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<html><head><meta charset="utf-8"><title>Categories.Adjoint.Monadic</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="Comment">-- Monadic Adjunctions</a>
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<a id="60" class="Comment">-- https://ncatlab.org/nlab/show/monadic+adjunction</a>
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<a id="112" class="Keyword">module</a> <a id="119" href="Categories.Adjoint.Monadic.html" class="Module">Categories.Adjoint.Monadic</a> <a id="146" class="Keyword">where</a>
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<a id="153" class="Keyword">open</a> <a id="158" class="Keyword">import</a> <a id="165" href="Level.html" class="Module">Level</a>
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<a id="172" class="Keyword">open</a> <a id="177" class="Keyword">import</a> <a id="184" href="Categories.Adjoint.html" class="Module">Categories.Adjoint</a>
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<a id="203" class="Keyword">open</a> <a id="208" class="Keyword">import</a> <a id="215" href="Categories.Adjoint.Properties.html" class="Module">Categories.Adjoint.Properties</a>
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<a id="245" class="Keyword">open</a> <a id="250" class="Keyword">import</a> <a id="257" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="277" class="Keyword">open</a> <a id="282" class="Keyword">import</a> <a id="289" href="Categories.Category.Equivalence.html" class="Module">Categories.Category.Equivalence</a>
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<a id="321" class="Keyword">open</a> <a id="326" class="Keyword">import</a> <a id="333" href="Categories.Functor.html" class="Module">Categories.Functor</a>
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<a id="352" class="Keyword">open</a> <a id="357" class="Keyword">import</a> <a id="364" href="Categories.Monad.html" class="Module">Categories.Monad</a>
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<a id="382" class="Keyword">open</a> <a id="387" class="Keyword">import</a> <a id="394" href="Categories.Category.Construction.EilenbergMoore.html" class="Module">Categories.Category.Construction.EilenbergMoore</a>
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<a id="442" class="Keyword">open</a> <a id="447" class="Keyword">import</a> <a id="454" href="Categories.Category.Construction.Properties.EilenbergMoore.html" class="Module">Categories.Category.Construction.Properties.EilenbergMoore</a>
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<a id="514" class="Keyword">private</a>
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<a id="524" class="Keyword">variable</a>
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<a id="537" href="Categories.Adjoint.Monadic.html#537" class="Generalizable">o</a> <a id="539" href="Categories.Adjoint.Monadic.html#539" class="Generalizable">ℓ</a> <a id="541" href="Categories.Adjoint.Monadic.html#541" class="Generalizable">e</a> <a id="543" class="Symbol">:</a> <a id="545" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="555" href="Categories.Adjoint.Monadic.html#555" class="Generalizable">𝒞</a> <a id="557" href="Categories.Adjoint.Monadic.html#557" class="Generalizable">𝒟</a> <a id="559" class="Symbol">:</a> <a id="561" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="570" href="Categories.Adjoint.Monadic.html#537" class="Generalizable">o</a> <a id="572" href="Categories.Adjoint.Monadic.html#539" class="Generalizable">ℓ</a> <a id="574" href="Categories.Adjoint.Monadic.html#541" class="Generalizable">e</a>
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<a id="577" class="Comment">-- An adjunction is monadic if the adjunction "comes from" the induced monad in some sense.</a>
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<a id="669" class="Keyword">record</a> <a id="IsMonadicAdjunction"></a><a id="676" href="Categories.Adjoint.Monadic.html#676" class="Record">IsMonadicAdjunction</a> <a id="696" class="Symbol">{</a><a id="697" href="Categories.Adjoint.Monadic.html#697" class="Bound">L</a> <a id="699" class="Symbol">:</a> <a id="701" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="709" href="Categories.Adjoint.Monadic.html#555" class="Generalizable">𝒞</a> <a id="711" href="Categories.Adjoint.Monadic.html#557" class="Generalizable">𝒟</a><a id="712" class="Symbol">}</a> <a id="714" class="Symbol">{</a><a id="715" href="Categories.Adjoint.Monadic.html#715" class="Bound">R</a> <a id="717" class="Symbol">:</a> <a id="719" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="727" href="Categories.Adjoint.Monadic.html#557" class="Generalizable">𝒟</a> <a id="729" href="Categories.Adjoint.Monadic.html#555" class="Generalizable">𝒞</a><a id="730" class="Symbol">}</a> <a id="732" class="Symbol">(</a><a id="733" href="Categories.Adjoint.Monadic.html#733" class="Bound">adjoint</a> <a id="741" class="Symbol">:</a> <a id="743" href="Categories.Adjoint.Monadic.html#697" class="Bound">L</a> <a id="745" href="Categories.Adjoint.html#7972" class="Function Operator">⊣</a> <a id="747" href="Categories.Adjoint.Monadic.html#715" class="Bound">R</a><a id="748" class="Symbol">)</a> <a id="750" class="Symbol">:</a> <a id="752" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="756" class="Symbol">(</a><a id="757" href="Level.html#602" class="Function">levelOfTerm</a> <a id="769" href="Categories.Adjoint.Monadic.html#709" class="Bound">𝒞</a> <a id="771" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="773" href="Level.html#602" class="Function">levelOfTerm</a> <a id="785" href="Categories.Adjoint.Monadic.html#711" class="Bound">𝒟</a><a id="786" class="Symbol">)</a> <a id="788" class="Keyword">where</a>
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<a id="796" class="Keyword">private</a>
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<a id="IsMonadicAdjunction.T"></a><a id="808" href="Categories.Adjoint.Monadic.html#808" class="Function">T</a> <a id="810" class="Symbol">:</a> <a id="812" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="818" href="Categories.Adjoint.Monadic.html#709" class="Bound">𝒞</a>
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<a id="824" href="Categories.Adjoint.Monadic.html#808" class="Function">T</a> <a id="826" class="Symbol">=</a> <a id="828" href="Categories.Adjoint.Properties.html#9436" class="Function">adjoint⇒monad</a> <a id="842" href="Categories.Adjoint.Monadic.html#733" class="Bound">adjoint</a>
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<a id="853" class="Keyword">field</a>
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<a id="IsMonadicAdjunction.Comparison⁻¹"></a><a id="863" href="Categories.Adjoint.Monadic.html#863" class="Field">Comparison⁻¹</a> <a id="876" class="Symbol">:</a> <a id="878" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="886" class="Symbol">(</a><a id="887" href="Categories.Category.Construction.EilenbergMoore.html#734" class="Function">EilenbergMoore</a> <a id="902" href="Categories.Adjoint.Monadic.html#808" class="Function">T</a><a id="903" class="Symbol">)</a> <a id="905" href="Categories.Adjoint.Monadic.html#711" class="Bound">𝒟</a>
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<a id="IsMonadicAdjunction.comparison-equiv"></a><a id="911" href="Categories.Adjoint.Monadic.html#911" class="Field">comparison-equiv</a> <a id="928" class="Symbol">:</a> <a id="930" href="Categories.Category.Equivalence.html#713" class="Record">WeakInverse</a> <a id="942" class="Symbol">(</a><a id="943" href="Categories.Category.Construction.Properties.EilenbergMoore.html#1167" class="Function">ComparisonF</a> <a id="955" href="Categories.Adjoint.Monadic.html#733" class="Bound">adjoint</a><a id="962" class="Symbol">)</a> <a id="964" href="Categories.Adjoint.Monadic.html#863" class="Field">Comparison⁻¹</a>
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</pre></body></html>
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