CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> multicategory in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> multicategory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/17607/#Item_4" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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"> <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="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/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></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></li> <ul> <li><a href='#in_components'>In components</a></li> <li><a href='#in_terms_of_cartesian_monads'>In terms of cartesian monads</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_operads'>Relation to operads</a></li> <li><a href='#relation_to_monoidal_categories'>Relation to monoidal categories</a></li> <li><a href='#exponentiability'>Exponentiability</a></li> </ul> <li><a href='#examples_and_special_cases'>Examples and special cases</a></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>Recall that a <a class="existingWikiWord" href="/nlab/show/category">category</a> consists of a collection of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> each having a single <a class="existingWikiWord" href="/nlab/show/object">object</a> as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A <em>multicategory</em> is like a category, except that one allows multiple inputs and a single output.</p> <p>Another term for <em>multicategory</em> is <em>coloured <a class="existingWikiWord" href="/nlab/show/operad">operad</a></em>.</p> <h2 id="definition">Definition</h2> <h3 id="in_components">In components</h3> <p>A <strong>multicategory</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> consists of</p> <ul> <li>A collection of <em>objects</em>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math>.</li> <li>A collection of <em>multimorphisms</em>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math>.</li> <li>A source map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">s: C_1 \to (C_0)*</annotation></semantics></math> to the collection of finite, possibly empty <a class="existingWikiWord" href="/nlab/show/list">list</a>s of objects (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">(C_0)*</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math>), and a target map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t: C_1 \to C_0</annotation></semantics></math>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: c_1, \ldots, c_n \to c</annotation></semantics></math> to indicate the source and target of a multimorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</li> <li>Identity and composition laws. The identity law is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mo lspace="verythinmathspace" rspace="0em">−</mo></msub><mo>:</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">1_{-}: C_0 \to C_1</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>c</mi></msub><mo>:</mo><mi>c</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">1_c: c \to c</annotation></semantics></math>. The composition law assigns, to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: c_1, \ldots, c_n \to c</annotation></semantics></math> together with an <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>-tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mi>i</mi></msub><mo>→</mo><msub><mi>c</mi> <mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle</annotation></semantics></math>, a composite<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c</annotation></semantics></math></div> <p>where the source is obtained by concatenating lists in the evident way.</p> </li> </ul> <p>These operations are subject to <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/identity">identity</a> axioms which the reader can probably figure out, but see for example (<a href="#Leinster">Leinster, page 35 ff.</a>), for details.</p> <p>Many people (especially non-category theorists) use the word <em>multicategory</em> or the word <em>colored <a class="existingWikiWord" href="/nlab/show/operad">operad</a></em> to mean what we would call a <em><a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a></em> / <em><a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a></em>. These are multicategories equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> on the multimorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">c_1, \ldots, c_n \to c</annotation></semantics></math> such that the composition is equivariant with respect to these actions.</p> <h3 id="in_terms_of_cartesian_monads">In terms of cartesian monads</h3> <p>An efficient abstract method for defining multicategories and related structures is through the formalism of <a class="existingWikiWord" href="/nlab/show/cartesian+monads">cartesian monads</a>. For ordinary categories, one uses the identity monad on <a class="existingWikiWord" href="/nlab/show/Set">Set</a>; for ordinary multicategories, one uses the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a> monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>*</mo><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">(-)*: Set \to Set</annotation></semantics></math>. This is a special case of the yet more general notion of <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>.</p> <p>We summarize here how the theory applies to the case of a <a class="existingWikiWord" href="/nlab/show/cartesian+monad">cartesian monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> on a category with <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>; see <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a> for the fully general context.</p> <ul> <li>First, a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span</strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/span">span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, that is, a diagram<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mover><mo>←</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>P</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">T X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y</annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span is often written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">p: X &#8696; Y</annotation></semantics></math>.</p> </li> </ul> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free monoid monad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to itself is called a <em>multigraph</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans are the 1-cells of a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>. A 2-cell between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">e, f: X &#8696; Y</annotation></semantics></math> is a 2-cell between ordinary spans from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. To horizontally compose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">e: X &#8696; Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mi>⇸</mi><mi>Z</mi></mrow><annotation encoding="application/x-tex">f: Y &#8696; Z</annotation></semantics></math>, take the ordinary span composite of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mover><mo>←</mo><mrow><mi>m</mi><mi>X</mi></mrow></mover><msup><mi>T</mi> <mn>2</mn></msup><mi>X</mi><mover><mo>←</mo><mrow><mi>T</mi><msub><mi>e</mi> <mn>1</mn></msub></mrow></mover><mi>T</mi><mi>E</mi><mover><mo>→</mo><mrow><mi>T</mi><msub><mi>e</mi> <mn>2</mn></msub></mrow></mover><mi>T</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>T</mi><mi>Y</mi><mover><mo>←</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><mi>F</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><msup><mi>T</mi> <mn>2</mn></msup><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">m: T^2 \to T</annotation></semantics></math> is the monad multiplication. The identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to itself is the span</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mover><mo>←</mo><mrow><mi>u</mi><mi>X</mi></mrow></mover><mi>X</mi><mover><mo>→</mo><mrow><msub><mn>1</mn> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">T X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">u: I \to T</annotation></semantics></math> is the monad unit. The verification of the bicategory axioms uses the cartesianness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in concert with the corresponding axioms on the bicategory of spans.</p> </li> <li> <p>A <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory</em> is defined to be a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> in the bicategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans.</p> </li> </ul> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free monoid monad on sets, then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory is a multicategory as defined above. For more examples and generalizations, see <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_operads">Relation to operads</h3> <p>A <a class="existingWikiWord" href="/nlab/show/nonpermutative+operad">nonpermutative</a> (or Stasheff-) <a class="existingWikiWord" href="/nlab/show/operad">operad</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> may be defined as an ordinary multicategory with exactly one object. Likewise, a <a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a> in any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is equivalent to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> multicategory with one object.</p> <p>More generally, the notion of <em>multi-colored <a class="existingWikiWord" href="/nlab/show/planar+operad">planar operad</a></em> is equivalent to that of multicategory, and the notion of <em>multi-colored <a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a></em> is equivalent to that of <a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a>.</p> <p>Fully generally, for each cartesian monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, there is a corresponding notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-operad, namely a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory whose underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span has the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mi>⇸</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">1 &#8696; 1</annotation></semantics></math>.</p> <p>For example, in Batanin’s approach to (weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-category">categories</a>, a <a class="existingWikiWord" href="/nlab/show/globular+operad">globular operad</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-operad, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free (strict) <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>-<a class="existingWikiWord" href="/nlab/show/strict+omega-category">category</a> monad on the category of <a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a>s.</p> <p>Ordinary (permutative/symmetric) operads, and their generalization to <a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategories</a>, can also be treated in the framework of <a class="existingWikiWord" href="/nlab/show/generalized+multicategories">generalized multicategories</a>, but they require a framework more general than that of cartesian monads.</p> <h3 id="relation_to_monoidal_categories">Relation to monoidal categories</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/2-adjunction">2-adjunction</a> between the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of multicategories and the 2-category of <a class="existingWikiWord" href="/nlab/show/strict+monoidal+categories">strict monoidal categories</a>, <a class="existingWikiWord" href="/nlab/show/strict+monoidal+functors">strict monoidal functors</a>, and <a class="existingWikiWord" href="/nlab/show/monoidal+natural+transformations">monoidal natural transformations</a>. <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="234.081pt" height="38.612pt" viewBox="0 0 234.081 38.612" version="1.2"> <defs> <g> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-1"> <path style="stroke:none;" d="M 7.890625 -1.875 L 4.328125 -9.84375 C 4.203125 -10.15625 4.0625 -10.15625 3.765625 -10.15625 L 0.609375 -10.15625 L 0.609375 -9.515625 L 2.171875 -9.515625 L 2.171875 -1.078125 C 2.171875 -0.75 2.15625 -0.734375 1.78125 -0.6875 C 1.515625 -0.65625 1.21875 -0.640625 0.953125 -0.640625 L 0.609375 -0.640625 L 0.609375 0 C 0.953125 -0.03125 2.09375 -0.03125 2.515625 -0.03125 C 2.9375 -0.03125 4.09375 -0.03125 4.4375 0 L 4.4375 -0.640625 L 4.09375 -0.640625 C 3.703125 -0.640625 3.671875 -0.640625 3.3125 -0.6875 C 2.890625 -0.734375 2.875 -0.75 2.875 -1.078125 L 2.875 -9.34375 L 2.890625 -9.34375 L 6.90625 -0.3125 C 6.984375 -0.140625 7.078125 0 7.328125 0 C 7.578125 0 7.671875 -0.140625 7.734375 -0.3125 L 11.84375 -9.515625 L 11.859375 -9.515625 L 11.859375 -0.640625 L 10.28125 -0.640625 L 10.28125 0 C 10.6875 -0.03125 12.21875 -0.03125 12.71875 -0.03125 C 13.234375 -0.03125 14.765625 -0.03125 15.171875 0 L 15.171875 -0.640625 L 13.59375 -0.640625 L 13.59375 -9.515625 L 15.171875 -9.515625 L 15.171875 -10.15625 L 12.015625 -10.15625 C 11.703125 -10.15625 11.578125 -10.15625 11.4375 -9.84375 Z M 7.890625 -1.875 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-2"> <path style="stroke:none;" d="M 5.21875 -6.546875 L 5.21875 -5.90625 C 6.140625 -5.90625 6.25 -5.90625 6.25 -5.328125 L 6.25 -2.421875 C 6.25 -1.296875 5.578125 -0.40625 4.421875 -0.40625 C 3.3125 -0.40625 3.25 -0.765625 3.25 -1.578125 L 3.25 -6.65625 L 0.625 -6.546875 L 0.625 -5.90625 C 1.53125 -5.90625 1.640625 -5.90625 1.640625 -5.328125 L 1.640625 -1.8125 C 1.640625 -0.34375 2.640625 0.09375 4.21875 0.09375 C 4.578125 0.09375 5.671875 0.09375 6.296875 -1.109375 L 6.3125 -1.109375 L 6.3125 0.09375 L 8.875 0 L 8.875 -0.640625 C 7.953125 -0.640625 7.859375 -0.640625 7.859375 -1.21875 L 7.859375 -6.65625 Z M 5.21875 -6.546875 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-3"> <path style="stroke:none;" d="M 3.21875 -10.265625 L 0.65625 -10.15625 L 0.65625 -9.515625 C 1.5625 -9.515625 1.671875 -9.515625 1.671875 -8.9375 L 1.671875 -0.640625 L 0.65625 -0.640625 L 0.65625 0 C 0.984375 -0.03125 2.046875 -0.03125 2.4375 -0.03125 C 2.84375 -0.03125 3.890625 -0.03125 4.234375 0 L 4.234375 -0.640625 L 3.21875 -0.640625 Z M 3.21875 -10.265625 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-4"> <path style="stroke:none;" d="M 3.125 -5.9375 L 5.25 -5.9375 L 5.25 -6.5625 L 3.125 -6.5625 L 3.125 -9.390625 L 2.5 -9.390625 C 2.484375 -7.9375 1.78125 -6.46875 0.3125 -6.421875 L 0.3125 -5.9375 L 1.515625 -5.9375 L 1.515625 -1.8125 C 1.515625 -0.25 2.71875 0.09375 3.78125 0.09375 C 4.890625 0.09375 5.53125 -0.734375 5.53125 -1.828125 L 5.53125 -2.625 L 4.90625 -2.625 L 4.90625 -1.84375 C 4.90625 -0.875 4.46875 -0.4375 4.015625 -0.4375 C 3.125 -0.4375 3.125 -1.4375 3.125 -1.78125 Z M 3.125 -5.9375 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-5"> <path style="stroke:none;" d="M 3.296875 -9.21875 C 3.296875 -9.828125 2.8125 -10.28125 2.234375 -10.28125 C 1.625 -10.28125 1.171875 -9.78125 1.171875 -9.21875 C 1.171875 -8.65625 1.625 -8.15625 2.234375 -8.15625 C 2.8125 -8.15625 3.296875 -8.609375 3.296875 -9.21875 Z M 0.703125 -6.546875 L 0.703125 -5.90625 C 1.5625 -5.90625 1.671875 -5.90625 1.671875 -5.328125 L 1.671875 -0.640625 L 0.65625 -0.640625 L 0.65625 0 C 0.984375 -0.03125 2.015625 -0.03125 2.40625 -0.03125 C 2.828125 -0.03125 3.765625 -0.03125 4.125 0 L 4.125 -0.640625 L 3.21875 -0.640625 L 3.21875 -6.65625 Z M 0.703125 -6.546875 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-6"> <path style="stroke:none;" d="M 6.921875 -1.734375 C 6.921875 -1.90625 6.765625 -1.90625 6.609375 -1.90625 C 6.375 -1.90625 6.359375 -1.890625 6.28125 -1.71875 C 6.1875 -1.4375 5.78125 -0.4375 4.421875 -0.4375 C 2.34375 -0.4375 2.34375 -2.6875 2.34375 -3.359375 C 2.34375 -4.234375 2.359375 -6.171875 4.296875 -6.171875 C 4.390625 -6.171875 5.265625 -6.140625 5.265625 -6.0625 C 5.265625 -6.046875 5.25 -6.03125 5.21875 -6.015625 C 5.171875 -5.984375 4.984375 -5.75 4.984375 -5.390625 C 4.984375 -4.765625 5.484375 -4.515625 5.859375 -4.515625 C 6.171875 -4.515625 6.734375 -4.703125 6.734375 -5.40625 C 6.734375 -6.609375 5.03125 -6.703125 4.234375 -6.703125 C 1.59375 -6.703125 0.5625 -4.984375 0.5625 -3.28125 C 0.5625 -1.265625 1.96875 0.09375 4.140625 0.09375 C 6.46875 0.09375 6.921875 -1.625 6.921875 -1.734375 Z M 6.921875 -1.734375 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-7"> <path style="stroke:none;" d="M 6.8125 -4.421875 C 6.8125 -5.78125 5.71875 -6.703125 3.640625 -6.703125 C 2.8125 -6.703125 1.0625 -6.625 1.0625 -5.390625 C 1.0625 -4.765625 1.53125 -4.5 1.9375 -4.5 C 2.375 -4.5 2.8125 -4.8125 2.8125 -5.375 C 2.8125 -5.65625 2.703125 -5.921875 2.453125 -6.078125 C 2.9375 -6.21875 3.296875 -6.21875 3.578125 -6.21875 C 4.59375 -6.21875 5.1875 -5.65625 5.1875 -4.4375 L 5.1875 -3.84375 C 2.859375 -3.84375 0.5 -3.203125 0.5 -1.546875 C 0.5 -0.203125 2.21875 0.09375 3.234375 0.09375 C 4.375 0.09375 5.109375 -0.53125 5.40625 -1.1875 C 5.40625 -0.640625 5.40625 0 6.890625 0 L 7.65625 0 C 7.953125 0 8.078125 0 8.078125 -0.328125 C 8.078125 -0.640625 7.953125 -0.640625 7.734375 -0.640625 C 6.8125 -0.65625 6.8125 -0.890625 6.8125 -1.234375 Z M 5.1875 -2.0625 C 5.1875 -0.671875 3.953125 -0.40625 3.484375 -0.40625 C 2.75 -0.40625 2.140625 -0.890625 2.140625 -1.5625 C 2.140625 -2.921875 3.75 -3.375 5.1875 -3.453125 Z M 5.1875 -2.0625 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-8"> <path style="stroke:none;" d="M 3.765625 -6.515625 C 2.296875 -6.84375 2.25 -7.90625 2.25 -8.109375 C 2.25 -8.859375 2.734375 -9.734375 4.171875 -9.734375 C 5.609375 -9.734375 6.890625 -9.046875 7.203125 -7.109375 C 7.25 -6.796875 7.25 -6.78125 7.53125 -6.78125 C 7.84375 -6.78125 7.84375 -6.828125 7.84375 -7.15625 L 7.84375 -9.9375 C 7.84375 -10.21875 7.84375 -10.328125 7.625 -10.328125 C 7.515625 -10.328125 7.46875 -10.3125 7.34375 -10.171875 L 6.703125 -9.4375 C 6.328125 -9.75 5.59375 -10.328125 4.15625 -10.328125 C 2.046875 -10.328125 0.921875 -9 0.921875 -7.390625 C 0.921875 -6.390625 1.453125 -5.703125 1.65625 -5.484375 C 2.421875 -4.71875 2.84375 -4.625 4.375 -4.296875 C 5.859375 -3.96875 6.03125 -3.9375 6.359375 -3.625 C 6.5625 -3.4375 6.984375 -3.015625 6.984375 -2.265625 C 6.984375 -1.625 6.671875 -0.453125 5.015625 -0.453125 C 3.71875 -0.453125 1.65625 -0.90625 1.53125 -3.0625 C 1.53125 -3.328125 1.53125 -3.375 1.234375 -3.375 C 0.921875 -3.375 0.921875 -3.328125 0.921875 -2.984375 L 0.921875 -0.203125 C 0.921875 0.078125 0.921875 0.171875 1.140625 0.171875 C 1.25 0.171875 1.25 0.15625 1.40625 0.03125 C 1.5625 -0.140625 1.703125 -0.328125 2.078125 -0.703125 C 2.9375 -0.015625 4.140625 0.171875 5.03125 0.171875 C 7.34375 0.171875 8.3125 -1.390625 8.3125 -2.953125 C 8.3125 -4.34375 7.40625 -5.6875 5.78125 -6.046875 Z M 3.765625 -6.515625 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-9"> <path style="stroke:none;" d="M 3.078125 -3.328125 C 3.078125 -3.8125 3.1875 -6.171875 4.90625 -6.171875 C 4.703125 -6 4.609375 -5.75 4.609375 -5.484375 C 4.609375 -4.890625 5.09375 -4.625 5.453125 -4.625 C 5.828125 -4.625 6.3125 -4.890625 6.3125 -5.484375 C 6.3125 -6.25 5.546875 -6.65625 4.84375 -6.65625 C 3.671875 -6.65625 3.171875 -5.640625 2.953125 -5.015625 L 2.9375 -5.015625 L 2.9375 -6.65625 L 0.515625 -6.546875 L 0.515625 -5.90625 C 1.4375 -5.90625 1.53125 -5.90625 1.53125 -5.328125 L 1.53125 -0.640625 L 0.515625 -0.640625 L 0.515625 0 C 0.859375 -0.03125 1.96875 -0.03125 2.375 -0.03125 C 2.8125 -0.03125 4.015625 -0.03125 4.359375 0 L 4.359375 -0.640625 L 3.078125 -0.640625 Z M 3.078125 -3.328125 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-10"> <path style="stroke:none;" d="M 7.84375 -3.234375 C 7.84375 -5.234375 6.484375 -6.703125 4.15625 -6.703125 C 1.75 -6.703125 0.453125 -5.171875 0.453125 -3.234375 C 0.453125 -1.296875 1.8125 0.09375 4.140625 0.09375 C 6.5625 0.09375 7.84375 -1.375 7.84375 -3.234375 Z M 4.15625 -0.4375 C 2.234375 -0.4375 2.234375 -2.171875 2.234375 -3.390625 C 2.234375 -4.078125 2.234375 -4.828125 2.515625 -5.359375 C 2.84375 -5.953125 3.5 -6.21875 4.140625 -6.21875 C 4.984375 -6.21875 5.515625 -5.8125 5.765625 -5.40625 C 6.0625 -4.875 6.0625 -4.09375 6.0625 -3.390625 C 6.0625 -2.15625 6.0625 -0.4375 4.15625 -0.4375 Z M 4.15625 -0.4375 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-11"> <path style="stroke:none;" d="M 7.859375 -4.53125 C 7.859375 -5.9375 7.25 -6.65625 5.625 -6.65625 C 4.5625 -6.65625 3.640625 -6.140625 3.140625 -5.03125 L 3.125 -5.03125 L 3.125 -6.65625 L 0.625 -6.546875 L 0.625 -5.90625 C 1.53125 -5.90625 1.640625 -5.90625 1.640625 -5.328125 L 1.640625 -0.640625 L 0.625 -0.640625 L 0.625 0 C 0.96875 -0.03125 2.03125 -0.03125 2.4375 -0.03125 C 2.859375 -0.03125 3.9375 -0.03125 4.28125 0 L 4.28125 -0.640625 L 3.25 -0.640625 L 3.25 -3.78125 C 3.25 -5.390625 4.4375 -6.171875 5.390625 -6.171875 C 5.9375 -6.171875 6.25 -5.8125 6.25 -4.6875 L 6.25 -0.640625 L 5.21875 -0.640625 L 5.21875 0 C 5.5625 -0.03125 6.625 -0.03125 7.046875 -0.03125 C 7.453125 -0.03125 8.53125 -0.03125 8.875 0 L 8.875 -0.640625 L 7.859375 -0.640625 Z M 7.859375 -4.53125 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph0-12"> <path style="stroke:none;" d="M 11.078125 -9.9375 C 11.078125 -10.28125 11.078125 -10.328125 10.75 -10.328125 L 9.640625 -9.265625 C 8.8125 -9.96875 7.828125 -10.328125 6.734375 -10.328125 C 3.203125 -10.328125 0.921875 -8.1875 0.921875 -5.078125 C 0.921875 -2.03125 3.125 0.171875 6.75 0.171875 C 9.375 0.171875 11.078125 -1.625 11.078125 -3.375 C 11.078125 -3.625 11 -3.640625 10.765625 -3.640625 C 10.625 -3.640625 10.46875 -3.640625 10.46875 -3.484375 C 10.328125 -1.234375 8.421875 -0.453125 7.09375 -0.453125 C 6.125 -0.453125 4.828125 -0.71875 4 -1.65625 C 3.484375 -2.21875 3 -3.03125 3 -5.078125 C 3 -6.53125 3.21875 -7.609375 3.921875 -8.4375 C 4.875 -9.546875 6.34375 -9.6875 7.0625 -9.6875 C 8.171875 -9.6875 9.953125 -9.078125 10.40625 -6.46875 C 10.4375 -6.328125 10.5625 -6.328125 10.75 -6.328125 C 11.078125 -6.328125 11.078125 -6.359375 11.078125 -6.71875 Z M 11.078125 -9.9375 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph1-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph1-1"> <path style="stroke:none;" d="M 3.96875 -3.703125 C 3.75 -3.671875 3.546875 -3.484375 3.546875 -3.25 C 3.546875 -3.0625 3.65625 -2.9375 3.875 -2.9375 C 4.015625 -2.9375 4.328125 -3.046875 4.328125 -3.484375 C 4.328125 -4.09375 3.6875 -4.34375 3.078125 -4.34375 C 1.75 -4.34375 1.34375 -3.40625 1.34375 -2.90625 C 1.34375 -2.8125 1.34375 -2.453125 1.703125 -2.171875 C 1.9375 -2 2.09375 -1.96875 2.609375 -1.875 C 2.953125 -1.796875 3.515625 -1.703125 3.515625 -1.1875 C 3.515625 -0.9375 3.328125 -0.609375 3.0625 -0.421875 C 2.6875 -0.1875 2.203125 -0.171875 2.046875 -0.171875 C 1.8125 -0.171875 1.140625 -0.21875 0.890625 -0.609375 C 1.40625 -0.625 1.46875 -1.03125 1.46875 -1.15625 C 1.46875 -1.453125 1.203125 -1.515625 1.078125 -1.515625 C 0.921875 -1.515625 0.515625 -1.40625 0.515625 -0.859375 C 0.515625 -0.28125 1.140625 0.09375 2.046875 0.09375 C 3.765625 0.09375 4.125 -1.109375 4.125 -1.53125 C 4.125 -2.421875 3.171875 -2.609375 2.796875 -2.671875 C 2.328125 -2.765625 1.9375 -2.828125 1.9375 -3.25 C 1.9375 -3.421875 2.109375 -4.078125 3.0625 -4.078125 C 3.4375 -4.078125 3.828125 -3.96875 3.96875 -3.703125 Z M 3.96875 -3.703125 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph1-2"> <path style="stroke:none;" d="M 3.109375 -3.1875 L 4.140625 -3.1875 C 4.953125 -3.1875 4.984375 -3.03125 4.984375 -2.75 C 4.984375 -2.6875 4.984375 -2.578125 4.921875 -2.328125 C 4.90625 -2.28125 4.90625 -2.203125 4.90625 -2.171875 C 4.90625 -2.171875 4.90625 -2.046875 5.046875 -2.046875 C 5.171875 -2.046875 5.203125 -2.140625 5.21875 -2.265625 L 5.75 -4.375 C 5.765625 -4.390625 5.78125 -4.53125 5.78125 -4.53125 C 5.78125 -4.578125 5.765625 -4.671875 5.625 -4.671875 C 5.515625 -4.671875 5.484375 -4.578125 5.46875 -4.46875 C 5.265625 -3.703125 5.03125 -3.515625 4.15625 -3.515625 L 3.203125 -3.515625 L 3.828125 -6 C 3.921875 -6.359375 3.921875 -6.375 4.328125 -6.375 L 5.78125 -6.375 C 6.953125 -6.375 7.28125 -6.125 7.28125 -5.28125 C 7.28125 -5.078125 7.234375 -4.828125 7.234375 -4.640625 C 7.234375 -4.515625 7.3125 -4.46875 7.390625 -4.46875 C 7.515625 -4.46875 7.53125 -4.5625 7.546875 -4.71875 L 7.734375 -6.40625 C 7.734375 -6.453125 7.734375 -6.515625 7.734375 -6.5625 C 7.734375 -6.703125 7.625 -6.703125 7.4375 -6.703125 L 2.390625 -6.703125 C 2.203125 -6.703125 2.078125 -6.703125 2.078125 -6.53125 C 2.078125 -6.375 2.203125 -6.375 2.359375 -6.375 C 2.4375 -6.375 2.578125 -6.375 2.734375 -6.359375 C 2.953125 -6.34375 2.984375 -6.3125 2.984375 -6.21875 C 2.984375 -6.171875 2.96875 -6.125 2.9375 -6.03125 L 1.625 -0.78125 C 1.53125 -0.40625 1.515625 -0.328125 0.78125 -0.328125 C 0.59375 -0.328125 0.484375 -0.328125 0.484375 -0.140625 C 0.484375 -0.09375 0.515625 0 0.640625 0 C 0.84375 0 1.078125 -0.03125 1.296875 -0.03125 L 2.65625 -0.03125 C 2.84375 -0.015625 3.203125 0 3.390625 0 C 3.453125 0 3.59375 0 3.59375 -0.1875 C 3.59375 -0.328125 3.484375 -0.328125 3.265625 -0.328125 C 3.0625 -0.328125 2.984375 -0.328125 2.765625 -0.34375 C 2.5 -0.375 2.46875 -0.40625 2.46875 -0.515625 C 2.46875 -0.53125 2.46875 -0.59375 2.5 -0.734375 Z M 3.109375 -3.1875 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph1-3"> <path style="stroke:none;" d="M 6.578125 -5.65625 C 6.6875 -6.140625 6.921875 -6.375 7.625 -6.40625 C 7.71875 -6.40625 7.796875 -6.46875 7.796875 -6.59375 C 7.796875 -6.65625 7.75 -6.734375 7.65625 -6.734375 C 7.59375 -6.734375 7.390625 -6.703125 6.671875 -6.703125 C 5.890625 -6.703125 5.765625 -6.734375 5.671875 -6.734375 C 5.515625 -6.734375 5.484375 -6.625 5.484375 -6.546875 C 5.484375 -6.421875 5.609375 -6.40625 5.703125 -6.40625 C 6.296875 -6.390625 6.296875 -6.125 6.296875 -5.984375 C 6.296875 -5.921875 6.296875 -5.875 6.28125 -5.8125 C 6.265625 -5.75 5.4375 -2.4375 5.40625 -2.3125 C 5.046875 -0.890625 3.8125 -0.125 2.828125 -0.125 C 2.15625 -0.125 1.53125 -0.515625 1.53125 -1.421875 C 1.53125 -1.59375 1.5625 -1.859375 1.609375 -2.015625 L 2.59375 -5.96875 C 2.6875 -6.3125 2.703125 -6.40625 3.421875 -6.40625 C 3.625 -6.40625 3.734375 -6.40625 3.734375 -6.59375 C 3.734375 -6.609375 3.734375 -6.734375 3.5625 -6.734375 C 3.375 -6.734375 3.140625 -6.71875 2.953125 -6.703125 L 2.34375 -6.703125 C 1.390625 -6.703125 1.140625 -6.734375 1.0625 -6.734375 C 1.03125 -6.734375 0.875 -6.734375 0.875 -6.546875 C 0.875 -6.40625 1 -6.40625 1.15625 -6.40625 C 1.484375 -6.40625 1.765625 -6.40625 1.765625 -6.25 C 1.765625 -6.1875 1.671875 -5.84375 1.609375 -5.640625 L 1.40625 -4.765625 L 0.890625 -2.6875 C 0.75 -2.140625 0.703125 -1.984375 0.703125 -1.71875 C 0.703125 -0.578125 1.609375 0.203125 2.796875 0.203125 C 4.125 0.203125 5.390625 -0.921875 5.703125 -2.203125 Z M 6.578125 -5.65625 "></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph2-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="7bECpJPquESxys7ihUfevwz7UZA=-glyph2-1"> <path style="stroke:none;" d="M 6.46875 5.8125 C 6.625 5.8125 6.84375 5.8125 6.84375 5.578125 C 6.84375 5.359375 6.625 5.359375 6.453125 5.359375 L 3.640625 5.359375 L 3.640625 0.96875 C 3.640625 0.796875 3.640625 0.578125 3.421875 0.578125 C 3.1875 0.578125 3.1875 0.796875 3.1875 0.96875 L 3.1875 5.359375 L 0.390625 5.359375 C 0.21875 5.359375 0 5.359375 0 5.578125 C 0 5.8125 0.21875 5.8125 0.375 5.8125 Z M 6.46875 5.8125 "></path> </symbol> </g> </defs> <g id="7bECpJPquESxys7ihUfevwz7UZA=-surface1"> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-1" x="7.079464" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-2" x="22.867557" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-3" x="32.114636" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-4" x="36.738176" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-5" x="43.211131" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-6" x="47.83467" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-7" x="55.232333" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-4" x="63.323897" y="22.812762"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-8" x="138.167204" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-4" x="147.414283" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-9" x="153.887239" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-1" x="160.685691" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-10" x="176.473784" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-11" x="184.796155" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-12" x="194.043234" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-7" x="206.064437" y="22.812762"></use> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph0-4" x="214.156" y="22.812762"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph1-1" x="220.628576" y="25.031714"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -40.57468 5.433035 L 14.638685 5.433035 " transform="matrix(0.990051,0,0,-0.990051,116.585076,18.785233)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486582 2.868435 C -2.032849 1.148196 -1.018855 0.335422 -0.000915103 0.000054756 C -1.018855 -0.335313 -2.032849 -1.148087 -2.486582 -2.868326 " transform="matrix(0.990051,0,0,-0.990051,131.313406,13.406304)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph1-2" x="99.938354" y="9.925264"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 15.116091 -6.095724 L -40.097274 -6.095724 " transform="matrix(0.990051,0,0,-0.990051,116.585076,18.785233)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.488452 2.8709 C -2.030774 1.146715 -1.020725 0.333942 0.0011598 -0.00142589 C -1.020725 -0.332848 -2.030774 -1.149567 -2.488452 -2.869806 " transform="matrix(-0.990051,0,0,0.990051,76.649586,24.821724)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph1-3" x="99.911623" y="35.042865"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#7bECpJPquESxys7ihUfevwz7UZA=-glyph2-1" x="100.557136" y="15.912104"></use> </g> </g> </svg> The left adjoint sends a multicategory to its <strong>monoidal envelope</strong>. The right adjoint sends a strict monoidal category to the multicategory with the same objects, and multimorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A_1, \ldots, A_n \to B</annotation></semantics></math> given by morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A_1 \otimes \cdots \otimes A_n \to B</annotation></semantics></math> (which is a <a class="existingWikiWord" href="/nlab/show/representable+multicategory">representable multicategory</a>).</p> <p>The right adjoint is <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a> and <span class="newWikiWord">locally fully faithful<a href="/nlab/new/locally+fully+faithful">?</a></span>, but not full (in general, the functors of multicategories correspond to <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functors">lax monoidal functors</a>). Furthermore, this 2-adjunction is <a class="existingWikiWord" href="/nlab/show/2-monadic">2-monadic</a> and <a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-adjunction">lax-idempotent</a>, and the unit is compontentwise <a class="existingWikiWord" href="/nlab/show/full+and+faithful">full and faithful</a>, i.e. for each multicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>→</mo><mi>UFM</mi></mrow><annotation encoding="application/x-tex">M \to UFM</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a>. The <a class="existingWikiWord" href="/nlab/show/pseudoalgebras">pseudoalgebras</a> for the induced <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a> are precisely the <a class="existingWikiWord" href="/nlab/show/representable+multicategories">representable multicategories</a>, equivalently the (non-strict) <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>.</p> <p>The adjunction is also <a class="existingWikiWord" href="/nlab/show/comonadic">comonadic</a>, and is consequently an example of a <a class="existingWikiWord" href="/nlab/show/nuclear+adjunction">nuclear adjunction</a>.</p> <p>There is also a <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a> 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>StrMonCat</mi> <mi>l</mi></msub><mo>→</mo><mi>Multicat</mi></mrow><annotation encoding="application/x-tex">StrMonCat_l \to Multicat</annotation></semantics></math> (abstractly, this is the forgetful functor from the 2-category of <span class="newWikiWord">strict algebras<a href="/nlab/new/strict+algebras">?</a></span> and <a class="existingWikiWord" href="/nlab/show/lax+morphisms">lax morphisms</a> for the induced 2-monad).</p> <p>For details on some of this material see (<a href="Hermida2000">Hermida 2000</a>).</p> <h3 id="exponentiability">Exponentiability</h3> <p>A multicategory is <a class="existingWikiWord" href="/nlab/show/exponentiable+object">exponentiable</a> if and only if it is <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal</a> (Proposition 2.8 of <a href="#Pisani14">Pisani 2014</a>). In particular, <a class="existingWikiWord" href="/nlab/show/representable+multicategories">representable multicategories</a> and <a class="existingWikiWord" href="/nlab/show/sequential+multicategories">sequential multicategories</a> are exponentiable. This gives an abstract construction of the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> tensor product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C, D]</annotation></semantics></math> for any <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>: it is precisely the (representable) multicategory structure on the functor multicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">D^C</annotation></semantics></math>.</p> <h2 id="examples_and_special_cases">Examples and special cases</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/permutative+categories">permutative categories</a> with multi-linear functors between them form a multicategory <a class="existingWikiWord" href="/nlab/show/PermCat">PermCat</a></li> </ul> <p>See also the examples at <em><a class="existingWikiWord" href="/nlab/show/operad">operad</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/multimorphism">multimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multifunctor">multifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polycategory">polycategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/properad">properad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+of+multicategories">fibration of multicategories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+multicategory">cartesian multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequential+multicategory">sequential multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>, <a class="existingWikiWord" href="/nlab/show/operad">operad</a></p> </li> </ul> <h2 id="references">References</h2> <p>Multicategories were introduced in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joachim+Lambek">Joachim Lambek</a>, <em>Deductive systems and categories II. Standard constructions and closed categories</em>, Category Theory, Homology Theory and their Applications I: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968 Volume One. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.</li> </ul> <p>and developed further in:</p> <ul> <li>Joachim Lambek, <em>Multicategories revisited</em>, Contemp. Math 92 (1989): 217-239.</li> </ul> <p>Hermida constructed a monadic 2-adjunction between the 2-category of nonsymmetric colored operads in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Set</mi><mo>,</mo><mo>×</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Set, \times)</annotation></semantics></math> (which he calls <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a>) and the 2-category of strict monoidal categories in Theorem 7.2 of this paper:</p> <ul> <li id="Hermida2000"><a class="existingWikiWord" href="/nlab/show/Claudio+Hermida">Claudio Hermida</a>, <em>Representable multicategories</em>, Advances in Mathematics, Volume 151, Issue 2, 2000, 164-225. (<a href="https://www.sciencedirect.com/science/article/pii/S0001870899918777/">pdf</a>)</li> </ul> <p>See also:</p> <ul id="Leinster"> <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> <ul> <li id="Pisani14"><a class="existingWikiWord" href="/nlab/show/Claudio+Pisani">Claudio Pisani</a>, <em>Sequential multicategories</em>, Theory and Applications of Categories 29.19 (2014), <a href="https://arxiv.org/abs/1402.0253">arXiv:1402.0253</a></li> </ul> <p>In this paper Elmendorf and Mandell use “multicategory” to mean what others call a <em>symmetric</em> multicategory, or symmetric colored operad:</p> <ul> <li id="ElmendorfMandell2007">A. D. Elmendorf and M. A. Mandell, <em>Permutative categories, multicategories, and algebraic K-theory</em> Algebraic & Geometric Topology 9.4 (2009): 2391-2441. (<a href="https://arxiv.org/abs/0710.0082">arXiv:0710.0082</a>)</li> </ul> <p>For more, see <a class="existingWikiWord" href="/nlab/show/operad#relation_to_symmetric_monoidal_categories">Operad: relation to symmetric monoidal categories</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on December 19, 2024 at 06:02:06. See the <a href="/nlab/history/multicategory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/multicategory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/17607/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/multicategory/38" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/multicategory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/multicategory" accesskey="S" class="navlink" id="history" rel="nofollow">History (38 revisions)</a> <a href="/nlab/show/multicategory/cite" style="color: black">Cite</a> <a href="/nlab/print/multicategory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/multicategory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>