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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Object.Initial.html
Leon Vatthauer 2bf4655eae
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<html><head><meta charset="utf-8"><title>Categories.Object.Initial</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="1344" class="Symbol">;</a> <a id="1346" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1355" class="Symbol">=</a> <a id="1357" class="Symbol">λ</a> <a id="1359" href="Categories.Object.Initial.html#1359" class="Bound">h</a> <a id="1361" class="Symbol">→</a> <a id="1364" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1376" href="Categories.Object.Initial.html#478" class="Function">!</a> <a id="1378" href="Categories.Object.Initial.html#1252" class="Bound">i</a> <a id="1380" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1382" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="1392" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1395" href="Categories.Object.Initial.html#506" class="Function">!-unique</a> <a id="1404" href="Categories.Object.Initial.html#1252" class="Bound">i</a> <a id="1406" class="Symbol">(</a><a id="1407" href="Categories.Object.Initial.html#1359" class="Bound">h</a> <a id="1409" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1411" href="Categories.Morphism.html#2006" class="Field">from</a><a id="1415" class="Symbol">)</a> <a id="1417" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="1426" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1434" class="Symbol">(</a><a id="1435" href="Categories.Object.Initial.html#1359" class="Bound">h</a> <a id="1437" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1439" href="Categories.Morphism.html#2006" class="Field">from</a><a id="1443" class="Symbol">)</a> <a id="1445" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="1447" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="1450" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1453" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="1461" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="1466" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1474" href="Categories.Object.Initial.html#1359" class="Bound">h</a> <a id="1489" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="1495" class="Symbol">}</a>
<a id="1499" class="Symbol">}</a>
<a id="1503" class="Keyword">where</a> <a id="1509" class="Keyword">open</a> <a id="1514" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="1518" href="Categories.Object.Initial.html#1258" class="Bound">i≅X</a>
<a id="up-to-iso-unique"></a><a id="1523" href="Categories.Object.Initial.html#1523" class="Function">up-to-iso-unique</a> <a id="1540" class="Symbol">:</a> <a id="1542" class="Symbol">∀</a> <a id="1544" href="Categories.Object.Initial.html#1544" class="Bound">i</a> <a id="1546" href="Categories.Object.Initial.html#1546" class="Bound">i′</a> <a id="1549" class="Symbol">→</a> <a id="1551" class="Symbol">(</a><a id="1552" href="Categories.Object.Initial.html#1552" class="Bound">iso</a> <a id="1556" class="Symbol">:</a> <a id="1558" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1560" href="Categories.Object.Initial.html#1544" class="Bound">i</a> <a id="1562" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1564" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1566" href="Categories.Object.Initial.html#1546" class="Bound">i′</a><a id="1568" class="Symbol">)</a> <a id="1570" class="Symbol">→</a> <a id="1572" href="Categories.Object.Initial.html#1011" class="Function">up-to-iso</a> <a id="1582" href="Categories.Object.Initial.html#1544" class="Bound">i</a> <a id="1584" href="Categories.Object.Initial.html#1546" class="Bound">i′</a> <a id="1587" href="Categories.Morphism.IsoEquiv.html#1761" class="Record Operator">≃</a> <a id="1589" href="Categories.Object.Initial.html#1552" class="Bound">iso</a>
<a id="1593" href="Categories.Object.Initial.html#1523" class="Function">up-to-iso-unique</a> <a id="1610" href="Categories.Object.Initial.html#1610" class="Bound">i</a> <a id="1612" href="Categories.Object.Initial.html#1612" class="Bound">i′</a> <a id="1615" href="Categories.Object.Initial.html#1615" class="Bound">iso</a> <a id="1619" class="Symbol">=</a> <a id="1621" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌞</a> <a id="1623" href="Categories.Object.Initial.html#506" class="Function">!-unique</a> <a id="1632" href="Categories.Object.Initial.html#1610" class="Bound">i</a> <a id="1634" class="Symbol">_</a> <a id="1636" href="Categories.Morphism.IsoEquiv.html#1807" class="InductiveConstructor Operator">⌟</a>
<a id="up-to-iso-invˡ"></a><a id="1639" href="Categories.Object.Initial.html#1639" class="Function">up-to-iso-invˡ</a> <a id="1654" class="Symbol">:</a> <a id="1656" class="Symbol">∀</a> <a id="1658" class="Symbol">{</a><a id="1659" href="Categories.Object.Initial.html#1659" class="Bound">t</a> <a id="1661" href="Categories.Object.Initial.html#1661" class="Bound">X</a><a id="1662" class="Symbol">}</a> <a id="1664" class="Symbol">{</a><a id="1665" href="Categories.Object.Initial.html#1665" class="Bound">i</a> <a id="1667" class="Symbol">:</a> <a id="1669" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1671" href="Categories.Object.Initial.html#1659" class="Bound">t</a> <a id="1673" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="1675" href="Categories.Object.Initial.html#1661" class="Bound">X</a><a id="1676" class="Symbol">}</a> <a id="1678" class="Symbol">→</a> <a id="1680" href="Categories.Object.Initial.html#1011" class="Function">up-to-iso</a> <a id="1690" href="Categories.Object.Initial.html#1659" class="Bound">t</a> <a id="1692" class="Symbol">(</a><a id="1693" href="Categories.Object.Initial.html#1174" class="Function">transport-by-iso</a> <a id="1710" href="Categories.Object.Initial.html#1659" class="Bound">t</a> <a id="1712" href="Categories.Object.Initial.html#1665" class="Bound">i</a><a id="1713" class="Symbol">)</a> <a id="1715" href="Categories.Morphism.IsoEquiv.html#1761" class="Record Operator">≃</a> <a id="1717" href="Categories.Object.Initial.html#1665" class="Bound">i</a>
<a id="1719" href="Categories.Object.Initial.html#1639" class="Function">up-to-iso-invˡ</a> <a id="1734" class="Symbol">{</a><a id="1735" href="Categories.Object.Initial.html#1735" class="Bound">t</a><a id="1736" class="Symbol">}</a> <a id="1738" class="Symbol">{</a><a id="1739" class="Argument">i</a> <a id="1741" class="Symbol">=</a> <a id="1743" href="Categories.Object.Initial.html#1743" class="Bound">i</a><a id="1744" class="Symbol">}</a> <a id="1746" class="Symbol">=</a> <a id="1748" href="Categories.Object.Initial.html#1523" class="Function">up-to-iso-unique</a> <a id="1765" href="Categories.Object.Initial.html#1735" class="Bound">t</a> <a id="1767" class="Symbol">(</a><a id="1768" href="Categories.Object.Initial.html#1174" class="Function">transport-by-iso</a> <a id="1785" href="Categories.Object.Initial.html#1735" class="Bound">t</a> <a id="1787" href="Categories.Object.Initial.html#1743" class="Bound">i</a><a id="1788" class="Symbol">)</a> <a id="1790" href="Categories.Object.Initial.html#1743" class="Bound">i</a>
<a id="up-to-iso-invʳ"></a><a id="1793" href="Categories.Object.Initial.html#1793" class="Function">up-to-iso-invʳ</a> <a id="1808" class="Symbol">:</a> <a id="1810" class="Symbol">∀</a> <a id="1812" class="Symbol">{</a><a id="1813" href="Categories.Object.Initial.html#1813" class="Bound">t</a> <a id="1815" href="Categories.Object.Initial.html#1815" class="Bound">t′</a><a id="1817" class="Symbol">}</a> <a id="1819" class="Symbol">→</a> <a id="1821" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1823" class="Symbol">(</a><a id="1824" href="Categories.Object.Initial.html#1174" class="Function">transport-by-iso</a> <a id="1841" href="Categories.Object.Initial.html#1813" class="Bound">t</a> <a id="1843" class="Symbol">(</a><a id="1844" href="Categories.Object.Initial.html#1011" class="Function">up-to-iso</a> <a id="1854" href="Categories.Object.Initial.html#1813" class="Bound">t</a> <a id="1856" href="Categories.Object.Initial.html#1815" class="Bound">t′</a><a id="1858" class="Symbol">))</a> <a id="1861" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1863" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1865" href="Categories.Object.Initial.html#1815" class="Bound">t′</a>
<a id="1868" href="Categories.Object.Initial.html#1793" class="Function">up-to-iso-invʳ</a> <a id="1883" class="Symbol">{</a><a id="1884" href="Categories.Object.Initial.html#1884" class="Bound">t</a><a id="1885" class="Symbol">}</a> <a id="1887" class="Symbol">{</a><a id="1888" href="Categories.Object.Initial.html#1888" class="Bound">t′</a><a id="1890" class="Symbol">}</a> <a id="1892" class="Symbol">=</a> <a id="1894" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">≡.refl</a>
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