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| <html><head><meta charset="utf-8"><title>Categories.Category.Monoidal.Symmetric.Properties</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> | ||
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| <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">using</a> <a id="75" class="Symbol">(</a><a id="76" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="84" class="Symbol">)</a> | ||
| <a id="86" class="Keyword">open</a> <a id="91" class="Keyword">import</a> <a id="98" href="Categories.Category.Monoidal.html" class="Module">Categories.Category.Monoidal</a> <a id="127" class="Keyword">using</a> <a id="133" class="Symbol">(</a><a id="134" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a><a id="142" class="Symbol">)</a> | ||
| <a id="144" class="Keyword">open</a> <a id="149" class="Keyword">import</a> <a id="156" href="Categories.Category.Monoidal.Symmetric.html" class="Module">Categories.Category.Monoidal.Symmetric</a> <a id="195" class="Keyword">using</a> <a id="201" class="Symbol">(</a><a id="202" href="Categories.Category.Monoidal.Symmetric.html#997" class="Record">Symmetric</a><a id="211" class="Symbol">)</a> | ||
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| <a id="214" class="Keyword">module</a> <a id="221" href="Categories.Category.Monoidal.Symmetric.Properties.html" class="Module">Categories.Category.Monoidal.Symmetric.Properties</a> | ||
| <a id="273" class="Symbol">{</a><a id="274" href="Categories.Category.Monoidal.Symmetric.Properties.html#274" class="Bound">o</a> <a id="276" href="Categories.Category.Monoidal.Symmetric.Properties.html#276" class="Bound">ℓ</a> <a id="278" href="Categories.Category.Monoidal.Symmetric.Properties.html#278" class="Bound">e</a><a id="279" class="Symbol">}</a> <a id="281" class="Symbol">{</a><a id="282" href="Categories.Category.Monoidal.Symmetric.Properties.html#282" class="Bound">C</a> <a id="284" class="Symbol">:</a> <a id="286" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="295" href="Categories.Category.Monoidal.Symmetric.Properties.html#274" class="Bound">o</a> <a id="297" href="Categories.Category.Monoidal.Symmetric.Properties.html#276" class="Bound">ℓ</a> <a id="299" href="Categories.Category.Monoidal.Symmetric.Properties.html#278" class="Bound">e</a><a id="300" class="Symbol">}</a> <a id="302" class="Symbol">{</a><a id="303" href="Categories.Category.Monoidal.Symmetric.Properties.html#303" class="Bound">M</a> <a id="305" class="Symbol">:</a> <a id="307" href="Categories.Category.Monoidal.Core.html#1287" class="Record">Monoidal</a> <a id="316" href="Categories.Category.Monoidal.Symmetric.Properties.html#282" class="Bound">C</a><a id="317" class="Symbol">}</a> <a id="319" class="Symbol">(</a><a id="320" href="Categories.Category.Monoidal.Symmetric.Properties.html#320" class="Bound">SM</a> <a id="323" class="Symbol">:</a> <a id="325" href="Categories.Category.Monoidal.Symmetric.html#997" class="Record">Symmetric</a> <a id="335" href="Categories.Category.Monoidal.Symmetric.Properties.html#303" class="Bound">M</a><a id="336" class="Symbol">)</a> <a id="338" class="Keyword">where</a> | ||
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| <a id="345" class="Keyword">open</a> <a id="350" class="Keyword">import</a> <a id="357" href="Data.Product.html" class="Module">Data.Product</a> <a id="370" class="Keyword">using</a> <a id="376" class="Symbol">(</a><a id="377" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="380" class="Symbol">)</a> | ||
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| <a id="383" class="Keyword">import</a> <a id="390" href="Categories.Category.Monoidal.Braided.Properties.html" class="Module">Categories.Category.Monoidal.Braided.Properties</a> <a id="438" class="Symbol">as</a> <a id="441" class="Module">BraidedProperties</a> | ||
| <a id="459" class="Keyword">open</a> <a id="464" class="Keyword">import</a> <a id="471" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="501" href="Categories.Category.Monoidal.Symmetric.Properties.html#282" class="Bound">C</a> | ||
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| <a id="504" class="Keyword">open</a> <a id="509" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="518" href="Categories.Category.Monoidal.Symmetric.Properties.html#282" class="Bound">C</a> | ||
| <a id="520" class="Keyword">open</a> <a id="525" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a> | ||
| <a id="538" class="Keyword">open</a> <a id="543" href="Categories.Category.Monoidal.Symmetric.html#997" class="Module">Symmetric</a> <a id="553" href="Categories.Category.Monoidal.Symmetric.Properties.html#320" class="Bound">SM</a> | ||
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| <a id="557" class="Comment">-- Shorthands for the braiding</a> | ||
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| <a id="589" class="Keyword">open</a> <a id="594" href="Categories.Category.Monoidal.Braided.Properties.html" class="Module">BraidedProperties</a> <a id="612" href="Categories.Category.Monoidal.Symmetric.html#1047" class="Field">braided</a> <a id="620" class="Keyword">public</a> <a id="627" class="Keyword">using</a> <a id="633" class="Symbol">(</a><a id="634" class="Keyword">module</a> <a id="641" href="Categories.Category.Monoidal.Braided.Properties.html#1156" class="Module">Shorthands</a><a id="651" class="Symbol">)</a> | ||
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| <a id="654" class="Comment">-- Extra properties of the braiding in a symmetric monoidal category</a> | ||
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| <a id="braiding-selfInverse"></a><a id="724" href="Categories.Category.Monoidal.Symmetric.Properties.html#724" class="Function">braiding-selfInverse</a> <a id="745" class="Symbol">:</a> <a id="747" class="Symbol">∀</a> <a id="749" class="Symbol">{</a><a id="750" href="Categories.Category.Monoidal.Symmetric.Properties.html#750" class="Bound">X</a> <a id="752" href="Categories.Category.Monoidal.Symmetric.Properties.html#752" class="Bound">Y</a><a id="753" class="Symbol">}</a> <a id="755" class="Symbol">→</a> <a id="757" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇐.η</a> <a id="770" class="Symbol">(</a><a id="771" href="Categories.Category.Monoidal.Symmetric.Properties.html#750" class="Bound">X</a> <a id="773" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="775" href="Categories.Category.Monoidal.Symmetric.Properties.html#752" class="Bound">Y</a><a id="776" class="Symbol">)</a> <a id="778" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="780" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇒.η</a> <a id="793" class="Symbol">(</a><a id="794" href="Categories.Category.Monoidal.Symmetric.Properties.html#752" class="Bound">Y</a> <a id="796" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="798" href="Categories.Category.Monoidal.Symmetric.Properties.html#750" class="Bound">X</a><a id="799" class="Symbol">)</a> | ||
| <a id="801" href="Categories.Category.Monoidal.Symmetric.Properties.html#724" class="Function">braiding-selfInverse</a> <a id="822" class="Symbol">=</a> <a id="824" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="831" href="Categories.Category.Monoidal.Symmetric.html#1218" class="Field">commutative</a> <a id="843" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="845" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="853" class="Symbol">(</a><a id="854" href="Categories.Morphism.html#1586" class="Function">braiding.iso.isoˡ</a> <a id="872" class="Symbol">_)</a> | ||
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| <a id="inv-commutative"></a><a id="876" href="Categories.Category.Monoidal.Symmetric.Properties.html#876" class="Function">inv-commutative</a> <a id="892" class="Symbol">:</a> <a id="894" class="Symbol">∀</a> <a id="896" class="Symbol">{</a><a id="897" href="Categories.Category.Monoidal.Symmetric.Properties.html#897" class="Bound">X</a> <a id="899" href="Categories.Category.Monoidal.Symmetric.Properties.html#899" class="Bound">Y</a><a id="900" class="Symbol">}</a> <a id="902" class="Symbol">→</a> <a id="904" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇐.η</a> <a id="917" class="Symbol">(</a><a id="918" href="Categories.Category.Monoidal.Symmetric.Properties.html#897" class="Bound">X</a> <a id="920" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="922" href="Categories.Category.Monoidal.Symmetric.Properties.html#899" class="Bound">Y</a><a id="923" class="Symbol">)</a> <a id="925" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="927" href="Categories.NaturalTransformation.Core.html#783" class="Function">braiding.⇐.η</a> <a id="940" class="Symbol">(</a><a id="941" href="Categories.Category.Monoidal.Symmetric.Properties.html#899" class="Bound">Y</a> <a id="943" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="945" href="Categories.Category.Monoidal.Symmetric.Properties.html#897" class="Bound">X</a><a id="946" class="Symbol">)</a> <a id="948" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="950" href="Categories.Category.Core.html#630" class="Function">id</a> | ||
| <a id="953" href="Categories.Category.Monoidal.Symmetric.Properties.html#876" class="Function">inv-commutative</a> <a id="969" class="Symbol">=</a> <a id="971" href="Categories.Category.Core.html#1438" class="Function">∘-resp-≈</a> <a id="980" href="Categories.Category.Monoidal.Symmetric.Properties.html#724" class="Function">braiding-selfInverse</a> <a id="1001" href="Categories.Category.Monoidal.Symmetric.Properties.html#724" class="Function">braiding-selfInverse</a> <a id="1022" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1024" href="Categories.Category.Monoidal.Symmetric.html#1218" class="Field">commutative</a> | ||
| </pre></body></html> |
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