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<html><head><meta charset="utf-8"><title>Categories.Adjoint.Construction.EilenbergMoore</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.Core.html" class="Module">Categories.Category.Core</a>
<a id="74" class="Keyword">open</a> <a id="79" class="Keyword">import</a> <a id="86" href="Categories.Monad.html" class="Module">Categories.Monad</a>
<a id="104" class="Keyword">module</a> <a id="111" href="Categories.Adjoint.Construction.EilenbergMoore.html" class="Module">Categories.Adjoint.Construction.EilenbergMoore</a> <a id="158" class="Symbol">{</a><a id="159" href="Categories.Adjoint.Construction.EilenbergMoore.html#159" class="Bound">o</a> <a id="161" href="Categories.Adjoint.Construction.EilenbergMoore.html#161" class="Bound">ℓ</a> <a id="163" href="Categories.Adjoint.Construction.EilenbergMoore.html#163" class="Bound">e</a><a id="164" class="Symbol">}</a> <a id="166" class="Symbol">{</a><a id="167" href="Categories.Adjoint.Construction.EilenbergMoore.html#167" class="Bound">C</a> <a id="169" class="Symbol">:</a> <a id="171" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="180" href="Categories.Adjoint.Construction.EilenbergMoore.html#159" class="Bound">o</a> <a id="182" href="Categories.Adjoint.Construction.EilenbergMoore.html#161" class="Bound">ℓ</a> <a id="184" href="Categories.Adjoint.Construction.EilenbergMoore.html#163" class="Bound">e</a><a id="185" class="Symbol">}</a> <a id="187" class="Symbol">(</a><a id="188" href="Categories.Adjoint.Construction.EilenbergMoore.html#188" class="Bound">M</a> <a id="190" class="Symbol">:</a> <a id="192" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="198" href="Categories.Adjoint.Construction.EilenbergMoore.html#167" class="Bound">C</a><a id="199" class="Symbol">)</a> <a id="201" class="Keyword">where</a>
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<a id="270" class="Keyword">open</a> <a id="275" class="Keyword">import</a> <a id="282" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
<a id="301" class="Keyword">open</a> <a id="306" class="Keyword">import</a> <a id="313" href="Categories.Functor.html" class="Module">Categories.Functor</a>
<a id="332" class="Keyword">open</a> <a id="337" class="Keyword">import</a> <a id="344" href="Categories.Functor.Properties.html" class="Module">Categories.Functor.Properties</a>
<a id="374" class="Keyword">open</a> <a id="379" class="Keyword">import</a> <a id="386" href="Categories.NaturalTransformation.Core.html" class="Module">Categories.NaturalTransformation.Core</a>
<a id="424" class="Keyword">open</a> <a id="429" class="Keyword">import</a> <a id="436" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="488" class="Keyword">using</a> <a id="494" class="Symbol">(</a><a id="495" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="498" class="Symbol">)</a>
<a id="500" class="Keyword">open</a> <a id="505" class="Keyword">import</a> <a id="512" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="542" href="Categories.Adjoint.Construction.EilenbergMoore.html#167" class="Bound">C</a>
<a id="545" class="Keyword">private</a>
<a id="555" class="Keyword">module</a> <a id="C"></a><a id="562" href="Categories.Adjoint.Construction.EilenbergMoore.html#562" class="Module">C</a> <a id="564" class="Symbol">=</a> <a id="566" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="575" href="Categories.Adjoint.Construction.EilenbergMoore.html#167" class="Bound">C</a>
<a id="579" class="Keyword">module</a> <a id="M"></a><a id="586" href="Categories.Adjoint.Construction.EilenbergMoore.html#586" class="Module">M</a> <a id="588" class="Symbol">=</a> <a id="590" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="596" href="Categories.Adjoint.Construction.EilenbergMoore.html#188" class="Bound">M</a>
<a id="600" class="Keyword">open</a> <a id="605" href="Categories.Monad.html#630" class="Module">M.F</a>
<a id="611" class="Keyword">open</a> <a id="616" href="Categories.Adjoint.Construction.EilenbergMoore.html#562" class="Module">C</a>
<a id="620" class="Keyword">open</a> <a id="625" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="640" class="Keyword">open</a> <a id="645" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="Forgetful"></a><a id="652" href="Categories.Adjoint.Construction.EilenbergMoore.html#652" class="Function">Forgetful</a> <a id="662" class="Symbol">:</a> <a id="664" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="672" href="Categories.Category.Construction.EilenbergMoore.html#734" class="Function">EilenbergMoore</a> <a id="687" href="Categories.Adjoint.Construction.EilenbergMoore.html#167" class="Bound">C</a>
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<a id="710" class="Symbol">{</a> <a id="712" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="725" class="Symbol">=</a> <a id="727" href="Categories.Category.Construction.EilenbergMoore.html#412" class="Field">Module.A</a>
<a id="738" class="Symbol">;</a> <a id="740" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="753" class="Symbol">=</a> <a id="755" href="Categories.Category.Construction.EilenbergMoore.html#664" class="Field">Module⇒.arr</a>
<a id="769" class="Symbol">;</a> <a id="771" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="784" class="Symbol">=</a> <a id="786" href="Relation.Binary.Structures.html#1577" class="Function">refl</a>
<a id="793" class="Symbol">;</a> <a id="795" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="808" class="Symbol">=</a> <a id="810" href="Relation.Binary.Structures.html#1577" class="Function">refl</a>
<a id="817" class="Symbol">;</a> <a id="819" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="832" class="Symbol">=</a> <a id="834" class="Symbol">λ</a> <a id="836" href="Categories.Adjoint.Construction.EilenbergMoore.html#836" class="Bound">eq</a> <a id="839" class="Symbol">→</a> <a id="841" href="Categories.Adjoint.Construction.EilenbergMoore.html#836" class="Bound">eq</a>
<a id="846" class="Symbol">}</a>
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<a id="881" href="Categories.Adjoint.Construction.EilenbergMoore.html#849" class="Function">Free</a> <a id="886" class="Symbol">=</a> <a id="888" class="Keyword">record</a>
<a id="897" class="Symbol">{</a> <a id="899" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="912" class="Symbol">=</a> <a id="914" class="Symbol">λ</a> <a id="916" href="Categories.Adjoint.Construction.EilenbergMoore.html#916" class="Bound">A</a> <a id="918" class="Symbol">→</a> <a id="920" class="Keyword">record</a>
<a id="931" class="Symbol">{</a> <a id="933" href="Categories.Category.Construction.EilenbergMoore.html#412" class="Field">A</a> <a id="942" class="Symbol">=</a> <a id="944" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="947" href="Categories.Adjoint.Construction.EilenbergMoore.html#916" class="Bound">A</a>
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<a id="978" class="Symbol">;</a> <a id="980" href="Categories.Category.Construction.EilenbergMoore.html#455" class="Field">commute</a> <a id="989" class="Symbol">=</a> <a id="991" href="Categories.Monad.html#759" class="Field">M.assoc</a>
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<a id="1032" class="Symbol">}</a>
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<a id="1070" class="Symbol">{</a> <a id="1072" href="Categories.Category.Construction.EilenbergMoore.html#664" class="Field">arr</a> <a id="1080" class="Symbol">=</a> <a id="1082" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1085" href="Categories.Adjoint.Construction.EilenbergMoore.html#1055" class="Bound">f</a>
<a id="1091" class="Symbol">;</a> <a id="1093" href="Categories.Category.Construction.EilenbergMoore.html#688" class="Field">commute</a> <a id="1101" class="Symbol">=</a> <a id="1103" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1105" class="Symbol">(</a><a id="1106" href="Categories.NaturalTransformation.Core.html#827" class="Function">M.μ.commute</a> <a id="1118" href="Categories.Adjoint.Construction.EilenbergMoore.html#1055" class="Bound">f</a><a id="1119" class="Symbol">)</a>
<a id="1125" class="Symbol">}</a>
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<a id="1157" class="Symbol">;</a> <a id="1159" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="1172" class="Symbol">=</a> <a id="1174" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a>
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<a id="1217" class="Symbol">}</a>
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<a id="1432" class="Symbol">}</a>
<a id="1436" class="Symbol">;</a> <a id="1438" href="Categories.NaturalTransformation.NaturalIsomorphism.html#927" class="Field">F⇐G</a> <a id="1442" class="Symbol">=</a> <a id="1444" class="Keyword">record</a>
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<a id="1485" class="Symbol">;</a> <a id="1487" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="1495" class="Symbol">=</a> <a id="1497" class="Symbol">λ</a> <a id="1499" href="Categories.Adjoint.Construction.EilenbergMoore.html#1499" class="Bound">_</a> <a id="1501" class="Symbol">→</a> <a id="1503" href="Categories.Functor.Properties.html#2676" class="Function Operator">[</a> <a id="1505" href="Categories.Monad.html#525" class="Field">M.F</a> <a id="1509" href="Categories.Functor.Properties.html#2676" class="Function Operator">]-resp-square</a> <a id="1523" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a>
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<a id="1593" class="Symbol">}</a>
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<a id="1622" class="Symbol">{</a> <a id="1624" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="1629" class="Symbol">=</a> <a id="1631" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">elimˡ</a> <a id="1637" href="Categories.Functor.Core.html#511" class="Function">identity</a> <a id="1646" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1648" href="Categories.Functor.Core.html#511" class="Function">identity</a>
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<a id="1700" class="Symbol">}</a>
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<a id="2014" class="Symbol">;</a> <a id="2016" href="Categories.Category.Construction.EilenbergMoore.html#688" class="Field">commute</a> <a id="2024" class="Symbol">=</a> <a id="2026" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2028" href="Categories.Category.Construction.EilenbergMoore.html#455" class="Field">X.commute</a>
<a id="2046" class="Symbol">}</a>
<a id="2052" class="Symbol">;</a> <a id="2054" href="Categories.NaturalTransformation.Core.html#827" class="Field">commute</a> <a id="2066" class="Symbol">=</a> <a id="2068" class="Symbol">λ</a> <a id="2070" href="Categories.Adjoint.Construction.EilenbergMoore.html#2070" class="Bound">f</a> <a id="2072" class="Symbol">→</a> <a id="2074" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2076" class="Symbol">(</a><a id="2077" href="Categories.Category.Construction.EilenbergMoore.html#688" class="Field">Module⇒.commute</a> <a id="2093" href="Categories.Adjoint.Construction.EilenbergMoore.html#2070" class="Bound">f</a><a id="2094" class="Symbol">)</a>
<a id="2100" class="Symbol">;</a> <a id="2102" href="Categories.NaturalTransformation.Core.html#1043" class="Field">sym-commute</a> <a id="2114" class="Symbol">=</a> <a id="2116" href="Categories.Category.Construction.EilenbergMoore.html#688" class="Field">Module⇒.commute</a>
<a id="2136" class="Symbol">}</a>
<a id="2140" class="Symbol">;</a> <a id="2142" href="Categories.Adjoint.html#1715" class="Field">zig</a> <a id="2149" class="Symbol">=</a> <a id="2151" href="Categories.Monad.html#899" class="Field">M.identityˡ</a>
<a id="2165" class="Symbol">;</a> <a id="2167" href="Categories.Adjoint.html#1788" class="Field">zag</a> <a id="2174" class="Symbol">=</a> <a id="2176" class="Symbol">λ</a> <a id="2178" class="Symbol">{</a><a id="2179" href="Categories.Adjoint.Construction.EilenbergMoore.html#2179" class="Bound">B</a><a id="2180" class="Symbol">}</a> <a id="2182" class="Symbol">→</a> <a id="2184" href="Categories.Category.Construction.EilenbergMoore.html#508" class="Field">Module.identity</a> <a id="2200" href="Categories.Adjoint.Construction.EilenbergMoore.html#2179" class="Bound">B</a>
<a id="2204" class="Symbol">}</a>
</pre></body></html>
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