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<html><head><meta charset="utf-8"><title>Categories.Category.Monoidal.Braided</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="69" class="Keyword">open</a> <a id="74" class="Keyword">import</a> <a id="81" href="Categories.Category.Monoidal.Core.html" class="Module">Categories.Category.Monoidal.Core</a> <a id="115" class="Keyword">using</a> <a id="121" class="Symbol">(</a><a id="122" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a><a id="130" class="Symbol">)</a>
<a id="133" class="Keyword">module</a> <a id="140" href="Categories.Category.Monoidal.Braided.html" class="Module">Categories.Category.Monoidal.Braided</a> <a id="177" class="Symbol">{</a><a id="178" href="Categories.Category.Monoidal.Braided.html#178" class="Bound">o</a> <a id="180" href="Categories.Category.Monoidal.Braided.html#180" class="Bound">ℓ</a> <a id="182" href="Categories.Category.Monoidal.Braided.html#182" class="Bound">e</a><a id="183" class="Symbol">}</a> <a id="185" class="Symbol">{</a><a id="186" href="Categories.Category.Monoidal.Braided.html#186" class="Bound">C</a> <a id="188" class="Symbol">:</a> <a id="190" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="199" href="Categories.Category.Monoidal.Braided.html#178" class="Bound">o</a> <a id="201" href="Categories.Category.Monoidal.Braided.html#180" class="Bound">ℓ</a> <a id="203" href="Categories.Category.Monoidal.Braided.html#182" class="Bound">e</a><a id="204" class="Symbol">}</a> <a id="206" class="Symbol">(</a><a id="207" href="Categories.Category.Monoidal.Braided.html#207" class="Bound">M</a> <a id="209" class="Symbol">:</a> <a id="211" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="220" href="Categories.Category.Monoidal.Braided.html#186" class="Bound">C</a><a id="221" class="Symbol">)</a> <a id="223" class="Keyword">where</a>
<a id="230" class="Keyword">open</a> <a id="235" class="Keyword">import</a> <a id="242" href="Level.html" class="Module">Level</a>
<a id="249" class="Keyword">open</a> <a id="254" class="Keyword">import</a> <a id="261" href="Data.Product.html" class="Module">Data.Product</a> <a id="274" class="Keyword">using</a> <a id="280" class="Symbol">(</a><a id="281" href="Agda.Builtin.Sigma.html#148" class="Record">Σ</a><a id="282" class="Symbol">;</a> <a id="284" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="287" class="Symbol">)</a>
<a id="290" class="Keyword">open</a> <a id="295" class="Keyword">import</a> <a id="302" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
<a id="331" class="Keyword">open</a> <a id="336" class="Keyword">import</a> <a id="343" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a>
<a id="396" class="Keyword">open</a> <a id="401" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="410" href="Categories.Category.Monoidal.Braided.html#186" class="Bound">C</a>
<a id="412" class="Keyword">open</a> <a id="417" href="Categories.Category.html#1298" class="Module">Commutation</a> <a id="429" href="Categories.Category.Monoidal.Braided.html#186" class="Bound">C</a>
<a id="432" class="Keyword">private</a>
<a id="442" class="Keyword">variable</a>
<a id="455" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="457" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="459" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="461" class="Symbol">:</a> <a id="463" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="468" class="Comment">-- braided monoidal category</a>
<a id="497" class="Comment">-- it has a braiding natural isomorphism has two hexagon identities.</a>
<a id="566" class="Comment">-- these two identities are directly expressed in the morphism level.</a>
<a id="636" class="Keyword">record</a> <a id="Braided"></a><a id="643" href="Categories.Category.Monoidal.Braided.html#643" class="Record">Braided</a> <a id="651" class="Symbol">:</a> <a id="653" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="657" class="Symbol">(</a><a id="658" href="Level.html#602" class="Function">levelOfTerm</a> <a id="670" href="Categories.Category.Monoidal.Braided.html#207" class="Bound">M</a><a id="671" class="Symbol">)</a> <a id="673" class="Keyword">where</a>
<a id="681" class="Keyword">open</a> <a id="686" href="Categories.Category.Monoidal.Core.html#1287" class="Module">Monoidal</a> <a id="695" href="Categories.Category.Monoidal.Braided.html#207" class="Bound">M</a> <a id="697" class="Keyword">public</a>
<a id="707" class="Keyword">field</a>
<a id="Braided.braiding"></a><a id="717" href="Categories.Category.Monoidal.Braided.html#717" class="Field">braiding</a> <a id="726" class="Symbol">:</a> <a id="728" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a> <a id="747" href="Categories.Category.Monoidal.Core.html#1355" class="Function">⊗</a> <a id="749" class="Symbol">(</a><a id="750" href="Categories.Functor.Bifunctor.html#2113" class="Function">flip-bifunctor</a> <a id="765" href="Categories.Category.Monoidal.Core.html#1355" class="Function">⊗</a><a id="766" class="Symbol">)</a>
<a id="771" class="Keyword">module</a> <a id="Braided.braiding"></a><a id="778" href="Categories.Category.Monoidal.Braided.html#778" class="Module">braiding</a> <a id="787" class="Symbol">=</a> <a id="789" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a> <a id="808" href="Categories.Category.Monoidal.Braided.html#717" class="Field">braiding</a>
<a id="820" class="Keyword">private</a>
<a id="Braided.B"></a><a id="832" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="834" class="Symbol">:</a> <a id="836" class="Symbol">∀</a> <a id="838" class="Symbol">{</a><a id="839" href="Categories.Category.Monoidal.Braided.html#839" class="Bound">X</a> <a id="841" href="Categories.Category.Monoidal.Braided.html#841" class="Bound">Y</a><a id="842" class="Symbol">}</a> <a id="844" class="Symbol">→</a> <a id="846" href="Categories.Category.Monoidal.Braided.html#839" class="Bound">X</a> <a id="848" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="851" href="Categories.Category.Monoidal.Braided.html#841" class="Bound">Y</a> <a id="853" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="855" href="Categories.Category.Monoidal.Braided.html#841" class="Bound">Y</a> <a id="857" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="860" href="Categories.Category.Monoidal.Braided.html#839" class="Bound">X</a>
<a id="866" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="868" class="Symbol">{</a><a id="869" href="Categories.Category.Monoidal.Braided.html#869" class="Bound">X</a><a id="870" class="Symbol">}</a> <a id="872" class="Symbol">{</a><a id="873" href="Categories.Category.Monoidal.Braided.html#873" class="Bound">Y</a><a id="874" class="Symbol">}</a> <a id="876" class="Symbol">=</a> <a id="878" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇒.η</a> <a id="891" class="Symbol">(</a><a id="892" href="Categories.Category.Monoidal.Braided.html#869" class="Bound">X</a> <a id="894" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="896" href="Categories.Category.Monoidal.Braided.html#873" class="Bound">Y</a><a id="897" class="Symbol">)</a>
<a id="902" class="Keyword">field</a>
<a id="Braided.hexagon₁"></a><a id="912" href="Categories.Category.Monoidal.Braided.html#912" class="Field">hexagon₁</a> <a id="921" class="Symbol">:</a> <a id="923" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="925" class="Symbol">(</a><a id="926" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="928" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="931" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a><a id="932" class="Symbol">)</a> <a id="934" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="937" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="939" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="941" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="943" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="946" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="948" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="951" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="953" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="973" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="976" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="979" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1001" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1004" class="Symbol">(</a><a id="1005" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1007" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1010" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a><a id="1011" class="Symbol">)</a> <a id="1013" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1016" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1018" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1037" href="Categories.Morphism.html#2006" class="Function">associator.from</a> <a id="1065" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1068" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1070" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1073" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1075" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1078" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1080" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1099" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1102" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1105" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a>
<a id="1122" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="1124" href="Categories.Morphism.html#2006" class="Function">associator.from</a> <a id="1152" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1155" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1157" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1160" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1162" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1165" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1167" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1186" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="1214" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1217" class="Symbol">(</a><a id="1218" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1220" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1223" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a><a id="1224" class="Symbol">)</a> <a id="1226" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1229" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1231" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1250" href="Categories.Morphism.html#2006" class="Function">associator.from</a>
<a id="1281" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
<a id="Braided.hexagon₂"></a><a id="1287" href="Categories.Category.Monoidal.Braided.html#1287" class="Field">hexagon₂</a> <a id="1296" class="Symbol">:</a> <a id="1298" href="Categories.Category.html#1379" class="Function Operator">[</a> <a id="1300" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1302" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1305" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1307" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1310" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1312" href="Categories.Category.html#1379" class="Function Operator">⇒</a> <a id="1314" class="Symbol">(</a><a id="1315" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1317" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1320" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a><a id="1321" class="Symbol">)</a> <a id="1323" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1326" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1328" href="Categories.Category.html#1379" class="Function Operator">]⟨</a>
<a id="1348" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1351" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1354" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="1376" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1379" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1381" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1384" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1386" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1389" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1391" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1410" class="Symbol">(</a><a id="1411" href="Categories.Morphism.html#2023" class="Function">associator.to</a> <a id="1438" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1441" class="Symbol">(</a><a id="1442" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1444" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1447" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a><a id="1448" class="Symbol">)</a> <a id="1450" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1453" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1455" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1474" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="1476" href="Categories.Category.Monoidal.Core.html#1536" class="Function Operator">⊗₁</a> <a id="1479" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1481" class="Symbol">)</a>
<a id="1498" href="Categories.Category.html#1379" class="Function Operator">≈</a> <a id="1500" href="Categories.Morphism.html#2023" class="Function">associator.to</a> <a id="1528" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1531" class="Symbol">(</a><a id="1532" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1534" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1537" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a><a id="1538" class="Symbol">)</a> <a id="1540" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1543" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1545" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1564" class="Symbol">(</a><a id="1565" href="Categories.Category.Monoidal.Braided.html#832" class="Function">B</a> <a id="1592" href="Categories.Category.html#1481" class="Function">⇒⟨</a> <a id="1595" href="Categories.Category.Monoidal.Braided.html#459" class="Generalizable">Z</a> <a id="1597" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1600" href="Categories.Category.Monoidal.Braided.html#455" class="Generalizable">X</a> <a id="1602" href="Categories.Category.Monoidal.Core.html#1435" class="Function Operator">⊗₀</a> <a id="1605" href="Categories.Category.Monoidal.Braided.html#457" class="Generalizable">Y</a> <a id="1607" href="Categories.Category.html#1481" class="Function">⟩</a>
<a id="1626" href="Categories.Morphism.html#2023" class="Function">associator.to</a><a id="1639" class="Symbol">)</a>
<a id="1656" href="Categories.Category.html#1379" class="Function Operator">⟩</a>
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