leonv/bsc-leon-vatthauer
1
0
Fork 0
mirror of https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git synced 2024-05-31 07:28:34 +02:00
Code Issues Projects Releases Packages Wiki Activity
bsc-leon-vatthauer/agda/bsc-thesis/Categories.Object.Cokernel.html

57 lines
21 KiB
HTML
Raw Normal View History

update index
2024-02-09 17:53:52 +01:00
<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Categories.Object.Cokernel</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="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="69" class="Keyword">open</a> <a id="74" class="Keyword">import</a> <a id="81" href="Categories.Object.Zero.html" class="Module">Categories.Object.Zero</a>
<a id="105" class="Comment">-- Cokernels of morphisms.</a>
<a id="132" class="Comment">-- https://ncatlab.org/nlab/show/cokernel</a>
<a id="174" class="Keyword">module</a> <a id="181" href="Categories.Object.Cokernel.html" class="Module">Categories.Object.Cokernel</a> <a id="208" class="Symbol">{</a><a id="209" href="Categories.Object.Cokernel.html#209" class="Bound">o</a> <a id="211" href="Categories.Object.Cokernel.html#211" class="Bound">ℓ</a> <a id="213" href="Categories.Object.Cokernel.html#213" class="Bound">e</a><a id="214" class="Symbol">}</a> <a id="216" class="Symbol">{</a><a id="217" href="Categories.Object.Cokernel.html#217" class="Bound">𝒞</a> <a id="219" class="Symbol">:</a> <a id="221" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="230" href="Categories.Object.Cokernel.html#209" class="Bound">o</a> <a id="232" href="Categories.Object.Cokernel.html#211" class="Bound">ℓ</a> <a id="234" href="Categories.Object.Cokernel.html#213" class="Bound">e</a><a id="235" class="Symbol">}</a> <a id="237" class="Symbol">(</a><a id="238" href="Categories.Object.Cokernel.html#238" class="Bound">𝟎</a> <a id="240" class="Symbol">:</a> <a id="242" href="Categories.Object.Zero.html#940" class="Record">Zero</a> <a id="247" href="Categories.Object.Cokernel.html#217" class="Bound">𝒞</a><a id="248" class="Symbol">)</a> <a id="250" class="Keyword">where</a>
<a id="257" class="Keyword">open</a> <a id="262" class="Keyword">import</a> <a id="269" href="Level.html" class="Module">Level</a>
<a id="276" class="Keyword">open</a> <a id="281" class="Keyword">import</a> <a id="288" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="308" href="Categories.Object.Cokernel.html#217" class="Bound">𝒞</a>
<a id="310" class="Keyword">open</a> <a id="315" class="Keyword">import</a> <a id="322" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="352" href="Categories.Object.Cokernel.html#217" class="Bound">𝒞</a>
<a id="356" class="Keyword">hiding</a> <a id="363" class="Symbol">(</a><a id="364" href="Categories.Morphism.Reasoning.Core.html#4290" class="Function">glue</a><a id="368" class="Symbol">)</a>
<a id="371" class="Keyword">open</a> <a id="376" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="385" href="Categories.Object.Cokernel.html#217" class="Bound">𝒞</a>
<a id="387" class="Keyword">open</a> <a id="392" href="Categories.Object.Zero.html#940" class="Module">Zero</a> <a id="397" href="Categories.Object.Cokernel.html#238" class="Bound">𝟎</a>
<a id="400" class="Keyword">open</a> <a id="405" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="419" class="Keyword">private</a>
<a id="429" class="Keyword">variable</a>
<a id="442" href="Categories.Object.Cokernel.html#442" class="Generalizable">A</a> <a id="444" href="Categories.Object.Cokernel.html#444" class="Generalizable">B</a> <a id="446" href="Categories.Object.Cokernel.html#446" class="Generalizable">X</a> <a id="448" class="Symbol">:</a> <a id="450" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="458" href="Categories.Object.Cokernel.html#458" class="Generalizable">f</a> <a id="460" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="462" href="Categories.Object.Cokernel.html#462" class="Generalizable">i</a> <a id="464" href="Categories.Object.Cokernel.html#464" class="Generalizable">j</a> <a id="466" href="Categories.Object.Cokernel.html#466" class="Generalizable">k</a> <a id="468" class="Symbol">:</a> <a id="470" href="Categories.Object.Cokernel.html#442" class="Generalizable">A</a> <a id="472" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="474" href="Categories.Object.Cokernel.html#444" class="Generalizable">B</a>
<a id="477" class="Keyword">record</a> <a id="IsCokernel"></a><a id="484" href="Categories.Object.Cokernel.html#484" class="Record">IsCokernel</a> <a id="495" class="Symbol">{</a><a id="496" href="Categories.Object.Cokernel.html#496" class="Bound">A</a> <a id="498" href="Categories.Object.Cokernel.html#498" class="Bound">B</a> <a id="500" href="Categories.Object.Cokernel.html#500" class="Bound">K</a><a id="501" class="Symbol">}</a> <a id="503" class="Symbol">(</a><a id="504" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="506" class="Symbol">:</a> <a id="508" href="Categories.Object.Cokernel.html#496" class="Bound">A</a> <a id="510" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="512" href="Categories.Object.Cokernel.html#498" class="Bound">B</a><a id="513" class="Symbol">)</a> <a id="515" class="Symbol">(</a><a id="516" href="Categories.Object.Cokernel.html#516" class="Bound">k</a> <a id="518" class="Symbol">:</a> <a id="520" href="Categories.Object.Cokernel.html#498" class="Bound">B</a> <a id="522" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="524" href="Categories.Object.Cokernel.html#500" class="Bound">K</a><a id="525" class="Symbol">)</a> <a id="527" class="Symbol">:</a> <a id="529" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="533" class="Symbol">(</a><a id="534" href="Categories.Object.Cokernel.html#209" class="Bound">o</a> <a id="536" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="538" href="Categories.Object.Cokernel.html#211" class="Bound">ℓ</a> <a id="540" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="542" href="Categories.Object.Cokernel.html#213" class="Bound">e</a><a id="543" class="Symbol">)</a> <a id="545" class="Keyword">where</a>
<a id="553" class="Keyword">field</a>
<a id="IsCokernel.commute"></a><a id="563" href="Categories.Object.Cokernel.html#563" class="Field">commute</a> <a id="573" class="Symbol">:</a> <a id="575" href="Categories.Object.Cokernel.html#516" class="Bound">k</a> <a id="577" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="579" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="581" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="583" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a>
<a id="IsCokernel.universal"></a><a id="593" href="Categories.Object.Cokernel.html#593" class="Field">universal</a> <a id="603" class="Symbol">:</a> <a id="605" class="Symbol">∀</a> <a id="607" class="Symbol">{</a><a id="608" href="Categories.Object.Cokernel.html#608" class="Bound">X</a><a id="609" class="Symbol">}</a> <a id="611" class="Symbol">{</a><a id="612" href="Categories.Object.Cokernel.html#612" class="Bound">h</a> <a id="614" class="Symbol">:</a> <a id="616" href="Categories.Object.Cokernel.html#498" class="Bound">B</a> <a id="618" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="620" href="Categories.Object.Cokernel.html#608" class="Bound">X</a><a id="621" class="Symbol">}</a> <a id="623" class="Symbol">→</a> <a id="625" href="Categories.Object.Cokernel.html#612" class="Bound">h</a> <a id="627" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="629" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="631" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="633" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a> <a id="639" class="Symbol">→</a> <a id="641" href="Categories.Object.Cokernel.html#500" class="Bound">K</a> <a id="643" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="645" href="Categories.Object.Cokernel.html#608" class="Bound">X</a>
<a id="IsCokernel.factors"></a><a id="651" href="Categories.Object.Cokernel.html#651" class="Field">factors</a> <a id="661" class="Symbol">:</a> <a id="663" class="Symbol">∀</a> <a id="665" class="Symbol">{</a><a id="666" href="Categories.Object.Cokernel.html#666" class="Bound">eq</a> <a id="669" class="Symbol">:</a> <a id="671" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="673" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="675" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="677" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="679" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a><a id="684" class="Symbol">}</a> <a id="686" class="Symbol">→</a> <a id="688" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="690" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="692" href="Categories.Object.Cokernel.html#593" class="Field">universal</a> <a id="702" href="Categories.Object.Cokernel.html#666" class="Bound">eq</a> <a id="705" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="707" href="Categories.Object.Cokernel.html#516" class="Bound">k</a>
<a id="IsCokernel.unique"></a><a id="713" href="Categories.Object.Cokernel.html#713" class="Field">unique</a> <a id="723" class="Symbol">:</a> <a id="725" class="Symbol">∀</a> <a id="727" class="Symbol">{</a><a id="728" href="Categories.Object.Cokernel.html#728" class="Bound">eq</a> <a id="731" class="Symbol">:</a> <a id="733" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="735" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="737" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="739" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="741" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a><a id="746" class="Symbol">}</a> <a id="748" class="Symbol">→</a> <a id="750" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="752" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="754" href="Categories.Object.Cokernel.html#462" class="Generalizable">i</a> <a id="756" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="758" href="Categories.Object.Cokernel.html#516" class="Bound">k</a> <a id="760" class="Symbol">→</a> <a id="762" href="Categories.Object.Cokernel.html#462" class="Generalizable">i</a> <a id="764" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="766" href="Categories.Object.Cokernel.html#593" class="Field">universal</a> <a id="776" href="Categories.Object.Cokernel.html#728" class="Bound">eq</a>
<a id="IsCokernel.universal-resp-≈"></a><a id="782" href="Categories.Object.Cokernel.html#782" class="Function">universal-resp-≈</a> <a id="799" class="Symbol">:</a> <a id="801" class="Symbol">∀</a> <a id="803" class="Symbol">{</a><a id="804" href="Categories.Object.Cokernel.html#804" class="Bound">eq</a> <a id="807" class="Symbol">:</a> <a id="809" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="811" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="813" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="815" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="817" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a><a id="822" class="Symbol">}</a> <a id="824" class="Symbol">{</a><a id="825" href="Categories.Object.Cokernel.html#825" class="Bound">eq′</a> <a id="829" class="Symbol">:</a> <a id="831" href="Categories.Object.Cokernel.html#462" class="Generalizable">i</a> <a id="833" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="835" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="837" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="839" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a><a id="844" class="Symbol">}</a> <a id="846" class="Symbol">→</a>
<a id="852" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="854" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="856" href="Categories.Object.Cokernel.html#462" class="Generalizable">i</a> <a id="858" class="Symbol">→</a> <a id="860" href="Categories.Object.Cokernel.html#593" class="Field">universal</a> <a id="870" href="Categories.Object.Cokernel.html#804" class="Bound">eq</a> <a id="873" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="875" href="Categories.Object.Cokernel.html#593" class="Field">universal</a> <a id="885" href="Categories.Object.Cokernel.html#825" class="Bound">eq′</a>
<a id="891" href="Categories.Object.Cokernel.html#782" class="Function">universal-resp-≈</a> <a id="908" href="Categories.Object.Cokernel.html#908" class="Bound">h≈i</a> <a id="912" class="Symbol">=</a> <a id="914" href="Categories.Object.Cokernel.html#713" class="Field">unique</a> <a id="921" class="Symbol">(</a><a id="922" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="924" href="Categories.Object.Cokernel.html#908" class="Bound">h≈i</a> <a id="928" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="930" href="Categories.Object.Cokernel.html#651" class="Field">factors</a><a id="937" class="Symbol">)</a>
<a id="IsCokernel.universal-∘"></a><a id="942" href="Categories.Object.Cokernel.html#942" class="Function">universal-∘</a> <a id="954" class="Symbol">:</a> <a id="956" href="Categories.Object.Cokernel.html#460" class="Generalizable">h</a> <a id="958" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="960" href="Categories.Object.Cokernel.html#516" class="Bound">k</a> <a id="962" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="964" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="966" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="968" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a>
<a id="976" href="Categories.Object.Cokernel.html#942" class="Function">universal-∘</a> <a id="988" class="Symbol">{</a><a id="989" class="Argument">h</a> <a id="991" class="Symbol">=</a> <a id="993" href="Categories.Object.Cokernel.html#993" class="Bound">h</a><a id="994" class="Symbol">}</a> <a id="996" class="Symbol">=</a> <a id="998" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1008" href="Categories.Object.Cokernel.html#993" class="Bound">h</a> <a id="1010" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1012" href="Categories.Object.Cokernel.html#516" class="Bound">k</a> <a id="1014" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1016" href="Categories.Object.Cokernel.html#504" class="Bound">f</a> <a id="1018" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1021" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1029" href="Categories.Object.Cokernel.html#563" class="Field">commute</a> <a id="1037" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1043" href="Categories.Object.Cokernel.html#993" class="Bound">h</a> <a id="1045" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1047" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a> <a id="1053" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1056" href="Categories.Object.Zero.html#723" class="Function">zero-∘ˡ</a> <a id="1064" href="Categories.Object.Cokernel.html#993" class="Bound">h</a> <a id="1066" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1072" href="Categories.Object.Zero.html#674" class="Function">zero⇒</a> <a id="1082" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="1085" class="Keyword">record</a> <a id="Cokernel"></a><a id="1092" href="Categories.Object.Cokernel.html#1092" class="Record">Cokernel</a> <a id="1101" class="Symbol">{</a><a id="1102" href="Categories.Object.Cokernel.html#1102" class="Bound">A</a> <a id="1104" href="Categories.Object.Cokernel.html#1104" class="Bound">B</a><a id="1105" class="Symbol">}</a> <a id="1107" class="Symbol">(</a><a id="1108" href="Categories.Object.Cokernel.html#1108" class="Bound">f</a> <a id="1110" class="Symbol">:</a> <a id="1112" href="Categories.Object.Cokernel.html#1102" class="Bound">A</a> <a id="1114" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1116" href="Categories.Object.Cokernel.html#1104" class="Bound">B</a><a id="1117" class="Symbol">)</a> <a id="1119" class="Symbol">:</a> <a id="1121" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1125" class="Symbol">(</a><a id="1126" href="Categories.Object.Cokernel.html#209" class="Bound">o</a> <a id="1128" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1130" href="Categories.Object.Cokernel.html#211" class="Bound">ℓ</a> <a id="1132" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1134" href="Categories.Object.Cokernel.html#213" class="Bound">e</a><a id="1135" class="Symbol">)</a> <a id="1137" class="Keyword">where</a>
<a id="1145" class="Keyword">field</a>
<a id="1155" class="Symbol">{</a><a id="Cokernel.cokernel"></a><a id="1156" href="Categories.Object.Cokernel.html#1156" class="Field">cokernel</a><a id="1164" class="Symbol">}</a> <a id="1166" class="Symbol">:</a> <a id="1168" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="Cokernel.cokernel⇒"></a><a id="1176" href="Categories.Object.Cokernel.html#1176" class="Field">cokernel⇒</a> <a id="1187" class="Symbol">:</a> <a id="1189" href="Categories.Object.Cokernel.html#1104" class="Bound">B</a> <a id="1191" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1193" href="Categories.Object.Cokernel.html#1156" class="Field">cokernel</a>
<a id="Cokernel.isCokernel"></a><a id="1206" href="Categories.Object.Cokernel.html#1206" class="Field">isCokernel</a> <a id="1217" class="Symbol">:</a> <a id="1219" href="Categories.Object.Cokernel.html#484" class="Record">IsCokernel</a> <a id="1230" href="Categories.Object.Cokernel.html#1108" class="Bound">f</a> <a id="1232" href="Categories.Object.Cokernel.html#1176" class="Field">cokernel⇒</a>
<a id="1245" class="Keyword">open</a> <a id="1250" href="Categories.Object.Cokernel.html#484" class="Module">IsCokernel</a> <a id="1261" href="Categories.Object.Cokernel.html#1206" class="Field">isCokernel</a> <a id="1272" class="Keyword">public</a>
<a id="IsCokernel⇒Cokernel"></a><a id="1280" href="Categories.Object.Cokernel.html#1280" class="Function">IsCokernel⇒Cokernel</a> <a id="1300" class="Symbol">:</a> <a id="1302" href="Categories.Object.Cokernel.html#484" class="Record">IsCokernel</a> <a id="1313" href="Categories.Object.Cokernel.html#458" class="Generalizable">f</a> <a id="1315" href="Categories.Object.Cokernel.html#466" class="Generalizable">k</a> <a id="1317" class="Symbol">→</a> <a id="1319" href="Categories.Object.Cokernel.html#1092" class="Record">Cokernel</a> <a id="1328" href="Categories.Object.Cokernel.html#458" class="Generalizable">f</a>
<a id="1330" href="Categories.Object.Cokernel.html#1280" class="Function">IsCokernel⇒Cokernel</a> <a id="1350" class="Symbol">{</a><a id="1351" class="Argument">k</a> <a id="1353" class="Symbol">=</a> <a id="1355" href="Categories.Object.Cokernel.html#1355" class="Bound">k</a><a id="1356" class="Symbol">}</a> <a id="1358" href="Categories.Object.Cokernel.html#1358" class="Bound">isCokernel</a> <a id="1369" class="Symbol">=</a> <a id="1371" class="Keyword">record</a>
<a id="1380" class="Symbol">{</a> <a id="1382" href="Categories.Object.Cokernel.html#1176" class="Field">cokernel⇒</a> <a id="1392" class="Symbol">=</a> <a id="1394" href="Categories.Object.Cokernel.html#1355" class="Bound">k</a>
<a id="1398" class="Symbol">;</a> <a id="1400" href="Categories.Object.Cokernel.html#1206" class="Field">isCokernel</a> <a id="1411" class="Symbol">=</a> <a id="1413" href="Categories.Object.Cokernel.html#1358" class="Bound">isCokernel</a>
<a id="1426" class="Symbol">}</a>
</pre></body></html>
Reference in a new issue Copy permalink