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32 lines
12 KiB
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<html><head><meta charset="utf-8"><title>Categories.Monad</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="36" class="Keyword">module</a> <a id="43" href="Categories.Monad.html" class="Module">Categories.Monad</a> <a id="60" class="Keyword">where</a>
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<a id="67" class="Keyword">open</a> <a id="72" class="Keyword">import</a> <a id="79" href="Level.html" class="Module">Level</a>
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<a id="86" class="Keyword">open</a> <a id="91" class="Keyword">import</a> <a id="98" href="Categories.Category.html" class="Module">Categories.Category</a> <a id="118" class="Keyword">using</a> <a id="124" class="Symbol">(</a><a id="125" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="133" class="Symbol">)</a>
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<a id="135" class="Keyword">open</a> <a id="140" class="Keyword">import</a> <a id="147" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="166" class="Keyword">using</a> <a id="172" class="Symbol">(</a><a id="173" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="180" class="Symbol">;</a> <a id="182" href="Categories.Functor.html#283" class="Function">Endofunctor</a><a id="193" class="Symbol">;</a> <a id="195" href="Categories.Functor.html#747" class="Function Operator">_∘F_</a><a id="199" class="Symbol">)</a> <a id="201" class="Keyword">renaming</a> <a id="210" class="Symbol">(</a><a id="211" href="Categories.Functor.html#349" class="Function">id</a> <a id="214" class="Symbol">to</a> <a id="217" class="Function">idF</a><a id="220" class="Symbol">)</a>
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<a id="222" class="Keyword">open</a> <a id="227" class="Keyword">import</a> <a id="234" href="Categories.NaturalTransformation.html" class="Module">Categories.NaturalTransformation</a> <a id="267" class="Keyword">renaming</a> <a id="276" class="Symbol">(</a><a id="277" href="Categories.NaturalTransformation.Core.html#2132" class="Function">id</a> <a id="280" class="Symbol">to</a> <a id="283" class="Function">idN</a><a id="286" class="Symbol">)</a>
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<a id="288" class="Keyword">open</a> <a id="293" class="Keyword">import</a> <a id="300" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a> <a id="352" class="Keyword">hiding</a> <a id="359" class="Symbol">(</a><a id="360" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3600" class="Function Operator">_≃_</a><a id="363" class="Symbol">)</a>
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<a id="365" class="Keyword">open</a> <a id="370" class="Keyword">import</a> <a id="377" href="Categories.NaturalTransformation.Equivalence.html" class="Module">Categories.NaturalTransformation.Equivalence</a>
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<a id="422" class="Keyword">open</a> <a id="427" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Module">NaturalIsomorphism</a>
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<a id="447" class="Keyword">record</a> <a id="Monad"></a><a id="454" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="460" class="Symbol">{</a><a id="461" href="Categories.Monad.html#461" class="Bound">o</a> <a id="463" href="Categories.Monad.html#463" class="Bound">ℓ</a> <a id="465" href="Categories.Monad.html#465" class="Bound">e</a><a id="466" class="Symbol">}</a> <a id="468" class="Symbol">(</a><a id="469" href="Categories.Monad.html#469" class="Bound">C</a> <a id="471" class="Symbol">:</a> <a id="473" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="482" href="Categories.Monad.html#461" class="Bound">o</a> <a id="484" href="Categories.Monad.html#463" class="Bound">ℓ</a> <a id="486" href="Categories.Monad.html#465" class="Bound">e</a><a id="487" class="Symbol">)</a> <a id="489" class="Symbol">:</a> <a id="491" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="495" class="Symbol">(</a><a id="496" href="Categories.Monad.html#461" class="Bound">o</a> <a id="498" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="500" href="Categories.Monad.html#463" class="Bound">ℓ</a> <a id="502" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="504" href="Categories.Monad.html#465" class="Bound">e</a><a id="505" class="Symbol">)</a> <a id="507" class="Keyword">where</a>
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<a id="515" class="Keyword">field</a>
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<a id="Monad.F"></a><a id="525" href="Categories.Monad.html#525" class="Field">F</a> <a id="527" class="Symbol">:</a> <a id="529" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="541" href="Categories.Monad.html#469" class="Bound">C</a>
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<a id="Monad.η"></a><a id="547" href="Categories.Monad.html#547" class="Field">η</a> <a id="549" class="Symbol">:</a> <a id="551" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="573" href="Categories.Monad.html#217" class="Function">idF</a> <a id="577" href="Categories.Monad.html#525" class="Field">F</a>
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<a id="Monad.μ"></a><a id="583" href="Categories.Monad.html#583" class="Field">μ</a> <a id="585" class="Symbol">:</a> <a id="587" href="Categories.NaturalTransformation.Core.html#466" class="Record">NaturalTransformation</a> <a id="609" class="Symbol">(</a><a id="610" href="Categories.Monad.html#525" class="Field">F</a> <a id="612" href="Categories.Functor.html#747" class="Function Operator">∘F</a> <a id="615" href="Categories.Monad.html#525" class="Field">F</a><a id="616" class="Symbol">)</a> <a id="618" href="Categories.Monad.html#525" class="Field">F</a>
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<a id="623" class="Keyword">module</a> <a id="Monad.F"></a><a id="630" href="Categories.Monad.html#630" class="Module">F</a> <a id="632" class="Symbol">=</a> <a id="634" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="642" href="Categories.Monad.html#525" class="Field">F</a>
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<a id="646" class="Keyword">module</a> <a id="Monad.η"></a><a id="653" href="Categories.Monad.html#653" class="Module">η</a> <a id="655" class="Symbol">=</a> <a id="657" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="679" href="Categories.Monad.html#547" class="Field">η</a>
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<a id="683" class="Keyword">module</a> <a id="Monad.μ"></a><a id="690" href="Categories.Monad.html#690" class="Module">μ</a> <a id="692" class="Symbol">=</a> <a id="694" href="Categories.NaturalTransformation.Core.html#466" class="Module">NaturalTransformation</a> <a id="716" href="Categories.Monad.html#583" class="Field">μ</a>
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<a id="721" class="Keyword">open</a> <a id="726" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="735" href="Categories.Monad.html#469" class="Bound">C</a>
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<a id="739" class="Keyword">open</a> <a id="744" href="Categories.Monad.html#630" class="Module">F</a>
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<a id="749" class="Keyword">field</a>
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<a id="Monad.assoc"></a><a id="759" href="Categories.Monad.html#759" class="Field">assoc</a> <a id="769" class="Symbol">:</a> <a id="771" class="Symbol">∀</a> <a id="773" class="Symbol">{</a><a id="774" href="Categories.Monad.html#774" class="Bound">X</a> <a id="776" class="Symbol">:</a> <a id="778" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="781" class="Symbol">}</a> <a id="783" class="Symbol">→</a> <a id="785" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="789" href="Categories.Monad.html#774" class="Bound">X</a> <a id="791" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="793" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="796" class="Symbol">(</a><a id="797" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="801" href="Categories.Monad.html#774" class="Bound">X</a><a id="802" class="Symbol">)</a> <a id="804" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="806" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="810" href="Categories.Monad.html#774" class="Bound">X</a> <a id="812" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="814" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="818" class="Symbol">(</a><a id="819" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="822" href="Categories.Monad.html#774" class="Bound">X</a><a id="823" class="Symbol">)</a>
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<a id="Monad.sym-assoc"></a><a id="829" href="Categories.Monad.html#829" class="Field">sym-assoc</a> <a id="839" class="Symbol">:</a> <a id="841" class="Symbol">∀</a> <a id="843" class="Symbol">{</a><a id="844" href="Categories.Monad.html#844" class="Bound">X</a> <a id="846" class="Symbol">:</a> <a id="848" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="851" class="Symbol">}</a> <a id="853" class="Symbol">→</a> <a id="855" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="859" href="Categories.Monad.html#844" class="Bound">X</a> <a id="861" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="863" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="867" class="Symbol">(</a><a id="868" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="871" href="Categories.Monad.html#844" class="Bound">X</a><a id="872" class="Symbol">)</a> <a id="874" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="876" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="880" href="Categories.Monad.html#844" class="Bound">X</a> <a id="882" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="884" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="887" class="Symbol">(</a><a id="888" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="892" href="Categories.Monad.html#844" class="Bound">X</a><a id="893" class="Symbol">)</a>
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<a id="Monad.identityˡ"></a><a id="899" href="Categories.Monad.html#899" class="Field">identityˡ</a> <a id="909" class="Symbol">:</a> <a id="911" class="Symbol">∀</a> <a id="913" class="Symbol">{</a><a id="914" href="Categories.Monad.html#914" class="Bound">X</a> <a id="916" class="Symbol">:</a> <a id="918" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="921" class="Symbol">}</a> <a id="923" class="Symbol">→</a> <a id="925" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="929" href="Categories.Monad.html#914" class="Bound">X</a> <a id="931" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="933" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="936" class="Symbol">(</a><a id="937" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="941" href="Categories.Monad.html#914" class="Bound">X</a><a id="942" class="Symbol">)</a> <a id="944" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="946" href="Categories.Category.Core.html#630" class="Function">id</a>
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<a id="Monad.identityʳ"></a><a id="953" href="Categories.Monad.html#953" class="Field">identityʳ</a> <a id="963" class="Symbol">:</a> <a id="965" class="Symbol">∀</a> <a id="967" class="Symbol">{</a><a id="968" href="Categories.Monad.html#968" class="Bound">X</a> <a id="970" class="Symbol">:</a> <a id="972" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="975" class="Symbol">}</a> <a id="977" class="Symbol">→</a> <a id="979" href="Categories.NaturalTransformation.Core.html#783" class="Function">μ.η</a> <a id="983" href="Categories.Monad.html#968" class="Bound">X</a> <a id="985" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="987" href="Categories.NaturalTransformation.Core.html#783" class="Function">η.η</a> <a id="991" class="Symbol">(</a><a id="992" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="995" href="Categories.Monad.html#968" class="Bound">X</a><a id="996" class="Symbol">)</a> <a id="998" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1000" href="Categories.Category.Core.html#630" class="Function">id</a>
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</pre></body></html>
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