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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Opetopes</em> are one of the <a class="existingWikiWord" href="/nlab/show/geometric+shapes+for+higher+structures">geometric shapes</a> of cells in the approach to the <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> of <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-categories">∞-categories</a> put forward in (<a href="#BaezDolan97">Baez-Dolan 97</a>) and developed by (<a href="#Makkai">Makkai</a>) and others: <em><a class="existingWikiWord" href="/nlab/show/opetopic+%E2%88%9E-categories">opetopic ∞-categories</a></em>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/syntax">syntactic</a> formalization of <a class="existingWikiWord" href="/nlab/show/opetopic+%E2%88%9E-categories">opetopic ∞-categories</a> in the <a href="#opetopic+omega-category#DefinitionByPalm">variant by Palm</a> is <em><a class="existingWikiWord" href="/nlab/show/opetopic+type+theory">opetopic type theory</a></em> (<a href="#Finster12">Finster 12</a>).</p> <p> <div class='num_remark' id='Etymology'> <h6>Remark</h6> <p><strong>(etymology)</strong> Judging from the abstract of <a href="#BaezDolan97">Baez & Dolan 1997</a>, the word “opetope” seems to derive from <em><a class="existingWikiWord" href="/nlab/show/operad">operad</a></em>/<em>operation</em> + <em><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a></em>. The paper notes that the first two syllables are meant to be pronounced as in “operation”.</p> </div> </p> <h2 id="references">References</h2> <p>An overview is in <a href="http://cheng.staff.shef.ac.uk/guidebook/guidebook-new.pdf#page=63">chapter 4</a> of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, <a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, <em>Higher dimensional categories: an illustrated guidebook</em> (<a href="http://eugeniacheng.com/wp-content/uploads/2017/02/cheng-lauda-guidebook.pdf">pdf</a>)</li> </ul> <p>and in chapter 7 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Higher operads, higher categories</em>, London Math. Soc. Lec. Note Series <strong>298</strong>, <a href="http://arxiv.org/abs/math.CT/0305049">math.CT/0305049</a></li> </ul> <p>Opetopes were introduced here:</p> <ul> <li id="BaezDolan97"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/James+Dolan">James Dolan</a>, Higher-dimensional algebra III: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories and the algebra of opetopes, <em>Adv. Math.</em> <strong>135</strong> (1998), 145–206. (<a href="http://arxiv.org/abs/q-alg/9702014">arXiv:q-alg/9702014</a>)</li> </ul> <p>Some mistakes were corrected in subsequent papers:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, The category of opetopes and the category of opetopic sets,</p> <p><em>Th. Appl. Cat.</em> <strong>11</strong> (2003), 353–374. <a href="http://arxiv.org/abs/math/0304284">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, Structures in higher-dimensional</p> <p>category theory. (<a href="http://arxiv.org/abs/math/0109021">arXiv</a>)</p> </li> </ul> <p>Makkai and collaborators introduced a slight variation they called ‘multitopes’:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Claudio+Hermida">Claudio Hermida</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Makkai">Michael Makkai</a>, <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <em>On weak higher-dimensional categories I, II</em> <em>Jour. Pure Appl. Alg.</em> <strong>157</strong> (2001), 221–277 (<a href="http://www.sciencedirect.com/science/article/pii/S0022404999001796">journal</a>, <a href="http://www.math.mcgill.ca/makkai/multitopicsets/">ps.gz files</a>)</p> </li> <li id="Makkai"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Makkai">Michael Makkai</a>, The multitopic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-category of all multitopic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories.</p> <p>(<a href="http://www.math.mcgill.ca/makkai/mltomcat04/">web</a>)</p> </li> </ul> <p>Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, Weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories: opetopic and multitopic foundations, <em>Jour. Pure Appl. Alg.</em> <strong>186</strong> (2004), 109–137.(<a href="http://arxiv.org/abs/math/0304277">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, Weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories: comparing opetopic foundations, <em>Jour. Pure Appl. Alg.</em> <strong>186</strong> (2004), 219–231.</p> <p>(<a href="http://arxiv.org/abs/math/0304279">arXiv</a>)</p> </li> </ul> <p>She has also shown that opetopic <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a> are “the same” as the ordinary kind:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, Opetopic bicategories: comparison with the classical theory. (<a href="http://arxiv.org/abs/math/0304285">arXiv</a>)</li> </ul> <p>A higher dimensional <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-notation for opetopes was introduced (as “zoom complexes” in section 1.1) in</p> <ul> <li id="KockJoyalBataninMascari07"><a class="existingWikiWord" href="/nlab/show/Joachim+Kock">Joachim Kock</a>, <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Batanin">Michael Batanin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Fran%C3%A7ois+Mascari">Jean-François Mascari</a>, <em>Polynomial functors and opetopes</em> (<a href="http://arxiv.org/abs/0706.1033">arXiv:0706.1033</a>)</li> </ul> <p>Animated exposition of this higher-dimensional string-diagram notation is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <em>Opetopic Diagrams 1 - Basics</em> (<a href="http://www.youtube.com/watch?v=OANwLohwJqk">video</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <em>Opetopic Diagrams 2 - Geometry</em> (<a href="http://www.youtube.com/watch?v=E7OvuA1jRKM">video</a>)</p> </li> </ul> <p>The variant of <a href="opetopic+omega-category#DefinitionByPalm">Palm opetopic omega-categories</a> is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thorsten+Palm">Thorsten Palm</a>, …</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/syntax">syntactic</a> formalization of <a class="existingWikiWord" href="/nlab/show/opetopic+omega-categories">opetopic omega-categories</a> in terms of <a class="existingWikiWord" href="/nlab/show/opetopic+type+theory">opetopic type theory</a> is in</p> <ul> <li id="Finster12"><a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <em>Type theory and the opetopes</em>, talk at HDACT Ljubljana, June 2012 (<a href="http://sma.epfl.ch/~finster/opetope/types-and-opetopes.pdf">pdf</a>)</li> </ul> <p>Something like an implementation of aspects of opetopic type theory <em>within</em> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is described in</p> <ul> <li id="Finster18"><a class="existingWikiWord" href="/nlab/show/Eric+Finster">Eric Finster</a>, <em><a class="existingWikiWord" href="/homotopytypetheory/show/Eric+Finster%2C+Towards+Higher+Universal+Algebra+in+Type+Theory">Towards Higher Universal Algebra in Type Theory</a></em>, <a href="https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html">Homotopy Type Theory Electronic Seminar</a> 2018 (<a href="https://www.youtube.com/watch?v=hlCVHVtAlqQ">recording</a>, <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> <a href="https://github.com/ericfinster/higher-alg">code</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 30, 2024 at 16:38:43. 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