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70 lines
31 KiB
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<html><head><meta charset="utf-8"><title>Categories.Category.Instance.Properties.Setoids.Limits.Canonical</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="37" class="Comment">-- A "canonical" presentation of limits in Setoid.</a>
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<a id="88" class="Comment">--</a>
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<a id="91" class="Comment">-- These limits are obviously isomorphic to those created by</a>
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<a id="152" class="Comment">-- the Completeness proof, but are far less unweildy to work with.</a>
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<a id="219" class="Comment">-- This isomorphism is witnessed by Categories.Diagram.Pullback.up-to-iso</a>
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<a id="294" class="Keyword">module</a> <a id="301" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html" class="Module">Categories.Category.Instance.Properties.Setoids.Limits.Canonical</a> <a id="366" class="Keyword">where</a>
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<a id="373" class="Keyword">open</a> <a id="378" class="Keyword">import</a> <a id="385" href="Level.html" class="Module">Level</a> <a id="391" class="Keyword">using</a> <a id="397" class="Symbol">(</a><a id="398" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="403" class="Symbol">;</a> <a id="405" href="Agda.Primitive.html#961" class="Primitive Operator">_⊔_</a><a id="408" class="Symbol">)</a>
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<a id="410" class="Keyword">open</a> <a id="415" class="Keyword">import</a> <a id="422" href="Data.Product.html" class="Module">Data.Product</a> <a id="435" class="Keyword">using</a> <a id="441" class="Symbol">(</a><a id="442" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="445" class="Symbol">;</a> <a id="447" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="450" class="Symbol">;</a> <a id="452" href="Data.Product.Base.html#2173" class="Function">map</a><a id="455" class="Symbol">;</a> <a id="457" href="Data.Product.Base.html#2747" class="Function">zip</a><a id="460" class="Symbol">;</a> <a id="462" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="467" class="Symbol">;</a> <a id="469" href="Data.Product.Base.html#650" class="Field">proj₂</a><a id="474" class="Symbol">;</a> <a id="476" href="Data.Product.Base.html#2000" class="Function Operator"><_,_></a><a id="481" class="Symbol">)</a>
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<a id="483" class="Keyword">open</a> <a id="488" class="Keyword">import</a> <a id="495" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="512" class="Keyword">using</a> <a id="518" class="Symbol">(</a><a id="519" href="Function.Bundles.html#2043" class="Record">Func</a><a id="523" class="Symbol">;</a> <a id="525" href="Function.Bundles.html#15133" class="Function Operator">_⟨$⟩_</a><a id="530" class="Symbol">)</a>
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<a id="532" class="Keyword">open</a> <a id="537" class="Keyword">import</a> <a id="544" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a> <a id="568" class="Keyword">using</a> <a id="574" class="Symbol">(</a><a id="575" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="581" class="Symbol">)</a>
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<a id="583" class="Keyword">import</a> <a id="590" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="623" class="Symbol">as</a> <a id="626" class="Module">SR</a>
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<a id="630" class="Keyword">open</a> <a id="635" class="Keyword">import</a> <a id="642" href="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a> <a id="679" class="Keyword">using</a> <a id="685" class="Symbol">(</a><a id="686" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a><a id="693" class="Symbol">)</a>
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<a id="695" class="Keyword">open</a> <a id="700" class="Keyword">import</a> <a id="707" href="Categories.Diagram.Pullback.html" class="Module">Categories.Diagram.Pullback</a> <a id="735" class="Keyword">using</a> <a id="741" class="Symbol">(</a><a id="742" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a><a id="750" class="Symbol">)</a>
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<a id="753" class="Keyword">open</a> <a id="758" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a> <a id="765" class="Keyword">renaming</a> <a id="774" class="Symbol">(</a><a id="775" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="779" class="Symbol">to</a> <a id="782" class="Field Operator">[_][_≈_]</a><a id="790" class="Symbol">)</a>
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<a id="792" class="Keyword">open</a> <a id="797" href="Function.Bundles.html#2043" class="Module">Func</a>
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<a id="803" class="Comment">--------------------------------------------------------------------------------</a>
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<a id="884" class="Comment">-- Pullbacks</a>
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<a id="898" class="Keyword">record</a> <a id="FiberProduct"></a><a id="905" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#905" class="Record">FiberProduct</a> <a id="918" class="Symbol">{</a><a id="919" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#919" class="Bound">o</a> <a id="921" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#921" class="Bound">ℓ</a><a id="922" class="Symbol">}</a> <a id="924" class="Symbol">{</a><a id="925" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#925" class="Bound">X</a> <a id="927" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#927" class="Bound">Y</a> <a id="929" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#929" class="Bound">Z</a> <a id="931" class="Symbol">:</a> <a id="933" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="940" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#919" class="Bound">o</a> <a id="942" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#921" class="Bound">ℓ</a><a id="943" class="Symbol">}</a> <a id="945" class="Symbol">(</a><a id="946" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#946" class="Bound">f</a> <a id="948" class="Symbol">:</a> <a id="950" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="955" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#925" class="Bound">X</a> <a id="957" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#929" class="Bound">Z</a><a id="958" class="Symbol">)</a> <a id="960" class="Symbol">(</a><a id="961" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#961" class="Bound">g</a> <a id="963" class="Symbol">:</a> <a id="965" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="970" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#927" class="Bound">Y</a> <a id="972" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#929" class="Bound">Z</a><a id="973" class="Symbol">)</a> <a id="975" class="Symbol">:</a> <a id="977" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="981" class="Symbol">(</a><a id="982" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#919" class="Bound">o</a> <a id="984" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="986" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#921" class="Bound">ℓ</a><a id="987" class="Symbol">)</a> <a id="989" class="Keyword">where</a>
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<a id="997" class="Keyword">constructor</a> <a id="mk-×"></a><a id="1009" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1009" class="InductiveConstructor">mk-×</a>
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<a id="1016" class="Keyword">field</a>
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<a id="FiberProduct.elem₁"></a><a id="1026" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1032" class="Symbol">:</a> <a id="1034" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="1042" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#925" class="Bound">X</a>
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<a id="FiberProduct.elem₂"></a><a id="1048" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1054" class="Symbol">:</a> <a id="1056" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="1064" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#927" class="Bound">Y</a>
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<a id="FiberProduct.commute"></a><a id="1070" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1070" class="Field">commute</a> <a id="1078" class="Symbol">:</a> <a id="1080" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">[</a> <a id="1082" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#929" class="Bound">Z</a> <a id="1084" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">][</a> <a id="1087" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#946" class="Bound">f</a> <a id="1089" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="1093" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1099" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">≈</a> <a id="1101" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#961" class="Bound">g</a> <a id="1103" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="1107" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1113" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">]</a>
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<a id="1116" class="Keyword">open</a> <a id="1121" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#905" class="Module">FiberProduct</a>
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<a id="FP-≈"></a><a id="1135" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1135" class="Function">FP-≈</a> <a id="1140" class="Symbol">:</a> <a id="1142" class="Symbol">∀</a> <a id="1144" class="Symbol">{</a><a id="1145" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1145" class="Bound">o</a> <a id="1147" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1147" class="Bound">ℓ</a><a id="1148" class="Symbol">}</a> <a id="1150" class="Symbol">{</a><a id="1151" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1151" class="Bound">X</a> <a id="1153" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1153" class="Bound">Y</a> <a id="1155" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1155" class="Bound">Z</a> <a id="1157" class="Symbol">:</a> <a id="1159" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="1166" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1145" class="Bound">o</a> <a id="1168" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1147" class="Bound">ℓ</a><a id="1169" class="Symbol">}</a> <a id="1171" class="Symbol">{</a><a id="1172" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1172" class="Bound">f</a> <a id="1174" class="Symbol">:</a> <a id="1176" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="1181" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1151" class="Bound">X</a> <a id="1183" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1155" class="Bound">Z</a><a id="1184" class="Symbol">}</a> <a id="1186" class="Symbol">{</a><a id="1187" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1187" class="Bound">g</a> <a id="1189" class="Symbol">:</a> <a id="1191" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="1196" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1153" class="Bound">Y</a> <a id="1198" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1155" class="Bound">Z</a><a id="1199" class="Symbol">}</a> <a id="1201" class="Symbol">→</a> <a id="1203" class="Symbol">(</a><a id="1204" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1204" class="Bound">fb₁</a> <a id="1208" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1208" class="Bound">fb₂</a> <a id="1212" class="Symbol">:</a> <a id="1214" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#905" class="Record">FiberProduct</a> <a id="1227" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1172" class="Bound">f</a> <a id="1229" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1187" class="Bound">g</a><a id="1230" class="Symbol">)</a> <a id="1232" class="Symbol">→</a> <a id="1234" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1238" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1147" class="Bound">ℓ</a>
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<a id="1240" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1135" class="Function">FP-≈</a> <a id="1245" class="Symbol">{</a><a id="1246" class="Argument">X</a> <a id="1248" class="Symbol">=</a> <a id="1250" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1250" class="Bound">X</a><a id="1251" class="Symbol">}</a> <a id="1253" class="Symbol">{</a><a id="1254" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1254" class="Bound">Y</a><a id="1255" class="Symbol">}</a> <a id="1257" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1257" class="Bound">p</a> <a id="1259" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1259" class="Bound">q</a> <a id="1261" class="Symbol">=</a> <a id="1263" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">[</a> <a id="1265" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1250" class="Bound">X</a> <a id="1267" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">][</a> <a id="1270" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1276" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1257" class="Bound">p</a> <a id="1278" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">≈</a> <a id="1280" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1286" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1259" class="Bound">q</a> <a id="1288" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">]</a> <a id="1290" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="1292" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">[</a> <a id="1294" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1254" class="Bound">Y</a> <a id="1296" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">][</a> <a id="1299" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1305" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1257" class="Bound">p</a> <a id="1307" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">≈</a> <a id="1309" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1315" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1259" class="Bound">q</a> <a id="1317" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#782" class="Field Operator">]</a>
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<a id="pullback"></a><a id="1320" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1320" class="Function">pullback</a> <a id="1329" class="Symbol">:</a> <a id="1331" class="Symbol">∀</a> <a id="1333" class="Symbol">(</a><a id="1334" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1334" class="Bound">o</a> <a id="1336" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1336" class="Bound">ℓ</a> <a id="1338" class="Symbol">:</a> <a id="1340" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="1345" class="Symbol">)</a> <a id="1347" class="Symbol">{</a><a id="1348" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1348" class="Bound">X</a> <a id="1350" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1350" class="Bound">Y</a> <a id="1352" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1352" class="Bound">Z</a> <a id="1354" class="Symbol">:</a> <a id="1356" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="1363" class="Symbol">(</a><a id="1364" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1334" class="Bound">o</a> <a id="1366" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1368" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1336" class="Bound">ℓ</a><a id="1369" class="Symbol">)</a> <a id="1371" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1336" class="Bound">ℓ</a><a id="1372" class="Symbol">}</a> <a id="1374" class="Symbol">→</a> <a id="1376" class="Symbol">(</a><a id="1377" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1377" class="Bound">f</a> <a id="1379" class="Symbol">:</a> <a id="1381" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="1386" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1348" class="Bound">X</a> <a id="1388" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1352" class="Bound">Z</a><a id="1389" class="Symbol">)</a> <a id="1391" class="Symbol">→</a> <a id="1393" class="Symbol">(</a><a id="1394" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1394" class="Bound">g</a> <a id="1396" class="Symbol">:</a> <a id="1398" href="Function.Bundles.html#2043" class="Record">Func</a> <a id="1403" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1350" class="Bound">Y</a> <a id="1405" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1352" class="Bound">Z</a><a id="1406" class="Symbol">)</a> <a id="1408" class="Symbol">→</a> <a id="1410" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="1419" class="Symbol">(</a><a id="1420" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="1428" class="Symbol">(</a><a id="1429" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1334" class="Bound">o</a> <a id="1431" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1433" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1336" class="Bound">ℓ</a><a id="1434" class="Symbol">)</a> <a id="1436" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1336" class="Bound">ℓ</a><a id="1437" class="Symbol">)</a> <a id="1439" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1377" class="Bound">f</a> <a id="1441" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1394" class="Bound">g</a>
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<a id="1443" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1320" class="Function">pullback</a> <a id="1452" class="Symbol">_</a> <a id="1454" class="Symbol">_</a> <a id="1456" class="Symbol">{</a><a id="1457" class="Argument">X</a> <a id="1459" class="Symbol">=</a> <a id="1461" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1461" class="Bound">X</a><a id="1462" class="Symbol">}</a> <a id="1464" class="Symbol">{</a><a id="1465" class="Argument">Y</a> <a id="1467" class="Symbol">=</a> <a id="1469" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1469" class="Bound">Y</a><a id="1470" class="Symbol">}</a> <a id="1472" class="Symbol">{</a><a id="1473" class="Argument">Z</a> <a id="1475" class="Symbol">=</a> <a id="1477" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1477" class="Bound">Z</a><a id="1478" class="Symbol">}</a> <a id="1480" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1480" class="Bound">f</a> <a id="1482" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1482" class="Bound">g</a> <a id="1484" class="Symbol">=</a> <a id="1486" class="Keyword">record</a>
|
|||
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<a id="1495" class="Symbol">{</a> <a id="1497" href="Categories.Diagram.Pullback.html#2099" class="Field">P</a> <a id="1499" class="Symbol">=</a> <a id="1501" class="Keyword">record</a>
|
|||
|
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<a id="1512" class="Symbol">{</a> <a id="1514" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="1522" class="Symbol">=</a> <a id="1524" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#905" class="Record">FiberProduct</a> <a id="1537" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1480" class="Bound">f</a> <a id="1539" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1482" class="Bound">g</a>
|
|||
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<a id="1545" class="Symbol">;</a> <a id="1547" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="1551" class="Symbol">=</a> <a id="1554" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1135" class="Function">FP-≈</a>
|
|||
|
|
<a id="1563" class="Symbol">;</a> <a id="1565" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="1579" class="Symbol">=</a> <a id="1581" class="Keyword">record</a>
|
|||
|
|
<a id="1594" class="Symbol">{</a> <a id="1596" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="1601" class="Symbol">=</a> <a id="1603" href="Relation.Binary.Structures.html#1596" class="Function">X.refl</a> <a id="1610" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1612" href="Relation.Binary.Structures.html#1596" class="Function">Y.refl</a>
|
|||
|
|
<a id="1625" class="Symbol">;</a> <a id="1627" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="1631" class="Symbol">=</a> <a id="1633" href="Data.Product.Base.html#2173" class="Function">map</a> <a id="1637" href="Relation.Binary.Structures.html#1200" class="Function">X.sym</a> <a id="1643" href="Relation.Binary.Structures.html#1200" class="Function">Y.sym</a>
|
|||
|
|
<a id="1655" class="Symbol">;</a> <a id="1657" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="1663" class="Symbol">=</a> <a id="1665" href="Data.Product.Base.html#2747" class="Function">zip</a> <a id="1669" href="Relation.Binary.Structures.html#1226" class="Function">X.trans</a> <a id="1677" href="Relation.Binary.Structures.html#1226" class="Function">Y.trans</a>
|
|||
|
|
<a id="1691" class="Symbol">}</a>
|
|||
|
|
<a id="1697" class="Symbol">}</a>
|
|||
|
|
<a id="1703" class="Symbol">;</a> <a id="1705" href="Categories.Diagram.Pullback.html#2112" class="Field">p₁</a> <a id="1708" class="Symbol">=</a> <a id="1710" class="Keyword">record</a> <a id="1717" class="Symbol">{</a> <a id="1719" href="Function.Bundles.html#2094" class="Field">to</a> <a id="1722" class="Symbol">=</a> <a id="1724" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1730" class="Symbol">;</a> <a id="1732" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="1737" class="Symbol">=</a> <a id="1739" href="Data.Product.Base.html#636" class="Field">proj₁</a> <a id="1745" class="Symbol">}</a>
|
|||
|
|
<a id="1751" class="Symbol">;</a> <a id="1753" href="Categories.Diagram.Pullback.html#2128" class="Field">p₂</a> <a id="1756" class="Symbol">=</a> <a id="1758" class="Keyword">record</a> <a id="1765" class="Symbol">{</a> <a id="1767" href="Function.Bundles.html#2094" class="Field">to</a> <a id="1770" class="Symbol">=</a> <a id="1772" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1778" class="Symbol">;</a> <a id="1780" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="1785" class="Symbol">=</a> <a id="1787" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="1793" class="Symbol">}</a>
|
|||
|
|
<a id="1799" class="Symbol">;</a> <a id="1801" href="Categories.Diagram.Pullback.html#2144" class="Field">isPullback</a> <a id="1812" class="Symbol">=</a> <a id="1814" class="Keyword">record</a>
|
|||
|
|
<a id="1827" class="Symbol">{</a> <a id="1829" href="Categories.Diagram.Pullback.html#854" class="Field">commute</a> <a id="1837" class="Symbol">=</a> <a id="1839" class="Symbol">λ</a> <a id="1841" class="Symbol">{</a> <a id="1843" class="Symbol">{</a><a id="1844" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1844" class="Bound">fg</a><a id="1846" class="Symbol">}</a> <a id="1848" class="Symbol">→</a> <a id="1850" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1070" class="Field">commute</a> <a id="1858" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1844" class="Bound">fg</a><a id="1860" class="Symbol">}</a>
|
|||
|
|
<a id="1868" class="Symbol">;</a> <a id="1870" href="Categories.Diagram.Pullback.html#886" class="Field">universal</a> <a id="1880" class="Symbol">=</a> <a id="1882" class="Symbol">λ</a> <a id="1884" class="Symbol">{</a><a id="1885" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1885" class="Bound">A</a><a id="1886" class="Symbol">}</a> <a id="1888" class="Symbol">{</a><a id="1889" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1889" class="Bound">h₁</a><a id="1891" class="Symbol">}</a> <a id="1893" class="Symbol">{</a><a id="1894" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1894" class="Bound">h₂</a><a id="1896" class="Symbol">}</a> <a id="1898" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1898" class="Bound">eq</a> <a id="1901" class="Symbol">→</a> <a id="1903" class="Keyword">record</a>
|
|||
|
|
<a id="1918" class="Symbol">{</a> <a id="1920" href="Function.Bundles.html#2094" class="Field">to</a> <a id="1923" class="Symbol">=</a> <a id="1925" class="Symbol">λ</a> <a id="1927" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1927" class="Bound">a</a> <a id="1929" class="Symbol">→</a> <a id="1931" class="Keyword">record</a>
|
|||
|
|
<a id="1948" class="Symbol">{</a> <a id="1950" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1026" class="Field">elem₁</a> <a id="1956" class="Symbol">=</a> <a id="1958" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1889" class="Bound">h₁</a> <a id="1961" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="1965" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1927" class="Bound">a</a>
|
|||
|
|
<a id="1977" class="Symbol">;</a> <a id="1979" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1048" class="Field">elem₂</a> <a id="1985" class="Symbol">=</a> <a id="1987" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1894" class="Bound">h₂</a> <a id="1990" href="Function.Bundles.html#15133" class="Function Operator">⟨$⟩</a> <a id="1994" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1927" class="Bound">a</a>
|
|||
|
|
<a id="2006" class="Symbol">;</a> <a id="2008" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1070" class="Field">commute</a> <a id="2016" class="Symbol">=</a> <a id="2018" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1898" class="Bound">eq</a>
|
|||
|
|
<a id="2031" class="Symbol">}</a>
|
|||
|
|
<a id="2041" class="Symbol">;</a> <a id="2043" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="2048" class="Symbol">=</a> <a id="2050" href="Data.Product.Base.html#2000" class="Function Operator"><</a> <a id="2052" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="2057" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1889" class="Bound">h₁</a> <a id="2060" href="Data.Product.Base.html#2000" class="Function Operator">,</a> <a id="2062" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="2067" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1894" class="Bound">h₂</a> <a id="2070" href="Data.Product.Base.html#2000" class="Function Operator">></a>
|
|||
|
|
<a id="2080" class="Symbol">}</a>
|
|||
|
|
<a id="2088" class="Symbol">;</a> <a id="2090" href="Categories.Diagram.Pullback.html#956" class="Field">unique</a> <a id="2097" class="Symbol">=</a> <a id="2099" class="Symbol">λ</a> <a id="2101" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2101" class="Bound">eq₁</a> <a id="2105" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2105" class="Bound">eq₂</a> <a id="2109" class="Symbol">→</a> <a id="2111" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2101" class="Bound">eq₁</a> <a id="2115" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2117" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2105" class="Bound">eq₂</a>
|
|||
|
|
<a id="2127" class="Symbol">;</a> <a id="2129" href="Categories.Diagram.Pullback.html#1081" class="Field">p₁∘universal≈h₁</a> <a id="2145" class="Symbol">=</a> <a id="2147" href="Relation.Binary.Structures.html#1596" class="Function">X.refl</a>
|
|||
|
|
<a id="2160" class="Symbol">;</a> <a id="2162" href="Categories.Diagram.Pullback.html#1179" class="Field">p₂∘universal≈h₂</a> <a id="2178" class="Symbol">=</a> <a id="2180" href="Relation.Binary.Structures.html#1596" class="Function">Y.refl</a>
|
|||
|
|
<a id="2193" class="Symbol">}</a>
|
|||
|
|
<a id="2199" class="Symbol">}</a>
|
|||
|
|
<a id="2205" class="Keyword">where</a>
|
|||
|
|
<a id="2217" class="Keyword">module</a> <a id="2224" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2224" class="Module">X</a> <a id="2226" class="Symbol">=</a> <a id="2228" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a> <a id="2235" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1461" class="Bound">X</a>
|
|||
|
|
<a id="2243" class="Keyword">module</a> <a id="2250" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#2250" class="Module">Y</a> <a id="2252" class="Symbol">=</a> <a id="2254" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a> <a id="2261" href="Categories.Category.Instance.Properties.Setoids.Limits.Canonical.html#1469" class="Bound">Y</a>
|
|||
|
|
</pre></body></html>
|