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</pre>-->
<h1 id="the-monad-k">The monad K</h1>
<pre class="Agda"><a id="515" class="Keyword">module</a> <a id="522" href="Monad.Instance.K.html" class="Module">Monad.Instance.K</a> <a id="539" class="Symbol">{</a><a id="540" href="Monad.Instance.K.html#540" class="Bound">o</a> <a id="542" href="Monad.Instance.K.html#542" class="Bound">ℓ</a> <a id="544" href="Monad.Instance.K.html#544" class="Bound">e</a><a id="545" class="Symbol">}</a> <a id="547" class="Symbol">(</a><a id="548" href="Monad.Instance.K.html#548" class="Bound">ambient</a> <a id="556" class="Symbol">:</a> <a id="558" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="566" href="Monad.Instance.K.html#540" class="Bound">o</a> <a id="568" href="Monad.Instance.K.html#542" class="Bound">ℓ</a> <a id="570" href="Monad.Instance.K.html#544" class="Bound">e</a><a id="571" class="Symbol">)</a> <a id="573" class="Keyword">where</a>
<a id="581" class="Keyword">open</a> <a id="586" href="Category.Ambient.html#1681" class="Module">Ambient</a> <a id="594" href="Monad.Instance.K.html#548" class="Bound">ambient</a>
<a id="604" class="Keyword">open</a> <a id="609" class="Keyword">import</a> <a id="616" href="Category.Construction.ElgotAlgebras.html" class="Module">Category.Construction.ElgotAlgebras</a> <a id="652" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>
<a id="666" class="Keyword">open</a> <a id="671" class="Keyword">import</a> <a id="678" href="Algebra.Elgot.html" class="Module">Algebra.Elgot</a> <a id="692" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a> <a id="704" class="Keyword">using</a> <a id="710" class="Symbol">(</a><a id="711" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a><a id="724" class="Symbol">)</a>
<a id="728" class="Keyword">open</a> <a id="733" class="Keyword">import</a> <a id="740" href="Algebra.Elgot.Free.html" class="Module">Algebra.Elgot.Free</a> <a id="759" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a> <a id="771" class="Keyword">using</a> <a id="777" class="Symbol">(</a><a id="778" href="Algebra.Elgot.Free.html#1441" class="Function">FreeElgotAlgebra</a><a id="794" class="Symbol">;</a> <a id="796" href="Algebra.Elgot.Free.html#1197" class="Function">elgotForgetfulF</a><a id="811" class="Symbol">)</a>
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<a id="933" class="Keyword">open</a> <a id="938" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
</pre>
<h2 id="definition">Definition</h2>
<p>The monad is defined by existence of free uniform-iteration algebras.
Since free objects yield and adjunctions, this yields a monad.</p>
<pre class="Agda"> <a id="1116" class="Keyword">record</a> <a id="MonadK"></a><a id="1123" href="Monad.Instance.K.html#1123" class="Record">MonadK</a> <a id="1130" class="Symbol">:</a> <a id="1132" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1136" class="Symbol">(</a><a id="1137" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1141" href="Monad.Instance.K.html#540" class="Bound">o</a> <a id="1143" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1145" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1149" href="Monad.Instance.K.html#542" class="Bound">ℓ</a> <a id="1151" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1153" href="Agda.Primitive.html#931" class="Primitive">suc</a> <a id="1157" href="Monad.Instance.K.html#544" class="Bound">e</a><a id="1158" class="Symbol">)</a> <a id="1160" class="Keyword">where</a>
<a id="1170" class="Keyword">field</a>
<a id="MonadK.freealgebras"></a><a id="1182" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1195" class="Symbol">:</a> <a id="1197" class="Symbol">∀</a> <a id="1199" href="Monad.Instance.K.html#1199" class="Bound">X</a> <a id="1201" class="Symbol">→</a> <a id="1203" href="Algebra.Elgot.Free.html#1441" class="Function">FreeElgotAlgebra</a> <a id="1220" href="Monad.Instance.K.html#1199" class="Bound">X</a>
<a id="MonadK.stable"></a><a id="1228" href="Monad.Instance.K.html#1228" class="Field">stable</a> <a id="1235" class="Symbol">:</a> <a id="1237" class="Symbol">∀</a> <a id="1239" href="Monad.Instance.K.html#1239" class="Bound">X</a> <a id="1241" class="Symbol">→</a> <a id="1243" href="Algebra.Elgot.Stable.html#2437" class="Record">IsStableFreeElgotAlgebra</a> <a id="1268" class="Symbol">(</a><a id="1269" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1282" href="Monad.Instance.K.html#1239" class="Bound">X</a><a id="1283" class="Symbol">)</a>
<a id="1290" class="Comment">-- helper for accessing elgot algebras</a>
<a id="MonadK.algebras"></a><a id="1333" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="1342" class="Symbol">:</a> <a id="1344" class="Symbol">∀</a> <a id="1346" class="Symbol">(</a><a id="1347" href="Monad.Instance.K.html#1347" class="Bound">X</a> <a id="1349" class="Symbol">:</a> <a id="1351" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1354" class="Symbol">)</a> <a id="1356" class="Symbol">→</a> <a id="1358" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
<a id="1376" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="1385" href="Monad.Instance.K.html#1385" class="Bound">X</a> <a id="1387" class="Symbol">=</a> <a id="1389" href="Categories.FreeObjects.Free.html#838" class="Field">FreeObject.FX</a> <a id="1403" class="Symbol">(</a><a id="1404" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1417" href="Monad.Instance.K.html#1385" class="Bound">X</a><a id="1418" class="Symbol">)</a>
<a id="MonadK.freeF"></a><a id="1425" href="Monad.Instance.K.html#1425" class="Function">freeF</a> <a id="1431" class="Symbol">:</a> <a id="1433" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1441" href="Category.Ambient.html#1757" class="Field">C</a> <a id="1443" href="Category.Construction.ElgotAlgebras.html#939" class="Function">Elgot-Algebras</a>
<a id="1462" href="Monad.Instance.K.html#1425" class="Function">freeF</a> <a id="1468" class="Symbol">=</a> <a id="1470" href="Categories.FreeObjects.Free.html#1758" class="Function">FO⇒Functor</a> <a id="1481" href="Algebra.Elgot.Free.html#1197" class="Function">elgotForgetfulF</a> <a id="1497" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a>
<a id="MonadK.adjoint"></a><a id="1519" href="Monad.Instance.K.html#1519" class="Function">adjoint</a> <a id="1527" class="Symbol">:</a> <a id="1529" href="Monad.Instance.K.html#1425" class="Function">freeF</a> <a id="1535" href="Categories.Adjoint.html#7818" class="Function Operator">⊣</a> <a id="1537" href="Algebra.Elgot.Free.html#1197" class="Function">elgotForgetfulF</a>
<a id="1557" href="Monad.Instance.K.html#1519" class="Function">adjoint</a> <a id="1565" class="Symbol">=</a> <a id="1567" href="Categories.FreeObjects.Free.html#7309" class="Function">FO⇒LAdj</a> <a id="1575" href="Algebra.Elgot.Free.html#1197" class="Function">elgotForgetfulF</a> <a id="1591" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a>
<a id="MonadK.monadK"></a><a id="1609" href="Monad.Instance.K.html#1609" class="Function">monadK</a> <a id="1616" class="Symbol">:</a> <a id="1618" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="1624" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="1630" href="Monad.Instance.K.html#1609" class="Function">monadK</a> <a id="1637" class="Symbol">=</a> <a id="1639" href="Categories.Adjoint.Properties.html#9436" class="Function">adjoint⇒monad</a> <a id="1653" href="Monad.Instance.K.html#1519" class="Function">adjoint</a>
<a id="1665" class="Keyword">module</a> <a id="MonadK.monadK"></a><a id="1672" href="Monad.Instance.K.html#1672" class="Module">monadK</a> <a id="1679" class="Symbol">=</a> <a id="1681" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="1687" href="Monad.Instance.K.html#1609" class="Function">monadK</a>
<a id="MonadK.kleisliK"></a><a id="1699" href="Monad.Instance.K.html#1699" class="Function">kleisliK</a> <a id="1708" class="Symbol">:</a> <a id="1710" href="Categories.Monad.Construction.Kleisli.html#882" class="Function">KleisliTriple</a> <a id="1724" href="Category.Ambient.html#1757" class="Field">C</a>
<a id="1730" href="Monad.Instance.K.html#1699" class="Function">kleisliK</a> <a id="1739" class="Symbol">=</a> <a id="1741" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="1755" href="Category.Ambient.html#1757" class="Field">C</a> <a id="1757" href="Monad.Instance.K.html#1609" class="Function">monadK</a>
<a id="1768" class="Keyword">module</a> <a id="MonadK.kleisliK"></a><a id="1775" href="Monad.Instance.K.html#1775" class="Module">kleisliK</a> <a id="1784" class="Symbol">=</a> <a id="1786" href="Monad.Instance.K.html#380" class="Module">RMonad</a> <a id="1793" href="Monad.Instance.K.html#1699" class="Function">kleisliK</a>
<a id="1807" class="Keyword">module</a> <a id="MonadK.K"></a><a id="1814" href="Monad.Instance.K.html#1814" class="Module">K</a> <a id="1816" class="Symbol">=</a> <a id="1818" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1826" href="Categories.Monad.html#525" class="Function">monadK.F</a>
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