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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical category formed by reversing morphisms</div> <p>In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>opposite category</b> or <b>dual category</b> <i>C</i><sup>op</sup> of a given <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> <i>C</i> is formed by reversing the <a href="/wiki/Morphism" title="Morphism">morphisms</a>, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, <span class="mwe-math-element"><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^{\text{op}})^{\text{op}}=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C^{\text{op}})^{\text{op}}=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abf77ad2a317f74e84b2ff2864cf0139342ab7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.408ex; height:2.843ex;" alt="{\displaystyle (C^{\text{op}})^{\text{op}}=C}"></span>. </p> <meta property="mw:PageProp/toc" /> <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=Opposite_category&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>An example comes from reversing the direction of inequalities in a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>. So if <i>X</i> is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> and ≤ a partial order relation, we can define a new partial order relation ≤<sup>op</sup> by</li></ul> <dl><dd><dl><dd><i>x</i> ≤<sup>op</sup> <i>y</i> if and only if <i>y</i> ≤ <i>x</i>.</dd></dl></dd></dl> <dl><dd>The new order is commonly called dual order of ≤, and is mostly denoted by ≥. Therefore, <a href="/wiki/Order_theory#Duality" title="Order theory">duality</a> plays an important role in order theory and every purely order theoretic concept has a dual. For example, there are opposite pairs child/parent, descendant/ancestor, <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a>/<a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>, <a href="/wiki/Down-set" class="mw-redirect" title="Down-set">down-set</a>/<a href="/wiki/Upper_set" title="Upper set">up-set</a>, <a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideal</a>/<a href="/wiki/Filter_(mathematics)" title="Filter (mathematics)">filter</a> etc. This order theoretic duality is in turn a special case of the construction of opposite categories as every ordered set can be <a href="/wiki/Order_theory#Category_theory" title="Order theory">understood</a> as a category.</dd></dl> <ul><li>Given a <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> (<i>S</i>, ·), one usually defines the opposite semigroup as (<i>S</i>, ·)<sup>op</sup> = (<i>S</i>, *) where <i>x</i>*<i>y</i> ≔ <i>y</i>·<i>x</i> for all <i>x</i>,<i>y</i> in <i>S</i>. So also for semigroups there is a strong duality principle. Clearly, the same construction works for groups, as well, and is known in <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>, too, where it is applied to the multiplicative semigroup of the ring to give the opposite ring. Again this process can be described by completing a semigroup to a monoid, taking the <a href="/wiki/Monoid#Relation_to_category_theory" title="Monoid">corresponding</a> opposite category, and then possibly removing the unit from that monoid.</li> <li>The category of <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a> and Boolean <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a> is <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to the opposite of the category of <a href="/wiki/Stone_space" title="Stone space">Stone spaces</a> and <a href="/wiki/Continuous_function#Continuous_functions_between_topological_spaces" title="Continuous function">continuous functions</a>.</li> <li>The category of <a href="/wiki/Affine_scheme" class="mw-redirect" title="Affine scheme">affine schemes</a> is <a href="/wiki/Equivalence_(category_theory)" class="mw-redirect" title="Equivalence (category theory)">equivalent</a> to the opposite of the category of <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>.</li> <li>The <a href="/wiki/Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a> restricts to an equivalence between the category of <a href="/wiki/Compact_space#Definition" title="Compact space">compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Topological_group" title="Topological group">topological groups</a> and the opposite of the category of (discrete) abelian groups.</li> <li>By the Gelfand–Naimark theorem, the category of localizable <a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">measurable spaces</a> (with <a href="/wiki/Measurable_function" title="Measurable function">measurable maps</a>) is equivalent to the category of commutative <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebras</a> (with <a href="/wiki/Normal_operator" title="Normal operator">normal</a> <a href="/wiki/Unital_map" title="Unital map">unital</a> homomorphisms of <a href="/wiki/*-algebra" title="*-algebra">*-algebras</a>).<sup id="cite_ref-MO1_1-0" class="reference"><a href="#cite_note-MO1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Opposite_category&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Opposite preserves products: </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 (C\times D)^{\text{op}}\cong C^{\text{op}}\times D^{\text{op}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>C</mi> <mo>×</mo> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>≅</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>×</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (C\times D)^{\text{op}}\cong C^{\text{op}}\times D^{\text{op}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b044a9cfe7e95e38acd79a1886fc0ff180d84a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.906ex; height:2.843ex;" alt="{\displaystyle (C\times D)^{\text{op}}\cong C^{\text{op}}\times D^{\text{op}}}"></span> (see <a href="/wiki/Product_category" title="Product category">product category</a>)</dd></dl> <p>Opposite preserves <a href="/wiki/Functor" title="Functor">functors</a>: </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 (\mathrm {Funct} (C,D))^{\text{op}}\cong \mathrm {Funct} (C^{\text{op}},D^{\text{op}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>,</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathrm {Funct} (C,D))^{\text{op}}\cong \mathrm {Funct} (C^{\text{op}},D^{\text{op}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674b02ef3efcf443e23ef556a3984b92d8d54fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.991ex; height:2.843ex;" alt="{\displaystyle (\mathrm {Funct} (C,D))^{\text{op}}\cong \mathrm {Funct} (C^{\text{op}},D^{\text{op}})}"></span><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><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 <a href="/wiki/Functor_category" title="Functor category">functor category</a>, <a href="/wiki/Opposite_functor" class="mw-redirect" title="Opposite functor">opposite functor</a>)</dd></dl> <p>Opposite preserves slices: </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 (F\downarrow G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">↓</mo> <mi>G</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo>≅</mo> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo stretchy="false">↓</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>op</mtext> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F\downarrow G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684054f8a5129c5dfc133240cb52185a20c06e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.736ex; height:2.843ex;" alt="{\displaystyle (F\downarrow G)^{\text{op}}\cong (G^{\text{op}}\downarrow F^{\text{op}})}"></span> (see <a href="/wiki/Comma_category" title="Comma category">comma category</a>)</dd></dl> <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=Opposite_category&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dual_object" title="Dual object">Dual object</a></li> <li><a href="/wiki/Dual_(category_theory)" title="Dual (category theory)">Dual (category theory)</a></li> <li><a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">Duality (mathematics)</a></li> <li><a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">Adjoint functor</a></li> <li><a href="/wiki/Contravariant_functor" class="mw-redirect" title="Contravariant functor">Contravariant functor</a></li> <li><a href="/wiki/Opposite_functor" class="mw-redirect" title="Opposite functor">Opposite functor</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=Opposite_category&action=edit&section=4" 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-MO1-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-MO1_1-0">^</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://mathoverflow.net/q/20740">"Is there an introduction to probability theory from a structuralist/categorical perspective?"</a>. MathOverflow<span class="reference-accessdate">. Retrieved <span class="nowrap">25 October</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Is+there+an+introduction+to+probability+theory+from+a+structuralist%2Fcategorical+perspective%3F&rft.pub=MathOverflow&rft_id=https%3A%2F%2Fmathoverflow.net%2Fq%2F20740&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOpposite+category" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">(<a href="#CITEREFHerrlichStrecker1979">Herrlich & Strecker 1979</a>, p. 99)</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">O. Wyler, <i>Lecture Notes on Topoi and Quasitopoi</i>, World Scientific, 1991, p. 8.</span> </li> </ol></div></div> <ul><li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/opposite+category">Opposite category</a> at the <a href="/wiki/NLab" title="NLab"><i>n</i>Lab</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDanilov2001" class="citation cs2">Danilov, V.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Dual_Category">"Dual Category"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dual+Category&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Danilov&rft.aufirst=V.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDual_Category&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOpposite+category" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1978" class="citation book cs1">Mac Lane, Saunders (1978). <i>Categories for the Working Mathematician</i> (Second ed.). New York, NY: Springer New York. p. 33. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1441931236" title="Special:BookSources/1441931236"><bdi>1441931236</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/851741862">851741862</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories+for+the+Working+Mathematician&rft.place=New+York%2C+NY&rft.pages=33&rft.edition=Second&rft.pub=Springer+New+York&rft.date=1978&rft_id=info%3Aoclcnum%2F851741862&rft.isbn=1441931236&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOpposite+category" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAwodey2010" class="citation book cs1">Awodey, Steve (2010). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/categorytheoryse00awod"><i>Category theory</i></a></span> (2nd ed.). Oxford: Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/categorytheoryse00awod/page/n69">53</a>–55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0199237180" title="Special:BookSources/978-0199237180"><bdi>978-0199237180</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/740446073">740446073</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+theory&rft.place=Oxford&rft.pages=53-55&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=2010&rft_id=info%3Aoclcnum%2F740446073&rft.isbn=978-0199237180&rft.aulast=Awodey&rft.aufirst=Steve&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcategorytheoryse00awod&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOpposite+category" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerrlichStrecker1979" class="citation book cs1">Herrlich, Horst; Strecker, George E. (1979). <a rel="nofollow" class="external text" href="https://www.heldermann.de/SSPM/SSPM01/sspm01.htm"><i>Category Theory</i></a>. SSPM (Sigma Series in Pure Mathematics) 01. Heldermann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-88538-001-6" title="Special:BookSources/978-3-88538-001-6"><bdi>978-3-88538-001-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory&rft.series=SSPM+%28Sigma+Series+in+Pure+Mathematics%29+01&rft.pub=Heldermann&rft.date=1979&rft.isbn=978-3-88538-001-6&rft.aulast=Herrlich&rft.aufirst=Horst&rft.au=Strecker%2C+George+E.&rft_id=https%3A%2F%2Fwww.heldermann.de%2FSSPM%2FSSPM01%2Fsspm01.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOpposite+category" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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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 class="mw-selflink selflink">Opposite category</a></li> <li><a href="/wiki/Quotient_category" title="Quotient category">Quotient category</a></li> <li><a href="/wiki/Product_category" title="Product category">Product category</a></li> <li><a href="/wiki/Comma_category" title="Comma category">Comma category</a></li> <li><a href="/wiki/Subcategory" title="Subcategory">Subcategory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Commutative_diagram" title="Commutative diagram"><img alt="A simple triangular commutative diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/60px-Commutative_diagram_for_morphism.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/90px-Commutative_diagram_for_morphism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Commutative_diagram_for_morphism.svg/120px-Commutative_diagram_for_morphism.svg.png 2x" data-file-width="100" data-file-height="100" /></a></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#e5e5ff;"><div id="Higher_category_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Higher_category_theory" title="Higher category theory">Higher category theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li class="mw-empty-elt"></li></ul></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Categorification" title="Categorification">Categorification</a></li> <li><a href="/wiki/Enriched_category" title="Enriched category">Enriched category</a></li> <li><a href="/wiki/Higher-dimensional_algebra" title="Higher-dimensional algebra">Higher-dimensional algebra</a></li> <li><a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a></li> <li><a href="/wiki/Model_category" title="Model category">Model category</a></li> <li><a href="/wiki/Simplex_category" title="Simplex category">Simplex category</a></li> <li><a href="/wiki/String_diagram" title="String diagram">String diagram</a></li> <li><a href="/wiki/Topos" title="Topos">Topos</a></li> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">n-categories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Weak_n-category" title="Weak n-category">Weak <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bicategory" title="Bicategory">Bicategory</a> (<a href="/wiki/Pseudo-functor" title="Pseudo-functor">pseudofunctor</a>)</li> <li><a href="/wiki/Tricategory" title="Tricategory">Tricategory</a></li> <li><a href="/wiki/Tetracategory" title="Tetracategory">Tetracategory</a></li> <li><a href="/wiki/Quasi-category" title="Quasi-category">Kan complex</a></li> <li><a href="/wiki/%E2%88%9E-groupoid" title="∞-groupoid">∞-groupoid</a></li> <li><a href="/wiki/%E2%88%9E-topos" title="∞-topos">∞-topos</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strict_n-category" class="mw-redirect" title="Strict n-category">Strict <var style="padding-right: 1px;">n</var>-categories</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strict_2-category" title="Strict 2-category">2-category</a> (<a href="/wiki/2-functor" title="2-functor">2-functor</a>)</li> <li><a href="/wiki/3-category" class="mw-redirect" title="3-category">3-category</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Categorification" title="Categorification">Categorified</a> concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/2-group" title="2-group">2-group</a></li> <li><a href="/wiki/2-ring" title="2-ring">2-ring</a></li> <li><a href="/wiki/En-ring" title="En-ring"><i>E<sub>n</sub></i>-ring</a></li> <li>(<a href="/wiki/Traced_monoidal_category" title="Traced monoidal category">Traced</a>)(<a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">Symmetric</a>) <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a></li> <li><a href="/wiki/N-group_(category_theory)" title="N-group (category theory)">n-group</a></li> <li><a href="/wiki/N-monoid" title="N-monoid">n-monoid</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Category_theory" title="Category:Category theory">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Outline_of_category_theory" title="Outline of category theory">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" 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