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<html><head><meta charset="utf-8"><title>Categories.Category.Construction.F-Algebras</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="1412" class="Symbol">(</a><a id="1413" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1415" href="Categories.Category.Construction.F-Algebras.html#1392" class="Bound">α₁</a> <a id="1418" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1420" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1422" href="Categories.Category.Construction.F-Algebras.html#1395" class="Bound">α₂</a><a id="1424" class="Symbol">)</a> <a id="1426" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1428" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="1430" href="Categories.Category.Construction.F-Algebras.html#1381" class="Bound">A</a> <a id="1443" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1446" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1452" class="Symbol">(</a><a id="1453" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="1462" href="Categories.Category.Construction.F-Algebras.html#1395" class="Bound">α₂</a><a id="1464" class="Symbol">)</a> <a id="1466" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1474" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1476" href="Categories.Category.Construction.F-Algebras.html#1392" class="Bound">α₁</a> <a id="1479" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1481" class="Symbol">(</a><a id="1482" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="1484" href="Categories.Category.Construction.F-Algebras.html#1385" class="Bound">B</a> <a id="1486" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1488" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1491" class="Symbol">(</a><a id="1492" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1494" href="Categories.Category.Construction.F-Algebras.html#1395" class="Bound">α₂</a><a id="1496" class="Symbol">))</a> <a id="1505" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1508" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1514" class="Symbol">(</a><a id="1515" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="1524" href="Categories.Category.Construction.F-Algebras.html#1392" class="Bound">α₁</a><a id="1526" class="Symbol">)</a> <a id="1528" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1536" class="Symbol">(</a><a id="1537" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="1539" href="Categories.Category.Construction.F-Algebras.html#1389" class="Bound">C</a> <a id="1541" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1543" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1546" class="Symbol">(</a><a id="1547" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1549" href="Categories.Category.Construction.F-Algebras.html#1392" class="Bound">α₁</a><a id="1551" class="Symbol">))</a> <a id="1554" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1556" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1559" class="Symbol">(</a><a id="1560" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1562" href="Categories.Category.Construction.F-Algebras.html#1395" class="Bound">α₂</a><a id="1564" class="Symbol">)</a> <a id="1567" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1570" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1576" class="Symbol">(</a><a id="1577" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1579" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a><a id="1591" class="Symbol">)</a> <a id="1593" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1601" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="1603" href="Categories.Category.Construction.F-Algebras.html#1389" class="Bound">C</a> <a id="1605" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1607" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1610" class="Symbol">(</a><a id="1611" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1613" href="Categories.Category.Construction.F-Algebras.html#1392" class="Bound">α₁</a> <a id="1616" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1618" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="1620" href="Categories.Category.Construction.F-Algebras.html#1395" class="Bound">α₂</a><a id="1622" class="Symbol">)</a> <a id="1626" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="1630" class="Keyword">module</a> <a id="Lambek"></a><a id="1637" href="Categories.Category.Construction.F-Algebras.html#1637" class="Module">Lambek</a> <a id="1644" class="Symbol">{</a><a id="1645" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a> <a id="1647" class="Symbol">:</a> <a id="1649" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="1658" href="Categories.Category.Construction.F-Algebras.html#424" class="Generalizable">o</a> <a id="1660" href="Categories.Category.Construction.F-Algebras.html#426" class="Generalizable">ℓ</a> <a id="1662" href="Categories.Category.Construction.F-Algebras.html#428" class="Generalizable">e</a><a id="1663" class="Symbol">}</a> <a id="1665" class="Symbol">{</a><a id="1666" href="Categories.Category.Construction.F-Algebras.html#1666" class="Bound">F</a> <a id="1668" class="Symbol">:</a> <a id="1670" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="1682" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a><a id="1683" class="Symbol">}</a> <a id="1685" class="Symbol">(</a><a id="1686" href="Categories.Category.Construction.F-Algebras.html#1686" class="Bound">I</a> <a id="1688" class="Symbol">:</a> <a id="1690" href="Categories.Object.Initial.html#760" class="Record">Initial</a> <a id="1698" class="Symbol">(</a><a id="1699" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a> <a id="1710" href="Categories.Category.Construction.F-Algebras.html#1666" class="Bound">F</a><a id="1711" class="Symbol">))</a> <a id="1714" class="Keyword">where</a>
<a id="1722" class="Keyword">open</a> <a id="1727" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1736" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a>
<a id="1740" class="Keyword">open</a> <a id="1745" href="Categories.Category.html#824" class="Module">Definitions</a> <a id="1757" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a>
<a id="1761" class="Keyword">open</a> <a id="1766" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1774" href="Categories.Category.Construction.F-Algebras.html#1666" class="Bound">F</a>
<a id="1778" class="Keyword">open</a> <a id="1783" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="1793" class="Keyword">using</a> <a id="1799" class="Symbol">(</a><a id="1800" href="Categories.Functor.Algebra.html#587" class="Field">α</a><a id="1801" class="Symbol">)</a>
<a id="1806" class="Keyword">open</a> <a id="1811" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1814" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a> <a id="1816" class="Keyword">using</a> <a id="1822" class="Symbol">(</a><a id="1823" href="Categories.Morphism.Reasoning.Core.html#4290" class="Function">glue</a><a id="1827" class="Symbol">)</a>
<a id="1831" class="Keyword">open</a> <a id="1836" href="Categories.Morphism.html" class="Module">Mor</a> <a id="1840" href="Categories.Category.Construction.F-Algebras.html#1645" class="Bound">𝒞</a>
<a id="1844" class="Keyword">open</a> <a id="1849" href="Categories.Object.Initial.html#760" class="Module">Initial</a> <a id="1857" href="Categories.Category.Construction.F-Algebras.html#1686" class="Bound">I</a> <a id="1859" class="Comment">-- so ⊥ is an F-Algebra, which is initial</a>
<a id="1904" class="Comment">-- While an expert might be able to decipher the proof at the nLab</a>
<a id="1973" class="Comment">-- (https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor)</a>
<a id="2046" class="Comment">-- I (JC) have found that the notes at</a>
<a id="2087" class="Comment">-- http://www.cs.ru.nl/~jrot/coalgebra/ak-algebras.pdf</a>
<a id="2146" class="Comment">-- are easier to follow, and lead to the full proof below.</a>
<a id="2207" class="Keyword">private</a>
<a id="2219" class="Keyword">module</a> <a id="Lambek.⊥"></a><a id="2226" href="Categories.Category.Construction.F-Algebras.html#2226" class="Module">⊥</a> <a id="2228" class="Symbol">=</a> <a id="2230" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="2240" href="Categories.Object.Initial.html#804" class="Field">⊥</a>
<a id="Lambek.A"></a><a id="2246" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a> <a id="2248" class="Symbol">=</a> <a id="2250" href="Categories.Functor.Algebra.html#573" class="Function">⊥.A</a>
<a id="Lambek.a"></a><a id="2258" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="2260" class="Symbol">:</a> <a id="2262" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="2265" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a> <a id="2267" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2269" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a>
<a id="2275" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="2277" class="Symbol">=</a> <a id="2279" href="Categories.Functor.Algebra.html#587" class="Function">⊥.α</a>
<a id="2288" class="Comment">-- The F-Algebra structure that will make things work</a>
<a id="Lambek.F⊥"></a><a id="2346" href="Categories.Category.Construction.F-Algebras.html#2346" class="Function">F⊥</a> <a id="2349" class="Symbol">:</a> <a id="2351" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="2361" href="Categories.Category.Construction.F-Algebras.html#1666" class="Bound">F</a>
<a id="2367" href="Categories.Category.Construction.F-Algebras.html#2346" class="Function">F⊥</a> <a id="2370" class="Symbol">=</a> <a id="2372" href="Categories.Functor.Algebra.html#859" class="Function">iterate</a> <a id="2380" href="Categories.Object.Initial.html#804" class="Field">⊥</a>
<a id="2387" class="Comment">-- By initiality, we get the following morphism</a>
<a id="Lambek.f"></a><a id="2439" href="Categories.Category.Construction.F-Algebras.html#2439" class="Function">f</a> <a id="2441" class="Symbol">:</a> <a id="2443" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a> <a id="2462" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="2464" href="Categories.Category.Construction.F-Algebras.html#2346" class="Function">F⊥</a>
<a id="2471" href="Categories.Category.Construction.F-Algebras.html#2439" class="Function">f</a> <a id="2473" class="Symbol">=</a> <a id="2475" href="Categories.Object.Initial.html#478" class="Function">!</a>
<a id="2482" class="Keyword">module</a> <a id="Lambek.FAM"></a><a id="2489" href="Categories.Category.Construction.F-Algebras.html#2489" class="Module">FAM</a> <a id="2493" class="Symbol">=</a> <a id="2495" href="Categories.Functor.Algebra.html#1307" class="Module">F-Algebra-Morphism</a> <a id="2514" href="Categories.Category.Construction.F-Algebras.html#2439" class="Function">f</a>
<a id="Lambek.i"></a><a id="2521" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a> <a id="2523" class="Symbol">:</a> <a id="2525" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a> <a id="2527" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2529" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="2532" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a>
<a id="2538" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a> <a id="2540" class="Symbol">=</a> <a id="2542" href="Categories.Functor.Algebra.html#1366" class="Function">FAM.f</a>
<a id="Lambek.a∘f"></a><a id="2553" href="Categories.Category.Construction.F-Algebras.html#2553" class="Function">a∘f</a> <a id="2557" class="Symbol">:</a> <a id="2559" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a> <a id="2578" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="2580" href="Categories.Object.Initial.html#804" class="Field">⊥</a>
<a id="2586" href="Categories.Category.Construction.F-Algebras.html#2553" class="Function">a∘f</a> <a id="2590" class="Symbol">=</a> <a id="2592" class="Keyword">record</a>
<a id="2605" class="Symbol">{</a> <a id="2607" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="2609" class="Symbol">=</a> <a id="2611" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="2613" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2615" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a>
<a id="2623" class="Symbol">;</a> <a id="2625" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="2634" class="Symbol">=</a> <a id="2636" href="Categories.Morphism.Reasoning.Core.html#4290" class="Function">glue</a> <a id="2641" href="Categories.Category.Construction.F-Algebras.html#2757" class="Function">triv</a> <a id="2646" href="Categories.Functor.Algebra.html#1388" class="Function">FAM.commutes</a> <a id="2659" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2661" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="2671" class="Symbol">(</a><a id="2672" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2674" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a><a id="2686" class="Symbol">)</a>
<a id="2694" class="Symbol">}</a>
<a id="2702" class="Keyword">where</a>
<a id="2716" class="Keyword">open</a> <a id="2721" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a> <a id="2734" class="Keyword">using</a> <a id="2740" class="Symbol">(</a><a id="2741" href="Categories.Category.Core.html#3061" class="Function Operator">_○_</a><a id="2744" class="Symbol">;</a> <a id="2746" href="Categories.Category.Core.html#3005" class="Function">⟺</a><a id="2747" class="Symbol">)</a>
<a id="2757" href="Categories.Category.Construction.F-Algebras.html#2757" class="Function">triv</a> <a id="2762" class="Symbol">:</a> <a id="2764" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="2782" class="Symbol">(</a><a id="2783" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="2785" href="Categories.Category.Construction.F-Algebras.html#2346" class="Function">F⊥</a><a id="2787" class="Symbol">)</a> <a id="2789" class="Symbol">(</a><a id="2790" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="2792" href="Categories.Category.Construction.F-Algebras.html#2346" class="Function">F⊥</a><a id="2794" class="Symbol">)</a> <a id="2796" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="2798" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a>
<a id="2808" href="Categories.Category.Construction.F-Algebras.html#2757" class="Function">triv</a> <a id="2813" class="Symbol">=</a> <a id="2815" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a>
<a id="Lambek.lambek"></a><a id="2829" href="Categories.Category.Construction.F-Algebras.html#2829" class="Function">lambek</a> <a id="2836" class="Symbol">:</a> <a id="2838" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a> <a id="2840" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="2842" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="2845" href="Categories.Category.Construction.F-Algebras.html#2246" class="Function">A</a>
<a id="2849" href="Categories.Category.Construction.F-Algebras.html#2829" class="Function">lambek</a> <a id="2856" class="Symbol">=</a> <a id="2858" class="Keyword">record</a>
<a id="2869" class="Symbol">{</a> <a id="2871" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2876" class="Symbol">=</a> <a id="2878" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a>
<a id="2884" class="Symbol">;</a> <a id="2886" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2889" class="Symbol">=</a> <a id="2891" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a>
<a id="2897" class="Symbol">;</a> <a id="2899" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="2903" class="Symbol">=</a> <a id="2905" class="Keyword">record</a>
<a id="2918" class="Symbol">{</a> <a id="2920" href="Categories.Morphism.html#1586" class="Field">isoˡ</a> <a id="2925" class="Symbol">=</a> <a id="2927" href="Categories.Object.Initial.html#698" class="Function">⊥-id</a> <a id="2932" href="Categories.Category.Construction.F-Algebras.html#2553" class="Function">a∘f</a>
<a id="2942" class="Symbol">;</a> <a id="2944" href="Categories.Morphism.html#1612" class="Field">isoʳ</a> <a id="2949" class="Symbol">=</a> <a id="2951" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2965" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a> <a id="2967" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2969" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="2977" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2980" href="Categories.Functor.Algebra.html#1388" class="Field">F-Algebra-Morphism.commutes</a> <a id="3008" href="Categories.Category.Construction.F-Algebras.html#2439" class="Function">f</a> <a id="3010" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3020" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="3023" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="3025" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3027" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="3030" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a> <a id="3032" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3036" href="Categories.Functor.Core.html#565" class="Function">homomorphism</a> <a id="3049" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="3059" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="3062" class="Symbol">(</a><a id="3063" href="Categories.Category.Construction.F-Algebras.html#2258" class="Function">a</a> <a id="3065" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3067" href="Categories.Category.Construction.F-Algebras.html#2521" class="Function">i</a><a id="3068" class="Symbol">)</a> <a id="3071" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3074" href="Categories.Functor.Core.html#696" class="Function">F-resp-≈</a> <a id="3083" class="Symbol">(</a><a id="3084" href="Categories.Object.Initial.html#698" class="Function">⊥-id</a> <a id="3089" href="Categories.Category.Construction.F-Algebras.html#2553" class="Function">a∘f</a><a id="3092" class="Symbol">)</a> <a id="3094" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3104" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="3107" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3116" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3119" href="Categories.Functor.Core.html#511" class="Function">identity</a> <a id="3128" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3138" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3141" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="3149" class="Symbol">}</a>
<a id="3155" class="Symbol">}</a>
<a id="3161" class="Keyword">where</a>
<a id="3173" class="Keyword">open</a> <a id="3178" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
</pre></body></html>
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