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Merge pull request #11 from rahulc29/6-tripos-to-topos-construction
Realizability Topos of an Arbitrary Total Combinatory Algebra
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docs/Cubical.Categories.Constructions.Slice.Properties.html
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| <!DOCTYPE HTML> | ||
| <html><head><meta charset="utf-8"><title>Cubical.Categories.Constructions.Slice</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">--safe</a> <a id="20" class="Symbol">#-}</a> | ||
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| <a id="25" class="Keyword">open</a> <a id="30" class="Keyword">import</a> <a id="37" href="Cubical.Foundations.Prelude.html" class="Module">Cubical.Foundations.Prelude</a> | ||
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| <a id="66" class="Keyword">open</a> <a id="71" class="Keyword">import</a> <a id="78" href="Cubical.Categories.Category.html" class="Module">Cubical.Categories.Category</a> | ||
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| <a id="107" class="Keyword">open</a> <a id="112" href="Cubical.Categories.Category.Base.html#334" class="Module">Category</a> | ||
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| <a id="122" class="Keyword">module</a> <a id="129" href="Cubical.Categories.Constructions.Slice.html" class="Module">Cubical.Categories.Constructions.Slice</a> | ||
| <a id="170" class="Symbol">{</a><a id="171" href="Cubical.Categories.Constructions.Slice.html#171" class="Bound">ℓ</a> <a id="173" href="Cubical.Categories.Constructions.Slice.html#173" class="Bound">ℓ'</a> <a id="176" class="Symbol">:</a> <a id="178" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="183" class="Symbol">}</a> | ||
| <a id="187" class="Symbol">(</a><a id="188" href="Cubical.Categories.Constructions.Slice.html#188" class="Bound">C</a> <a id="190" class="Symbol">:</a> <a id="192" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="201" href="Cubical.Categories.Constructions.Slice.html#171" class="Bound">ℓ</a> <a id="203" href="Cubical.Categories.Constructions.Slice.html#173" class="Bound">ℓ'</a><a id="205" class="Symbol">)</a> | ||
| <a id="209" class="Symbol">(</a><a id="210" href="Cubical.Categories.Constructions.Slice.html#210" class="Bound">c</a> <a id="212" class="Symbol">:</a> <a id="214" href="Cubical.Categories.Constructions.Slice.html#188" class="Bound">C</a> <a id="216" class="Symbol">.</a><a id="217" href="Cubical.Categories.Category.Base.html#452" class="Field">ob</a><a id="219" class="Symbol">)</a> | ||
| <a id="223" class="Keyword">where</a> | ||
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| <a id="230" class="Keyword">open</a> <a id="235" class="Keyword">import</a> <a id="242" href="Cubical.Categories.Constructions.Slice.Base.html" class="Module">Cubical.Categories.Constructions.Slice.Base</a> <a id="286" href="Cubical.Categories.Constructions.Slice.html#188" class="Bound">C</a> <a id="288" href="Cubical.Categories.Constructions.Slice.html#210" class="Bound">c</a> <a id="290" class="Keyword">public</a> | ||
| <a id="297" class="Keyword">open</a> <a id="302" class="Keyword">import</a> <a id="309" href="Cubical.Categories.Constructions.Slice.Properties.html" class="Module">Cubical.Categories.Constructions.Slice.Properties</a> <a id="359" href="Cubical.Categories.Constructions.Slice.html#188" class="Bound">C</a> <a id="361" href="Cubical.Categories.Constructions.Slice.html#210" class="Bound">c</a> <a id="363" class="Keyword">public</a> | ||
| </pre></body></html> |
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| <!DOCTYPE HTML> | ||
| <html><head><meta charset="utf-8"><title>Cubical.Categories.Equivalence.Base</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">--safe</a> <a id="20" class="Symbol">#-}</a> | ||
| <a id="24" class="Keyword">module</a> <a id="31" href="Cubical.Categories.Equivalence.Base.html" class="Module">Cubical.Categories.Equivalence.Base</a> <a id="67" class="Keyword">where</a> | ||
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| <a id="74" class="Keyword">open</a> <a id="79" class="Keyword">import</a> <a id="86" href="Cubical.Foundations.Prelude.html" class="Module">Cubical.Foundations.Prelude</a> | ||
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| <a id="115" class="Keyword">open</a> <a id="120" class="Keyword">import</a> <a id="127" href="Cubical.HITs.PropositionalTruncation.html" class="Module">Cubical.HITs.PropositionalTruncation</a> | ||
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| <a id="165" class="Keyword">open</a> <a id="170" class="Keyword">import</a> <a id="177" href="Cubical.Categories.Category.html" class="Module">Cubical.Categories.Category</a> | ||
| <a id="205" class="Keyword">open</a> <a id="210" class="Keyword">import</a> <a id="217" href="Cubical.Categories.Functor.html" class="Module">Cubical.Categories.Functor</a> | ||
| <a id="244" class="Keyword">open</a> <a id="249" class="Keyword">import</a> <a id="256" href="Cubical.Categories.NaturalTransformation.html" class="Module">Cubical.Categories.NaturalTransformation</a> | ||
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| <a id="298" class="Keyword">open</a> <a id="303" href="Cubical.Categories.Category.Base.html#334" class="Module">Category</a> | ||
| <a id="312" class="Keyword">open</a> <a id="317" href="Cubical.Categories.Functor.Base.html#441" class="Module">Functor</a> | ||
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| <a id="326" class="Keyword">private</a> | ||
| <a id="336" class="Keyword">variable</a> | ||
| <a id="349" href="Cubical.Categories.Equivalence.Base.html#349" class="Generalizable">ℓC</a> <a id="352" href="Cubical.Categories.Equivalence.Base.html#352" class="Generalizable">ℓC'</a> <a id="356" href="Cubical.Categories.Equivalence.Base.html#356" class="Generalizable">ℓD</a> <a id="359" href="Cubical.Categories.Equivalence.Base.html#359" class="Generalizable">ℓD'</a> <a id="363" class="Symbol">:</a> <a id="365" href="Agda.Primitive.html#742" class="Postulate">Level</a> | ||
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| <a id="372" class="Keyword">record</a> <a id="WeakInverse"></a><a id="379" href="Cubical.Categories.Equivalence.Base.html#379" class="Record">WeakInverse</a> <a id="391" class="Symbol">{</a><a id="392" href="Cubical.Categories.Equivalence.Base.html#392" class="Bound">C</a> <a id="394" class="Symbol">:</a> <a id="396" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="405" href="Cubical.Categories.Equivalence.Base.html#349" class="Generalizable">ℓC</a> <a id="408" href="Cubical.Categories.Equivalence.Base.html#352" class="Generalizable">ℓC'</a><a id="411" class="Symbol">}</a> <a id="413" class="Symbol">{</a><a id="414" href="Cubical.Categories.Equivalence.Base.html#414" class="Bound">D</a> <a id="416" class="Symbol">:</a> <a id="418" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="427" href="Cubical.Categories.Equivalence.Base.html#356" class="Generalizable">ℓD</a> <a id="430" href="Cubical.Categories.Equivalence.Base.html#359" class="Generalizable">ℓD'</a><a id="433" class="Symbol">}</a> | ||
| <a id="456" class="Symbol">(</a><a id="457" href="Cubical.Categories.Equivalence.Base.html#457" class="Bound">func</a> <a id="462" class="Symbol">:</a> <a id="464" href="Cubical.Categories.Functor.Base.html#441" class="Record">Functor</a> <a id="472" href="Cubical.Categories.Equivalence.Base.html#392" class="Bound">C</a> <a id="474" href="Cubical.Categories.Equivalence.Base.html#414" class="Bound">D</a><a id="475" class="Symbol">)</a> <a id="477" class="Symbol">:</a> <a id="479" href="Agda.Primitive.html#388" class="Primitive">Type</a> <a id="484" class="Symbol">(</a><a id="485" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="491" class="Symbol">(</a><a id="492" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="498" href="Cubical.Categories.Equivalence.Base.html#405" class="Bound">ℓC</a> <a id="501" href="Cubical.Categories.Equivalence.Base.html#408" class="Bound">ℓC'</a><a id="504" class="Symbol">)</a> <a id="506" class="Symbol">(</a><a id="507" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="513" href="Cubical.Categories.Equivalence.Base.html#427" class="Bound">ℓD</a> <a id="516" href="Cubical.Categories.Equivalence.Base.html#430" class="Bound">ℓD'</a><a id="519" class="Symbol">))</a> <a id="522" class="Keyword">where</a> | ||
| <a id="530" class="Keyword">field</a> | ||
| <a id="WeakInverse.invFunc"></a><a id="540" href="Cubical.Categories.Equivalence.Base.html#540" class="Field">invFunc</a> <a id="548" class="Symbol">:</a> <a id="550" href="Cubical.Categories.Functor.Base.html#441" class="Record">Functor</a> <a id="558" href="Cubical.Categories.Equivalence.Base.html#414" class="Bound">D</a> <a id="560" href="Cubical.Categories.Equivalence.Base.html#392" class="Bound">C</a> | ||
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| <a id="WeakInverse.η"></a><a id="567" href="Cubical.Categories.Equivalence.Base.html#567" class="Field">η</a> <a id="569" class="Symbol">:</a> <a id="571" href="Cubical.Categories.Functor.Base.html#4025" class="Function Operator">𝟙⟨</a> <a id="574" href="Cubical.Categories.Equivalence.Base.html#392" class="Bound">C</a> <a id="576" href="Cubical.Categories.Functor.Base.html#4025" class="Function Operator">⟩</a> <a id="578" href="Cubical.Categories.NaturalTransformation.Base.html#2633" class="Function Operator">≅ᶜ</a> <a id="581" href="Cubical.Categories.Equivalence.Base.html#540" class="Field">invFunc</a> <a id="589" href="Cubical.Categories.Functor.Base.html#4767" class="Function Operator">∘F</a> <a id="592" href="Cubical.Categories.Equivalence.Base.html#457" class="Bound">func</a> | ||
| <a id="WeakInverse.ε"></a><a id="601" href="Cubical.Categories.Equivalence.Base.html#601" class="Field">ε</a> <a id="603" class="Symbol">:</a> <a id="605" href="Cubical.Categories.Equivalence.Base.html#457" class="Bound">func</a> <a id="610" href="Cubical.Categories.Functor.Base.html#4767" class="Function Operator">∘F</a> <a id="613" href="Cubical.Categories.Equivalence.Base.html#540" class="Field">invFunc</a> <a id="621" href="Cubical.Categories.NaturalTransformation.Base.html#2633" class="Function Operator">≅ᶜ</a> <a id="624" href="Cubical.Categories.Functor.Base.html#4025" class="Function Operator">𝟙⟨</a> <a id="627" href="Cubical.Categories.Equivalence.Base.html#414" class="Bound">D</a> <a id="629" href="Cubical.Categories.Functor.Base.html#4025" class="Function Operator">⟩</a> | ||
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| <a id="632" class="Comment">-- I don't know of a good alternative representation of isEquivalence that</a> | ||
| <a id="707" class="Comment">-- avoids truncation in the general case. If the categories are univalent, then</a> | ||
| <a id="788" class="Comment">-- an adjoint equivalence might be enough.</a> | ||
| <a id="isEquivalence"></a><a id="831" href="Cubical.Categories.Equivalence.Base.html#831" class="Function">isEquivalence</a> <a id="845" class="Symbol">:</a> <a id="847" class="Symbol">{</a><a id="848" href="Cubical.Categories.Equivalence.Base.html#848" class="Bound">C</a> <a id="850" class="Symbol">:</a> <a id="852" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="861" href="Cubical.Categories.Equivalence.Base.html#349" class="Generalizable">ℓC</a> <a id="864" href="Cubical.Categories.Equivalence.Base.html#352" class="Generalizable">ℓC'</a><a id="867" class="Symbol">}</a> <a id="869" class="Symbol">{</a><a id="870" href="Cubical.Categories.Equivalence.Base.html#870" class="Bound">D</a> <a id="872" class="Symbol">:</a> <a id="874" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="883" href="Cubical.Categories.Equivalence.Base.html#356" class="Generalizable">ℓD</a> <a id="886" href="Cubical.Categories.Equivalence.Base.html#359" class="Generalizable">ℓD'</a><a id="889" class="Symbol">}</a> | ||
| <a id="905" class="Symbol">→</a> <a id="907" class="Symbol">(</a><a id="908" href="Cubical.Categories.Equivalence.Base.html#908" class="Bound">func</a> <a id="913" class="Symbol">:</a> <a id="915" href="Cubical.Categories.Functor.Base.html#441" class="Record">Functor</a> <a id="923" href="Cubical.Categories.Equivalence.Base.html#848" class="Bound">C</a> <a id="925" href="Cubical.Categories.Equivalence.Base.html#870" class="Bound">D</a><a id="926" class="Symbol">)</a> <a id="928" class="Symbol">→</a> <a id="930" href="Agda.Primitive.html#388" class="Primitive">Type</a> <a id="935" class="Symbol">(</a><a id="936" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="942" class="Symbol">(</a><a id="943" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="949" href="Cubical.Categories.Equivalence.Base.html#349" class="Generalizable">ℓC</a> <a id="952" href="Cubical.Categories.Equivalence.Base.html#352" class="Generalizable">ℓC'</a><a id="955" class="Symbol">)</a> <a id="957" class="Symbol">(</a><a id="958" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="964" href="Cubical.Categories.Equivalence.Base.html#356" class="Generalizable">ℓD</a> <a id="967" href="Cubical.Categories.Equivalence.Base.html#359" class="Generalizable">ℓD'</a><a id="970" class="Symbol">))</a> | ||
| <a id="973" href="Cubical.Categories.Equivalence.Base.html#831" class="Function">isEquivalence</a> <a id="987" href="Cubical.Categories.Equivalence.Base.html#987" class="Bound">func</a> <a id="992" class="Symbol">=</a> <a id="994" href="Cubical.HITs.PropositionalTruncation.Base.html#249" class="Datatype Operator">∥</a> <a id="996" href="Cubical.Categories.Equivalence.Base.html#379" class="Record">WeakInverse</a> <a id="1008" href="Cubical.Categories.Equivalence.Base.html#987" class="Bound">func</a> <a id="1013" href="Cubical.HITs.PropositionalTruncation.Base.html#249" class="Datatype Operator">∥₁</a> | ||
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| <a id="1017" class="Keyword">record</a> <a id="_≃ᶜ_"></a><a id="1024" href="Cubical.Categories.Equivalence.Base.html#1024" class="Record Operator">_≃ᶜ_</a> <a id="1029" class="Symbol">(</a><a id="1030" href="Cubical.Categories.Equivalence.Base.html#1030" class="Bound">C</a> <a id="1032" class="Symbol">:</a> <a id="1034" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="1043" href="Cubical.Categories.Equivalence.Base.html#349" class="Generalizable">ℓC</a> <a id="1046" href="Cubical.Categories.Equivalence.Base.html#352" class="Generalizable">ℓC'</a><a id="1049" class="Symbol">)</a> <a id="1051" class="Symbol">(</a><a id="1052" href="Cubical.Categories.Equivalence.Base.html#1052" class="Bound">D</a> <a id="1054" class="Symbol">:</a> <a id="1056" href="Cubical.Categories.Category.Base.html#334" class="Record">Category</a> <a id="1065" href="Cubical.Categories.Equivalence.Base.html#356" class="Generalizable">ℓD</a> <a id="1068" href="Cubical.Categories.Equivalence.Base.html#359" class="Generalizable">ℓD'</a><a id="1071" class="Symbol">)</a> <a id="1073" class="Symbol">:</a> | ||
| <a id="1090" href="Agda.Primitive.html#388" class="Primitive">Type</a> <a id="1095" class="Symbol">(</a><a id="1096" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="1102" class="Symbol">(</a><a id="1103" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="1109" href="Cubical.Categories.Equivalence.Base.html#1043" class="Bound">ℓC</a> <a id="1112" href="Cubical.Categories.Equivalence.Base.html#1046" class="Bound">ℓC'</a><a id="1115" class="Symbol">)</a> <a id="1117" class="Symbol">(</a><a id="1118" href="Agda.Primitive.html#961" class="Primitive">ℓ-max</a> <a id="1124" href="Cubical.Categories.Equivalence.Base.html#1065" class="Bound">ℓD</a> <a id="1127" href="Cubical.Categories.Equivalence.Base.html#1068" class="Bound">ℓD'</a><a id="1130" class="Symbol">))</a> <a id="1133" class="Keyword">where</a> | ||
| <a id="1141" class="Keyword">constructor</a> <a id="equivᶜ"></a><a id="1153" href="Cubical.Categories.Equivalence.Base.html#1153" class="InductiveConstructor">equivᶜ</a> | ||
| <a id="1162" class="Keyword">field</a> | ||
| <a id="_≃ᶜ_.func"></a><a id="1172" href="Cubical.Categories.Equivalence.Base.html#1172" class="Field">func</a> <a id="1177" class="Symbol">:</a> <a id="1179" href="Cubical.Categories.Functor.Base.html#441" class="Record">Functor</a> <a id="1187" href="Cubical.Categories.Equivalence.Base.html#1030" class="Bound">C</a> <a id="1189" href="Cubical.Categories.Equivalence.Base.html#1052" class="Bound">D</a> | ||
| <a id="_≃ᶜ_.isEquiv"></a><a id="1195" href="Cubical.Categories.Equivalence.Base.html#1195" class="Field">isEquiv</a> <a id="1203" class="Symbol">:</a> <a id="1205" href="Cubical.Categories.Equivalence.Base.html#831" class="Function">isEquivalence</a> <a id="1219" href="Cubical.Categories.Equivalence.Base.html#1172" class="Field">func</a> | ||
| </pre></body></html> |
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