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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="knot_theory">Knot theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/knot">knot</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/link">link</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+complement">knot complement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+diagrams">knot diagrams</a>, <a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reidemeister+move">Reidemeister move</a></p> </li> </ul> <p><strong>Examples/classes:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trefoil+knot">trefoil knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/torus+knot">torus knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+knot">singular knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+knot">hyperbolic knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borromean+link">Borromean link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+link">Whitehead link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+link">Hopf link</a></p> </li> </ul> <p><strong>Types</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+knot">prime knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mutant+knot">mutant knot</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+invariants">knot invariants</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/crossing+number">crossing number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bridge+number">bridge number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unknotting+number">unknotting number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colorability">colorability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+group">knot group</a></p> </li> <li> <p><span class="newWikiWord">knot genus<a href="/nlab/new/knot+genus">?</a></span></p> </li> <li> <p>polynomial knot invariants</p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jones+polynomial">Jones polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HOMFLY+polynomial">HOMFLY polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+polynomial">Alexander polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+invariants">Reshetikhin-Turaev invariants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+knot+invariants">Vassiliev knot invariants</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">pertrubative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a></p> </li> <li> <p><span class="newWikiWord">Kauffman bracket<a href="/nlab/new/Kauffman+bracket">?</a></span></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/link+invariants">link invariants</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Milnor+mu-bar+invariants">Milnor mu-bar invariants</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linking+number">linking number</a></p> </li> </ul> <p><strong>Related concepts:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+skein+relation">Vassiliev skein relation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seifert+surface">Seifert surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+knot+theory">virtual knot theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a>, <a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/volume+conjecture">volume conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/knot+theory">knot theory</a></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> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_tangles'>Relation to tangles</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>A <strong>ribbon category</strong> [<a href="#ReshetikhinTuraev90">Reshetikhin & Turaev (1990)</a>] (also called a <strong>tortile category</strong> [<a href="#JoyalStreet93">Joyal & Street (1993)</a>, <a href="#Shum94">Shum 1994</a>, <a href="#Selinger11">Selinger 2011 §4.7</a>] or <strong>balanced rigid braided tensor category</strong>) 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><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>𝟙</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, \mathbb{1}, \alpha, l, r)</annotation></semantics></math> equipped with <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>β</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\beta=\{\beta_{X,Y}\}</annotation></semantics></math>, twist <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>θ</mi> <mi>X</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\theta=\{\theta_X\}</annotation></semantics></math> and duality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>∨</mo><mo>,</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vee, b, d)</annotation></semantics></math> that satisfy some compatibility conditions.</p> <h2 id="definition">Definition</h2> <p>Recall that:</p> <p> <div class='num_defn' id='BraidedMonoidalCategory'> <h6>Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></strong> 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/braiding">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>, which is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo lspace="verythinmathspace">:</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">\beta_{X,Y}\colon X \otimes Y \to Y \otimes X</annotation></semantics></math> obeying the <a class="existingWikiWord" href="/nlab/show/hexagon+identities">hexagon identities</a>.</p> </div> </p> <p> <div class='num_defn' id='RigidMonoidalCategory'> <h6>Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a></strong> is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> (Def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a>) where for every <a class="existingWikiWord" href="/nlab/show/object">object</a> <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>, there exist objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>∨</mo></msup></mrow><annotation encoding="application/x-tex">X^{\vee}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><msup><mo></mo><mo>∨</mo></msup></mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">{^{\vee}}X</annotation></semantics></math> (called its <a class="existingWikiWord" href="/nlab/show/dualisable+object">right and left dual</a>) and associated morphisms</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> <mi>X</mi></msub><mo>:</mo><mi>𝟙</mi><mo>→</mo><msup><mi>X</mi> <mo>∨</mo></msup><mo>⊗</mo><mi>X</mi><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>⊗</mo><msup><mi>X</mi> <mo>∨</mo></msup><mo>→</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">b_X:\mathbb{1}\to X^{\vee}\otimes X, d_X: X\otimes X^{\vee}\to \mathbb{1}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>X</mi></msub><mo>:</mo><mi>𝟙</mi><mo>→</mo><mi>X</mi><mo>⊗</mo><mrow><msup><mo></mo><mo>∨</mo></msup></mrow><mi>X</mi><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo>:</mo><mrow><msup><mo></mo><mo>∨</mo></msup></mrow><mi>X</mi><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">b_X:\mathbb{1}\to X\otimes {^{\vee}}X, d_X: {^{\vee}}X\otimes X\to \mathbb{1}</annotation></semantics></math></div> <p>obeying the <a class="existingWikiWord" href="/nlab/show/zig-zag+identities">zig-zag identities</a>:</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><msub><mi>d</mi> <mi>X</mi></msub><mo>⊗</mo><msub><mtext>id</mtext> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mtext>id</mtext> <mi>X</mi></msub><mo>⊗</mo><msub><mi>b</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mtext>id</mtext> <mi>X</mi></msub><mo>,</mo></mrow><annotation encoding="application/x-tex">(d_X\otimes \text{id}_X)\circ (\text{id}_X\otimes b_{X})=\text{id}_{X},</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mtext>id</mtext> <mrow><msup><mi>X</mi> <mo>∨</mo></msup></mrow></msub><mo>⊗</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mi>X</mi></msub><mo>⊗</mo><msub><mtext>id</mtext> <mrow><msup><mi>X</mi> <mo>∨</mo></msup></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mtext>id</mtext> <mrow><msup><mi>X</mi> <mo>∨</mo></msup></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">(\text{id}_{X^{\vee}}\otimes d_{X})\circ ( b_{X}\otimes \text{id}_{X^{\vee}})=\text{id}_{X^{\vee}}.</annotation></semantics></math></div> <p></p> </div> </p> <p>Now:</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <strong>twist</strong> on rigid braided monoidal category (Def. <a class="maruku-ref" href="#RigidMonoidalCategory"></a>) is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> from the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> to itself, with components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\theta_X \colon X \to X</annotation></semantics></math> for which</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mrow><mi>X</mi><mo>⊗</mo><mi>Y</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>β</mi> <mrow><mi>Y</mi><mo>,</mo><mi>X</mi></mrow></msub><mo>∘</mo><msub><mi>β</mi> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo>∘</mo><msub><mi>θ</mi> <mi>X</mi></msub><mo>⊗</mo><msub><mi>θ</mi> <mi>Y</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \theta_{X\otimes Y} \;=\; \beta_{Y,X} \circ \beta_{X,Y} \circ \theta_{X}\otimes \theta_{Y} \,, </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>𝟙</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">id</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \theta_{\mathbb{1}} \;=\; \mathrm{id} \,, </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mrow><msup><mi>X</mi> <mo>∨</mo></msup></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>θ</mi> <mi>X</mi> <mo>∨</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \theta_{X^{\vee}} \;=\; \theta_{X}^{\vee} \,. </annotation></semantics></math></div> <p>A <strong>ribbon category</strong> (<em>tortile category</em>) is a rigid braided monoidal category equipped with such a twist.</p> </div> (e.g. <a href="#Shum94">Shum 1994 Def. 1.3</a>).</p> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between ribbon categories is a <strong>ribbon functor</strong> (<em>tortile functor</em>) if it preserves all this structure up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_tangles">Relation to tangles</h3> <p> <div class='num_prop' id='ShumTheorem'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Shum%27s+theorem">Shum's theorem</a>)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/category+of+framed+oriented+tangles">category of framed oriented tangles</a> is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalently</a> the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a> generated by a single <a class="existingWikiWord" href="/nlab/show/object">object</a>.</p> </div> (<a href="#Shum94">Shum 1994</a>, <a href="#Yetter01">Yetter 2001 Thm. 9.1</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></li> </ul> <h2 id="references">References</h2> <ul> <li id="ReshetikhinTuraev90"> <p><a class="existingWikiWord" href="/nlab/show/Nicolai+Reshetikhin">Nicolai Reshetikhin</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, <em>Ribbon graphs and their invariants derived from quantum groups</em>, Commun. Math. Phys. <strong>127</strong> 1 (1990) [<a href="https://doi.org/10.1007/BF02096491">doi:10.1007/BF02096491</a>]</p> </li> <li id="JoyalStreet93"> <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>, Advances in Mathematics <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> <li id="Turaev94"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Turaev">Vladimir Turaev</a>, §I.1 in: <em>Quantum invariants of knots and 3-manifolds</em>, de Gruyter Studies in Mathematics <strong>18</strong>, de Gruyter & Co. (1994) [<a href="https://doi.org/10.1515/9783110435221">doi:10.1515/9783110435221</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/turaev5.pdf">pdf</a>]</p> </li> <li id="Shum94"> <p><a class="existingWikiWord" href="/nlab/show/Mei+Chee+Shum">Mei Chee Shum</a>, <em>Tortile tensor categories</em>, Journal of Pure and Applied Algebra <strong>93</strong> 1 (1994) 57-110 [<a href="https://doi.org/10.1016/0022-4049(92)00039-T">10.1016/0022-4049(92)00039-T</a>]</p> </li> <li id="Yetter01"> <p><a class="existingWikiWord" href="/nlab/show/David+N.+Yetter">David N. Yetter</a>: <em>Functorial Knot Theory – Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants</em>, Series on Knots and Everything <strong>26</strong>, World Scientific (2001) [<a href="https://doi.org/10.1142/4542">doi:10.1142/4542</a>]</p> </li> <li id="Selinger11"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Selinger">Peter Selinger</a>, §4.7 in: <em>A survey of graphical languages for monoidal categories</em>, Springer Lecture Notes in Physics <strong>813</strong> (2011) 289-355 [<a href="https://arxiv.org/abs/0908.3347">arXiv:0908.3347</a>, <a href="https://doi.org/10.1007/978-3-642-12821-9_4">doi:10.1007/978-3-642-12821-9_4</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Bump">Daniel Bump</a>: <em>Ribbon categories</em>, lecture 4 of <em><a href="http://sporadic.stanford.edu/quantum/">Quantum Groups</a></em> (2022) [slides: <a href="http://sporadic.stanford.edu/quantum/lecture4.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Bump-RibbonCategories.pdf" title="pdf">pdf</a>, notes: <a href="http://sporadic.stanford.edu/quantum/Lecture-10-13-22.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 31, 2024 at 19:41:12. 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