe89d4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.92ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ob\ } {\mathcal {C}}}"></span> 间自然地存在三个映射：<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Id} \colon \ X\mapsto 1_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>:</mo> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">↦</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Id} \colon \ X\mapsto 1_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3606481fcdcd118db96ce87859de1d8e0e7a0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.136ex; height:2.509ex;" alt="{\displaystyle \mathrm {Id} \colon \ X\mapsto 1_{X}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Dom} \colon \ f\mapsto A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo>:</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Dom} \colon \ f\mapsto A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b34d3afd13fdefb7e6a416c77d7c332c1e6c500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.124ex; height:2.509ex;" alt="{\displaystyle \mathrm {Dom} \colon \ f\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 \mathrm {Cod} \colon \ f\mapsto B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>:</mo> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Cod} \colon \ f\mapsto B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d0dc6aadef631f03955e7a731b7aaf1572d1ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.404ex; height:2.509ex;" alt="{\displaystyle \mathrm {Cod} \colon \ f\mapsto B}"></span>，如图 1 所示。 </p> <div class="mw-heading mw-heading3"><h3 id="小范畴和局部小范畴"><span id=".E5.B0.8F.E8.8C.83.E7.95.B4.E5.92.8C.E5.B1.80.E9.83.A8.E5.B0.8F.E8.8C.83.E7.95.B4"></span>小范畴和局部小范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=3" title="编辑章节：小范畴和局部小范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个范畴 <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> 称作<b>小范畴</b>（Small Category），当且仅当其态射类 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> </mrow> <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 \mathrm {Mor\ } {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6980f2e294d88a954a10fe14c3150af9c6830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mor\ } {\mathcal {C}}}"></span> 比<a href="/wiki/%E7%9C%9F%E7%B1%BB" class="mw-redirect" title="真类">真类</a>小，即仅有集合那么大。 </p><p>一个范畴 <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> 称作<b>局部小范畴</b>（Locally Small Category），当且仅当对任意对象对 <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,B)\in (\mathrm {Ob} \ {\mathcal {C}})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)\in (\mathrm {Ob} \ {\mathcal {C}})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380f385e908db0f20071024e6b1fab661bcbaacd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.975ex; height:3.176ex;" alt="{\displaystyle (A,B)\in (\mathrm {Ob} \ {\mathcal {C}})^{2}}"></span>，其对应的的态射类 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895cbf53566e59cb8b510e61b2e4cb29619b4ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.881ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} _{\mathcal {C}}\ (A,B)}"></span> 均为非真类的集合。 </p><p>数学研究中，许多重要的范畴（例如集合的范畴），通常即使非小，也是局部小的。 </p> <div class="mw-heading mw-heading2"><h2 id="范畴举例"><span id=".E8.8C.83.E7.95.B4.E4.B8.BE.E4.BE.8B"></span>范畴举例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=4" title="编辑章节：范畴举例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>每一範疇都可由其物件、態射和態射複合來表示。 </p> <ul><li>所有<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>的<a href="/wiki/%E9%9B%86%E5%90%88%E8%8C%83%E7%95%B4" title="集合范畴">范畴</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 {\mathsf {Set}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Set}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31488c9fed260a93d36653220c2bd75b771cc440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.165ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Set}}}"></span>，其態射為集合間的<a href="/wiki/%E5%87%BD%E6%95%B8" class="mw-redirect" title="函數">函數</a>，而態射複合則為一般的函數複合。<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>注释 2<span class="cite-bracket">]</span></a></sup> <ul><li>所有<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%97%9C%E4%BF%82" class="mw-redirect" title="預序關係">預序關係</a>的<a href="/wiki/%E9%A2%84%E5%BA%8F%E8%8C%83%E7%95%B4" title="预序范畴">范畴</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 {\mathsf {Ord}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">O</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Ord}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ed0d5ba050bed964f1143f3960dcd7fa0c6d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.708ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Ord}}}"></span>，其態射為<a href="/wiki/%E5%8D%95%E8%B0%83%E5%87%BD%E6%95%B0" title="单调函数">單調函數</a>。</li> <li>所有<a href="/wiki/%E5%8E%9F%E7%BE%A4" title="原群">原群</a>的<a href="/w/index.php?title=%E5%8E%9F%E7%BE%A4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="原群范畴（页面不存在）">范畴</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 {\mathsf {Mag}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Mag}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badb329e34b0ce1e021f332f47b0abb6d820fb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.314ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Mag}}}"></span>，其態射為原群間的<a href="/wiki/%E5%90%8C%E6%80%81" title="同态">同態</a>。</li> <li>所有<a href="/wiki/%E7%BE%A4" title="群">群</a>的<a href="/wiki/%E7%BE%A4%E7%AF%84%E7%96%87" title="群範疇">范畴</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 {\mathsf {Group}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">G</mi> <mi mathvariant="sans-serif">r</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">u</mi> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Group}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba0d21de83f93ca9182485cfd5d36480328a424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.912ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Group}}}"></span>，其態射為<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%85%8B" title="群同態">群同態</a>。</li> <li>所有<a href="/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%BE%A4" class="mw-redirect" title="阿貝爾群">阿貝爾群</a>的<a href="/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%AF%84%E7%96%87" title="阿貝爾範疇">範疇</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 {\mathsf {Ab}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Ab}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15eb8608ba950f032e1e3e94f08cb75d30818295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.752ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Ab}}}"></span>，其態射為<a href="/wiki/%E7%BE%A4%E5%90%8C%E6%85%8B" title="群同態">群同態</a>。</li> <li>所有<a href="/wiki/%E7%8E%AF_(%E4%BB%A3%E6%95%B0)" title="环 (代数)">環</a>的<a href="/w/index.php?title=%E7%8E%AF%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="环范畴（页面不存在）">范畴</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 {\mathsf {Ring}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Ring}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78c9ceb36afa0269f4088e97466109f2de8c27a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.422ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Ring}}}"></span>，其態射為<a href="/wiki/%E7%8E%AF%E5%90%8C%E6%80%81" title="环同态">環同態</a>。<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>注释 3<span class="cite-bracket">]</span></a></sup></li> <li>所有於<a href="/wiki/%E5%9F%9F_(%E6%95%B8%E5%AD%B8)" class="mw-redirect" title="域 (數學)">體</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 \mathbb {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ff8b374f407dec6b1b2f35866c8e8cd0fb150d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {k} }"></span>（維持固定）上的<a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93" class="mw-redirect" title="向量空間">向量空間</a>的<a href="/w/index.php?title=%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="线性空间范畴（页面不存在）">范畴</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 {\mathsf {Vect}}_{\mathbb {k} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">V</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">k</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Vect}}_{\mathbb {k} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d68ef91d9f3d332b33bda033ebf3ffc81f272aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.601ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Vect}}_{\mathbb {k} }}"></span>，其態射為<a href="/wiki/%E7%B7%9A%E6%80%A7%E6%98%A0%E5%B0%84" class="mw-redirect" title="線性映射">線性映射</a>。</li> <li>所有<a href="/wiki/%E6%8B%93%E6%A8%B8%E7%A9%BA%E9%96%93" class="mw-redirect" title="拓樸空間">拓樸空間</a>的<a href="/w/index.php?title=%E6%8B%93%E6%89%91%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="拓扑范畴（页面不存在）">范畴</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 {\mathsf {Top}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Top}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66a6c1bbef1cb427eff3d718be6e2109b1ccdff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.947ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Top}}}"></span>，其態射為<a href="/wiki/%E9%80%A3%E7%BA%8C%E5%87%BD%E6%95%B8_(%E6%8B%93%E6%92%B2%E5%AD%B8)" title="連續函數 (拓撲學)">連續函數</a>。</li> <li>所有<a href="/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间">度量空間</a>的<a href="/w/index.php?title=%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="度量空间范畴（页面不存在）">范畴</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 {\mathsf {Met}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Met}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f287d84a4d5d18b2cab891d6d2fd7d527d2fc6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.905ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Met}}}"></span>，其態射為<a href="/w/index.php?title=%E5%BA%A6%E9%87%8F%E6%98%A0%E5%B0%84&action=edit&redlink=1" class="new" title="度量映射（页面不存在）">度量映射</a>。</li> <li>所有<a href="/wiki/%E4%B8%80%E8%87%B4%E7%A9%BA%E9%96%93" class="mw-redirect" title="一致空間">一致空間</a>的<a href="/w/index.php?title=%E4%B8%80%E8%87%B4%E7%A9%BA%E9%97%B4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="一致空间范畴（页面不存在）">范畴</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 {\mathsf {Uni}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">U</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">i</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Uni}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90225cfad49f654e000644d3ecdec6b7e7426365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.357ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Uni}}}"></span>，其態射為<a href="/wiki/%E4%B8%80%E8%87%B4%E9%80%A3%E7%BA%8C" class="mw-redirect" title="一致連續">一致連續函數</a>。</li> <li>所有光滑<a href="/wiki/%E6%B5%81%E5%BD%A2" title="流形">流形</a>的<a href="/w/index.php?title=%E6%B5%81%E5%BD%A2%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="流形范畴（页面不存在）">范畴</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 {\mathsf {Man}}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">n</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Man}}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15a09e80943f958dfa9b1154e72976b661e3dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.413ex; height:2.343ex;" alt="{\displaystyle {\mathsf {Man}}^{p}}"></span>，其態射為 <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> 次連續可微映射。</li></ul></li> <li>所有小範疇的<a href="/w/index.php?title=%E5%B0%8F%E8%8C%83%E7%95%B4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="小范畴范畴（页面不存在）">范畴</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 {\mathsf {Cat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Cat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94f8d5051c63c3b8c6cc75b7e55e72e8e0da9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.443ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Cat}}}"></span>，其態射為<a href="/wiki/%E5%87%BD%E5%AD%90" title="函子">函子</a>。</li> <li>所有局部小范畴的<a href="/w/index.php?title=%E5%B1%80%E9%83%A8%E5%B0%8F%E8%8C%83%E7%95%B4%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="局部小范畴范畴（页面不存在）">范畴</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 {\mathsf {CAT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CAT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6c05b5127f935dd5e535c7cb3dd8c05c0efeba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.618ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CAT}}}"></span>。<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>注释 4<span class="cite-bracket">]</span></a></sup></li> <li>所有<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>的<a href="/wiki/%E9%97%9C%E4%BF%82%E7%AF%84%E7%96%87" title="關係範疇">关系范畴</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 {\mathsf {Rel}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">l</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Rel}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55f34eddfa4853c62c34afaeff4b6223802a6cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.09ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Rel}}}"></span>，其態射為<a href="/wiki/%E5%85%B3%E7%B3%BB_(%E6%95%B0%E5%AD%A6)" title="关系 (数学)">關係</a>。</li> <li>任一<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%97%9C%E4%BF%82" class="mw-redirect" title="預序關係">預序集</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 (P,\preceq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>⪯</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P,\preceq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b078b4e5b6256bc6842c56e8afec3ce3e943d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.397ex; height:2.843ex;" alt="{\displaystyle (P,\preceq )}"></span> 均蕴含一個小範疇，其对象為 <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> 的元，态射为有序对 <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 (p,q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9769c58523b9b639866a2d48e657d9c26911143a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}"></span> 使得 <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 p\preceq q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⪯</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\preceq q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e363c1fe9b0d2edad68eafcdeebd93d8c60c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.343ex;" alt="{\displaystyle p\preceq q}"></span>。<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>注释 5<span class="cite-bracket">]</span></a></sup></li> <li>任一<a href="/wiki/%E4%B9%88%E5%8D%8A%E7%BE%A4" class="mw-redirect" title="么半群">幺半群</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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> 均蕴含一个携唯一一个对象 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 的小范畴 <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 {\mathsf {B}}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {B}}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0202e00ed56ee78be33b33f5589c95cdbd2998c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.993ex; height:2.176ex;" alt="{\displaystyle {\mathsf {B}}M}"></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 {\mathsf {B}}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {B}}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0202e00ed56ee78be33b33f5589c95cdbd2998c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.993ex; height:2.176ex;" alt="{\displaystyle {\mathsf {B}}M}"></span> 以 <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> 中的元作为态射，每个态射各自表示 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 上一个不同的自同态，而态射复合由 <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> 的乘法给出。<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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> 的幺元 <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 e\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef354bbee2ecdd82295a98cd2ec23ffbef54abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.366ex; height:2.176ex;" alt="{\displaystyle e\in M}"></span> 也作为 <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 {\mathsf {B}}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {B}}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0202e00ed56ee78be33b33f5589c95cdbd2998c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.993ex; height:2.176ex;" alt="{\displaystyle {\mathsf {B}}M}"></span> 这唯一一个对象的单位态射存在。可以将范畴这一概念视作幺半群之延伸概念。</li> <li>任意<a href="/wiki/%E6%9C%89%E5%90%91%E5%9B%BE" class="mw-redirect" title="有向图">有向图</a>蕴含一个自然的小范畴，以图的<a href="/wiki/%E9%A1%B6%E7%82%B9_(%E5%9B%BE%E8%AE%BA)" title="顶点 (图论)">顶点</a>为对象，有向路径为态射，路径串联为态射复合。这被称作由有向图产生的「自由范畴」。</li> <li>若<i>I</i>是一個<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>，「在<i>I</i>上的<a href="/wiki/%E5%85%B7%E9%AB%94%E7%AF%84%E7%96%87" title="具體範疇">具體範疇</a>」會是個小範疇，其物件為<i>I</i>的元素，而態射則只有單位態射。當然，其態射複合的公理是必然滿足的。</li></ul> <div class="mw-heading mw-heading2"><h2 id="态射类型"><span id=".E6.80.81.E5.B0.84.E7.B1.BB.E5.9E.8B"></span>态射类型</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=5" title="编辑章节：态射类型"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>一个<a href="/wiki/%E6%80%81%E5%B0%84" title="态射">态射</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \ a\to b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mtext> </mtext> <mi>a</mi> <mo stretchy="false">→</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \ a\to b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d603aa942d1bf73c08f8812787ebb2ef32bfb8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.735ex; height:2.509ex;" alt="{\displaystyle f\colon \ a\to b}"></span> 被称为： </p> <ul><li>同构（<b>Iso</b>morphism），当且仅当存在态射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \ b\to c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mtext> </mtext> <mi>b</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon \ b\to c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3998f7db2b97cb96b99a57a6d8e37eb2b2cfdd49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.349ex; height:2.509ex;" alt="{\displaystyle g\colon \ b\to c}"></span>，满足 <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=1_{a},\,fg=1_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>f</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mi>g</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gf=1_{a},\,fg=1_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad9b678ca7cab82bc4941e727581bddb7447f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.772ex; height:2.509ex;" alt="{\displaystyle gf=1_{a},\,fg=1_{b}}"></span>，换言之，存在逆；</li> <li>自态射（<b>End</b>omorphism），当且仅当 <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=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2e4888d97a754d4bfa4da297b226788a73c6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b=a}"></span>，即 <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> 是从 <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> 到 <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> 自身的态射；</li> <li>自同构（<b>Aut</b>omorphism），当且仅当 <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> 同时为同构与自态射；</li> <li><a href="/wiki/%E5%96%AE%E6%85%8B%E5%B0%84" title="單態射">单态射</a>（<b>Mono</b>morphism），当且仅当对任意态射 <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,k\in \mathrm {Hom} \ (x,\,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h,k\in \mathrm {Hom} \ (x,\,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08db4ca4f8ce7caa262ba31889fe9d5db9c8597e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.637ex; height:2.843ex;" alt="{\displaystyle h,k\in \mathrm {Hom} \ (x,\,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 fh=fk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>h</mi> <mo>=</mo> <mi>f</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fh=fk}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451670e71b1b6dd8cbed43922ad8612c75ba7b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.206ex; height:2.509ex;" alt="{\displaystyle fh=fk}"></span> 均蕴含 <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=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3dd95ec422d9fe5b53c7858b94a0314e7f0d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.649ex; height:2.176ex;" alt="{\displaystyle h=k}"></span>；</li> <li><a href="/w/index.php?title=%E6%BB%A1%E6%80%81%E5%B0%84&action=edit&redlink=1" class="new" title="满态射（页面不存在）">满态射</a>（<b>Epi</b>morphism），当且仅当对任意态射 <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,k\in \mathrm {Hom} \ (b,\,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h,k\in \mathrm {Hom} \ (b,\,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc5c457e3dadd299ff35184ff8630451d948635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.404ex; height:2.843ex;" alt="{\displaystyle h,k\in \mathrm {Hom} \ (b,\,x)}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle hf=kf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi>f</mi> <mo>=</mo> <mi>k</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle hf=kf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dd3f40ecdb548d514974087a2d6c0da7016342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.206ex; height:2.509ex;" alt="{\displaystyle hf=kf}"></span> 均蕴含 <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=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3dd95ec422d9fe5b53c7858b94a0314e7f0d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.649ex; height:2.176ex;" alt="{\displaystyle h=k}"></span>；</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \ b\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mtext> </mtext> <mi>b</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon \ b\to a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067cbb1f7ca09e344ac8d3c0363383b3e117752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.572ex; height:2.509ex;" alt="{\displaystyle g\colon \ b\to a}"></span> 的截面（Section），当且仅当 <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=1_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <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 gf=1_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7c0aa414aeef6c1382085cb9a18896c4a54f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.757ex; height:2.509ex;" alt="{\displaystyle gf=1_{a}}"></span>，也称作 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> 的右逆（Right Reverse）或分裂单态射（Split Monomorphism）；</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \ b\to a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mtext> </mtext> <mi>b</mi> <mo stretchy="false">→</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon \ b\to a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067cbb1f7ca09e344ac8d3c0363383b3e117752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.572ex; height:2.509ex;" alt="{\displaystyle g\colon \ b\to a}"></span> 的收缩（Retraction），当且仅当 <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 fg=1_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg=1_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6224e6e22e26369cff76f7e906579ed0056a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.593ex; height:2.509ex;" alt="{\displaystyle fg=1_{b}}"></span>，也称作 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> 的左逆（Left Reverse）或分裂满态射（Split Epimorphism）；</li></ul> <p>也记 <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> 上的所有自态射构成类 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} \ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {End} \ a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb916aa20203ca7aa810ec684f95d64e55b5558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle \mathrm {End} \ a}"></span>，所有自同构构成类 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Aut} \ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> </mrow> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Aut} \ a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf203476a213ccc6584e711168f9bf462f56d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle \mathrm {Aut} \ a}"></span>。 </p><p>下述三个命题是等价的： </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> 是单态射且是收缩。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> 是满态射且是截面。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> 是同构。</li></ol> <p>态射之间的关系（例如 <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 fg=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd89b1e71cb771faf608a8370a32cfcc9f26cd7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.832ex; height:2.509ex;" alt="{\displaystyle fg=h}"></span>）可以非常方便地表示为<a href="/wiki/%E4%BA%A4%E6%8D%A2%E5%9B%BE%E8%A1%A8" title="交换图表">交换图表</a>，其中物件表示为点，态射表示为箭头。 </p> <div class="mw-heading mw-heading2"><h2 id="特别的范畴"><span id=".E7.89.B9.E5.88.AB.E7.9A.84.E8.8C.83.E7.95.B4"></span>特别的范畴</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=6" title="编辑章节：特别的范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="子范畴"><span id=".E5.AD.90.E8.8C.83.E7.95.B4"></span>子范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=7" title="编辑章节：子范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>给定一个范畴 <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>，称范畴 <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 {D}}}"> <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">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"></span> 为 <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> 之子范畴（Subcategory），当且仅当： </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ob} \ {\mathcal {D}}\subseteq \mathrm {Ob} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <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 \mathrm {Ob} \ {\mathcal {D}}\subseteq \mathrm {Ob} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a019b8f7465af472a2257d6e643d32292dab225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.492ex; height:2.343ex;" alt="{\displaystyle \mathrm {Ob} \ {\mathcal {D}}\subseteq \mathrm {Ob} \ {\mathcal {C}}}"></span>，</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor} \ {\mathcal {D}}\subseteq \mathrm {Mor} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <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 \mathrm {Mor} \ {\mathcal {D}}\subseteq \mathrm {Mor} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb8ea61e0147cdece8308a4c659010b05fee976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.701ex; height:2.343ex;" alt="{\displaystyle \mathrm {Mor} \ {\mathcal {D}}\subseteq \mathrm {Mor} \ {\mathcal {C}}}"></span>，</li> <li>同时，态射复合仍然保持。</li></ul> <div class="mw-heading mw-heading3"><h3 id="群胚"><span id=".E7.BE.A4.E8.83.9A"></span>群胚</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=8" title="编辑章节：群胚"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>称 <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> 为一群胚（Groupoid），当且仅当其中所有态射为同构。 </p> <ul><li>群可被定义作具唯一一个对象的群胚；</li></ul> <p>任意范畴 <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> 均内含一个最大群胚（Maximal Groupoid），为包含全部 <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> 的对象，而包含且仅包含全部自态射作为态射的子范畴。 </p> <div class="mw-heading mw-heading3"><h3 id="对偶范畴"><span id=".E5.AF.B9.E5.81.B6.E8.8C.83.E7.95.B4"></span>对偶范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=9" title="编辑章节：对偶范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>令 <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> 为一范畴，规定其<a href="/w/index.php?title=%E5%AF%B9%E5%81%B6%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="对偶范畴（页面不存在）">对偶范畴</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 {\mathcal {C}}^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2db750fcb75108f0e27b36b05caaaeece86f40f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.212ex; height:2.343ex;" alt="{\displaystyle {\mathcal {C}}^{\mathrm {op} }}"></span> 如下： </p> <ul><li>以 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ob} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <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 \mathrm {Ob} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5cece6411abdf0a0e3e5263bc58c1c5ad62d337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.92ex; height:2.176ex;" alt="{\displaystyle \mathrm {Ob} \ {\mathcal {C}}}"></span> 为 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ob} \ {\mathcal {C}}^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <msup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ob} \ {\mathcal {C}}^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/085d73246773a7bf46eb37e3ae4f5cb414475a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.893ex; height:2.343ex;" alt="{\displaystyle \mathrm {Ob} \ {\mathcal {C}}^{\mathrm {op} }}"></span>；</li> <li>由如下从 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <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 \mathrm {Mor} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e74b8c15137bc30a40bcb6410c4c181e32d56e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mor} \ {\mathcal {C}}}"></span> 到 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <msup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77be97155c15eea33489552bd5a5782fd10f36ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.997ex; height:2.343ex;" alt="{\displaystyle \mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }}"></span> 的一一对应<a href="/wiki/%E5%87%BD%E5%AD%90" title="函子">函子</a>完全生成后者：</li></ul> <p><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 {\begin{aligned}\mathrm {Mor} \ {\mathcal {C}}\qquad &\to &\mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }\\f\colon \ X\to Y\quad &\mapsto &f^{\mathrm {op} }\colon \ Y\to X\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mspace width="2em" /> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <msup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo>:</mo> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> <mspace width="1em" /> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> </mtd> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>:</mo> <mtext> </mtext> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Mor} \ {\mathcal {C}}\qquad &\to &\mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }\\f\colon \ X\to Y\quad &\mapsto &f^{\mathrm {op} }\colon \ Y\to X\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7292f5c514c40b318f1215a65dee9d99d890438f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.219ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Mor} \ {\mathcal {C}}\qquad &\to &\mathrm {Mor} \ {\mathcal {C}}^{\mathrm {op} }\\f\colon \ X\to Y\quad &\mapsto &f^{\mathrm {op} }\colon \ Y\to X\end{aligned}}}"></span> </p><p>其中满足：<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 \forall f,g\in \mathrm {Mor} \ {\mathcal {C}},(f\circ _{\mathcal {C}}g)^{\mathrm {op} }:=g^{\mathrm {op} }\circ _{{\mathcal {C}}^{\mathrm {op} }}f^{\mathrm {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <mi>f</mi> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <mo>:=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall f,g\in \mathrm {Mor} \ {\mathcal {C}},(f\circ _{\mathcal {C}}g)^{\mathrm {op} }:=g^{\mathrm {op} }\circ _{{\mathcal {C}}^{\mathrm {op} }}f^{\mathrm {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aec39adb5888d5acb459250210fbc380f457b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.096ex; height:2.843ex;" alt="{\displaystyle \forall f,g\in \mathrm {Mor} \ {\mathcal {C}},(f\circ _{\mathcal {C}}g)^{\mathrm {op} }:=g^{\mathrm {op} }\circ _{{\mathcal {C}}^{\mathrm {op} }}f^{\mathrm {op} }}"></span>。 </p><p>利用对偶范畴可证明如下的<a href="/w/index.php?title=%E5%AF%B9%E5%81%B6%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="对偶定理（页面不存在）">对偶定理</a>： </p><p>定理：下列三条定理等价： </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \ x\to y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">→</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \ x\to y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0e4bc66ae1bd1e7e3d0b87e79f239d588f33b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.992ex; height:2.509ex;" alt="{\displaystyle f\colon \ x\to y}"></span> 为范畴 <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> 中的一个同构（双态射）；</li> <li>对所有对象 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <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 c\in \mathrm {Ob} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbaeb68fd78fa58db3a6ae96d6de575c13aeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.768ex; height:2.176ex;" alt="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> 上的后复合定义了双射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}\colon \ \mathrm {Hom} (c,x)\to \mathrm {Hom} (c,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}\colon \ \mathrm {Hom} (c,x)\to \mathrm {Hom} (c,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b13a0f5562968c93c902047a91ed436f373387c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.29ex; height:2.843ex;" alt="{\displaystyle f_{*}\colon \ \mathrm {Hom} (c,x)\to \mathrm {Hom} (c,y)}"></span>；</li> <li>对所有对象 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <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 c\in \mathrm {Ob} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbaeb68fd78fa58db3a6ae96d6de575c13aeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.768ex; height:2.176ex;" alt="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> 上的前复合定义了双射 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\colon \ \mathrm {Hom} (y,c)\to \mathrm {Hom} (x,c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}\colon \ \mathrm {Hom} (y,c)\to \mathrm {Hom} (x,c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5a013666c37504d3675d360fcd0a4c1885f2a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.472ex; height:2.843ex;" alt="{\displaystyle f^{*}\colon \ \mathrm {Hom} (y,c)\to \mathrm {Hom} (x,c)}"></span>；</li></ol> <div class="mw-heading mw-heading3"><h3 id="积范畴"><span id=".E7.A7.AF.E8.8C.83.E7.95.B4"></span>积范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=10" title="编辑章节：积范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对任意范畴 <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> 和 <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 {D}}}"> <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">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"></span>，定义其积范畴 <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}}\times {\mathcal {D}}}"> <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>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\times {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236254c30e1c33836cc907e03ae5aa5ee1bb327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.871ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\times {\mathcal {D}}}"></span> 如下： </p> <ul><li>以形如 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c,\,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c,\,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0958df10671d204dd61eab67cc097dfd6ae8841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.453ex; height:2.843ex;" alt="{\displaystyle (c,\,d)}"></span> 的<a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">有序对</a>为对象，其中 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}},\,d\in \mathrm {Ob} \ {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}},\,d\in \mathrm {Ob} \ {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b6ca3df737a33c3ecfe90c9491404c469b58c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.719ex; height:2.509ex;" alt="{\displaystyle c\in \mathrm {Ob} \ {\mathcal {C}},\,d\in \mathrm {Ob} \ {\mathcal {D}}}"></span>，</li> <li>以形如 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,\,g)\colon \ (c,\,d)\to (c',\,d')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,\,g)\colon \ (c,\,d)\to (c',\,d')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef535427b532021278ad445e075d22aa796b8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.131ex; height:3.009ex;" alt="{\displaystyle (f,\,g)\colon \ (c,\,d)\to (c',\,d')}"></span> 的有序对为态射，同时</li> <li>结合律与单位态射也如此被逐分量定义。</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:The_Commutative_Graph_between_d,_e,_f_in_Comma_Category_F_downarrow_G.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg/220px-The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg/330px-The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg/440px-The_Commutative_Graph_between_d%2C_e%2C_f_in_Comma_Category_F_downarrow_G.svg.png 2x" data-file-width="130" data-file-height="80" /></a><figcaption>图 2：逗号范畴之态射</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="逗号范畴"><span id=".E9.80.97.E5.8F.B7.E8.8C.83.E7.95.B4"></span>逗号范畴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=11" title="编辑章节：逗号范畴"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>给定<a href="/wiki/%E5%87%BD%E5%AD%90" title="函子">函子</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon \ {\mathcal {D}}\to {\mathcal {C}},\,G\colon \ {\mathcal {E}}\to {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </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>,</mo> <mspace width="thinmathspace" /> <mi>G</mi> <mo>:</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\colon \ {\mathcal {D}}\to {\mathcal {C}},\,G\colon \ {\mathcal {E}}\to {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9898bf5fed6380a17412a7bc4b0946694b72adf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.027ex; height:2.509ex;" alt="{\displaystyle F\colon \ {\mathcal {D}}\to {\mathcal {C}},\,G\colon \ {\mathcal {E}}\to {\mathcal {C}}}"></span>，定义其<a href="/w/index.php?title=%E9%80%97%E5%8F%B7%E8%8C%83%E7%95%B4&action=edit&redlink=1" class="new" title="逗号范畴（页面不存在）">逗号范畴</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\downarrow G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">↓</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\downarrow G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd73e3b5c162c28cb59debaac9fc886e07a7f5cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.02ex; height:2.509ex;" alt="{\displaystyle F\downarrow G}"></span> 如下： </p> <ul><li>以有序三元组 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d,\,e,\,f\colon \ Fd\to Ge)\in \mathrm {Ob} \ {\mathcal {D}}\times \mathrm {Ob} \ {\mathcal {E}}\times \mathrm {Mor} \ {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>e</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>:</mo> <mtext> </mtext> <mi>F</mi> <mi>d</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mi>e</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <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 (d,\,e,\,f\colon \ Fd\to Ge)\in \mathrm {Ob} \ {\mathcal {D}}\times \mathrm {Ob} \ {\mathcal {E}}\times \mathrm {Mor} \ {\mathcal {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5036d7a8591f271de3788b18a9edd82ed0875d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.337ex; height:2.843ex;" alt="{\displaystyle (d,\,e,\,f\colon \ Fd\to Ge)\in \mathrm {Ob} \ {\mathcal {D}}\times \mathrm {Ob} \ {\mathcal {E}}\times \mathrm {Mor} \ {\mathcal {C}}}"></span> 为对象，</li></ul> <ul><li>以有序对 <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\colon \ d\to d',\,k\colon \ e\to e')\in \mathrm {Mor} \ {\mathcal {D}}\times \mathrm {Mor} \ {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>:</mo> <mtext> </mtext> <mi>d</mi> <mo stretchy="false">→</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo>:</mo> <mtext> </mtext> <mi>e</mi> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h\colon \ d\to d',\,k\colon \ e\to e')\in \mathrm {Mor} \ {\mathcal {D}}\times \mathrm {Mor} \ {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6552a2c718b76008d4ed30eb209e9c0efe2228a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.564ex; height:3.009ex;" alt="{\displaystyle (h\colon \ d\to d',\,k\colon \ e\to e')\in \mathrm {Mor} \ {\mathcal {D}}\times \mathrm {Mor} \ {\mathcal {E}}}"></span> 为态射，使得对于每个 <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,\,k)\colon \ (d,\,e,\,f)\to (d',\,e',\,f')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>e</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h,\,k)\colon \ (d,\,e,\,f)\to (d',\,e',\,f')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10193bb0b032691f52c000908def3e3f7f74e15a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.566ex; height:3.009ex;" alt="{\displaystyle (h,\,k)\colon \ (d,\,e,\,f)\to (d',\,e',\,f')}"></span>，图 2 在 <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> 中交换，即：使得 <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'\circ Fh=Gk\circ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∘</mo> <mi>F</mi> <mi>h</mi> <mo>=</mo> <mi>G</mi> <mi>k</mi> <mo>∘</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'\circ Fh=Gk\circ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1c1764457e9ed1474cfbd0d376b8252b96a26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.889ex; height:2.843ex;" alt="{\displaystyle f'\circ Fh=Gk\circ f}"></span>。</li></ul> <div class="mw-heading mw-heading2"><h2 id="範疇類型"><span id=".E7.AF.84.E7.96.87.E9.A1.9E.E5.9E.8B"></span>範疇類型</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=12" title="编辑章节：範疇類型"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>在许多范畴中，例如阿贝尔群范畴或向量空间范畴，态射集合 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} (a,\,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} (a,\,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f43678e947e811496183b61d978093f046aafa04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.299ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} (a,\,b)}"></span> 不仅是集合，而且还是<a href="/wiki/%E9%98%BF%E8%B4%9D%E5%B0%94%E7%BE%A4" title="阿贝尔群">阿贝尔群</a>，并且态射的复合与这些阿贝尔群之间的群结构兼容，即复合映射是<a href="/wiki/%E5%8F%8C%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="双线性映射">双线性的</a>。这种范畴称为<a href="/wiki/%E9%A0%90%E5%8F%AF%E5%8A%A0%E7%AF%84%E7%96%87" title="預可加範疇">预可加范畴</a>。如果在此基础上这个范畴还带有所有有限<a href="/wiki/%E7%A7%AF_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="积 (范畴论)">积</a>和<a href="/wiki/%E4%B8%8A%E7%A7%AF" title="上积">上积</a>，那么我们称之为<a href="/wiki/%E5%8F%AF%E5%8A%A0%E7%AF%84%E7%96%87" title="可加範疇">可加范畴</a>。如果更进一步地，所有态射都有核和上核，并且每个满态射都是上核而每个单态射都是核，那么我们称之为<a href="/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%AF%84%E7%96%87" title="阿貝爾範疇">阿贝尔范畴</a>。阿贝尔范畴的典型例子是阿贝尔群的范畴。</li> <li>范畴是完备的当其拥有所有<a href="/wiki/%E6%9E%81%E9%99%90_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="极限 (范畴论)">极限</a>。集合、阿贝尔群、拓扑空间的范畴都是完备的。</li> <li>范畴是<a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E9%97%AD%E8%8C%83%E7%95%B4" title="笛卡儿闭范畴">笛卡尔闭</a>的当其拥有所有有限直积、且有限积上的态射总是可由任一因子上的态射确定。笛卡尔闭范畴包括 <b><a href="/wiki/%E9%9B%86%E5%90%88%E8%8C%83%E7%95%B4" title="集合范畴">Set</a></b> 和 <b>CPO</b>，即<a href="/wiki/%E5%AE%8C%E5%85%A8%E5%81%8F%E5%BA%8F" title="完全偏序">完全偏序</a>和<a href="/wiki/%E6%96%AF%E7%A7%91%E7%89%B9%E8%BF%9E%E7%BB%AD%E6%80%A7" title="斯科特连续性">斯科特连续函数</a>组成的范畴。</li> <li><a href="/wiki/%E6%8B%93%E6%92%B2%E6%96%AF" title="拓撲斯">拓扑斯</a>是一种特定的笛卡尔闭范畴；所有数学内容都可以用拓扑斯的语言形式化（正如所有经典数学都可以用集合范畴的语言形式化一般）。拓扑斯也可用于表示逻辑理论。</li></ul> <div class="mw-heading mw-heading2"><h2 id="注释"><span id=".E6.B3.A8.E9.87.8A"></span>注释</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=13" title="编辑章节：注释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <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">此处并未限定是唯一一个。</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">此处及下列皆為<a href="/wiki/%E5%85%B7%E9%AB%94%E7%AF%84%E7%96%87" title="具體範疇">具體範疇</a>的例子，即：在 <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 {\mathsf {Set}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Set}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31488c9fed260a93d36653220c2bd75b771cc440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.165ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Set}}}"></span> 上加入一些結構，且要求態射為對應於此附加結構的函數，態射複合為簡單的一般函數複合。</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">部分作者习惯将一般环的范畴记作 <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 {\mathsf {Rng}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Rng}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f410762cf17552545223bc399580170add4b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.866ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Rng}}}"></span>，而将幺环的范畴记作 <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 {\mathsf {Ring}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">i</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Ring}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78c9ceb36afa0269f4088e97466109f2de8c27a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.422ex; height:2.509ex;" alt="{\displaystyle {\mathsf {Ring}}}"></span>。<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">由于 <a href="/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA" title="罗素悖论">Russell 悖论</a>，找到这样一个范畴使得 <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 {\mathsf {CAT}}\in \mathrm {Ob} \ {\mathsf {CAT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {CAT}}\in \mathrm {Ob} \ {\mathsf {CAT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74c4520fd5e54d5b89b6a2032f9820141bf65d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.759ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CAT}}\in \mathrm {Ob} \ {\mathsf {CAT}}}"></span> 并不可行，不过显然有 <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 {\mathsf {Cat}}\in \mathrm {Ob} \ {\mathsf {CAT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">b</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {Cat}}\in \mathrm {Ob} \ {\mathsf {CAT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa2c0ae07cfe644fc43760639368b979da34f3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.584ex; height:2.176ex;" alt="{\displaystyle {\mathsf {Cat}}\in \mathrm {Ob} \ {\mathsf {CAT}}}"></span>。<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></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">可以验证，这样的态射复合满足定义的公理。</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=14" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation journal">Bjorn Poonen. <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/1404.0135">Why all rings should have a 1</a>. <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>. 2014. <span class="plainlinks"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1404.0135"><span title="arXiv">arXiv:1404.0135</span></a> <span typeof="mw:File"><span title="可免费查阅"><img alt="可免费查阅" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29&rft.atitle=Why+all+rings+should+have+a+1&rft.au=Bjorn+Poonen&rft.date=2014&rft.genre=article&rft.jtitle=arXiv&rft_id=https%3A%2F%2Farxiv.org%2Fpdf%2F1404.0135&rft_id=info%3Aarxiv%2F1404.0135&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></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"><cite class="citation book">Emily Riehl. Category Theory in Context. USA: Aurora. 2014. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-486-80903-8" title="Special:网络书源/978-0-486-80903-8"><span title="国际标准书号">ISBN</span> 978-0-486-80903-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%AF%84%E7%96%87+%28%E6%95%B8%E5%AD%B8%29&rft.au=Emily+Riehl&rft.btitle=Category+Theory+in+Context&rft.date=2014&rft.genre=book&rft.isbn=978-0-486-80903-8&rft.place=USA&rft.pub=Aurora&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <ul><li>Adámek, Jiří, Herrlich, Horst, & Strecker, George E.（1990）. <a rel="nofollow" class="external text" href="http://katmat.math.uni-bremen.de/acc/acc.pdf"><i>Abstract and Concrete Categories</i></a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20150421081851/http://katmat.math.uni-bremen.de/acc/acc.pdf">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）. Originally publ. John Wiley & Sons. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0471609226" class="internal mw-magiclink-isbn">ISBN 0-471-60922-6</a>.（now free on-line edition）</li> <li>Asperti, Andrea, & Longo, Giuseppe (1991). <a rel="nofollow" class="external text" href="ftp://ftp.di.ens.fr/pub/users/longo/CategTypesStructures/book.pdf"><i>Categories, Types and Structures</i></a>. Originally publ. M.I.T. Press.</li> <li>Barr, Michael, & Wells, Charles (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100821021308/http://www.cwru.edu/artsci/math/wells/pub/ttt.html"><i>Toposes, Triples and Theories</i></a>.（revised and corrected free online version of <i>Grundlehren der mathematischen Wissenschaften（278）.</i> Springer-Verlag,1983)</li> <li>Borceux, Francis (1994). <i>Handbook of Categorical Algebra.</i>. Vols. 50-52 of <i>Encyclopedia of Mathematics and its Applications.</i> Cambridge: Cambridge University Press.</li> <li>Lawvere, William, & Schanuel, Steve.（1997）. <i>Conceptual Mathematics: A First Introduction to Categories</i>. Cambridge: Cambridge University Press.</li> <li>Mac Lane, Saunders (1998). <i>Categories for the Working Mathematician</i>（2nd ed.）. Graduate Texts in Mathematics 5. Springer. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0387984038" class="internal mw-magiclink-isbn">ISBN 0-387-98403-8</a>.</li> <li>Jean-Pierre Marquis, <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/category-theory/">"Category Theory"</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20211121231337/http://plato.stanford.edu/entries/category-theory/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） in <a rel="nofollow" class="external text" href="http://plato.stanford.edu/"><i>Stanford Encyclopedia of Philosophy</i></a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/19961227155812/http://plato.stanford.edu/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）, 2006</li></ul> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&action=edit&section=15" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20061014180302/http://www.mta.ca/~cat-dist/categories.html">Homepage of the Categories mailing list</a>, with extensive list of resources</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20020123032906/http://us.geocities.com/alex_stef/mylist.html"><i>Category Theory</i> section of Alexandre Stefanov's list of free online mathematics resources</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf;position:relative}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em;width:auto;padding-left:0.2em;position:absolute;left:1em}.mw-parser-output .navbox .mw-collapsible-toggle{margin-left:0.5em;position:absolute;right:1em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="范畴论" 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="collapsible-title navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%AF%84%E7%96%87%E8%AB%96" title="Template:範疇論"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E7%AF%84%E7%96%87%E8%AB%96" title="Template talk:範疇論"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%AF%84%E7%96%87%E8%AB%96" title="Special:编辑页面/Template:範疇論"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="范畴论" style="font-size:110%;margin:0 5em"><a href="/wiki/%E8%8C%83%E7%95%B4%E8%AE%BA" title="范畴论">范畴论</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";background:#e5e5ff;"><div id="基本概念" style="font-size:110%;margin:0 5em">基本概念</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">基本概念</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="mw-selflink selflink">範疇</a></li> <li><a href="/wiki/%E4%BC%B4%E9%9A%A8%E5%87%BD%E5%AD%90" title="伴隨函子">伴隨函子</a></li> <li><a href="/wiki/%E4%BA%A4%E6%8D%A2%E5%9B%BE%E8%A1%A8" title="交换图表">交换图表</a></li> <li><a href="/wiki/%E5%85%B7%E9%AB%94%E7%AF%84%E7%96%87" title="具體範疇">具體範疇</a></li> <li><a href="/wiki/%E5%87%BD%E5%AD%90" title="函子">函子</a></li> <li><a href="/wiki/%E6%80%81%E5%B0%84" title="态射">态射</a></li> <li><a href="/wiki/%E8%87%AA%E7%84%B6%E8%AE%8A%E6%8F%9B" title="自然變換">自然變換</a></li> <li><a href="/wiki/%E6%B3%9B%E6%80%A7%E8%B4%A8" title="泛性质">泛性质</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%B3%9B%E6%80%A7%E8%B4%A8" title="泛性质">泛構造</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 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/%E6%9E%81%E9%99%90_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="极限 (范畴论)">極限</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A7%8B%E5%AF%B9%E8%B1%A1%E5%92%8C%E7%BB%88%E5%AF%B9%E8%B1%A1" title="始对象和终对象">終對象</a></li> <li><a href="/wiki/%E7%A7%AF_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="积 (范畴论)">積</a></li> <li><a href="/wiki/%E7%AD%89%E5%8C%96%E5%AD%90" title="等化子">等化子</a> <ul><li><span class="ilh-all" data-orig-title="核 (范畴论)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kernel (category theory)"><span class="ilh-page"><a href="/w/index.php?title=%E6%A0%B8_(%E8%8C%83%E7%95%B4%E8%AE%BA)&action=edit&redlink=1" class="new" title="核 (范畴论)（页面不存在）">核</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kernel_(category_theory)" class="extiw" title="en:Kernel (category theory)"><span lang="en" dir="auto">Kernel (category theory)</span></a></span>）</span></span></li></ul></li> <li><a href="/wiki/%E6%8B%89%E5%9B%9E_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="拉回 (范畴论)">拉回</a></li> <li><span class="ilh-all" data-orig-title="逆極限" data-lang-code="en" data-lang-name="英语" data-foreign-title="Inverse limit"><span class="ilh-page"><a href="/w/index.php?title=%E9%80%86%E6%A5%B5%E9%99%90&action=edit&redlink=1" class="new" title="逆極限（页面不存在）">逆極限</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Inverse_limit" class="extiw" title="en:Inverse limit"><span lang="en" dir="auto">Inverse limit</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%9E%81%E9%99%90_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="极限 (范畴论)">餘極限</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A7%8B%E5%AF%B9%E8%B1%A1%E5%92%8C%E7%BB%88%E5%AF%B9%E8%B1%A1" title="始对象和终对象">始對象</a></li> <li><span class="ilh-all" data-orig-title="餘積" data-lang-code="en" data-lang-name="英语" data-foreign-title="Coproduct"><span class="ilh-page"><a href="/w/index.php?title=%E9%A4%98%E7%A9%8D&action=edit&redlink=1" class="new" title="餘積（页面不存在）">餘積</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Coproduct" class="extiw" title="en:Coproduct"><span lang="en" dir="auto">Coproduct</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="餘等化子" data-lang-code="en" data-lang-name="英语" data-foreign-title="Coequalizer"><span class="ilh-page"><a href="/w/index.php?title=%E9%A4%98%E7%AD%89%E5%8C%96%E5%AD%90&action=edit&redlink=1" class="new" title="餘等化子（页面不存在）">餘等化子</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Coequalizer" class="extiw" title="en:Coequalizer"><span lang="en" dir="auto">Coequalizer</span></a></span>）</span></span> <ul><li><a href="/wiki/%E4%BD%99%E6%A0%B8" title="余核">餘核</a></li></ul></li> <li><a href="/wiki/%E6%8E%A8%E5%87%BA_(%E8%8C%83%E7%95%B4%E8%AE%BA)" title="推出 (范畴论)">推出</a></li> <li><span class="ilh-all" data-orig-title="正極限" data-lang-code="en" data-lang-name="英语" data-foreign-title="Direct limit"><span class="ilh-page"><a href="/w/index.php?title=%E6%AD%A3%E6%A5%B5%E9%99%90&action=edit&redlink=1" class="new" title="正極限（页面不存在）">正極限</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Direct_limit" class="extiw" title="en:Direct limit"><span lang="en" dir="auto">Direct limit</span></a></span>）</span></span></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="ilh-all" data-orig-title="代數範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Variety (universal algebra)"><span class="ilh-page"><a href="/w/index.php?title=%E4%BB%A3%E6%95%B8%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="代數範疇（页面不存在）">代數範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Variety_(universal_algebra)" class="extiw" title="en:Variety (universal algebra)"><span lang="en" dir="auto">Variety (universal algebra)</span></a></span>）</span></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%9B%86%E5%90%88%E8%8C%83%E7%95%B4" title="集合范畴">集合</a></li> <li><a href="/wiki/%E9%97%9C%E4%BF%82%E7%AF%84%E7%96%87" title="關係範疇">關係</a></li> <li><a href="/w/index.php?title=%E5%8E%9F%E7%BE%A4%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="原群範疇（页面不存在）">原群</a></li> <li><a href="/wiki/%E7%BE%A4%E7%AF%84%E7%96%87" title="群範疇">群</a></li> <li><a href="/wiki/%E9%98%BF%E8%B4%9D%E5%B0%94%E7%BE%A4%E8%8C%83%E7%95%B4" class="mw-redirect" title="阿贝尔群范畴">阿貝爾群</a></li> <li><span class="ilh-all" data-orig-title="環範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Category of rings"><span class="ilh-page"><a href="/w/index.php?title=%E7%92%B0%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="環範疇（页面不存在）">環</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Category_of_rings" class="extiw" title="en:Category of rings"><span lang="en" dir="auto">Category of rings</span></a></span>）</span></span> <ul><li><a href="/w/index.php?title=%E5%9F%9F%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="域範疇（页面不存在）">域</a></li></ul></li> <li><span class="ilh-all" data-orig-title="模範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Category of modules"><span class="ilh-page"><a href="/w/index.php?title=%E6%A8%A1%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="模範疇（页面不存在）">模</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Category_of_modules" class="extiw" title="en:Category of modules"><span lang="en" dir="auto">Category of modules</span></a></span>）</span></span> <ul><li><a href="/w/index.php?title=%E5%90%91%E9%87%8F%E7%A9%BA%E9%96%93%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="向量空間範疇（页面不存在）">向量空間</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">範疇上的構造</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="自由範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Free category"><span class="ilh-page"><a href="/w/index.php?title=%E8%87%AA%E7%94%B1%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="自由範疇（页面不存在）">自由範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Free_category" class="extiw" title="en:Free category"><span lang="en" dir="auto">Free category</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%87%BD%E5%AD%90%E7%AF%84%E7%96%87" title="函子範疇">函子範疇</a></li> <li><span class="ilh-all" data-orig-title="商範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quotient category"><span class="ilh-page"><a href="/w/index.php?title=%E5%95%86%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="商範疇（页面不存在）">商範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quotient_category" class="extiw" title="en:Quotient category"><span lang="en" dir="auto">Quotient category</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%A9%8D%E7%AF%84%E7%96%87" title="積範疇">積範疇</a></li> <li><a href="/wiki/%E5%AD%90%E7%AF%84%E7%96%87" title="子範疇">子範疇</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:0px 0px 0px 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:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="collapsible-title navbox-title" colspan="2" style=";background:#e5e5ff;"><div id="高階範疇論（英语：-&#123;Higher_category_theory&#125;-）" style="font-size:110%;margin:0 5em"><span class="ilh-all" data-orig-title="高階範疇論" data-lang-code="en" data-lang-name="英语" data-foreign-title="Higher category theory"><span class="ilh-page"><a href="/w/index.php?title=%E9%AB%98%E9%9A%8E%E7%AF%84%E7%96%87%E8%AB%96&action=edit&redlink=1" class="new" title="高階範疇論（页面不存在）">高階範疇論</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Higher_category_theory" class="extiw" title="en:Higher category theory"><span lang="en" dir="auto">Higher category theory</span></a></span>）</span></span></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 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%">基本概念</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <li><a href="/wiki/%E8%8C%83%E7%95%B4%E5%8C%96" title="范畴化">範疇化</a></li> <li><span class="ilh-all" data-orig-title="濃縮範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Enriched category"><span class="ilh-page"><a href="/w/index.php?title=%E6%BF%83%E7%B8%AE%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="濃縮範疇（页面不存在）">濃縮範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Enriched_category" class="extiw" title="en:Enriched category"><span lang="en" dir="auto">Enriched category</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="高維代數" data-lang-code="en" data-lang-name="英语" data-foreign-title="Higher-dimensional algebra"><span class="ilh-page"><a href="/wiki/%E9%AB%98%E7%BB%B4%E4%BB%A3%E6%95%B0" title="高维代数">高維代數</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Higher-dimensional_algebra" class="extiw" title="en:Higher-dimensional algebra"><span lang="en" dir="auto">Higher-dimensional algebra</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="同倫假設" data-lang-code="en" data-lang-name="英语" data-foreign-title="Homotopy hypothesis"><span class="ilh-page"><a href="/w/index.php?title=%E5%90%8C%E5%80%AB%E5%81%87%E8%A8%AD&action=edit&redlink=1" class="new" title="同倫假設（页面不存在）">同倫假設</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Homotopy_hypothesis" class="extiw" title="en:Homotopy hypothesis"><span lang="en" dir="auto">Homotopy hypothesis</span></a></span>）</span></span></li> <li><a href="/wiki/%E6%A8%A1%E5%9E%8B%E8%8C%83%E7%95%B4" title="模型范畴">模型范畴</a></li> <li><a href="/wiki/%E5%8D%95%E7%BA%AF%E8%8C%83%E7%95%B4" title="单纯范畴">单纯范畴</a></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><var>n</var>-範疇</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span class="ilh-all" data-orig-title="弱 n-範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Weak n-categories"><span class="ilh-page"><a href="/w/index.php?title=%E5%BC%B1_n-%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="弱 n-範疇（页面不存在）">弱 n-範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Weak_n-categories" class="extiw" title="en:Weak n-categories"><span lang="en" dir="auto">Weak n-categories</span></a></span>）</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="雙範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Bicategory"><span class="ilh-page"><a href="/w/index.php?title=%E9%9B%99%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="雙範疇（页面不存在）">雙範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Bicategory" class="extiw" title="en:Bicategory"><span lang="en" dir="auto">Bicategory</span></a></span>）</span></span> <ul><li><span class="ilh-all" data-orig-title="偽函子" data-lang-code="en" data-lang-name="英语" data-foreign-title="Pseudo-functor"><span class="ilh-page"><a href="/w/index.php?title=%E5%81%BD%E5%87%BD%E5%AD%90&action=edit&redlink=1" class="new" title="偽函子（页面不存在）">偽函子</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Pseudo-functor" class="extiw" title="en:Pseudo-functor"><span lang="en" dir="auto">Pseudo-functor</span></a></span>）</span></span></li></ul></li> <li><span class="ilh-all" data-orig-title="三範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tricategory"><span class="ilh-page"><a href="/w/index.php?title=%E4%B8%89%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="三範疇（页面不存在）">三範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tricategory" class="extiw" title="en:Tricategory"><span lang="en" dir="auto">Tricategory</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="四範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Tetracategory"><span class="ilh-page"><a href="/w/index.php?title=%E5%9B%9B%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="四範疇（页面不存在）">四範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Tetracategory" class="extiw" title="en:Tetracategory"><span lang="en" dir="auto">Tetracategory</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="闞複形" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kan complex"><span class="ilh-page"><a href="/w/index.php?title=%E9%97%9E%E8%A4%87%E5%BD%A2&action=edit&redlink=1" class="new" title="闞複形（页面不存在）">闞複形</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kan_complex" class="extiw" title="en:Kan complex"><span lang="en" dir="auto">Kan complex</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="∞-廣群" data-lang-code="en" data-lang-name="英语" data-foreign-title="∞-groupoid"><span class="ilh-page"><a href="/w/index.php?title=%E2%88%9E-%E5%BB%A3%E7%BE%A4&action=edit&redlink=1" class="new" title="∞-廣群（页面不存在）">∞-廣群</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/%E2%88%9E-groupoid" class="extiw" title="en:∞-groupoid"><span lang="en" dir="auto">∞-groupoid</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="∞-拓撲斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="∞-topos"><span class="ilh-page"><a href="/w/index.php?title=%E2%88%9E-%E6%8B%93%E6%92%B2%E6%96%AF&action=edit&redlink=1" class="new" title="∞-拓撲斯（页面不存在）">∞-拓撲斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/%E2%88%9E-topos" class="extiw" title="en:∞-topos"><span lang="en" dir="auto">∞-topos</span></a></span>）</span></span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/N-%E7%AF%84%E7%96%87" title="N-範疇">強 <var>n</var>-範疇</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/2-%E7%AF%84%E7%96%87" title="2-範疇">2-範疇</a> <ul><li><span class="ilh-all" data-orig-title="2-函子" data-lang-code="en" data-lang-name="英语" data-foreign-title="2-functor"><span class="ilh-page"><a href="/w/index.php?title=2-%E5%87%BD%E5%AD%90&action=edit&redlink=1" class="new" title="2-函子（页面不存在）">2-函子</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/2-functor" class="extiw" title="en:2-functor"><span lang="en" dir="auto">2-functor</span></a></span>）</span></span></li></ul></li> <li><span class="ilh-all" data-orig-title="3-範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="3-category"><span class="ilh-page"><a href="/w/index.php?title=3-%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="3-範疇（页面不存在）">3-範疇</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/3-category" class="extiw" title="en:3-category"><span lang="en" dir="auto">3-category</span></a></span>）</span></span></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%8C%83%E7%95%B4%E5%8C%96" title="范畴化">範疇化</a>概念</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="ilh-all" data-orig-title="2-群" data-lang-code="en" data-lang-name="英语" data-foreign-title="2-group"><span class="ilh-page"><a href="/w/index.php?title=2-%E7%BE%A4&action=edit&redlink=1" class="new" title="2-群（页面不存在）">2-群</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/2-group" class="extiw" title="en:2-group"><span lang="en" dir="auto">2-group</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="2-環" data-lang-code="en" data-lang-name="英语" data-foreign-title="2-ring"><span class="ilh-page"><a href="/w/index.php?title=2-%E7%92%B0&action=edit&redlink=1" class="new" title="2-環（页面不存在）">2-環</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/2-ring" class="extiw" title="en:2-ring"><span lang="en" dir="auto">2-ring</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="En-環" data-lang-code="en" data-lang-name="英语" data-foreign-title="En-ring"><span class="ilh-page"><a href="/w/index.php?title=En-%E7%92%B0&action=edit&redlink=1" class="new" title="En-環（页面不存在）"><i>E<sub>n</sub></i>-環</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/En-ring" class="extiw" title="en:En-ring"><span lang="en" dir="auto">En-ring</span></a></span>）</span></span></li> <li>(<span class="ilh-all" data-orig-title="對稱么半範疇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Symmetric monoidal category"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E4%B9%88%E5%8D%8A%E7%AF%84%E7%96%87&action=edit&redlink=1" class="new" title="對稱么半範疇（页面不存在）">對稱</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Symmetric_monoidal_category" class="extiw" title="en:Symmetric monoidal category"><span lang="en" dir="auto">Symmetric monoidal category</span></a></span>）</span></span>) <a href="/wiki/%E4%B9%88%E5%8D%8A%E7%AF%84%E7%96%87" title="么半範疇">么半範疇</a></li> <li><span class="ilh-all" data-orig-title="n-群 (範疇論)" data-lang-code="en" data-lang-name="英语" data-foreign-title="N-group (category theory)"><span class="ilh-page"><a href="/w/index.php?title=N-%E7%BE%A4_(%E7%AF%84%E7%96%87%E8%AB%96)&action=edit&redlink=1" class="new" title="N-群 (範疇論)（页面不存在）"><i>n</i>-群</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/N-group_(category_theory)" class="extiw" title="en:N-group (category theory)"><span lang="en" dir="auto">N-group (category theory)</span></a></span>）</span></span></li> <li><span class="ilh-all" data-orig-title="n-么半群" data-lang-code="en" data-lang-name="英语" data-foreign-title="n-monoid"><span class="ilh-page"><a href="/w/index.php?title=N-%E4%B9%88%E5%8D%8A%E7%BE%A4&action=edit&redlink=1" class="new" title="N-么半群（页面不存在）"><i>n</i>-么半群</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/n-monoid" class="extiw" title="en:n-monoid"><span lang="en" dir="auto">n-monoid</span></a></span>）</span></span></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 typeof="mw:File"><span title="分类"><img alt="分类" src="//upload.wikimedia.org/wikipedia/commons/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/commons/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/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:%E8%8C%83%E7%95%B4%E8%AE%BA" title="Category:范畴论">分類</a></li> <li><span typeof="mw:File"><span title="主题"><img alt="主题" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Portal:%E6%95%B0%E5%AD%A6" title="Portal:数学">主題</a></li> <li><span typeof="mw:File"><span title="维基教科书页面"><img alt="维基教科书页面" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> <a href="https://zh.wikibooks.org/wiki/%E7%BA%AF%E7%B2%B9%E6%95%B0%E5%AD%A6/%E7%AF%84%E7%96%87%E8%AB%96" class="extiw" title="b:纯粹数学/範疇論">維基教科書</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=範疇_(數學)&oldid=85069390">https://zh.wikipedia.org/w/index.php?title=範疇_(數學)&oldid=85069390</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E8%8C%83%E7%95%B4%E8%AE%BA" title="Category:范畴论">范畴论</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">隐藏分类：<ul><li><a href="/wiki/Category:%E8%87%AA2018%E5%B9%B41%E6%9C%88%E7%BC%BA%E5%B0%91%E6%B3%A8%E8%84%9A%E7%9A%84%E6%9D%A1%E7%9B%AE" title="Category:自2018年1月缺少注脚的条目">自2018年1月缺少注脚的条目</a></li><li><a href="/wiki/Category:%E8%87%AA2024%E5%B9%B411%E6%9C%88%E6%9C%89%E6%9C%AA%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90%E8%AF%AD%E5%8F%A5%E7%9A%84%E6%9D%A1%E7%9B%AE" title="Category:自2024年11月有未列明来源语句的条目">自2024年11月有未列明来源语句的条目</a></li><li><a href="/wiki/Category:%E4%BD%BF%E7%94%A8ISBN%E9%AD%94%E6%9C%AF%E9%93%BE%E6%8E%A5%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:使用ISBN魔术链接的页面">使用ISBN魔术链接的页面</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> 本页面最后修订于2024年11月23日 (星期六) 08:55。</li> <li id="footer-info-copyright">本站的全部文字在<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.zh">知识共享署名-相同方式共享 4.0协议</a>之条款下提供，附加条款亦可能应用。（请参阅<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">使用条款</a>）<br /> Wikipedia®和维基百科标志是<a rel="nofollow" class="external text" href="https://wikimediafoundation.org/zh">维基媒体基金会</a>的注册商标；维基™是维基媒体基金会的商标。<br /> 维基媒体基金会是按美国国內稅收法501(c)(3)登记的<a class="external text" href="https://donate.wikimedia.org/wiki/Special:MyLanguage/Tax_deductibility">非营利慈善机构</a>。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隐私政策</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E">关于维基百科</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:%E5%85%8D%E8%B4%A3%E5%A3%B0%E6%98%8E">免责声明</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行为准则</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">开发者</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh.wikipedia.org">统计</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie声明</a></li> <li id="footer-places-mobileview"><a href="//zh.m.wikipedia.org/w/index.php?title=%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">手机版视图</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-vmhgq","wgBackendResponseTime":249,"wgPageParseReport":{"limitreport":{"cputime":"0.542","walltime":"0.796","ppvisitednodes":{"value":2719,"limit":1000000},"postexpandincludesize":{"value":172102,"limit":2097152},"templateargumentsize":{"value":18722,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":31,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":21384,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 404.591 1 -total"," 56.28% 227.686 1 Template:範疇論"," 55.57% 224.819 1 Template:Navbox_with_collapsible_groups"," 33.99% 137.501 4 Template:Navbox"," 21.61% 87.434 1 Template:No_footnotes"," 20.21% 81.757 1 Template:Ambox"," 17.90% 72.415 30 Template:Link-en"," 15.33% 62.038 2 Template:Reflist"," 11.39% 46.086 1 Template:Cite_journal"," 5.21% 21.091 3 Template:Icon"]},"scribunto":{"limitreport-timeusage":{"value":"0.190","limit":"10.000"},"limitreport-memusage":{"value":4383424,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-849f99967d-psbgh","timestamp":"20241123085525","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u7bc4\u7587 (\u6578\u5b78)","url":"https:\/\/zh.wikipedia.org\/wiki\/%E7%AF%84%E7%96%87_(%E6%95%B8%E5%AD%B8)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q719395","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q719395","author":{"@type":"Organization","name":"\u7ef4\u57fa\u5a92\u4f53\u9879\u76ee\u8d21\u732e\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-10-15T09:13:48Z","dateModified":"2024-11-23T08:55:12Z"}</script> </body> </html>