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<html><head><meta charset="utf-8"><title>Categories.Category.Complete.Finitely</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="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="70" class="Keyword">module</a> <a id="77" href="Categories.Category.Complete.Finitely.html" class="Module">Categories.Category.Complete.Finitely</a> <a id="115" class="Symbol">{</a><a id="116" href="Categories.Category.Complete.Finitely.html#116" class="Bound">o</a> <a id="118" href="Categories.Category.Complete.Finitely.html#118" class="Bound">ℓ</a> <a id="120" href="Categories.Category.Complete.Finitely.html#120" class="Bound">e</a><a id="121" class="Symbol">}</a> <a id="123" class="Symbol">(</a><a id="124" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a> <a id="126" class="Symbol">:</a> <a id="128" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="137" href="Categories.Category.Complete.Finitely.html#116" class="Bound">o</a> <a id="139" href="Categories.Category.Complete.Finitely.html#118" class="Bound">ℓ</a> <a id="141" href="Categories.Category.Complete.Finitely.html#120" class="Bound">e</a><a id="142" class="Symbol">)</a> <a id="144" class="Keyword">where</a>
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<a id="151" class="Keyword">open</a> <a id="156" class="Keyword">import</a> <a id="163" href="Level.html" class="Module">Level</a>
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<a id="170" class="Keyword">open</a> <a id="175" class="Keyword">import</a> <a id="182" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="217" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a> <a id="219" class="Keyword">using</a> <a id="225" class="Symbol">(</a><a id="226" href="Categories.Category.BinaryProducts.html#848" class="Record">BinaryProducts</a><a id="240" class="Symbol">)</a>
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<a id="242" class="Keyword">open</a> <a id="247" class="Keyword">import</a> <a id="254" href="Categories.Category.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="284" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a> <a id="286" class="Keyword">using</a> <a id="292" class="Symbol">(</a><a id="293" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="302" class="Symbol">)</a>
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<a id="304" class="Keyword">open</a> <a id="309" class="Keyword">import</a> <a id="316" href="Categories.Diagram.Equalizer.html" class="Module">Categories.Diagram.Equalizer</a> <a id="345" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a> <a id="347" class="Keyword">using</a> <a id="353" class="Symbol">(</a><a id="354" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a><a id="363" class="Symbol">)</a>
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<a id="365" class="Keyword">open</a> <a id="370" class="Keyword">import</a> <a id="377" href="Categories.Diagram.Pullback.html" class="Module">Categories.Diagram.Pullback</a> <a id="405" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a> <a id="407" class="Keyword">using</a> <a id="413" class="Symbol">(</a><a id="414" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a><a id="422" class="Symbol">;</a> <a id="424" href="Categories.Diagram.Pullback.html#5412" class="Function">Product×Equalizer⇒Pullback</a><a id="450" class="Symbol">)</a>
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<a id="453" class="Keyword">open</a> <a id="458" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="467" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a>
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<a id="470" class="Keyword">record</a> <a id="FinitelyComplete"></a><a id="477" href="Categories.Category.Complete.Finitely.html#477" class="Record">FinitelyComplete</a> <a id="494" class="Symbol">:</a> <a id="496" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="500" class="Symbol">(</a><a id="501" href="Level.html#602" class="Function">levelOfTerm</a> <a id="513" href="Categories.Category.Complete.Finitely.html#124" class="Bound">C</a><a id="514" class="Symbol">)</a> <a id="516" class="Keyword">where</a>
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<a id="524" class="Keyword">field</a>
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<a id="FinitelyComplete.cartesian"></a><a id="534" href="Categories.Category.Complete.Finitely.html#534" class="Field">cartesian</a> <a id="544" class="Symbol">:</a> <a id="546" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a>
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<a id="FinitelyComplete.equalizer"></a><a id="560" href="Categories.Category.Complete.Finitely.html#560" class="Field">equalizer</a> <a id="570" class="Symbol">:</a> <a id="572" class="Symbol">∀</a> <a id="574" class="Symbol">{</a><a id="575" href="Categories.Category.Complete.Finitely.html#575" class="Bound">A</a> <a id="577" href="Categories.Category.Complete.Finitely.html#577" class="Bound">B</a><a id="578" class="Symbol">}</a> <a id="580" class="Symbol">(</a><a id="581" href="Categories.Category.Complete.Finitely.html#581" class="Bound">f</a> <a id="583" href="Categories.Category.Complete.Finitely.html#583" class="Bound">g</a> <a id="585" class="Symbol">:</a> <a id="587" href="Categories.Category.Complete.Finitely.html#575" class="Bound">A</a> <a id="589" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="591" href="Categories.Category.Complete.Finitely.html#577" class="Bound">B</a><a id="592" class="Symbol">)</a> <a id="594" class="Symbol">→</a> <a id="596" href="Categories.Diagram.Equalizer.html#1948" class="Record">Equalizer</a> <a id="606" href="Categories.Category.Complete.Finitely.html#581" class="Bound">f</a> <a id="608" href="Categories.Category.Complete.Finitely.html#583" class="Bound">g</a>
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<a id="613" class="Keyword">module</a> <a id="FinitelyComplete.equalizer"></a><a id="620" href="Categories.Category.Complete.Finitely.html#620" class="Module">equalizer</a> <a id="630" class="Symbol">{</a><a id="631" href="Categories.Category.Complete.Finitely.html#631" class="Bound">A</a> <a id="633" href="Categories.Category.Complete.Finitely.html#633" class="Bound">B</a><a id="634" class="Symbol">}</a> <a id="636" class="Symbol">(</a><a id="637" href="Categories.Category.Complete.Finitely.html#637" class="Bound">f</a> <a id="639" href="Categories.Category.Complete.Finitely.html#639" class="Bound">g</a> <a id="641" class="Symbol">:</a> <a id="643" href="Categories.Category.Complete.Finitely.html#631" class="Bound">A</a> <a id="645" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="647" href="Categories.Category.Complete.Finitely.html#633" class="Bound">B</a><a id="648" class="Symbol">)</a> <a id="650" class="Symbol">=</a> <a id="652" href="Categories.Diagram.Equalizer.html#1948" class="Module">Equalizer</a> <a id="662" class="Symbol">(</a><a id="663" href="Categories.Category.Complete.Finitely.html#560" class="Field">equalizer</a> <a id="673" href="Categories.Category.Complete.Finitely.html#637" class="Bound">f</a> <a id="675" href="Categories.Category.Complete.Finitely.html#639" class="Bound">g</a><a id="676" class="Symbol">)</a>
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<a id="680" class="Keyword">open</a> <a id="685" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="695" href="Categories.Category.Complete.Finitely.html#534" class="Field">cartesian</a> <a id="705" class="Keyword">using</a> <a id="711" class="Symbol">(</a><a id="712" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="720" class="Symbol">)</a>
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<a id="FinitelyComplete.pullback"></a><a id="725" href="Categories.Category.Complete.Finitely.html#725" class="Function">pullback</a> <a id="734" class="Symbol">:</a> <a id="736" class="Symbol">∀</a> <a id="738" class="Symbol">{</a><a id="739" href="Categories.Category.Complete.Finitely.html#739" class="Bound">X</a> <a id="741" href="Categories.Category.Complete.Finitely.html#741" class="Bound">Y</a> <a id="743" href="Categories.Category.Complete.Finitely.html#743" class="Bound">Z</a><a id="744" class="Symbol">}</a> <a id="746" class="Symbol">(</a><a id="747" href="Categories.Category.Complete.Finitely.html#747" class="Bound">f</a> <a id="749" class="Symbol">:</a> <a id="751" href="Categories.Category.Complete.Finitely.html#739" class="Bound">X</a> <a id="753" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="755" href="Categories.Category.Complete.Finitely.html#743" class="Bound">Z</a><a id="756" class="Symbol">)</a> <a id="758" class="Symbol">(</a><a id="759" href="Categories.Category.Complete.Finitely.html#759" class="Bound">g</a> <a id="761" class="Symbol">:</a> <a id="763" href="Categories.Category.Complete.Finitely.html#741" class="Bound">Y</a> <a id="765" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="767" href="Categories.Category.Complete.Finitely.html#743" class="Bound">Z</a><a id="768" class="Symbol">)</a> <a id="770" class="Symbol">→</a> <a id="772" href="Categories.Diagram.Pullback.html#2029" class="Record">Pullback</a> <a id="781" href="Categories.Category.Complete.Finitely.html#747" class="Bound">f</a> <a id="783" href="Categories.Category.Complete.Finitely.html#759" class="Bound">g</a>
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<a id="787" href="Categories.Category.Complete.Finitely.html#725" class="Function">pullback</a> <a id="796" href="Categories.Category.Complete.Finitely.html#796" class="Bound">f</a> <a id="798" href="Categories.Category.Complete.Finitely.html#798" class="Bound">g</a> <a id="800" class="Symbol">=</a> <a id="802" href="Categories.Diagram.Pullback.html#5412" class="Function">Product×Equalizer⇒Pullback</a> <a id="829" class="Symbol">(</a><a id="830" href="Categories.Category.BinaryProducts.html#952" class="Field">BinaryProducts.product</a> <a id="853" href="Categories.Category.Cartesian.html#801" class="Function">products</a><a id="861" class="Symbol">)</a> <a id="863" class="Symbol">(</a><a id="864" href="Categories.Category.Complete.Finitely.html#560" class="Field">equalizer</a> <a id="874" class="Symbol">_</a> <a id="876" class="Symbol">_)</a>
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<a id="882" class="Keyword">module</a> <a id="FinitelyComplete.pullback"></a><a id="889" href="Categories.Category.Complete.Finitely.html#889" class="Module">pullback</a> <a id="898" class="Symbol">{</a><a id="899" href="Categories.Category.Complete.Finitely.html#899" class="Bound">X</a> <a id="901" href="Categories.Category.Complete.Finitely.html#901" class="Bound">Y</a> <a id="903" href="Categories.Category.Complete.Finitely.html#903" class="Bound">Z</a><a id="904" class="Symbol">}</a> <a id="906" class="Symbol">(</a><a id="907" href="Categories.Category.Complete.Finitely.html#907" class="Bound">f</a> <a id="909" class="Symbol">:</a> <a id="911" href="Categories.Category.Complete.Finitely.html#899" class="Bound">X</a> <a id="913" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="915" href="Categories.Category.Complete.Finitely.html#903" class="Bound">Z</a><a id="916" class="Symbol">)</a> <a id="918" class="Symbol">(</a><a id="919" href="Categories.Category.Complete.Finitely.html#919" class="Bound">g</a> <a id="921" class="Symbol">:</a> <a id="923" href="Categories.Category.Complete.Finitely.html#901" class="Bound">Y</a> <a id="925" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="927" href="Categories.Category.Complete.Finitely.html#903" class="Bound">Z</a><a id="928" class="Symbol">)</a> <a id="930" class="Symbol">=</a> <a id="932" href="Categories.Diagram.Pullback.html#2029" class="Module">Pullback</a> <a id="941" class="Symbol">(</a><a id="942" href="Categories.Category.Complete.Finitely.html#725" class="Function">pullback</a> <a id="951" href="Categories.Category.Complete.Finitely.html#907" class="Bound">f</a> <a id="953" href="Categories.Category.Complete.Finitely.html#919" class="Bound">g</a><a id="954" class="Symbol">)</a>
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