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<html><head><meta charset="utf-8"><title>Categories.Category.Instance.Cats</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="36" class="Keyword">module</a> <a id="43" href="Categories.Category.Instance.Cats.html" class="Module">Categories.Category.Instance.Cats</a> <a id="77" class="Keyword">where</a>
<a id="84" class="Comment">-- The (large) category of (small) categories.</a>
<a id="131" class="Comment">-- Even though Agda can figure out the levels, it is worth making them explicit,</a>
<a id="212" class="Comment">-- to see the large level jumps involved.</a>
<a id="255" class="Keyword">open</a> <a id="260" class="Keyword">import</a> <a id="267" href="Level.html" class="Module">Level</a>
<a id="273" class="Keyword">open</a> <a id="278" class="Keyword">import</a> <a id="285" href="Categories.Category.html" class="Module">Categories.Category</a> <a id="305" class="Keyword">using</a> <a id="311" class="Symbol">(</a><a id="312" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="320" class="Symbol">)</a>
<a id="322" class="Keyword">open</a> <a id="327" class="Keyword">import</a> <a id="334" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="353" class="Keyword">using</a> <a id="359" class="Symbol">(</a><a id="360" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="367" class="Symbol">;</a> <a id="369" href="Categories.Functor.html#349" class="Function">id</a><a id="371" class="Symbol">;</a> <a id="373" href="Categories.Functor.html#747" class="Function Operator">_∘F_</a><a id="377" class="Symbol">)</a>
<a id="379" class="Keyword">open</a> <a id="384" class="Keyword">import</a> <a id="391" href="Categories.NaturalTransformation.NaturalIsomorphism.html" class="Module">Categories.NaturalTransformation.NaturalIsomorphism</a>
<a id="445" class="Keyword">using</a> <a id="451" class="Symbol">(</a><a id="452" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a><a id="470" class="Symbol">;</a> <a id="472" href="Categories.NaturalTransformation.NaturalIsomorphism.html#7073" class="Function">associator</a><a id="482" class="Symbol">;</a> <a id="484" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6216" class="Function">unitorˡ</a><a id="491" class="Symbol">;</a> <a id="493" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6312" class="Function">unitorʳ</a><a id="500" class="Symbol">;</a> <a id="502" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6406" class="Function">unitor²</a><a id="509" class="Symbol">;</a> <a id="511" href="Categories.NaturalTransformation.NaturalIsomorphism.html#5279" class="Function">isEquivalence</a><a id="524" class="Symbol">;</a> <a id="526" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3893" class="Function Operator">_ⓘₕ_</a><a id="530" class="Symbol">;</a> <a id="532" href="Categories.NaturalTransformation.NaturalIsomorphism.html#4964" class="Function">sym</a><a id="535" class="Symbol">)</a>
<a id="537" class="Keyword">private</a>
<a id="547" class="Keyword">variable</a>
<a id="560" href="Categories.Category.Instance.Cats.html#560" class="Generalizable">o</a> <a id="562" href="Categories.Category.Instance.Cats.html#562" class="Generalizable">ℓ</a> <a id="564" href="Categories.Category.Instance.Cats.html#564" class="Generalizable">e</a> <a id="566" class="Symbol">:</a> <a id="568" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="578" href="Categories.Category.Instance.Cats.html#578" class="Generalizable">C</a> <a id="580" href="Categories.Category.Instance.Cats.html#580" class="Generalizable">D</a> <a id="582" href="Categories.Category.Instance.Cats.html#582" class="Generalizable">E</a> <a id="584" class="Symbol">:</a> <a id="586" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="595" href="Categories.Category.Instance.Cats.html#560" class="Generalizable">o</a> <a id="597" href="Categories.Category.Instance.Cats.html#562" class="Generalizable">ℓ</a> <a id="599" href="Categories.Category.Instance.Cats.html#564" class="Generalizable">e</a>
<a id="605" href="Categories.Category.Instance.Cats.html#605" class="Generalizable">F</a> <a id="607" href="Categories.Category.Instance.Cats.html#607" class="Generalizable">G</a> <a id="609" href="Categories.Category.Instance.Cats.html#609" class="Generalizable">H</a> <a id="611" href="Categories.Category.Instance.Cats.html#611" class="Generalizable">I</a> <a id="613" class="Symbol">:</a> <a id="615" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="623" href="Categories.Category.Instance.Cats.html#578" class="Generalizable">C</a> <a id="625" href="Categories.Category.Instance.Cats.html#580" class="Generalizable">D</a>
<a id="Cats"></a><a id="628" href="Categories.Category.Instance.Cats.html#628" class="Function">Cats</a> <a id="633" class="Symbol">:</a> <a id="635" class="Symbol">∀</a> <a id="637" href="Categories.Category.Instance.Cats.html#637" class="Bound">o</a> <a id="639" href="Categories.Category.Instance.Cats.html#639" class="Bound">ℓ</a> <a id="641" href="Categories.Category.Instance.Cats.html#641" class="Bound">e</a> <a id="643" class="Symbol">→</a> <a id="645" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="654" class="Symbol">(</a><a id="655" href="Agda.Primitive.html#774" class="Primitive">suc</a> <a id="659" class="Symbol">(</a><a id="660" href="Categories.Category.Instance.Cats.html#637" class="Bound">o</a> <a id="662" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="664" href="Categories.Category.Instance.Cats.html#639" class="Bound">ℓ</a> <a id="666" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="669" href="Categories.Category.Instance.Cats.html#641" class="Bound">e</a><a id="670" class="Symbol">))</a> <a id="673" class="Symbol">(</a><a id="674" href="Categories.Category.Instance.Cats.html#637" class="Bound">o</a> <a id="676" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="678" href="Categories.Category.Instance.Cats.html#639" class="Bound">ℓ</a> <a id="680" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="682" href="Categories.Category.Instance.Cats.html#641" class="Bound">e</a><a id="683" class="Symbol">)</a> <a id="685" class="Symbol">(</a><a id="686" href="Categories.Category.Instance.Cats.html#637" class="Bound">o</a> <a id="688" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="690" href="Categories.Category.Instance.Cats.html#639" class="Bound">ℓ</a> <a id="692" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="694" href="Categories.Category.Instance.Cats.html#641" class="Bound">e</a><a id="695" class="Symbol">)</a>
<a id="697" href="Categories.Category.Instance.Cats.html#628" class="Function">Cats</a> <a id="702" href="Categories.Category.Instance.Cats.html#702" class="Bound">o</a> <a id="704" href="Categories.Category.Instance.Cats.html#704" class="Bound">ℓ</a> <a id="706" href="Categories.Category.Instance.Cats.html#706" class="Bound">e</a> <a id="708" class="Symbol">=</a> <a id="710" class="Keyword">record</a>
<a id="719" class="Symbol">{</a> <a id="721" href="Categories.Category.Core.html#559" class="Field">Obj</a> <a id="731" class="Symbol">=</a> <a id="733" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="742" href="Categories.Category.Instance.Cats.html#702" class="Bound">o</a> <a id="744" href="Categories.Category.Instance.Cats.html#704" class="Bound">ℓ</a> <a id="746" href="Categories.Category.Instance.Cats.html#706" class="Bound">e</a>
<a id="750" class="Symbol">;</a> <a id="752" href="Categories.Category.Core.html#575" class="Field Operator">_⇒_</a> <a id="762" class="Symbol">=</a> <a id="764" href="Categories.Functor.Core.html#248" class="Record">Functor</a>
<a id="774" class="Symbol">;</a> <a id="776" href="Categories.Category.Core.html#595" class="Field Operator">_≈_</a> <a id="786" class="Symbol">=</a> <a id="788" href="Categories.NaturalTransformation.NaturalIsomorphism.html#651" class="Record">NaturalIsomorphism</a>
<a id="809" class="Symbol">;</a> <a id="811" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="821" class="Symbol">=</a> <a id="823" href="Categories.Functor.html#349" class="Function">id</a>
<a id="828" class="Symbol">;</a> <a id="830" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a> <a id="840" class="Symbol">=</a> <a id="842" href="Categories.Functor.html#747" class="Function Operator">_∘F_</a>
<a id="849" class="Symbol">;</a> <a id="851" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="861" class="Symbol">=</a> <a id="863" class="Symbol">λ</a> <a id="865" class="Symbol">{</a><a id="866" href="Categories.Category.Instance.Cats.html#866" class="Bound">_</a> <a id="868" href="Categories.Category.Instance.Cats.html#868" class="Bound">_</a> <a id="870" href="Categories.Category.Instance.Cats.html#870" class="Bound">_</a> <a id="872" href="Categories.Category.Instance.Cats.html#872" class="Bound">_</a> <a id="874" href="Categories.Category.Instance.Cats.html#874" class="Bound">F</a> <a id="876" href="Categories.Category.Instance.Cats.html#876" class="Bound">G</a> <a id="878" href="Categories.Category.Instance.Cats.html#878" class="Bound">H</a><a id="879" class="Symbol">}</a> <a id="881" class="Symbol">→</a> <a id="883" href="Categories.NaturalTransformation.NaturalIsomorphism.html#7073" class="Function">associator</a> <a id="894" href="Categories.Category.Instance.Cats.html#874" class="Bound">F</a> <a id="896" href="Categories.Category.Instance.Cats.html#876" class="Bound">G</a> <a id="898" href="Categories.Category.Instance.Cats.html#878" class="Bound">H</a>
<a id="902" class="Symbol">;</a> <a id="904" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="914" class="Symbol">=</a> <a id="916" class="Symbol">λ</a> <a id="918" class="Symbol">{</a><a id="919" href="Categories.Category.Instance.Cats.html#919" class="Bound">_</a> <a id="921" href="Categories.Category.Instance.Cats.html#921" class="Bound">_</a> <a id="923" href="Categories.Category.Instance.Cats.html#923" class="Bound">_</a> <a id="925" href="Categories.Category.Instance.Cats.html#925" class="Bound">_</a> <a id="927" href="Categories.Category.Instance.Cats.html#927" class="Bound">F</a> <a id="929" href="Categories.Category.Instance.Cats.html#929" class="Bound">G</a> <a id="931" href="Categories.Category.Instance.Cats.html#931" class="Bound">H</a><a id="932" class="Symbol">}</a> <a id="934" class="Symbol">→</a> <a id="936" href="Categories.NaturalTransformation.NaturalIsomorphism.html#4964" class="Function">sym</a> <a id="940" class="Symbol">(</a><a id="941" href="Categories.NaturalTransformation.NaturalIsomorphism.html#7073" class="Function">associator</a> <a id="952" href="Categories.Category.Instance.Cats.html#927" class="Bound">F</a> <a id="954" href="Categories.Category.Instance.Cats.html#929" class="Bound">G</a> <a id="956" href="Categories.Category.Instance.Cats.html#931" class="Bound">H</a><a id="957" class="Symbol">)</a>
<a id="961" class="Symbol">;</a> <a id="963" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="973" class="Symbol">=</a> <a id="975" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6216" class="Function">unitorˡ</a>
<a id="985" class="Symbol">;</a> <a id="987" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="997" class="Symbol">=</a> <a id="999" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6312" class="Function">unitorʳ</a>
<a id="1009" class="Symbol">;</a> <a id="1011" href="Categories.Category.Core.html#1339" class="Field">identity²</a> <a id="1021" class="Symbol">=</a> <a id="1023" href="Categories.NaturalTransformation.NaturalIsomorphism.html#6406" class="Function">unitor²</a>
<a id="1033" class="Symbol">;</a> <a id="1035" href="Categories.Category.Core.html#1384" class="Field">equiv</a> <a id="1045" class="Symbol">=</a> <a id="1047" href="Categories.NaturalTransformation.NaturalIsomorphism.html#5279" class="Function">isEquivalence</a>
<a id="1063" class="Symbol">;</a> <a id="1065" href="Categories.Category.Core.html#1438" class="Field">∘-resp-≈</a> <a id="1075" class="Symbol">=</a> <a id="1077" href="Categories.NaturalTransformation.NaturalIsomorphism.html#3893" class="Function Operator">_ⓘₕ_</a>
<a id="1084" class="Symbol">}</a>
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