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> braided monoidal category 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> braided monoidal category </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/14439/#Item_6" 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> </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_terms_of_higher_monoidal_structure'>In terms of higher monoidal structure</a></li> <li><a href='#the_2category_of_braided_monoidal_categories'>The 2-category of braided monoidal categories</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#tannaka_duality'>Tannaka duality</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Intuitively speaking, a braided monoidal category is a category with a tensor product and an isomorphism called the ‘braiding’ which lets us ‘switch’ two objects in a tensor product like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \otimes y</annotation></semantics></math>. Thus the tensor product is “commutative” in a sense, but not as coherently commutative as in a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>.</p> <p>A braided monoidal category is a special case of the notion of <a class="existingWikiWord" href="/nlab/show/braided+pseudomonoid">braided pseudomonoid</a> in a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a>.</p> <h2 id="Definition">Definition</h2> <div class="num_defn" id="BraidedMonoidalCategory"> <h6 id="definition_2">Definition</h6> <p>A <strong>braided monoidal category</strong>, or (“braided <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>”, but see there), is a <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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>:</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>→</mo><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> B_{x,y} : x \otimes y \to y \otimes x </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/braiding">braiding</a></strong>, such that the following two kinds of <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a> for all <a class="existingWikiWord" href="/nlab/show/objects">objects</a> involved (called the <strong>hexagon identities</strong> encoding the compatibility of the braiding with the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> for the tensor product):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>⊗</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>x</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⊗</mo><mi>Id</mi></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>a</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>a</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>y</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>Id</mi><mo>⊗</mo><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>y</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{B_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{B_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes B_{x,z}}{\to}& y \otimes (z \otimes x) } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>z</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>Id</mi><mo>⊗</mo><msub><mi>B</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>a</mi> <mrow><mi>z</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub><mo>⊗</mo><mi>Id</mi></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{B_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes B_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{B_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi><mo>→</mo><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)</annotation></semantics></math> denotes the components of the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^\otimes</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>If the braiding in def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a> “squares” to the identity in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>∘</mo><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>=</mo><msub><mi>id</mi> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_{y,x} \circ B_{x,y} = id_{x \otimes y}</annotation></semantics></math>, then the braided monoidal category is called a <em><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Intuitively speaking, the first hexagon identity in def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a> says we may braid <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> past <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>⊗</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y \otimes z</annotation></semantics></math> ‘all at once’ or in two steps. The second hexagon identity says that we may braid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \otimes y</annotation></semantics></math> past <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math> all at once or in two steps.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>From these axioms in def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a>, it follows that the braiding is compatible with the left and right <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>x</mi></msub><mo>:</mo><mi>I</mi><mo>⊗</mo><mi>x</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">l_x : I \otimes x \to x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>x</mi></msub><mo>:</mo><mi>x</mi><mo>⊗</mo><mi>I</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">r_x : x \otimes I \to x</annotation></semantics></math>. That is to say, for all objects <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> the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>I</mi><mo>⊗</mo><mi>x</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>B</mi> <mrow><mi>I</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mi>x</mi><mo>⊗</mo><mi>I</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msub><mi>l</mi> <mi>x</mi></msub></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><msub><mi>r</mi> <mi>x</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ I \otimes x &&\stackrel{B_{I,x}}{\to}&& x \otimes I \\ & {}_{l_x}\searrow && \swarrow_{r_x} \\ && x } </annotation></semantics></math></div> <p>commutes.</p> </div> <h3 id="in_terms_of_higher_monoidal_structure">In terms of higher monoidal structure</h3> <p>In terms of the language of <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-categories">k-tuply monoidal n-categories</a> a braided monoidal category is a <em>doubly monoidal 1-category</em> .</p> <p>Accordingly, by <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> twice, it may be identified with a <a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a> with a single <a class="existingWikiWord" href="/nlab/show/object">object</a> and a single 1-<a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>.</p> <p>However, unlike the definition of a monoidal category as a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> with one object, this identification is not trivial; a doubly-degenerate tricategory is literally a category with two monoidal structures that <a class="existingWikiWord" href="/nlab/show/interchange+law">interchange</a> up to isomorphism. It requires the <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a> to deduce an equivalence with braided monoidal categories.</p> <p>A commutative <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> is the same as a monoid <a class="existingWikiWord" href="/nlab/show/internalization">in the category of</a> monoids. Similarly, a braided monoidal category is equivalent to a monoidal-category object (that is, a <a class="existingWikiWord" href="/nlab/show/pseudomonoid">pseudomonoid</a>) in the monoidal 2-category of monoidal categories. This result goes back to the <a href="http://maths.mq.edu.au/~street/JS1.pdf">1986 paper by Joyal and Street</a>. (There is also a notion of <a class="existingWikiWord" href="/nlab/show/braided+pseudomonoid">braided pseudomonoid</a> that specializes directly in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> to braided monoidal categories.)</p> <p>A braided monoidal category is equivalently a category that is equipped with the structure of an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over</a> the <a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little 2-cubes operad</a>.</p> <p>Details are in example 1.2.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔼</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{E}[k]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Ek-Algebras">Algebras</a></li> </ul> <h3 id="the_2category_of_braided_monoidal_categories">The 2-category of braided monoidal categories</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a> BrMonCat with:</p> <ul> <li>braided monoidal categories as objects,</li> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functors</a> as morphisms,</li> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+natural+transformation">braided monoidal natural transformations</a> as 2-morphisms.</li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> is a braided monoidal category.</p> </li> <li> <p>The monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢ℳℴ𝒹</mi></mrow><annotation encoding="application/x-tex">\mathcal{GMod}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/graded+modules">graded modules</a> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (with the usual <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of graded modules</a>) can be made into a braided monoidal category in multiple ways, each one given by an invertible element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> of the base ring. Infact, all braidings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢ℳℴ𝒹</mi></mrow><annotation encoding="application/x-tex">\mathcal{GMod}</annotation></semantics></math> arise in this way (as in <a href="JoyalStreet86">Joyal, Street</a>). The braiding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mrow><mi>V</mi><mo>,</mo><mi>W</mi></mrow> <mi>u</mi></msubsup><mo>:</mo><mi>V</mi><mo>⊗</mo><mi>W</mi><mo>→</mo><mi>W</mi><mo>⊗</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">B^u_{V,W} : V \otimes W \to W \otimes V</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>↦</mo><msup><mi>u</mi> <mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></msup><mi>y</mi><mo>⊗</mo><mi>x</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> x \otimes y \mapsto u^{|x||y|} y \otimes x, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|y|</annotation></semantics></math> denote the degrees. It’s evident that the resulting braided monoidal category is symmetric if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">u^2 = 1</annotation></semantics></math>.</p> </li> <li> <p>The category of <a class="existingWikiWord" href="/nlab/show/crossed+G-set">crossed G-sets</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/braid+category">braid category</a> is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided</a> <a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a> on an object.</p> </li> <li> <p>The category of <a class="existingWikiWord" href="/nlab/show/vines">vines</a> is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided</a> <a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a> containing a <a class="existingWikiWord" href="/nlab/show/braided+monoid">braided monoid</a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="tannaka_duality">Tannaka duality</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+%28n%2Cr%29-category">k-tuply monoidal (n,r)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> </li> <li> <p><strong>braided monoidal category</strong>, <a class="existingWikiWord" href="/nlab/show/braided+monoidal+%28%E2%88%9E%2C1%29-category">braided monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+dagger+category">braided monoidal dagger category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> , <a class="existingWikiWord" href="/nlab/show/closed+monoidal+%28%E2%88%9E%2C1%29-category">closed monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-fold+monoidal+category">2-fold monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/duoidal+category">duoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+groupoid">braided monoidal groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+2-group">braided 2-group</a>, <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a></li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §XI of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li id="Borceux94"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 6.1 of: <em>Handbook of Categorical Algebra</em> Vol. 2: <em>Categories and Structures</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994)</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, Chapter 2 of: <em>Tensor Categories</em>, AMS Mathematical Surveys and Monographs <strong>205</strong> (2015) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://bookstore.ams.org/surv-205">ISBN:978-1-4704-3441-0</a>, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> <blockquote> <p>(focus on <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a>)</p> </blockquote> </li> </ul> <p>Exposition of basics of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>, Section 2: <em><a href="geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra">Basic notions of categorical algebra</a></em></li> </ul> <p>The original articles on braided monoidal categories (the published version does not completely supersede the <em>Macquarie Math Reports</em>, which have some extra results):</p> <ul> <li id="JoyalStreet85"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>: <em>Braided monoidal categories</em>, <em>Macquarie Math Reports</em> <strong>85-0067</strong> (1985) [<a href="http://web.science.mq.edu.au/~street/BMC850067.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalStreet-BraidedMonoidal85.pdf" title="pdf">pdf</a>]</p> </li> <li id="JoyalStreet86"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>: <em>Braided monoidal categories</em>, <em>Macquarie Math Reports</em> <strong>86-0081</strong> (1986) [<a href="http://maths.mq.edu.au/~street/JS1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JoyalStreet-BraidedMonoidal.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Braided tensor categories</em>, Adv. Math. <strong>102</strong> (1993) 20-78 [<a href="https://doi.org/10.1006/aima.1993.1055">doi:10.1006/aima.1993.1055</a>]</p> </li> </ul> <p>Around the same time the same definition was also proposed independently by <a class="existingWikiWord" href="/nlab/show/Lawrence+Breen">Lawrence Breen</a> in a letter to <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>:</p> <ul> <li id="Breen88"><a class="existingWikiWord" href="/nlab/show/Lawrence+Breen">Lawrence Breen</a>, <em>Une lettre à P. Deligne au sujet des <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-catégories tressées</em> (1988) (<a href="http://www.math.univ-paris13.fr/~breen/deligne.pdf">pdf</a>)</li> </ul> <p>Exposition of basic definitions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em><a href="http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf">Some definitions everyone should know</a></em> (2004)</li> </ul> <p>An elementary introduction using <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>:</p> <ul> <li id="BaezStay11"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>: <em>Physics, Topology, Logic and Computation: A Rosetta Stone</em>, in: <em>New Structures for Physics</em>, Lecture Notes in Physics <strong>813</strong> Springer (2011) 95-174 [<a href="http://arxiv.org/abs/0903.0340">arXiv:0903.0340</a>, <a href="https://doi.org/10.1007/978-3-642-12821-9_2">doi:10.1007/978-3-642-12821-9_2</a>]</li> </ul> <p>A generalisation of braidings to <em>lax braidings</em>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is not required to be invertible:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a>, <a class="existingWikiWord" href="/nlab/show/Elango+Panchadcharam">Elango Panchadcharam</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Lax braidings and the lax centre</em>, Contemporary Mathematics <strong>441</strong> 1 (2007) [<a href="http://science.mq.edu.au/~street/laxcentre.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DayPanchadcharamStreet-LaxBraidings.pdf" title="pdf">pdf</a>]</li> </ul> <p>On a construction of group-crossed tensor categories that depends on both a group action and a grading:</p> <ul> <li>Mizuki Oikawa. <em>Center construction for group-crossed tensor categories</em> (2024). (<a href="https://arxiv.org/abs/2404.09972">arXiv:2404.09972</a>).</li> </ul> <p>On a reformulation of braided monoidal categories in a manner closer to symmetric <a class="existingWikiWord" href="/nlab/show/closed+categories">closed categories</a> and braided <a class="existingWikiWord" href="/nlab/show/skew-monoidal+categories">skew-monoidal categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexei+Davydov">Alexei Davydov</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>: <em>An alternative description of braided monoidal categories</em>, Applied Categorical Structures <strong>23</strong> (2015) 279-309 [<a href="https://doi.org/10.1007/s10485-013-9338-3">doi:10.1007/s10485-013-9338-3</a>, <a href="https://www.math.uni-hamburg.de/home/runkel/PDF/bcat.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 1, 2024 at 12:47:30. See the <a href="/nlab/history/braided+monoidal+category" 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/braided+monoidal+category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14439/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/braided+monoidal+category/57" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/braided+monoidal+category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/braided+monoidal+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (57 revisions)</a> <a href="/nlab/show/braided+monoidal+category/cite" style="color: black">Cite</a> <a href="/nlab/print/braided+monoidal+category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/braided+monoidal+category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>