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<html><head><meta charset="utf-8"><title>Categories.Morphism.Duality</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="70" class="Keyword">module</a> <a id="77" href="Categories.Morphism.Duality.html" class="Module">Categories.Morphism.Duality</a> <a id="105" class="Symbol">{</a><a id="106" href="Categories.Morphism.Duality.html#106" class="Bound">o</a> <a id="108" href="Categories.Morphism.Duality.html#108" class="Bound">ℓ</a> <a id="110" href="Categories.Morphism.Duality.html#110" class="Bound">e</a><a id="111" class="Symbol">}</a> <a id="113" class="Symbol">(</a><a id="114" href="Categories.Morphism.Duality.html#114" class="Bound">C</a> <a id="116" class="Symbol">:</a> <a id="118" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="127" href="Categories.Morphism.Duality.html#106" class="Bound">o</a> <a id="129" href="Categories.Morphism.Duality.html#108" class="Bound">ℓ</a> <a id="131" href="Categories.Morphism.Duality.html#110" class="Bound">e</a><a id="132" class="Symbol">)</a> <a id="134" class="Keyword">where</a>
<a id="141" class="Keyword">open</a> <a id="146" class="Keyword">import</a> <a id="153" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a> <a id="191" class="Keyword">using</a> <a id="197" class="Symbol">(</a><a id="198" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="201" class="Symbol">;</a> <a id="203" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="207" class="Symbol">)</a>
<a id="210" class="Keyword">open</a> <a id="215" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="224" href="Categories.Morphism.Duality.html#114" class="Bound">C</a>
<a id="227" class="Keyword">import</a> <a id="234" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="254" class="Symbol">as</a> <a id="257" class="Module">M</a>
<a id="259" class="Keyword">private</a>
<a id="269" class="Keyword">module</a> <a id="Op"></a><a id="276" href="Categories.Morphism.Duality.html#276" class="Module">Op</a> <a id="279" class="Symbol">=</a> <a id="281" href="Categories.Morphism.html" class="Module">M</a> <a id="283" href="Categories.Category.Core.html#3132" class="Function">op</a>
<a id="286" class="Keyword">open</a> <a id="291" href="Categories.Morphism.html" class="Module">M</a> <a id="293" href="Categories.Morphism.Duality.html#114" class="Bound">C</a>
<a id="296" class="Keyword">open</a> <a id="301" class="Keyword">import</a> <a id="308" href="Categories.Morphism.Properties.html" class="Module">Categories.Morphism.Properties</a> <a id="339" href="Categories.Morphism.Duality.html#114" class="Bound">C</a>
<a id="342" class="Keyword">private</a>
<a id="352" class="Keyword">variable</a>
<a id="365" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="367" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a> <a id="369" href="Categories.Morphism.Duality.html#369" class="Generalizable">X</a> <a id="371" href="Categories.Morphism.Duality.html#371" class="Generalizable">Y</a> <a id="373" class="Symbol">:</a> <a id="375" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="383" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="385" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a> <a id="387" href="Categories.Morphism.Duality.html#387" class="Generalizable">h</a> <a id="389" class="Symbol">:</a> <a id="391" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="393" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="395" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a>
<a id="Mono⇒op-Epi"></a><a id="398" href="Categories.Morphism.Duality.html#398" class="Function">Mono⇒op-Epi</a> <a id="410" class="Symbol">:</a> <a id="412" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="417" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="419" class="Symbol">→</a> <a id="421" href="Categories.Morphism.html#875" class="Function">Op.Epi</a> <a id="428" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a>
<a id="430" href="Categories.Morphism.Duality.html#398" class="Function">Mono⇒op-Epi</a> <a id="442" href="Categories.Morphism.Duality.html#442" class="Bound">mono</a> <a id="447" class="Symbol">=</a> <a id="449" href="Categories.Morphism.Duality.html#442" class="Bound">mono</a>
<a id="Epi⇒op-Mono"></a><a id="455" href="Categories.Morphism.Duality.html#455" class="Function">Epi⇒op-Mono</a> <a id="467" class="Symbol">:</a> <a id="469" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="473" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="475" class="Symbol">→</a> <a id="477" href="Categories.Morphism.html#488" class="Function">Op.Mono</a> <a id="485" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a>
<a id="487" href="Categories.Morphism.Duality.html#455" class="Function">Epi⇒op-Mono</a> <a id="499" href="Categories.Morphism.Duality.html#499" class="Bound">epi</a> <a id="503" class="Symbol">=</a> <a id="505" href="Categories.Morphism.Duality.html#499" class="Bound">epi</a>
<a id="Iso⇒op-Iso"></a><a id="510" href="Categories.Morphism.Duality.html#510" class="Function">Iso⇒op-Iso</a> <a id="521" class="Symbol">:</a> <a id="523" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="527" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="529" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a> <a id="531" class="Symbol">→</a> <a id="533" href="Categories.Morphism.html#1528" class="Record">Op.Iso</a> <a id="540" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a> <a id="542" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a>
<a id="544" href="Categories.Morphism.Duality.html#510" class="Function">Iso⇒op-Iso</a> <a id="555" href="Categories.Morphism.Duality.html#555" class="Bound">iso</a> <a id="559" class="Symbol">=</a> <a id="561" class="Keyword">record</a>
<a id="570" class="Symbol">{</a> <a id="572" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="577" class="Symbol">=</a> <a id="579" href="Categories.Morphism.html#1586" class="Field">isoˡ</a>
<a id="586" class="Symbol">;</a> <a id="588" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="593" class="Symbol">=</a> <a id="595" href="Categories.Morphism.html#1612" class="Field">isoʳ</a>
<a id="602" class="Symbol">}</a>
<a id="606" class="Keyword">where</a> <a id="612" class="Keyword">open</a> <a id="617" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="621" href="Categories.Morphism.Duality.html#555" class="Bound">iso</a>
<a id="op-Iso⇒Iso"></a><a id="626" href="Categories.Morphism.Duality.html#626" class="Function">op-Iso⇒Iso</a> <a id="637" class="Symbol">:</a> <a id="639" href="Categories.Morphism.html#1528" class="Record">Op.Iso</a> <a id="646" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a> <a id="648" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="650" class="Symbol">→</a> <a id="652" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="656" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="658" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a>
<a id="660" href="Categories.Morphism.Duality.html#626" class="Function">op-Iso⇒Iso</a> <a id="671" href="Categories.Morphism.Duality.html#671" class="Bound">iso</a> <a id="675" class="Symbol">=</a> <a id="677" class="Keyword">record</a>
<a id="686" class="Symbol">{</a> <a id="688" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="693" class="Symbol">=</a> <a id="695" href="Categories.Morphism.html#1586" class="Field">isoˡ</a>
<a id="702" class="Symbol">;</a> <a id="704" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="709" class="Symbol">=</a> <a id="711" href="Categories.Morphism.html#1612" class="Field">isoʳ</a>
<a id="718" class="Symbol">}</a>
<a id="722" class="Keyword">where</a> <a id="728" class="Keyword">open</a> <a id="733" href="Categories.Morphism.html#1528" class="Module">Op.Iso</a> <a id="740" href="Categories.Morphism.Duality.html#671" class="Bound">iso</a>
<a id="≅⇒op-≅"></a><a id="745" href="Categories.Morphism.Duality.html#745" class="Function">≅⇒op-≅</a> <a id="752" class="Symbol">:</a> <a id="754" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="756" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="758" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a> <a id="760" class="Symbol">→</a> <a id="762" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="764" href="Categories.Morphism.html#1958" class="Record Operator">Op.≅</a> <a id="769" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a>
<a id="771" href="Categories.Morphism.Duality.html#745" class="Function">≅⇒op-≅</a> <a id="778" href="Categories.Morphism.Duality.html#778" class="Bound">A≅B</a> <a id="782" class="Symbol">=</a> <a id="784" class="Keyword">record</a>
<a id="793" class="Symbol">{</a> <a id="795" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="800" class="Symbol">=</a> <a id="802" href="Categories.Morphism.html#2023" class="Field">to</a>
<a id="807" class="Symbol">;</a> <a id="809" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="814" class="Symbol">=</a> <a id="816" href="Categories.Morphism.html#2006" class="Field">from</a>
<a id="823" class="Symbol">;</a> <a id="825" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="830" class="Symbol">=</a> <a id="832" href="Categories.Morphism.Duality.html#510" class="Function">Iso⇒op-Iso</a> <a id="843" href="Categories.Morphism.html#2040" class="Field">iso</a>
<a id="849" class="Symbol">}</a>
<a id="853" class="Keyword">where</a> <a id="859" class="Keyword">open</a> <a id="864" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="868" href="Categories.Morphism.Duality.html#778" class="Bound">A≅B</a>
<a id="op-≅⇒≅"></a><a id="873" href="Categories.Morphism.Duality.html#873" class="Function">op-≅⇒≅</a> <a id="880" class="Symbol">:</a> <a id="882" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="884" href="Categories.Morphism.html#1958" class="Record Operator">Op.≅</a> <a id="889" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a> <a id="891" class="Symbol">→</a> <a id="893" href="Categories.Morphism.Duality.html#365" class="Generalizable">A</a> <a id="895" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="897" href="Categories.Morphism.Duality.html#367" class="Generalizable">B</a>
<a id="899" href="Categories.Morphism.Duality.html#873" class="Function">op-≅⇒≅</a> <a id="906" href="Categories.Morphism.Duality.html#906" class="Bound">A≅B</a> <a id="910" class="Symbol">=</a> <a id="912" class="Keyword">record</a>
<a id="921" class="Symbol">{</a> <a id="923" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="928" class="Symbol">=</a> <a id="930" href="Categories.Morphism.html#2023" class="Field">to</a>
<a id="935" class="Symbol">;</a> <a id="937" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="942" class="Symbol">=</a> <a id="944" href="Categories.Morphism.html#2006" class="Field">from</a>
<a id="951" class="Symbol">;</a> <a id="953" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="958" class="Symbol">=</a> <a id="960" href="Categories.Morphism.Duality.html#626" class="Function">op-Iso⇒Iso</a> <a id="971" href="Categories.Morphism.html#2040" class="Field">iso</a>
<a id="977" class="Symbol">}</a>
<a id="981" class="Keyword">where</a> <a id="987" class="Keyword">open</a> <a id="992" href="Categories.Morphism.html#1958" class="Module Operator">Op._≅_</a> <a id="999" href="Categories.Morphism.Duality.html#906" class="Bound">A≅B</a>
<a id="1005" class="Keyword">module</a> <a id="MorphismDualityConversionProperties"></a><a id="1012" href="Categories.Morphism.Duality.html#1012" class="Module">MorphismDualityConversionProperties</a> <a id="1048" class="Keyword">where</a>
<a id="1056" class="Keyword">private</a>
<a id="MorphismDualityConversionProperties.op-Iso-involutive"></a><a id="1068" href="Categories.Morphism.Duality.html#1068" class="Function">op-Iso-involutive</a> <a id="1086" class="Symbol">:</a> <a id="1088" class="Symbol">∀(</a><a id="1090" href="Categories.Morphism.Duality.html#1090" class="Bound">iso</a> <a id="1094" class="Symbol">:</a> <a id="1096" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="1100" href="Categories.Morphism.Duality.html#383" class="Generalizable">f</a> <a id="1102" href="Categories.Morphism.Duality.html#385" class="Generalizable">g</a><a id="1103" class="Symbol">)</a> <a id="1105" class="Symbol">→</a> <a id="1107" href="Categories.Morphism.Duality.html#626" class="Function">op-Iso⇒Iso</a> <a id="1118" class="Symbol">(</a><a id="1119" href="Categories.Morphism.Duality.html#510" class="Function">Iso⇒op-Iso</a> <a id="1130" href="Categories.Morphism.Duality.html#1090" class="Bound">iso</a><a id="1133" class="Symbol">)</a> <a id="1135" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1137" href="Categories.Morphism.Duality.html#1090" class="Bound">iso</a>
<a id="1145" href="Categories.Morphism.Duality.html#1068" class="Function">op-Iso-involutive</a> <a id="1163" class="Symbol">_</a> <a id="1165" class="Symbol">=</a> <a id="1167" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="MorphismDualityConversionProperties.op-≅-involutive"></a><a id="1177" href="Categories.Morphism.Duality.html#1177" class="Function">op-≅-involutive</a> <a id="1193" class="Symbol">:</a> <a id="1195" class="Symbol">∀(</a><a id="1197" href="Categories.Morphism.Duality.html#1197" class="Bound">A′</a> <a id="1200" href="Categories.Morphism.Duality.html#1200" class="Bound">B′</a> <a id="1203" class="Symbol">:</a> <a id="1205" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="1208" class="Symbol">)</a> <a id="1210" class="Symbol">→</a> <a id="1212" class="Symbol">(</a><a id="1213" href="Categories.Morphism.Duality.html#1213" class="Bound">A′≅B′</a> <a id="1219" class="Symbol">:</a> <a id="1221" href="Categories.Morphism.Duality.html#1197" class="Bound">A′</a> <a id="1224" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1226" href="Categories.Morphism.Duality.html#1200" class="Bound">B′</a><a id="1228" class="Symbol">)</a>
<a id="1253" class="Symbol">→</a> <a id="1255" href="Categories.Morphism.Duality.html#873" class="Function">op-≅⇒≅</a> <a id="1262" class="Symbol">(</a><a id="1263" href="Categories.Morphism.Duality.html#745" class="Function">≅⇒op-≅</a> <a id="1270" href="Categories.Morphism.Duality.html#1213" class="Bound">A′≅B′</a><a id="1275" class="Symbol">)</a> <a id="1277" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1279" href="Categories.Morphism.Duality.html#1213" class="Bound">A′≅B′</a>
<a id="1289" href="Categories.Morphism.Duality.html#1177" class="Function">op-≅-involutive</a> <a id="1305" class="Symbol">_</a> <a id="1307" class="Symbol">_</a> <a id="1309" class="Symbol">_</a> <a id="1311" class="Symbol">=</a> <a id="1313" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
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