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id="mw-revision-name"><a href="/wiki/User:Smimram" class="mw-userlink" title="User:Smimram" data-mw-revid="146505258"><bdi>Smimram</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Smimram" class="mw-usertoollinks-talk" title="User talk:Smimram">talk</a> | <a href="/wiki/Special:Contributions/Smimram" class="mw-usertoollinks-contribs" title="Special:Contributions/Smimram">contribs</a>)</span></span> at <span id="mw-revision-date">10:59, 23 July 2007</span><span id="mw-revision-summary"> <span class="comment">(<span class="autocomment"><a href="#Categories_with_duals">→<bdi dir="ltr">Categories with duals</bdi></a></span>)</span></span>. The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this revision, which may differ significantly from the <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/wiki/Dual_object">current revision</a></span>.</b></p><div id="revision-info-plain" style="display: none;">Revision as of 10:59, 23 July 2007 by <a href="/wiki/User:Smimram" class="mw-userlink" title="User:Smimram" data-mw-revid="146505258"><bdi>Smimram</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Smimram" class="mw-usertoollinks-talk" title="User talk:Smimram">talk</a> | <a href="/wiki/Special:Contributions/Smimram" class="mw-usertoollinks-contribs" title="Special:Contributions/Smimram">contribs</a>)</span> <span class="comment">(<span class="autocomment"><a href="#Categories_with_duals">→<bdi dir="ltr">Categories with duals</bdi></a></span>)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Dual_object&diff=prev&oldid=146505258" title="Dual object">diff</a>) <a href="/w/index.php?title=Dual_object&direction=prev&oldid=146505258" title="Dual object">← Previous revision</a> | <a href="/wiki/Dual_object" title="Dual object">Latest revision</a> (<a href="/w/index.php?title=Dual_object&diff=cur&oldid=146505258" title="Dual object">diff</a>) | <a href="/w/index.php?title=Dual_object&direction=next&oldid=146505258" title="Dual object">Newer revision →</a> (<a href="/w/index.php?title=Dual_object&diff=next&oldid=146505258" title="Dual object">diff</a>)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>An object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> of a <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {C} ,\otimes ,I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>,</mo> <mo>⊗</mo> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {C} ,\otimes ,I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d3e8ef7179f9a1493fe6159740d09500af37ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.788ex; height:2.843ex;" alt="{\displaystyle (\mathbf {C} ,\otimes ,I)}"></span> admits a <b>dual</b> when there exists an object <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </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/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}"></span> together with two morphisms </p> <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 \eta _{A}:I\to A^{*}\otimes A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi>I</mi> <mo stretchy="false">→</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⊗</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{A}:I\to A^{*}\otimes A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc4c593cb99ec059659dba0fabe33afa3ab0430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.724ex; height:2.843ex;" alt="{\displaystyle \eta _{A}:I\to A^{*}\otimes A}"></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 \varepsilon _{A}:A\otimes A^{*}\to I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo>⊗</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{A}:A\otimes A^{*}\to I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13067f903c75e186619490366e6584725a2f7c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.652ex; height:2.676ex;" alt="{\displaystyle \varepsilon _{A}:A\otimes A^{*}\to I}"></span></dd></dl> <p>such that </p> <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 \lambda _{A}\circ (\varepsilon _{A}\otimes A)\circ \alpha _{A,A^{*},A}^{-1}\circ (A\otimes \eta _{A})\circ \rho _{A}^{-1}=\mathrm {id} _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>,</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊗</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∘</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{A}\circ (\varepsilon _{A}\otimes A)\circ \alpha _{A,A^{*},A}^{-1}\circ (A\otimes \eta _{A})\circ \rho _{A}^{-1}=\mathrm {id} _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d6de748856fa9be8059095885e6b2051e79213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:46.754ex; height:3.843ex;" alt="{\displaystyle \lambda _{A}\circ (\varepsilon _{A}\otimes A)\circ \alpha _{A,A^{*},A}^{-1}\circ (A\otimes \eta _{A})\circ \rho _{A}^{-1}=\mathrm {id} _{A}}"></span></dd></dl> <p>and </p> <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 \rho _{A^{*}}\circ (A^{*}\otimes \varepsilon _{A})\circ \alpha _{A^{*},A,A^{*}}\circ (\eta _{A}\otimes A^{*})\circ \lambda _{A^{*}}^{-1}=\mathrm {id} _{A^{*}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⊗</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∘</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <mi>A</mi> <mo>,</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∘</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{A^{*}}\circ (A^{*}\otimes \varepsilon _{A})\circ \alpha _{A^{*},A,A^{*}}\circ (\eta _{A}\otimes A^{*})\circ \lambda _{A^{*}}^{-1}=\mathrm {id} _{A^{*}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36b037f2fd128181fd236ac3ed91739bbb96729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.358ex; height:3.509ex;" alt="{\displaystyle \rho _{A^{*}}\circ (A^{*}\otimes \varepsilon _{A})\circ \alpha _{A^{*},A,A^{*}}\circ (\eta _{A}\otimes A^{*})\circ \lambda _{A^{*}}^{-1}=\mathrm {id} _{A^{*}}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Categories_with_duals">Categories with duals</h2></div> <p>A monoidal category where every object has a left (resp. right) dual is sometimes called a left (resp. right) <b>autonomous category</b>. A monoidal with both left and right duals is sometimes called an <i>autonomous category</i> or a <b>pivotal category</b>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <p><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAndré_Joyal_and_Ross_Street" class="citation journal cs1">André Joyal and Ross Street. "The Geometry of Tensor calculus II". <i>Synthese Lib</i>. <b>259</b>: 29–68.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Synthese+Lib&rft.atitle=The+Geometry+of+Tensor+calculus+II&rft.volume=259&rft.pages=29-68&rft.au=Andr%C3%A9+Joyal+and+Ross+Street&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADual+object" class="Z3988"></span> </p> <style data-mw-deduplicate="TemplateStyles:r1012311289">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/File:Associatividadecat.svg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Associatividadecat.svg/30px-Associatividadecat.svg.png" decoding="async" width="30" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Associatividadecat.svg/45px-Associatividadecat.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Associatividadecat.svg/60px-Associatividadecat.svg.png 2x" data-file-width="512" data-file-height="595" /></a></span></td><td><p class="asbox-body">This <a href="/wiki/Category_theory" title="Category theory">category theory</a>-related article is a <a href="/wiki/Wikipedia:Stub" title="Wikipedia:Stub">stub</a>. 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