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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #33 to #34: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></strong></p> <h2 id='sidebar_concepts'>Concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='sidebar_extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='duality'>Duality</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/duality'>duality</a></strong></p> <ul> <li> <p>abstract duality: <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a>,</p> <p><a class='existingWikiWord' href='/nlab/show/diff/Eckmann-Hilton+duality'>Eckmann-Hilton duality</a></p> </li> <li> <p>concrete duality: <a class='existingWikiWord' href='/nlab/show/diff/dualizable+object'>dual object</a>, <a class='existingWikiWord' href='/nlab/show/diff/dualizable+object'>dualizable object</a>, <a class='existingWikiWord' href='/nlab/show/diff/fully+dualizable+object'>fully dualizable object</a>, <a class='existingWikiWord' href='/nlab/show/diff/dualizing+object'>dualizing object</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/dual+vector+space'>dual vector space</a></li> </ul> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p>between <a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>higher geometry</a>/<a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gelfand+duality'>Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Langlands+program'>Langlands duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+Langlands+correspondence'>geometric Langlands duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/quantum+geometric+Langlands+correspondence'>quantum geometric Langlands duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin+dual'>Pontryagin duality</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cartier+duality'>Cartier duality</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+duality'>Poincaré duality</a> for <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifolds</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Koszul+duality'>Koszul duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Spanier-Whitehead+duality'>Spanier-Whitehead duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+duality'>Grothendieck duality</a></p> </li> </ul> <p><strong>In QFT and String theory</strong></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/duality+in+physics'>duality in physics</a></strong>, <strong><a class='existingWikiWord' href='/nlab/show/diff/duality+in+string+theory'>duality in string theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Seiberg+duality'>Seiberg duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/AGT+correspondence'>AGT conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/S-duality'>S-duality</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/electric-magnetic+duality'>electro-magnetic duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/S-duality'>Montonen-Olive duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+Langlands+correspondence'>geometric Langlands duality</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/T-duality'>T-duality</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/mirror+symmetry'>mirror symmetry</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/U-duality'>U-duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open%2Fclosed+string+duality'>open/closed string duality</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/AdS-CFT+correspondence'>AdS/CFT duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/KLT+relations'>KLT relations</a></p> </li> </ul> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#in_category_theory'>In category theory</a></li><li><a href='#opposite_functors'>Opposite functors</a></li><li><a href='#opposite_natural_transformations'>Opposite natural transformations</a></li><li><a href='#the_oppositization_1functor'>The oppositization 1-functor</a></li><li><a href='#TheOppositization2Functor'>The oppositization 2-functor</a></li><li><a href='#in_enriched_category_theory'>In enriched category theory</a></li><li><a href='#in_higher_category_theory'>In higher category theory</a></li></ul></li><li><a href='#the_nerve_of_the_opposite_category'>The nerve of the opposite category</a></li><li><a href='#classes_of_examples'>Classes of examples</a><ul><li><a href='#OppositeGroup'>Opposite group</a></li><li><a href='#opposite_of_the_opposite'>Opposite of the opposite</a></li><li><a href='#coalgebraic_structures'>Co-algebraic structures</a></li><li><a href='#duality_2'>Duality</a></li></ul></li><li><a href='#specific_examples'>Specific examples</a><ul><li><a href='#opposite_of__and_'>Opposite of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>FinSet</mi></mrow><annotation encoding='application/x-tex'>FinSet</annotation></semantics></math></a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>For a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, its <em>opposite category</em> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is the category obtained by formally reversing the direction of all its <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> (while retaining their original composition law).</p> <p>Categories generalize (are a <a class='existingWikiWord' href='/nlab/show/diff/horizontal+categorification'>horizontal categorification</a> of) <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoids</a>, <a class='existingWikiWord' href='/nlab/show/diff/group'>groups</a> and <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebras</a>, and forming the opposite category corresponds to forming the opposite of a group, of a monoid, of an algebra.</p> <h2 id='definition'>Definition</h2> <h3 id='in_category_theory'>In category theory</h3> <p>For a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, the <strong>opposite category</strong> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> has the same <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> as <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, but a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>f : x \to y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is the same as a morphism <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>y</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>f : y \to x</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and a composite of morphisms <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mi>f</mi></mrow><annotation encoding='application/x-tex'>g f</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is defined to be the composite <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mi>g</mi></mrow><annotation encoding='application/x-tex'>f g</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> <p>More precisely, <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi mathvariant='normal'>obj</mi></msub></mrow><annotation encoding='application/x-tex'>C_{\mathrm{obj}}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub></mrow><annotation encoding='application/x-tex'>C_{\mathrm{mor}}</annotation></semantics></math> are, respectively, the collections of <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> and of <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and if the structure maps of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> are</p> <ul> <li> <p>source and target: <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>s</mi> <mi>C</mi></msub><mo>,</mo><msub><mi>t</mi> <mi>C</mi></msub><mo>:</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><mo>→</mo><msub><mi>C</mi> <mi mathvariant='normal'>obj</mi></msub></mrow><annotation encoding='application/x-tex'>s_C,t_C : C_{\mathrm{mor}} \to C_{\mathrm{obj}}</annotation></semantics></math></p> </li> <li> <p>identity-assignment: <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>C</mi></msub><mo>:</mo><msub><mi>C</mi> <mi mathvariant='normal'>obj</mi></msub><mo>→</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub></mrow><annotation encoding='application/x-tex'>i_C : C_{\mathrm{obj}} \to C_{\mathrm{mor}}</annotation></semantics></math></p> </li> <li> <p>composition: <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mi>C</mi></msub><mo>:</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><msub><mo>×</mo> <mrow><msub><mi>C</mi> <mi mathvariant='normal'>obj</mi></msub></mrow></msub><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><mo>→</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub></mrow><annotation encoding='application/x-tex'>\circ_C : C_{\mathrm{mor}} \times_{C_{\mathrm{obj}}} C_{\mathrm{mor}} \to C_{\mathrm{mor}}</annotation></semantics></math></p> </li> </ul> <p>then <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is the category with</p> <ul> <li> <p>the same (isomorphic) collections of objects and morphisms</p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>C</mi> <mi>op</mi></msup><msub><mo stretchy='false'>)</mo> <mi mathvariant='normal'>obj</mi></msub><mo>:</mo><mo>=</mo><msub><mi>C</mi> <mi mathvariant='normal'>obj</mi></msub></mrow><annotation encoding='application/x-tex'>(C^{op})_{\mathrm{obj}} := C_{\mathrm{obj}}</annotation></semantics></math></p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>C</mi> <mi>op</mi></msup><msub><mo stretchy='false'>)</mo> <mi mathvariant='normal'>mor</mi></msub><mo>:</mo><mo>=</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub></mrow><annotation encoding='application/x-tex'>(C^{op})_{\mathrm{mor}} := C_{\mathrm{mor}}</annotation></semantics></math></p> </li> <li> <p>the same identity-assigning map</p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>i</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>i_{C^{op}} := i_C</annotation></semantics></math></p> </li> <li> <p><em>switched</em> source and target maps</p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>s</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>t</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>s_{C^{op}} := t_C</annotation></semantics></math></p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>s</mi> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>t_{C^{op}} := s_C</annotation></semantics></math></p> </li> <li> <p>the <em>same</em> composition operation,</p> <p><math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><mo>=</mo><msub><mo>∘</mo> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>\circ_{C^{op}} := \circ_{C}</annotation></semantics></math>.</p> <p>or more precisely, the composition operation of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∘</mo> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><msubsup><mi>C</mi> <mi mathvariant='normal'>mor</mi> <mi>op</mi></msubsup><msub><mrow /> <mrow><msup><mi>s</mi> <mi>op</mi></msup></mrow></msub><msub><mo>×</mo> <mrow><msup><mi>t</mi> <mi>op</mi></msup></mrow></msub><msubsup><mi>C</mi> <mi mathvariant='normal'>mor</mi> <mi>op</mi></msubsup><mo>=</mo><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><msub><mrow /> <mi>t</mi></msub><msub><mo>×</mo> <mi>s</mi></msub><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><msub><mrow /> <mi>s</mi></msub><msub><mo>×</mo> <mi>t</mi></msub><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><mover><mo>→</mo><mo>∘</mo></mover><msub><mi>C</mi> <mi mathvariant='normal'>mor</mi></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \circ_{C^{op}} : C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{op}} C_{\mathrm{mor}}^{op} = C_{\mathrm{mor}} {}_{t} \times_{s} C_{\mathrm{mor}} \stackrel{\simeq}{\to} C_{\mathrm{mor}} {}_{s} \times_{t} C_{\mathrm{mor}} \stackrel{\circ}{\to} C_{\mathrm{mor}} \,, </annotation></semantics></math></div> <p>where the isomorphism in the middle is the unique one induced from the universality of the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>.</p> </li> </ul> <p>Notice that hence the composition law does <em>not</em> change when passing to the opposite category. Only the interpretation of in which direction the arrows point does change. So forming the opposite category is a completely formal process. Nevertheless, due to the switch of source and target, the opposite category <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is usually far from being <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalent</a> to <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. See the examples below.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is a monoidal category, then <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>V</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>V^{op}</annotation></semantics></math> is equivalent to <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Σ</mi><mi>V</mi><msup><mo stretchy='false'>)</mo> <mi>co</mi></msup></mrow><annotation encoding='application/x-tex'>(\Sigma V)^{co}</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mi>V</mi></mrow><annotation encoding='application/x-tex'>\Sigma V</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>, i.e. <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> viewed as a one-object <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>co</mi></mrow><annotation encoding='application/x-tex'>{co}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/opposite+2-category'>opposite on 2-cells</a></p> <h3 id='opposite_functors'>Opposite functors</h3> <p>Given <a class='existingWikiWord' href='/nlab/show/diff/category'>categories</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, the <strong>opposite functor</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F:C\to D</annotation></semantics></math> is the functor <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mi>op</mi></msup><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>F^{op}:C^{op}\to D^{op}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>F</mi> <mi>obj</mi> <mi>op</mi></msubsup><mo>=</mo><msub><mi>F</mi> <mi>obj</mi></msub></mrow><annotation encoding='application/x-tex'>F^{op}_{obj}=F_{obj}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>F</mi> <mi>mor</mi> <mi>op</mi></msubsup><mo>=</mo><msub><mi>F</mi> <mi>mor</mi></msub></mrow><annotation encoding='application/x-tex'>F^{op}_{mor}=F_{mor}</annotation></semantics></math>.</p> <p>In the literature, <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>F^{op}</annotation></semantics></math> is often confused with <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>. This is unfortunate, since (for example) <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>G</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>F^{op}\to G^{op}</annotation></semantics></math> (of functors <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}\to D^{op}</annotation></semantics></math>) can be identified with natural transformations <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>G\to F</annotation></semantics></math> (and not <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>F\to G</annotation></semantics></math>).</p> <h3 id='opposite_natural_transformations'>Opposite natural transformations</h3> <p>Given <a class='existingWikiWord' href='/nlab/show/diff/category'>categories</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>F,G:C\to D</annotation></semantics></math>, the <strong>opposite natural transformation</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>t:F\to G</annotation></semantics></math> is the natural transformation <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>t</mi> <mi>op</mi></msup><mo>:</mo><msup><mi>G</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>F</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>t^{op}:G^{op}\to F^{op}</annotation></semantics></math>, induced by the same map <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>obj</mi></msub><mo>→</mo><msub><mi>D</mi> <mi>mor</mi></msub></mrow><annotation encoding='application/x-tex'>C_{obj}\to D_{mor}</annotation></semantics></math> as <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>.</p> <p>Again, in the literature <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>t</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>t^{op}</annotation></semantics></math> is often confused with <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math>.</p> <h3 id='the_oppositization_1functor'>The oppositization 1-functor</h3> <p>If we let <a class='existingWikiWord' href='/nlab/show/diff/Cat'>$Cat$</a> denote the <a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a> of <a class='existingWikiWord' href='/nlab/show/diff/strict+category'>strict</a> <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small categories</a> with <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> between them, then there is an <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>op</mi><mo lspace='verythinmathspace'>:</mo><mi>Cat</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>op \colon Cat \to Cat</annotation></semantics></math> sending each category to its opposite and each functor to its opposite, and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>op</mi> <mn>2</mn></msup><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><msub><mi>id</mi> <mi>Cat</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> op^2 \;=\; id_{Cat} \,. </annotation></semantics></math></div> <p>In fact, up to <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a>, there are only two functors <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><mi>Cat</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>F \colon Cat \to Cat</annotation></semantics></math> that are <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalences</a>: the <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity functor</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>id</mi> <mi><span><del class='diffmod'> CaT</del><ins class='diffmod'> Cat</ins></span></mi></msub></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> id_{CaT}</del><ins class='diffmod'> id_{Cat}</ins></span></annotation></semantics></math> and the oppositization functor <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>op</mi></mrow><annotation encoding='application/x-tex'>op</annotation></semantics></math>. In other words, the <a class='existingWikiWord' href='/nlab/show/diff/automorphism+2-group'>automorphism 2-group</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Aut</mi><mo stretchy='false'>(</mo><mi>Cat</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Aut(Cat)</annotation></semantics></math> of all <a class='existingWikiWord' href='/nlab/show/diff/autoequivalence+type'>autoequivalences</a> of the 1-category <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> is equivalent to the group <a class='existingWikiWord' href='/nlab/show/diff/group+of+order+2'>$\mathbb{Z}/2$</a> viewed as a 0-truncated <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a>.</p> <p>To see this, note that any autoequivalence of the 1-category <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> fixes the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math> up to unique <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>, and the <a class='existingWikiWord' href='/nlab/show/diff/arrow+category'>arrow category</a> 2 is the unique minimal generator (i.e. it is a generator and no proper <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a> is a generator) so it is also fixed up to isomorphism. Since every category is functorially a <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> of copies of 2, once we know whether the autoequivalence fixes or swaps the two maps <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>1 \to 2</annotation></semantics></math> the autoequivalence is determined up to a natural isomorphism, so every such autoequivalence is naturally isomorphic to either the identity or <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>op</mi></mrow><annotation encoding='application/x-tex'>op</annotation></semantics></math>.</p> <p>For this argument and related questions touching on higher category theory see <a href='#Toën05'>Toën (2005), Thm. 6.3</a>; <a href='#BarwickSchommerPries11'>Barwick & Schommer-Pries (2011,21), Rem. 13.16</a>; <a href='#AraGrothGutiérrez15'>Ara, Groth & Gutiérrez (2013, 15)</a> (cf. also <a href='#Campion15'>Campion 2015</a>).</p> <h3 id='TheOppositization2Functor'>The oppositization 2-functor</h3> <p>There are many advantages to treating <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math> as a 2-category with natural transformations as 2-morphisms. Then the above three constructions of the opposite category, opposite functor, and opposite natural transformation combine together to give the <strong>oppositization 2-functor</strong></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>op</mi><mo>:</mo><msup><mi>Cat</mi> <mi>co</mi></msup><mo>→</mo><mi>Cat</mi><mo>,</mo></mrow><annotation encoding='application/x-tex'>op: Cat^{co}\to Cat,</annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Cat</mi> <mi>co</mi></msup></mrow><annotation encoding='application/x-tex'>Cat^{co}</annotation></semantics></math> denotes the 2-cell dual of the <a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>, with the direction of 2-morphisms reversed and the direction of 1-morphisms preserved.</p> <h3 id='in_enriched_category_theory'>In enriched category theory</h3> <p>The definition has a direct generalization to <a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched category</a> the <strong>opposite <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched category</strong> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is defined to be the <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched category with the same objects as <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C^{op}(c,d) := C(d,c) </annotation></semantics></math></div> <p>and composition given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>σ</mi></mover><mi>C</mi><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mo>∘</mo> <mi>C</mi></msub></mrow></mover><msub><mi>C</mi> <mrow><mi>c</mi><mo>,</mo><mi>a</mi></mrow></msub><mo>=</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C^{op}(b,c)\otimes C^{op}(a,b) := C(c,b) \otimes C(b,a) \stackrel{\sigma}{\to} C(b,a) \otimes C(c,b) \stackrel{\circ_C}{\to} C_{c,a} =: C^{op}(a,c) \,. </annotation></semantics></math></div> <p>The unit maps <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>j</mi> <mi>a</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>j_a : I \to C^{op}(a,a)</annotation></semantics></math> are those of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> under the identification <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^{op}(a,a) = C(a,a)</annotation></semantics></math>.</p> <p>Note that the <a class='existingWikiWord' href='/nlab/show/diff/braiding'>braiding</a> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is used in defining composition for <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>. So, we cannot define the opposite of a <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched category if <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is merely a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a>, though <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched categories still make perfect sense in this case. If <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+category'>braided monoidal category</a> there are (at least) two ways to define “<math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>”, resulting in two different “opposite categories”: we can use either the braiding or the inverse braiding. If <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> these two definitions coincide.</p> <p>The opposite category can be regarded as a <a class='existingWikiWord' href='/nlab/show/diff/dualizable+object'>dual object</a> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal bicategory</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>Prof</mi></mrow><annotation encoding='application/x-tex'>V Prof</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-categories and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/profunctor'>profunctors</a>. (Note that this does not characterize <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> up to equivalence, but only up to <a class='existingWikiWord' href='/nlab/show/diff/Morita+equivalence'>Morita equivalence</a>, i.e. up to <a class='existingWikiWord' href='/nlab/show/diff/Cauchy+completion'>Cauchy completion</a>.) When <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is symmetric, then <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>Prof</mi></mrow><annotation encoding='application/x-tex'>V Prof</annotation></semantics></math> is also symmetric monoidal, so there is only one notion of dual object. When <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is braided, then <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mi>Prof</mi></mrow><annotation encoding='application/x-tex'>V Prof</annotation></semantics></math> is not symmetric and has two notions of dual: a left dual and a right dual. These are exactly the two different opposite categories referred to above (the “left opposite” and “right opposite”).</p> <h3 id='in_higher_category_theory'>In higher category theory</h3> <p>See</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+2-category'>opposite 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+%28infinity%2C1%29-category'>opposite (∞,1)-category</a></p> </li> </ul> <h2 id='the_nerve_of_the_opposite_category'>The nerve of the opposite category</h2> <p>The <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerve</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C^{op})</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> that is degreewise the same as <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C)</annotation></semantics></math>, but in each degree with the order of the face and the order of the degeneracy maps reversed. See <a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite quasi-category</a> for more details.</p> <h2 id='classes_of_examples'>Classes of examples</h2> <h3 id='OppositeGroup'>Opposite group</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mo>⋅</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G = (S, \cdot)</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> (or <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> or <a class='existingWikiWord' href='/nlab/show/diff/associative+unital+algebra'>associative algebra</a>, etc.) with product operation</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>⋅</mo> <mi>G</mi></msub><mo>:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'> \cdot_G : S \times S \to S </annotation></semantics></math></div> <p>the <strong>opposite group</strong> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>G</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>G^{op}</annotation></semantics></math> is the group whose underlying <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> (underlying object, underlying vector space, etc.) is the same as that of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>op</mi></msup><mo>:</mo><mo>=</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'> S^{op} := S </annotation></semantics></math></div> <p>but whose product operation is that of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> but combined with a switch of the order of the arguments:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>⋅</mo> <mrow><msup><mi>G</mi> <mi>op</mi></msup></mrow></msub><mo>:</mo><mi>S</mi><mo>×</mo><mi>S</mi><mover><mo>→</mo><mi>σ</mi></mover><mi>S</mi><mo>×</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mo>⋅</mo> <mi>G</mi></msub></mrow></mover><mi>S</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdot_{G^{op}} : S \times S \stackrel{\sigma}{\to} S \times S \stackrel{\cdot_G}{\to} S \,. </annotation></semantics></math></div> <p>So for <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>g,h \in S</annotation></semantics></math> two elements we have that their product in the opposite group is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><msub><mo>⋅</mo> <mrow><msup><mi>G</mi> <mi>op</mi></msup></mrow></msub><mi>h</mi><mo>:</mo><mo>=</mo><mi>h</mi><msub><mo>⋅</mo> <mi>G</mi></msub><mi>g</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> g \cdot_{G^{op}} h := h \cdot_{G} g \,. </annotation></semantics></math></div> <p>Now, the group <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> may be thought of as the pointed one-object <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}G</annotation></semantics></math> which is the groupoid with a single object, with <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> as its set of morphisms, and with <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>⋅</mo> <mi>G</mi></msub></mrow><annotation encoding='application/x-tex'>\cdot_G</annotation></semantics></math> its composition operation.</p> <p>Under this identification of groups with one-object categories, passing to the opposite category corresponds precisely to passing to the opposite group</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>=</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mo stretchy='false'>(</mo><msup><mi>G</mi> <mi>op</mi></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\mathbf{B}G)^{op} = \mathbf{B}(G^{op}) \,. </annotation></semantics></math></div> <h3 id='opposite_of_the_opposite'>Opposite of the opposite</h3> <p>The opposite of an opposite category is the original category:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>C</mi> <mi>op</mi></msup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>=</mo><mi>C</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (C^{op})^{op} = C \,. </annotation></semantics></math></div> <p>This is also true for <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched categories when <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is symmetric monoidal, but not when <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is merely braided. However, in the latter case we can say <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>C</mi> <mrow><mi>op</mi><mn>1</mn></mrow></msup><msup><mo stretchy='false'>)</mo> <mrow><mi>op</mi><mn>2</mn></mrow></msup><mo>=</mo><mi>C</mi><mo>=</mo><mo stretchy='false'>(</mo><msup><mi>C</mi> <mrow><mi>op</mi><mn>2</mn></mrow></msup><msup><mo stretchy='false'>)</mo> <mrow><mi>op</mi><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>(C^{op1})^{op2} = C = (C^{op2})^{op1}</annotation></semantics></math>, i.e. the two different notions of “opposite category” are inverse to each other (as is always the case for left and right dualization operations in a non-symmetric monoidal (bi)category).</p> <h3 id='coalgebraic_structures'>Co-algebraic structures</h3> <p>Every algebraic structure in a category, for instance the notion of <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> in a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, has a co-version, where in the original definition the direction of all morphisms is reversed – for instance the co-version of a monoid is a <a class='existingWikiWord' href='/nlab/show/diff/comonoid'>comonoid</a>. Of an <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a> its a <a class='existingWikiWord' href='/nlab/show/diff/coalgebra'>coalgebra</a>, etc.</p> <p>One may express this succinctly by saying that a co-structure in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an original structure in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>. For instance a <a class='existingWikiWord' href='/nlab/show/diff/comonoid'>comonoid</a> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> in <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>C^{op}</annotation></semantics></math>.</p> <h3 id='duality_2'>Duality</h3> <p>Passing to the opposite category is a realization of abstract <a class='existingWikiWord' href='/nlab/show/diff/duality'>duality</a>.</p> <p>This goes as far as <em>defining</em> some entities as objects in an opposite category–in particular, all generalizations of <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a> which characterize <a class='existingWikiWord' href='/nlab/show/diff/space+and+quantity'>spaces</a> in terms of <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebras</a>. The idea of <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+geometry'>noncommutative geometry</a> is essentially to define a category of <em>spaces</em> as the opposite category of a category of algebras. Similarly, a <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> is opposite to a <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>.</p> <div class='query'> <p>Are there examples where algebras are defined as dual to spaces?</p> </div> <p>Another example is the definition of the category of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid'>algebroids</a> as that opposite to quasi-free differential graded algebras, identifying every <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebra with its <a class='existingWikiWord' href='/nlab/show/diff/duality'>dual</a> <a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+algebra'>Chevalley-Eilenberg algebra</a>.</p> <h2 id='specific_examples'>Specific examples</h2> <h3 id='opposite_of__and_'>Opposite of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>FinSet</mi></mrow><annotation encoding='application/x-tex'>FinSet</annotation></semantics></math></h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/power+set'>power set</a>-<a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>Set</mi> <mi>op</mi></msup><mo>→</mo><mi>Bool</mi></mrow><annotation encoding='application/x-tex'> \mathcal{P} \;\colon\; Set^{op} \to Bool </annotation></semantics></math></div> <p>constitutes an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a> from the opposite category of <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> to that of <a class='existingWikiWord' href='/nlab/show/diff/complete+Boolean+algebra'>complete atomic Boolean algebras</a>. See at <em><a href='Set#OppositeCategory'>Set – Properties – Opposite category and Boolean algebras</a></em></p> <p>Restricted to <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite sets</a> this says that the opposite of the category <a class='existingWikiWord' href='/nlab/show/diff/FinSet'>FinSet</a> of <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite sets</a> is <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalent</a> to the category of finite <a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>boolean algebras</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>FinSet</mi> <mi>op</mi></msup><mo>≃</mo><mi>FinBoolAlg</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> FinSet^{op} \simeq FinBoolAlg \,. </annotation></semantics></math></div> <p>See at <em><a href='FinSet#OppositeCategory'>FinSet – Properties – Opposite category</a></em>. See at <em><a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a></em> for more.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+2-category'>opposite 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+model+structure'>opposite model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+%28infinity%2C1%29-category'>opposite (infinity,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+adjunction'>opposite adjunction</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The basic notion – §II.2 of:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]</li> </ul> <p>In the generality of <a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a> – p. 12 of:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Max+Kelly'>Max Kelly</a>, <em>Basic Concepts of Enriched Category Theory</em>, Lecture Notes in Mathematics <strong>64</strong>, Cambridge University Press (1982)</p> <p>Republished as: Reprints in Theory and Applications of Categories <strong>10</strong> (2005) 1-136 [[tac:10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), <a href='http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf'>pdf</a>]</p> </li> </ul> <p>On classification of the autoequivalences of <a class='existingWikiWord' href='/nlab/show/diff/Cat'>$Cat$</a> (and more generally of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29Cat'>$(\infty,1)Cat$</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29Cat'>$(\infinity,n)Cat$</a> and of <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29Operad'>$(\infty,1)Operad$</a>):</p> <ul> <li id='Toën05'> <p><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+To%C3%ABn'>Bertrand Toën</a>, <em>Vers une axiomatisation de la théorie des catégories supérieures</em>, K-Theory <strong>34</strong> 3 (2005) 233-263 [[arXiv:math/0409598](https://arxiv.org/abs/math/0409598), <a href='http://dx.doi.org/10.1007/s10977-005-4556-6'>doi:10.1007/s10977-005-4556-6</a>]</p> </li> <li id='BarwickSchommerPries11'> <p><a class='existingWikiWord' href='/nlab/show/diff/Clark+Barwick'>Clark Barwick</a>, <a class='existingWikiWord' href='/nlab/show/diff/Chris+Schommer-Pries'>Chris Schommer-Pries</a>, Rem. 13.16 in: <em>On the Unicity of the Homotopy Theory of Higher Categories</em>, J. Amer. Math. Soc. <strong>34</strong> (2021) 1011-1058 [[arXiv:1112.0040](http://arxiv.org/abs/1112.0040), <a href='http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/'>slides</a>, <a href='https://doi.org/10.1090/jams/972'>doi:10.1090/jams/972</a>]</p> </li> <li id='AraGrothGutiérrez15'> <p><a class='existingWikiWord' href='/nlab/show/diff/Dimitri+Ara'>Dimitri Ara</a>, <a class='existingWikiWord' href='/nlab/show/diff/Moritz+Groth'>Moritz Groth</a>, <a class='existingWikiWord' href='/nlab/show/diff/Javier+Guti%C3%A9rrez'>Javier J. Gutiérrez</a>: <em>On autoequivalences of the <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category of <math class='maruku-mathml' display='inline' id='mathml_1e5aafe91f607d94948ff3885b95e7b0f585858a_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-operads</em>, Mathematische Zeitschrift <strong>281</strong> 3 (2015) 807-848 [[arXiv:1312.4994](https://arxiv.org/abs/1312.4994), <a href='https://doi.org/10.1007/s00209-015-1509-5'>doi:10.1007/s00209-015-1509-5</a>]</p> </li> </ul> <p>and discussion for the case of <a class='existingWikiWord' href='/nlab/show/diff/Ho%28Cat%29'>Ho(Cat)</a>:</p> <ul> <li id='Campion15'><a class='existingWikiWord' href='/nlab/show/diff/Tim+Campion'>Tim Campion</a>, <em>Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences?</em> (2015) [[MO:q/223424](https://mathoverflow.net/q/223424/381)]</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 6, 2024 at 22:37:33. 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