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<html><head><meta charset="utf-8"><title>Categories.Category.Groupoid</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.Category.Groupoid.html" class="Module">Categories.Category.Groupoid</a> <a id="72" class="Keyword">where</a>
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<a id="79" class="Keyword">open</a> <a id="84" class="Keyword">import</a> <a id="91" href="Level.html" class="Module">Level</a> <a id="97" class="Keyword">using</a> <a id="103" class="Symbol">(</a><a id="104" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="109" class="Symbol">;</a> <a id="111" href="Agda.Primitive.html#774" class="Primitive">suc</a><a id="114" class="Symbol">;</a> <a id="116" href="Agda.Primitive.html#804" class="Primitive Operator">_⊔_</a><a id="119" class="Symbol">)</a>
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<a id="122" class="Keyword">open</a> <a id="127" class="Keyword">import</a> <a id="134" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="154" class="Keyword">import</a> <a id="161" href="Categories.Morphism.html" class="Module">Categories.Morphism</a>
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<a id="182" class="Keyword">record</a> <a id="IsGroupoid"></a><a id="189" href="Categories.Category.Groupoid.html#189" class="Record">IsGroupoid</a> <a id="200" class="Symbol">{</a><a id="201" href="Categories.Category.Groupoid.html#201" class="Bound">o</a> <a id="203" href="Categories.Category.Groupoid.html#203" class="Bound">ℓ</a> <a id="205" href="Categories.Category.Groupoid.html#205" class="Bound">e</a><a id="206" class="Symbol">}</a> <a id="208" class="Symbol">(</a><a id="209" href="Categories.Category.Groupoid.html#209" class="Bound">C</a> <a id="211" class="Symbol">:</a> <a id="213" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="222" href="Categories.Category.Groupoid.html#201" class="Bound">o</a> <a id="224" href="Categories.Category.Groupoid.html#203" class="Bound">ℓ</a> <a id="226" href="Categories.Category.Groupoid.html#205" class="Bound">e</a><a id="227" class="Symbol">)</a> <a id="229" class="Symbol">:</a> <a id="231" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="235" class="Symbol">(</a><a id="236" href="Categories.Category.Groupoid.html#201" class="Bound">o</a> <a id="238" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="240" href="Categories.Category.Groupoid.html#203" class="Bound">ℓ</a> <a id="242" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="244" href="Categories.Category.Groupoid.html#205" class="Bound">e</a><a id="245" class="Symbol">)</a> <a id="247" class="Keyword">where</a>
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<a id="255" class="Keyword">open</a> <a id="260" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="269" href="Categories.Category.Groupoid.html#209" class="Bound">C</a> <a id="271" class="Keyword">public</a>
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<a id="280" class="Keyword">open</a> <a id="285" href="Categories.Category.html#824" class="Module">Definitions</a> <a id="297" href="Categories.Category.Groupoid.html#209" class="Bound">C</a> <a id="299" class="Keyword">public</a>
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<a id="309" class="Keyword">open</a> <a id="314" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="334" href="Categories.Category.Groupoid.html#209" class="Bound">C</a>
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<a id="339" class="Keyword">infix</a> <a id="345" class="Number">10</a> <a id="348" href="Categories.Category.Groupoid.html#365" class="Field Operator">_⁻¹</a>
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<a id="355" class="Keyword">field</a>
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<a id="IsGroupoid._⁻¹"></a><a id="365" href="Categories.Category.Groupoid.html#365" class="Field Operator">_⁻¹</a> <a id="369" class="Symbol">:</a> <a id="371" class="Symbol">∀</a> <a id="373" class="Symbol">{</a><a id="374" href="Categories.Category.Groupoid.html#374" class="Bound">A</a> <a id="376" href="Categories.Category.Groupoid.html#376" class="Bound">B</a><a id="377" class="Symbol">}</a> <a id="379" class="Symbol">→</a> <a id="381" href="Categories.Category.Groupoid.html#374" class="Bound">A</a> <a id="383" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="385" href="Categories.Category.Groupoid.html#376" class="Bound">B</a> <a id="387" class="Symbol">→</a> <a id="389" href="Categories.Category.Groupoid.html#376" class="Bound">B</a> <a id="391" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="393" href="Categories.Category.Groupoid.html#374" class="Bound">A</a>
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<a id="IsGroupoid.iso"></a><a id="399" href="Categories.Category.Groupoid.html#399" class="Field">iso</a> <a id="403" class="Symbol">:</a> <a id="405" class="Symbol">∀</a> <a id="407" class="Symbol">{</a><a id="408" href="Categories.Category.Groupoid.html#408" class="Bound">A</a> <a id="410" href="Categories.Category.Groupoid.html#410" class="Bound">B</a><a id="411" class="Symbol">}</a> <a id="413" class="Symbol">{</a><a id="414" href="Categories.Category.Groupoid.html#414" class="Bound">f</a> <a id="416" class="Symbol">:</a> <a id="418" href="Categories.Category.Groupoid.html#408" class="Bound">A</a> <a id="420" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="422" href="Categories.Category.Groupoid.html#410" class="Bound">B</a><a id="423" class="Symbol">}</a> <a id="425" class="Symbol">→</a> <a id="427" href="Categories.Morphism.html#1528" class="Record">Iso</a> <a id="431" href="Categories.Category.Groupoid.html#414" class="Bound">f</a> <a id="433" class="Symbol">(</a><a id="434" href="Categories.Category.Groupoid.html#414" class="Bound">f</a> <a id="436" href="Categories.Category.Groupoid.html#365" class="Field Operator">⁻¹</a><a id="438" class="Symbol">)</a>
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<a id="443" class="Keyword">module</a> <a id="IsGroupoid.iso"></a><a id="450" href="Categories.Category.Groupoid.html#450" class="Module">iso</a> <a id="454" class="Symbol">{</a><a id="455" href="Categories.Category.Groupoid.html#455" class="Bound">A</a> <a id="457" href="Categories.Category.Groupoid.html#457" class="Bound">B</a> <a id="459" href="Categories.Category.Groupoid.html#459" class="Bound">f</a><a id="460" class="Symbol">}</a> <a id="462" class="Symbol">=</a> <a id="464" href="Categories.Morphism.html#1528" class="Module">Iso</a> <a id="468" class="Symbol">(</a><a id="469" href="Categories.Category.Groupoid.html#399" class="Field">iso</a> <a id="473" class="Symbol">{</a><a id="474" href="Categories.Category.Groupoid.html#455" class="Bound">A</a><a id="475" class="Symbol">}</a> <a id="477" class="Symbol">{</a><a id="478" href="Categories.Category.Groupoid.html#457" class="Bound">B</a><a id="479" class="Symbol">}</a> <a id="481" class="Symbol">{</a><a id="482" href="Categories.Category.Groupoid.html#459" class="Bound">f</a><a id="483" class="Symbol">})</a>
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<a id="IsGroupoid.equiv-obj"></a><a id="489" href="Categories.Category.Groupoid.html#489" class="Function">equiv-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.Groupoid.html#504" class="Bound">A</a> <a id="506" href="Categories.Category.Groupoid.html#506" class="Bound">B</a><a id="507" class="Symbol">}</a> <a id="509" class="Symbol">→</a> <a id="511" href="Categories.Category.Groupoid.html#504" class="Bound">A</a> <a id="513" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="515" href="Categories.Category.Groupoid.html#506" class="Bound">B</a> <a id="517" class="Symbol">→</a> <a id="519" href="Categories.Category.Groupoid.html#504" class="Bound">A</a> <a id="521" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="523" href="Categories.Category.Groupoid.html#506" class="Bound">B</a>
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<a id="527" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="537" href="Categories.Category.Groupoid.html#537" class="Bound">f</a> <a id="539" class="Symbol">=</a> <a id="541" class="Keyword">record</a>
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<a id="552" class="Symbol">{</a> <a id="554" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="559" class="Symbol">=</a> <a id="561" href="Categories.Category.Groupoid.html#537" class="Bound">f</a>
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<a id="567" class="Symbol">;</a> <a id="569" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="574" class="Symbol">=</a> <a id="576" class="Symbol">_</a>
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<a id="582" class="Symbol">;</a> <a id="584" href="Categories.Morphism.html#2040" class="Field">iso</a> <a id="589" class="Symbol">=</a> <a id="591" href="Categories.Category.Groupoid.html#399" class="Field">iso</a>
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<a id="599" class="Symbol">}</a>
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<a id="604" class="Comment">-- this definition doesn't seem to 'carry its weight'</a>
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<a id="IsGroupoid.equiv-obj-sym"></a><a id="660" href="Categories.Category.Groupoid.html#660" class="Function">equiv-obj-sym</a> <a id="674" class="Symbol">:</a> <a id="676" class="Symbol">∀</a> <a id="678" class="Symbol">{</a><a id="679" href="Categories.Category.Groupoid.html#679" class="Bound">A</a> <a id="681" href="Categories.Category.Groupoid.html#681" class="Bound">B</a><a id="682" class="Symbol">}</a> <a id="684" class="Symbol">→</a> <a id="686" href="Categories.Category.Groupoid.html#679" class="Bound">A</a> <a id="688" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="690" href="Categories.Category.Groupoid.html#681" class="Bound">B</a> <a id="692" class="Symbol">→</a> <a id="694" href="Categories.Category.Groupoid.html#681" class="Bound">B</a> <a id="696" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="698" href="Categories.Category.Groupoid.html#679" class="Bound">A</a>
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<a id="702" href="Categories.Category.Groupoid.html#660" class="Function">equiv-obj-sym</a> <a id="716" href="Categories.Category.Groupoid.html#716" class="Bound">f</a> <a id="718" class="Symbol">=</a> <a id="720" href="Relation.Binary.Structures.html#1603" class="Function">≅.sym</a> <a id="726" class="Symbol">(</a><a id="727" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="737" href="Categories.Category.Groupoid.html#716" class="Bound">f</a><a id="738" class="Symbol">)</a>
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<a id="741" class="Comment">-- A groupoid is a category that has a groupoid structure</a>
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<a id="800" class="Keyword">record</a> <a id="Groupoid"></a><a id="807" href="Categories.Category.Groupoid.html#807" class="Record">Groupoid</a> <a id="816" class="Symbol">(</a><a id="817" href="Categories.Category.Groupoid.html#817" class="Bound">o</a> <a id="819" href="Categories.Category.Groupoid.html#819" class="Bound">ℓ</a> <a id="821" href="Categories.Category.Groupoid.html#821" class="Bound">e</a> <a id="823" class="Symbol">:</a> <a id="825" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="830" class="Symbol">)</a> <a id="832" class="Symbol">:</a> <a id="834" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="838" class="Symbol">(</a><a id="839" href="Agda.Primitive.html#774" class="Primitive">suc</a> <a id="843" class="Symbol">(</a><a id="844" href="Categories.Category.Groupoid.html#817" class="Bound">o</a> <a id="846" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="848" href="Categories.Category.Groupoid.html#819" class="Bound">ℓ</a> <a id="850" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="852" href="Categories.Category.Groupoid.html#821" class="Bound">e</a><a id="853" class="Symbol">))</a> <a id="856" class="Keyword">where</a>
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<a id="864" class="Keyword">field</a>
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<a id="Groupoid.category"></a><a id="874" href="Categories.Category.Groupoid.html#874" class="Field">category</a> <a id="885" class="Symbol">:</a> <a id="887" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="896" href="Categories.Category.Groupoid.html#817" class="Bound">o</a> <a id="898" href="Categories.Category.Groupoid.html#819" class="Bound">ℓ</a> <a id="900" href="Categories.Category.Groupoid.html#821" class="Bound">e</a>
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<a id="Groupoid.isGroupoid"></a><a id="906" href="Categories.Category.Groupoid.html#906" class="Field">isGroupoid</a> <a id="917" class="Symbol">:</a> <a id="919" href="Categories.Category.Groupoid.html#189" class="Record">IsGroupoid</a> <a id="930" href="Categories.Category.Groupoid.html#874" class="Field">category</a>
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<a id="942" class="Keyword">open</a> <a id="947" href="Categories.Category.Groupoid.html#189" class="Module">IsGroupoid</a> <a id="958" href="Categories.Category.Groupoid.html#906" class="Field">isGroupoid</a> <a id="969" class="Keyword">public</a>
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</pre></body></html> |