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<span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Category_of_small_categories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Category_of_small_categories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Category of small categories</span> </div> </a> <ul id="toc-Category_of_small_categories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Doctrines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Doctrines"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Doctrines</span> </div> </a> <ul id="toc-Doctrines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li 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</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the notion of a weak 2-category, see <a href="/wiki/Bicategory" title="Bicategory">bicategory</a>.</div> <p>In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a <b>strict 2-category</b> is a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> with "<a href="/wiki/Morphism" title="Morphism">morphisms</a> between morphisms", that is, where each <a href="/wiki/Hom-set" class="mw-redirect" title="Hom-set">hom-set</a> itself carries the structure of a category. It can be formally defined as a category <a href="/wiki/Enriched_category" title="Enriched category">enriched</a> over <b>Cat</b> (the <a href="/wiki/Category_of_small_categories" title="Category of small categories">category of categories and functors</a>, with the <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal</a> structure given by <a href="/wiki/Product_category" title="Product category">product of categories</a>). </p><p>The concept of 2-category was first introduced by <a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a> in his work on <a href="/wiki/Enriched_categories" class="mw-redirect" title="Enriched categories">enriched categories</a> in 1965.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The more general concept of <a href="/wiki/Bicategory" title="Bicategory">bicategory</a> (or <i>weak</i> 2-<i>category</i>), where composition of morphisms is <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> only up to a 2-isomorphism, was introduced in 1968 by <a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A 2-category <b>C</b> consists of: </p> <ul><li>A <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</a> of 0-<i>cells</i> (or <i><a href="/wiki/Object_(category_theory)" class="mw-redirect" title="Object (category theory)">objects</a></i>) <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, ....</li> <li>For all objects <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, a category <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71321910d2547fb660523682add481bd5ebd2b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.282ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (A,B)}"></span>. The objects <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g:A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17911b4e8e6a2f58d32826d24e9f879918303975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.487ex; height:2.509ex;" alt="{\displaystyle f,g:A\to B}"></span> of this category are called 1-<i>cells</i> and its morphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha :f\Rightarrow g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">⇒</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha :f\Rightarrow g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a99ee50429a1cdffeb301f336f4b88c37afe772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.433ex; height:2.509ex;" alt="{\displaystyle \alpha :f\Rightarrow g}"></span> are called 2-<i>cells</i>; the composition in this category is usually written <span class="mwe-math-element"><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> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d99143cc5359509a4d0c7174513a47d3e4a4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{1}}"></span> and called <i>vertical composition</i> or <i>composition along a</i> 1-<i>cell</i>.</li> <li>For any object <span class="texhtml mvar" style="font-style:italic;">A</span> there is a <a href="/wiki/Functor" title="Functor">functor</a> from the <a href="/wiki/Terminal_object" class="mw-redirect" title="Terminal object">terminal</a> <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> (with one object and one arrow) to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (A,A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (A,A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1304ebd73801a5b60224a6a40bbfdcb4f5ddfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.261ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (A,A)}"></span> that picks out the <a href="/wiki/Identity_morphism" class="mw-redirect" title="Identity morphism">identity</a> 1-cell <span class="texhtml">id<sub><i>A</i></sub></span> on <span class="texhtml mvar" style="font-style:italic;">A</span> and its identity 2-cell <span class="texhtml">id<sub>id<sub><i>A</i></sub></sub></span>. In practice these two are often denoted simply by <span class="texhtml mvar" style="font-style:italic;">A</span>.</li> <li>For all objects <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span> and <span class="texhtml mvar" style="font-style:italic;">C</span>, there is a functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b96132faf11a2f9f9c095837493a86df18cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.575ex; height:2.843ex;" alt="{\displaystyle \circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)}"></span>, called <i>horizontal composition</i> or <i>composition along a 0-cell</i>, which is associative and admits<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="the meaning of "admits" is unclear in this context (March 2020)">clarification needed</span></a></i>]</sup> the identity 1 and 2-cells of <span class="texhtml">id<sub><i>A</i></sub></span> as identities. Here, associativity for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13dfb21dbf5672d4088d58559e1a723cecf0392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{0}}"></span> means that horizontally composing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78699a32d9114fdbf104ed288f9933494b6bfe57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.732ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)}"></span> twice to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (A,D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (A,D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d741766570522f96dd868d43c9a071312f50e60d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.442ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (A,D)}"></span> is independent of which of the two <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa85d1759f208b66bc45fcd825284aec98612fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.61ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (C,D)\times \mathbf {C} (B,C)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (B,C)\times \mathbf {C} (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (B,C)\times \mathbf {C} (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f49f5a225e7ee3c5160a5dceedd5580d9cd5885b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.427ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (B,C)\times \mathbf {C} (A,B)}"></span> are composed first. The composition symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13dfb21dbf5672d4088d58559e1a723cecf0392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{0}}"></span> is often omitted, the horizontal composite of 2-cells <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon f\Rightarrow g\colon A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">⇒</mo> <mi>g</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \colon f\Rightarrow g\colon A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93303d8b20b9934895557bfbec21533a2331b6b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.685ex; height:2.509ex;" alt="{\displaystyle \alpha \colon f\Rightarrow g\colon A\to B}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \colon f'\Rightarrow g'\colon B\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>:</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">⇒</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \colon f'\Rightarrow g'\colon B\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcfe52dac530e2ed7d620dccd8a65725de41ee10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.967ex; height:2.843ex;" alt="{\displaystyle \beta \colon f'\Rightarrow g'\colon B\to C}"></span> being written simply as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \alpha \colon f'f\Rightarrow g'g\colon A\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mi>α</mi> <mo>:</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mi>f</mi> <mo stretchy="false">⇒</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mi>g</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \alpha \colon f'f\Rightarrow g'g\colon A\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0f45c9b3394fc89246e41c78854a0b432bf0d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.828ex; height:2.843ex;" alt="{\displaystyle \beta \alpha \colon f'f\Rightarrow g'g\colon A\to C}"></span>.</li></ul> <p>The <i>0-cells</i>, <i>1-cells</i>, and <i>2-cells</i> terminology is replaced by <i>0-morphisms</i>, <i>1-morphisms</i>, and <i>2-morphisms</i> in some sources<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> (see also <a href="/wiki/Higher_category_theory" title="Higher category theory">Higher category theory</a>). </p><p>The notion of 2-category differs from the more general notion of a <a href="/wiki/Bicategory" title="Bicategory">bicategory</a> in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as <b>Cat</b>-enriched categories: </p> <ul><li>Vertical composition is associative and unital, the units being the identity 2-cells <span class="texhtml">id<sub><i>f</i></sub></span>.</li> <li>Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells <span class="texhtml">id<sub>id<sub><i>A</i></sub></sub></span> on the identity 1-cells <span class="texhtml">id<sub><i>A</i></sub></span>.</li> <li>The <a href="/w/index.php?title=Interchange_law&action=edit&redlink=1" class="new" title="Interchange law (page does not exist)">interchange law</a> holds; i.e. it is true that for composable 2-cells <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma ,\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma ,\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb7ee31949dcce33b443132ac97f42927549e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.232ex; height:2.843ex;" alt="{\displaystyle \alpha ,\beta ,\gamma ,\delta }"></span></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha \circ _{0}\beta )\circ _{1}(\gamma \circ _{0}\delta )=(\alpha \circ _{1}\gamma )\circ _{0}(\beta \circ _{1}\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>β</mi> <mo stretchy="false">)</mo> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ</mi> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>γ</mi> <mo stretchy="false">)</mo> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β</mi> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha \circ _{0}\beta )\circ _{1}(\gamma \circ _{0}\delta )=(\alpha \circ _{1}\gamma )\circ _{0}(\beta \circ _{1}\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caac30a7a03a9304307fe2a7fd0127be2c5f0dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.091ex; height:2.843ex;" alt="{\displaystyle (\alpha \circ _{0}\beta )\circ _{1}(\gamma \circ _{0}\delta )=(\alpha \circ _{1}\gamma )\circ _{0}(\beta \circ _{1}\delta )}"></span></dd></dl></dd></dl> <p>The interchange law follows from the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13dfb21dbf5672d4088d58559e1a723cecf0392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{0}}"></span> is a functor between hom categories. It can be drawn as a <a href="/w/index.php?title=Pasting_diagram&action=edit&redlink=1" class="new" title="Pasting diagram (page does not exist)">pasting diagram</a> as follows: </p> <table cellpadding="2" align="center"> <tbody><tr valign="center"> <td><span typeof="mw:File"><a href="/wiki/File:2-category_horizontal_composition_upper.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/2-category_horizontal_composition_upper.svg/240px-2-category_horizontal_composition_upper.svg.png" decoding="async" width="240" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/2-category_horizontal_composition_upper.svg/360px-2-category_horizontal_composition_upper.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/2-category_horizontal_composition_upper.svg/480px-2-category_horizontal_composition_upper.svg.png 2x" data-file-width="320" data-file-height="124" /></a></span> </td> <td rowspan="3" style="font-size:large"> =  </td> <td rowspan="3"><span typeof="mw:File"><a href="/wiki/File:2-category_double_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/2-category_double_composition.svg/240px-2-category_double_composition.svg.png" decoding="async" width="240" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/2-category_double_composition.svg/360px-2-category_double_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/2-category_double_composition.svg/480px-2-category_double_composition.svg.png 2x" data-file-width="320" data-file-height="224" /></a></span> </td> <td rowspan="3" style="font-size:large"> =  </td> <td rowspan="3"><span typeof="mw:File"><a href="/wiki/File:2-category_vertical_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/132px-2-category_vertical_composition.svg.png" decoding="async" width="132" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/198px-2-category_vertical_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/264px-2-category_vertical_composition.svg.png 2x" data-file-width="176" data-file-height="224" /></a></span> </td> <td rowspan="3" style="font-size:large"><span class="mwe-math-element"><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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13dfb21dbf5672d4088d58559e1a723cecf0392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{0}}"></span> </td> <td rowspan="3"><span typeof="mw:File"><a href="/wiki/File:2-category_vertical_composition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/132px-2-category_vertical_composition.svg.png" decoding="async" width="132" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/198px-2-category_vertical_composition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/2-category_vertical_composition.svg/264px-2-category_vertical_composition.svg.png 2x" data-file-width="176" data-file-height="224" /></a></span> </td></tr> <tr> <td style="font-size:large" align="center"><span class="mwe-math-element"><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 _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d99143cc5359509a4d0c7174513a47d3e4a4e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle \circ _{1}}"></span> </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:2-category_horizontal_composition_lower.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/2-category_horizontal_composition_lower.svg/240px-2-category_horizontal_composition_lower.svg.png" decoding="async" width="240" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/2-category_horizontal_composition_lower.svg/360px-2-category_horizontal_composition_lower.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/2-category_horizontal_composition_lower.svg/480px-2-category_horizontal_composition_lower.svg.png 2x" data-file-width="320" data-file-height="124" /></a></span> </td></tr></tbody></table> <p>Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both. The <i>2-cell</i> are drawn with double arrows ⇒, the <i>1-cell</i> with single arrows →, and the <i>0-cell</i> with points. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The category <a href="/wiki/Category_of_preordered_sets" title="Category of preordered sets"><b>Ord</b></a> (of preordered sets) is a 2-category since preordered sets can easily be interpreted as categories. </p> <div class="mw-heading mw-heading3"><h3 id="Category_of_small_categories">Category of small categories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=3" title="Edit section: Category of small categories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The archetypal 2-category is the <a href="/wiki/Category_of_small_categories" title="Category of small categories">category of small categories</a>, with natural transformations serving as 2-morphisms; typically 2-morphisms are given by Greek letters (such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> above) for this reason. </p><p>The objects (<i>0-cells</i>) are all small categories, and for all objects <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> the category <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} (A,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} (A,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71321910d2547fb660523682add481bd5ebd2b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.282ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} (A,B)}"></span> is a <a href="/wiki/Functor_category" title="Functor category">functor category</a>. In this context, vertical composition is<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> the composition of natural transformations. </p> <div class="mw-heading mw-heading2"><h2 id="Doctrines">Doctrines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=4" title="Edit section: Doctrines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematics, a <b>doctrine</b> is simply a 2-category which is heuristically regarded as a system of theories. For example, <a href="/wiki/Algebraic_theory" title="Algebraic theory">algebraic theories</a>, as invented by <a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a>, is an example of a doctrine, as are <a href="/w/index.php?title=Multi-sorted_theory&action=edit&redlink=1" class="new" title="Multi-sorted theory (page does not exist)">multi-sorted theories</a>, <a href="/wiki/Operad" title="Operad">operads</a>, <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a>, and <a href="/wiki/Topos_(mathematics)" class="mw-redirect" title="Topos (mathematics)">toposes</a>. </p><p>The objects of the 2-category are called <i>theories</i>, the 1-morphisms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon A\rightarrow 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\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867edf1a68f4da4b8c01de80e666173d070f5a95" 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\rightarrow B}"></span> are called <i>models</i> of the <span class="texhtml mvar" style="font-style:italic;">A</span> in <span class="texhtml mvar" style="font-style:italic;">B</span>, and the 2-morphisms are called <i>morphisms between models.</i> </p><p>The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms. It is this vocabulary that makes the theory of doctrines worth while. </p><p>For example, the 2-category <b>Cat</b> of categories, functors, and natural transformations is a doctrine. One sees immediately that all <a href="/wiki/Presheaf_category" class="mw-redirect" title="Presheaf category">presheaf categories</a> are categories of models. </p><p>As another example, one may take the subcategory of <b>Cat</b> consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object. </p><p>Doctrines were discovered by <a href="/wiki/Jonathan_Mock_Beck" title="Jonathan Mock Beck">Jonathan Mock Beck</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/N-category" class="mw-redirect" title="N-category"><i>n</i>-category</a></li> <li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/2-category">2-category</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><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"><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a>, Catégories et structures, Dunod, Paris 1965.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a>, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/2-category">"2-category in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=2-category+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2F2-category&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrict+2-category" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/vertical+composition">"vertical composition in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=vertical+composition+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fvertical%2Bcomposition&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrict+2-category" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strict_2-category&action=edit&section=7" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Generalised algebraic models</i>, by Claudia Centazzo.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#e5e5ff;"><div id="Key_concepts" style="font-size:114%;margin:0 4em">Key concepts</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Category</a> <ul><li><a href="/wiki/Abelian_category" title="Abelian category">Abelian</a></li> <li><a href="/wiki/Additive_category" title="Additive category">Additive</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a></li> <li><a href="/wiki/Pre-abelian_category" title="Pre-abelian category">Pre-abelian</a></li> <li><a href="/wiki/Preadditive_category" title="Preadditive category">Preadditive</a></li> <li><a href="/wiki/Bicategory" title="Bicategory">Bicategory</a></li></ul></li> <li><a href="/wiki/Adjoint_functors" title="Adjoint functors">Adjoint functors</a></li> <li><a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">CCC</a></li> <li><a href="/wiki/Commutative_diagram" title="Commutative diagram">Commutative diagram</a></li> <li><a href="/wiki/End_(category_theory)" title="End (category theory)">End</a></li> <li><a href="/wiki/Exponential_object" title="Exponential object">Exponential</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li> <li><a href="/wiki/Kan_extension" title="Kan extension">Kan extension</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Natural_transformation" title="Natural transformation">Natural transformation</a></li> <li><a href="/wiki/Universal_property" title="Universal property">Universal property</a></li> <li><a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Universal_construction" class="mw-redirect" title="Universal construction">Universal constructions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Initial_and_terminal_objects" title="Initial and terminal objects">Terminal objects</a></li> <li><a href="/wiki/Product_(category_theory)" title="Product (category theory)">Products</a></li> <li><a href="/wiki/Equaliser_(mathematics)" title="Equaliser (mathematics)">Equalizers</a> <ul><li><a href="/wiki/Kernel_(category_theory)" title="Kernel (category theory)">Kernels</a></li></ul></li> <li><a href="/wiki/Pullback_(category_theory)" title="Pullback (category theory)">Pullbacks</a></li> <li><a href="/wiki/Inverse_limit" title="Inverse limit">Inverse limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Colimit" class="mw-redirect" title="Colimit">Colimits</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Initial_and_terminal_objects" title="Initial and terminal objects">Initial objects</a></li> <li><a href="/wiki/Coproduct" title="Coproduct">Coproducts</a></li> <li><a href="/wiki/Coequalizer" title="Coequalizer">Coequalizers</a> <ul><li><a href="/wiki/Cokernel" title="Cokernel">Cokernels and quotients</a></li></ul></li> <li><a href="/wiki/Pushout_(category_theory)" title="Pushout (category theory)">Pushout</a></li> <li><a href="/wiki/Direct_limit" title="Direct limit">Direct limit</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebraic_category" class="mw-redirect" title="Algebraic category">Algebraic categories</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_of_sets" title="Category of sets">Sets</a></li> <li><a href="/wiki/Category_of_relations" title="Category of relations">Relations</a></li> <li><a href="/wiki/Category_of_magmas" class="mw-redirect" title="Category of magmas">Magmas</a></li> <li><a href="/wiki/Category_of_groups" title="Category of groups">Groups</a></li> <li><a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">Abelian groups</a></li> <li><a href="/wiki/Category_of_rings" title="Category of rings">Rings</a> (<a href="/wiki/Category_of_rings#Category_of_fields" title="Category of rings">Fields</a>)</li> <li><a href="/wiki/Category_of_modules" title="Category of modules">Modules</a> (<a href="/wiki/Category_of_modules#Example:_the_category_of_vector_spaces" title="Category of modules">Vector spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions on categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Free_category" title="Free category">Free category</a></li> <li><a href="/wiki/Functor_category" title="Functor category">Functor category</a></li> <li><a href="/wiki/Kleisli_category" title="Kleisli category">Kleisli category</a></li> <li><a href="/wiki/Opposite_category" title="Opposite category">Opposite category</a></li> <li><a href="/wiki/Quotient_category" title="Quotient category">Quotient category</a></li> <li><a href="/wiki/Product_category" title="Product category">Product category</a></li> <li><a href="/wiki/Comma_category" title="Comma category">Comma category</a></li> <li><a href="/wiki/Subcategory" title="Subcategory">Subcategory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><span class="skin-invert" typeof="mw:File"><a href="/wiki/Commutative_diagram" title="Commutative diagram"><img alt="A simple triangular commutative diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/60px-Commutative_diagram_for_morphism.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/90px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/120px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#e5e5ff;"><div id="Higher_category_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Higher_category_theory" title="Higher category theory">Higher category theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li class="mw-empty-elt"></li></ul></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Categorification" title="Categorification">Categorification</a></li> <li><a href="/wiki/Enriched_category" title="Enriched category">Enriched category</a></li> <li><a href="/wiki/Higher-dimensional_algebra" title="Higher-dimensional algebra">Higher-dimensional algebra</a></li> <li><a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a></li> <li><a href="/wiki/Model_category" title="Model category">Model category</a></li> <li><a href="/wiki/Simplex_category" title="Simplex category">Simplex category</a></li> <li><a href="/wiki/String_diagram" title="String diagram">String diagram</a></li> <li><a href="/wiki/Topos" title="Topos">Topos</a></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">n-categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Weak_n-category" title="Weak n-category">Weak <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bicategory" title="Bicategory">Bicategory</a> (<a href="/wiki/Pseudo-functor" title="Pseudo-functor">pseudofunctor</a>)</li> <li><a href="/wiki/Tricategory" title="Tricategory">Tricategory</a></li> <li><a href="/wiki/Tetracategory" title="Tetracategory">Tetracategory</a></li> <li><a href="/wiki/Quasi-category" title="Quasi-category">Kan complex</a></li> <li><a href="/wiki/%E2%88%9E-groupoid" title="∞-groupoid">∞-groupoid</a></li> <li><a href="/wiki/%E2%88%9E-topos" title="∞-topos">∞-topos</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strict_n-category" class="mw-redirect" title="Strict n-category">Strict <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">2-category</a> (<a href="/wiki/2-functor" title="2-functor">2-functor</a>)</li> <li><a href="/wiki/3-category" class="mw-redirect" title="3-category">3-category</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Categorification" title="Categorification">Categorified</a> concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2-group" title="2-group">2-group</a></li> <li><a href="/wiki/2-ring" title="2-ring">2-ring</a></li> <li><a href="/wiki/En-ring" title="En-ring"><i>E<sub>n</sub></i>-ring</a></li> <li>(<a href="/wiki/Traced_monoidal_category" title="Traced monoidal category">Traced</a>)(<a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">Symmetric</a>) <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a></li> <li><a href="/wiki/N-group_(category_theory)" title="N-group (category theory)">n-group</a></li> <li><a href="/wiki/N-monoid" title="N-monoid">n-monoid</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Category_theory" title="Category:Category theory">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Outline_of_category_theory" title="Outline of category theory">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Glossary_of_category_theory" title="Glossary of category theory">Glossary</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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