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<html><head><meta charset="utf-8"><title>Categories.Category.Construction.Path</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">open</a> <a id="41" class="Keyword">import</a> <a id="48" href="Level.html" class="Module">Level</a>
<a id="55" class="Keyword">open</a> <a id="60" class="Keyword">import</a> <a id="67" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="88" class="Keyword">module</a> <a id="95" href="Categories.Category.Construction.Path.html" class="Module">Categories.Category.Construction.Path</a> <a id="133" class="Symbol">{</a><a id="134" href="Categories.Category.Construction.Path.html#134" class="Bound">o</a> <a id="136" href="Categories.Category.Construction.Path.html#136" class="Bound">ℓ</a> <a id="138" href="Categories.Category.Construction.Path.html#138" class="Bound">e</a> <a id="140" class="Symbol">:</a> <a id="142" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="147" class="Symbol">}</a> <a id="149" class="Symbol">(</a><a id="150" href="Categories.Category.Construction.Path.html#150" class="Bound">C</a> <a id="152" class="Symbol">:</a> <a id="154" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="163" href="Categories.Category.Construction.Path.html#134" class="Bound">o</a> <a id="165" href="Categories.Category.Construction.Path.html#136" class="Bound">ℓ</a> <a id="167" href="Categories.Category.Construction.Path.html#138" class="Bound">e</a><a id="168" class="Symbol">)</a> <a id="170" class="Keyword">where</a>
<a id="177" class="Keyword">open</a> <a id="182" class="Keyword">import</a> <a id="189" href="Function.html" class="Module">Function</a> <a id="198" class="Keyword">using</a> <a id="204" class="Symbol">(</a><a id="205" href="Function.Base.html#1638" class="Function">flip</a><a id="209" class="Symbol">)</a>
<a id="211" class="Keyword">open</a> <a id="216" class="Keyword">import</a> <a id="223" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="239" class="Keyword">hiding</a> <a id="246" class="Symbol">(</a><a id="247" href="Relation.Binary.Core.html#1268" class="Function Operator">_⇒_</a><a id="250" class="Symbol">)</a>
<a id="252" class="Keyword">open</a> <a id="257" class="Keyword">import</a> <a id="264" href="Relation.Binary.Construct.Closure.Transitive.html" class="Module">Relation.Binary.Construct.Closure.Transitive</a>
<a id="310" class="Keyword">open</a> <a id="315" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="324" href="Categories.Category.Construction.Path.html#150" class="Bound">C</a>
<a id="327" class="Comment">-- Defining the Path Category</a>
<a id="∘-tc"></a><a id="357" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="362" class="Symbol">:</a> <a id="364" class="Symbol">{</a><a id="365" href="Categories.Category.Construction.Path.html#365" class="Bound">A</a> <a id="367" href="Categories.Category.Construction.Path.html#367" class="Bound">B</a> <a id="369" class="Symbol">:</a> <a id="371" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="374" class="Symbol">}</a> <a id="376" class="Symbol">→</a> <a id="378" href="Categories.Category.Construction.Path.html#365" class="Bound">A</a> <a id="380" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">[</a> <a id="382" href="Categories.Category.Core.html#575" class="Field Operator">_⇒_</a> <a id="386" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">]⁺</a> <a id="389" href="Categories.Category.Construction.Path.html#367" class="Bound">B</a> <a id="391" class="Symbol">→</a> <a id="393" href="Categories.Category.Construction.Path.html#365" class="Bound">A</a> <a id="395" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="397" href="Categories.Category.Construction.Path.html#367" class="Bound">B</a>
<a id="399" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="404" href="Relation.Binary.Construct.Closure.Transitive.html#2795" class="InductiveConstructor Operator">[</a> <a id="406" href="Categories.Category.Construction.Path.html#406" class="Bound">f</a> <a id="408" href="Relation.Binary.Construct.Closure.Transitive.html#2795" class="InductiveConstructor Operator">]</a> <a id="421" class="Symbol">=</a> <a id="423" href="Categories.Category.Construction.Path.html#406" class="Bound">f</a>
<a id="425" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="430" class="Symbol">(_</a> <a id="433" href="Relation.Binary.Construct.Closure.Transitive.html#2844" class="InductiveConstructor Operator">∼⁺⟨</a> <a id="437" href="Categories.Category.Construction.Path.html#437" class="Bound">f⁺</a> <a id="440" href="Relation.Binary.Construct.Closure.Transitive.html#2844" class="InductiveConstructor Operator">⟩</a> <a id="442" href="Categories.Category.Construction.Path.html#442" class="Bound">f⁺′</a><a id="445" class="Symbol">)</a> <a id="447" class="Symbol">=</a> <a id="449" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="454" href="Categories.Category.Construction.Path.html#442" class="Bound">f⁺′</a> <a id="458" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="460" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="465" href="Categories.Category.Construction.Path.html#437" class="Bound">f⁺</a>
<a id="469" class="Keyword">infix</a> <a id="475" class="Number">4</a> <a id="477" href="Categories.Category.Construction.Path.html#482" class="Function Operator">_≈⁺_</a>
<a id="_≈⁺_"></a><a id="482" href="Categories.Category.Construction.Path.html#482" class="Function Operator">_≈⁺_</a> <a id="487" class="Symbol">:</a> <a id="489" class="Symbol">{</a><a id="490" href="Categories.Category.Construction.Path.html#490" class="Bound">A</a> <a id="492" href="Categories.Category.Construction.Path.html#492" class="Bound">B</a> <a id="494" class="Symbol">:</a> <a id="496" href="Categories.Category.Core.html#559" class="Field">Obj</a><a id="499" class="Symbol">}</a> <a id="501" class="Symbol">→</a> <a id="503" class="Symbol">(</a><a id="504" href="Categories.Category.Construction.Path.html#504" class="Bound">i</a> <a id="506" href="Categories.Category.Construction.Path.html#506" class="Bound">j</a> <a id="508" class="Symbol">:</a> <a id="510" href="Categories.Category.Construction.Path.html#490" class="Bound">A</a> <a id="512" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">[</a> <a id="514" href="Categories.Category.Core.html#575" class="Field Operator">_⇒_</a> <a id="518" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">]⁺</a> <a id="521" href="Categories.Category.Construction.Path.html#492" class="Bound">B</a><a id="522" class="Symbol">)</a> <a id="524" class="Symbol">→</a> <a id="526" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="530" href="Categories.Category.Construction.Path.html#138" class="Bound">e</a>
<a id="532" href="Categories.Category.Construction.Path.html#532" class="Bound">f⁺</a> <a id="535" href="Categories.Category.Construction.Path.html#482" class="Function Operator">≈⁺</a> <a id="538" href="Categories.Category.Construction.Path.html#538" class="Bound">g⁺</a> <a id="541" class="Symbol">=</a> <a id="543" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="548" href="Categories.Category.Construction.Path.html#532" class="Bound">f⁺</a> <a id="551" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="553" href="Categories.Category.Construction.Path.html#357" class="Function">∘-tc</a> <a id="558" href="Categories.Category.Construction.Path.html#538" class="Bound">g⁺</a>
<a id="Path"></a><a id="562" href="Categories.Category.Construction.Path.html#562" class="Function">Path</a> <a id="567" class="Symbol">:</a> <a id="569" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="578" href="Categories.Category.Construction.Path.html#134" class="Bound">o</a> <a id="580" class="Symbol">(</a><a id="581" href="Categories.Category.Construction.Path.html#134" class="Bound">o</a> <a id="583" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="585" href="Categories.Category.Construction.Path.html#136" class="Bound">ℓ</a><a id="586" class="Symbol">)</a> <a id="588" href="Categories.Category.Construction.Path.html#138" class="Bound">e</a>
<a id="590" href="Categories.Category.Construction.Path.html#562" class="Function">Path</a> <a id="595" class="Symbol">=</a> <a id="597" class="Keyword">record</a>
<a id="606" class="Symbol">{</a> <a id="608" href="Categories.Category.Core.html#559" class="Field">Obj</a> <a id="618" class="Symbol">=</a> <a id="620" href="Categories.Category.Core.html#559" class="Field">Obj</a>
<a id="626" class="Symbol">;</a> <a id="628" href="Categories.Category.Core.html#575" class="Field Operator">_⇒_</a> <a id="638" class="Symbol">=</a> <a id="640" class="Symbol">λ</a> <a id="642" href="Categories.Category.Construction.Path.html#642" class="Bound">A</a> <a id="644" href="Categories.Category.Construction.Path.html#644" class="Bound">B</a> <a id="646" class="Symbol">→</a> <a id="648" href="Categories.Category.Construction.Path.html#642" class="Bound">A</a> <a id="650" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">[</a> <a id="652" href="Categories.Category.Core.html#575" class="Field Operator">_⇒_</a> <a id="656" href="Relation.Binary.Construct.Closure.Transitive.html#2738" class="Datatype">]⁺</a> <a id="659" href="Categories.Category.Construction.Path.html#644" class="Bound">B</a>
<a id="663" class="Symbol">;</a> <a id="665" href="Categories.Category.Core.html#595" class="Field Operator">_≈_</a> <a id="675" class="Symbol">=</a> <a id="677" href="Categories.Category.Construction.Path.html#482" class="Function Operator">_≈⁺_</a>
<a id="684" class="Symbol">;</a> <a id="686" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="696" class="Symbol">=</a> <a id="698" href="Relation.Binary.Construct.Closure.Transitive.html#2795" class="InductiveConstructor Operator">[</a> <a id="700" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="703" href="Relation.Binary.Construct.Closure.Transitive.html#2795" class="InductiveConstructor Operator">]</a>
<a id="707" class="Symbol">;</a> <a id="709" href="Categories.Category.Core.html#656" class="Field Operator">_∘_</a> <a id="719" class="Symbol">=</a> <a id="721" href="Function.Base.html#1638" class="Function">flip</a> <a id="726" class="Symbol">(_</a> <a id="729" href="Relation.Binary.Construct.Closure.Transitive.html#2844" class="InductiveConstructor Operator">∼⁺⟨_⟩_</a><a id="735" class="Symbol">)</a>
<a id="739" class="Symbol">;</a> <a id="741" href="Categories.Category.Core.html#715" class="Field">assoc</a> <a id="751" class="Symbol">=</a> <a id="753" href="Categories.Category.Core.html#715" class="Field">assoc</a>
<a id="761" class="Symbol">;</a> <a id="763" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a> <a id="773" class="Symbol">=</a> <a id="775" href="Categories.Category.Core.html#1004" class="Field">sym-assoc</a>
<a id="787" class="Symbol">;</a> <a id="789" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="799" class="Symbol">=</a> <a id="801" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a>
<a id="813" class="Symbol">;</a> <a id="815" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="825" class="Symbol">=</a> <a id="827" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a>
<a id="839" class="Symbol">;</a> <a id="841" href="Categories.Category.Core.html#1339" class="Field">identity²</a> <a id="851" class="Symbol">=</a> <a id="853" href="Categories.Category.Core.html#1339" class="Field">identity²</a>
<a id="865" class="Symbol">;</a> <a id="867" href="Categories.Category.Core.html#1384" class="Field">equiv</a> <a id="877" class="Symbol">=</a> <a id="879" class="Keyword">record</a>
<a id="890" class="Symbol">{</a> <a id="892" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="898" class="Symbol">=</a> <a id="900" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a>
<a id="915" class="Symbol">;</a> <a id="917" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="923" class="Symbol">=</a> <a id="925" href="Relation.Binary.Structures.html#1622" class="Function">Equiv.sym</a>
<a id="939" class="Symbol">;</a> <a id="941" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="947" class="Symbol">=</a> <a id="949" href="Relation.Binary.Structures.html#1648" class="Function">Equiv.trans</a>
<a id="965" class="Symbol">}</a>
<a id="969" class="Symbol">;</a> <a id="971" href="Categories.Category.Core.html#1438" class="Field">∘-resp-≈</a> <a id="981" class="Symbol">=</a> <a id="983" href="Categories.Category.Core.html#1438" class="Field">∘-resp-≈</a>
<a id="994" class="Symbol">}</a>
<a id="998" class="Keyword">where</a> <a id="1004" class="Keyword">open</a> <a id="1009" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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