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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Adjoint.Equivalence.html
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<html><head><meta charset="utf-8"><title>Categories.Adjoint.Equivalence</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="101" class="Keyword">open</a> <a id="106" class="Keyword">import</a> <a id="113" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
<a id="132" class="Keyword">open</a> <a id="137" class="Keyword">import</a> <a id="144" href="Categories.Adjoint.TwoSided.html" class="Module">Categories.Adjoint.TwoSided</a>
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<a id="489" class="Keyword">private</a>
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<a id="539" href="Categories.Adjoint.Equivalence.html#539" class="Generalizable">C</a> <a id="541" href="Categories.Adjoint.Equivalence.html#541" class="Generalizable">D</a> <a id="543" href="Categories.Adjoint.Equivalence.html#543" class="Generalizable">E</a> <a id="548" class="Symbol">:</a> <a id="550" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="559" href="Categories.Adjoint.Equivalence.html#512" class="Generalizable">o</a> <a id="561" href="Categories.Adjoint.Equivalence.html#514" class="Generalizable">ℓ</a> <a id="563" href="Categories.Adjoint.Equivalence.html#516" class="Generalizable">e</a>
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<a id="672" class="Keyword">field</a>
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<a id="845" href="Categories.Adjoint.Equivalence.html#821" class="Function">refl</a> <a id="850" class="Symbol">=</a> <a id="852" class="Keyword">record</a>
<a id="861" class="Symbol">{</a> <a id="863" href="Categories.Adjoint.Equivalence.html#682" class="Field">L</a> <a id="868" class="Symbol">=</a> <a id="870" href="Categories.Adjoint.Equivalence.html#343" class="Function">idF</a>
<a id="876" class="Symbol">;</a> <a id="878" href="Categories.Adjoint.Equivalence.html#705" class="Field">R</a> <a id="883" class="Symbol">=</a> <a id="885" href="Categories.Adjoint.Equivalence.html#343" class="Function">idF</a>
<a id="891" class="Symbol">;</a> <a id="893" href="Categories.Adjoint.Equivalence.html#728" class="Field">L⊣⊢R</a> <a id="898" class="Symbol">=</a> <a id="900" href="Categories.Adjoint.TwoSided.html#6204" class="Function">id⊣⊢id</a>
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<a id="954" href="Categories.Adjoint.Equivalence.html#912" class="Function">sym</a> <a id="958" href="Categories.Adjoint.Equivalence.html#958" class="Bound">e</a> <a id="960" class="Symbol">=</a> <a id="962" class="Keyword">record</a>
<a id="971" class="Symbol">{</a> <a id="973" href="Categories.Adjoint.Equivalence.html#682" class="Field">L</a> <a id="978" class="Symbol">=</a> <a id="980" href="Categories.Adjoint.Equivalence.html#705" class="Field">R</a>
<a id="984" class="Symbol">;</a> <a id="986" href="Categories.Adjoint.Equivalence.html#705" class="Field">R</a> <a id="991" class="Symbol">=</a> <a id="993" href="Categories.Adjoint.Equivalence.html#682" class="Field">L</a>
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<a id="1106" href="Categories.Adjoint.Equivalence.html#1043" class="Function">trans</a> <a id="1112" href="Categories.Adjoint.Equivalence.html#1112" class="Bound">e</a> <a id="1114" href="Categories.Adjoint.Equivalence.html#1114" class="Bound">e′</a> <a id="1117" class="Symbol">=</a> <a id="1119" class="Keyword">record</a>
<a id="1128" class="Symbol">{</a> <a id="1130" href="Categories.Adjoint.Equivalence.html#682" class="Field">L</a> <a id="1135" class="Symbol">=</a> <a id="1137" href="Categories.Adjoint.Equivalence.html#682" class="Field">e′.L</a> <a id="1142" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1145" href="Categories.Adjoint.Equivalence.html#682" class="Function">e.L</a>
<a id="1151" class="Symbol">;</a> <a id="1153" href="Categories.Adjoint.Equivalence.html#705" class="Field">R</a> <a id="1158" class="Symbol">=</a> <a id="1160" href="Categories.Adjoint.Equivalence.html#705" class="Function">e.R</a> <a id="1164" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="1167" href="Categories.Adjoint.Equivalence.html#705" class="Field">e′.R</a>
<a id="1174" class="Symbol">;</a> <a id="1176" href="Categories.Adjoint.Equivalence.html#728" class="Field">L⊣⊢R</a> <a id="1181" class="Symbol">=</a> <a id="1183" href="Categories.Adjoint.Equivalence.html#728" class="Function">e.L⊣⊢R</a> <a id="1190" href="Categories.Adjoint.TwoSided.Compose.html#722" class="Function Operator">∘⊣⊢</a> <a id="1194" href="Categories.Adjoint.Equivalence.html#728" class="Field">e′.L⊣⊢R</a>
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<a id="1268" class="Keyword">module</a> <a id="1275" href="Categories.Adjoint.Equivalence.html#1275" class="Module">e′</a> <a id="1278" class="Symbol">=</a> <a id="1280" href="Categories.Adjoint.Equivalence.html#573" class="Module">⊣Equivalence</a> <a id="1293" href="Categories.Adjoint.Equivalence.html#1114" class="Bound">e′</a> <a id="1296" class="Keyword">using</a> <a id="1302" class="Symbol">(</a><a id="1303" href="Categories.Adjoint.Equivalence.html#682" class="Field">L</a><a id="1304" class="Symbol">;</a> <a id="1306" href="Categories.Adjoint.Equivalence.html#705" class="Field">R</a><a id="1307" class="Symbol">;</a> <a id="1309" href="Categories.Adjoint.Equivalence.html#728" class="Field">L⊣⊢R</a><a id="1313" class="Symbol">)</a>
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<a id="1385" href="Categories.Adjoint.Equivalence.html#1316" class="Function">isEquivalence</a> <a id="1399" class="Symbol">=</a> <a id="1401" class="Keyword">record</a>
<a id="1410" class="Symbol">{</a> <a id="1412" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="1418" class="Symbol">=</a> <a id="1420" href="Categories.Adjoint.Equivalence.html#821" class="Function">refl</a>
<a id="1427" class="Symbol">;</a> <a id="1429" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="1435" class="Symbol">=</a> <a id="1437" href="Categories.Adjoint.Equivalence.html#912" class="Function">sym</a>
<a id="1443" class="Symbol">;</a> <a id="1445" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="1451" class="Symbol">=</a> <a id="1453" href="Categories.Adjoint.Equivalence.html#1043" class="Function">trans</a>
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<a id="setoid"></a><a id="1464" href="Categories.Adjoint.Equivalence.html#1464" class="Function">setoid</a> <a id="1471" class="Symbol">:</a> <a id="1473" class="Symbol">∀</a> <a id="1475" href="Categories.Adjoint.Equivalence.html#1475" class="Bound">o</a> <a id="1477" href="Categories.Adjoint.Equivalence.html#1477" class="Bound">ℓ</a> <a id="1479" href="Categories.Adjoint.Equivalence.html#1479" class="Bound">e</a> <a id="1481" class="Symbol">→</a> <a id="1483" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="1490" class="Symbol">_</a> <a id="1492" class="Symbol">_</a>
<a id="1494" href="Categories.Adjoint.Equivalence.html#1464" class="Function">setoid</a> <a id="1501" href="Categories.Adjoint.Equivalence.html#1501" class="Bound">o</a> <a id="1503" href="Categories.Adjoint.Equivalence.html#1503" class="Bound">ℓ</a> <a id="1505" href="Categories.Adjoint.Equivalence.html#1505" class="Bound">e</a> <a id="1507" class="Symbol">=</a> <a id="1509" class="Keyword">record</a>
<a id="1518" class="Symbol">{</a> <a id="1520" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="1534" class="Symbol">=</a> <a id="1536" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1545" href="Categories.Adjoint.Equivalence.html#1501" class="Bound">o</a> <a id="1547" href="Categories.Adjoint.Equivalence.html#1503" class="Bound">ℓ</a> <a id="1549" href="Categories.Adjoint.Equivalence.html#1505" class="Bound">e</a>
<a id="1553" class="Symbol">;</a> <a id="1555" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="1569" class="Symbol">=</a> <a id="1571" href="Categories.Adjoint.Equivalence.html#573" class="Record">⊣Equivalence</a>
<a id="1586" class="Symbol">;</a> <a id="1588" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="1602" class="Symbol">=</a> <a id="1604" href="Categories.Adjoint.Equivalence.html#1316" class="Function">isEquivalence</a>
<a id="1620" class="Symbol">}</a>
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