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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (GeneralizedMorphismsForCAP) - Chapter 4: Generalized Morphism Category by Three <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap4" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap4_mj.html">[MathJax on]</a></p> <p><a id="X7F39133179F83B50" name="X7F39133179F83B50"></a></p> <div class="ChapSects"><a href="chap4.html#X7F39133179F83B50">4 <span class="Heading">Gene <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X869A7C2A85C2743A">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E861596801CA565">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87934BAA84C6E654">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8247F41681D15233">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8347A40179DBD909">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81F0F4FA78F9F361">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8028877F819A0DBF">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B17B7587A449B8E">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87B0A0AF84F8AF7D">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CCBCD5385E55586">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X784C006B8525E205">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78A8A55883C06AB1">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80E329707BAA3B29">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8217014B870B3EC2">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78C1A22D86943E97">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A50AC43855B33B1">4 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8477D6E679E15474">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D60D9F97A2F7D5F">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F9E6B188755170A">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E0CF2A87AC08A8B">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8296606484DDDF6C">4 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8165802A8625467E">4 </div></div> </div> <h3>4 <span class="Heading">Generalized Morphism Category by Three Arrows</span></h3> <p><a id="X7D03633A7D98026B" name="X7D03633A7D98026B"></a></p> <h4>4.1 <span class="Heading">GAP Categories</span></h4> <p><a id="X869A7C2A85C2743A" name="X869A7C2A85C2743A"></a></p> <h5>4.1-1 IsGeneralizedMorphismCategoryByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The GAP category of the category of generalized morphisms by three arrows.</p> <p><a id="X7E861596801CA565" name="X7E861596801CA565"></a></p> <h5>4.1-2 IsGeneralizedMorphismCategoryByThreeArrowsObject</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The GAP category of objects in the generalized morphism category by three arrows.</p> <p><a id="X87934BAA84C6E654" name="X87934BAA84C6E654"></a></p> <h5>4.1-3 IsGeneralizedMorphismByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The GAP category of morphisms in the generalized morphism category by three arrows.</p> <p><a id="X871597447BB998A1" name="X871597447BB998A1"></a></p> <h4>4.2 <span class="Heading">Properties</span></h4> <p><a id="X8247F41681D15233" name="X8247F41681D15233"></a></p> <h5>4.2-1 HasIdentitiesAsReversedArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X8347A40179DBD909" name="X8347A40179DBD909"></a></p> <h5>4.2-2 HasIdentityAsSourceAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X81F0F4FA78F9F361" name="X81F0F4FA78F9F361"></a></p> <h5>4.2-3 HasIdentityAsRangeAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p> <h4>4.3 <span class="Heading">Attributes</span></h4> <p><a id="X8028877F819A0DBF" name="X8028877F819A0DBF"></a></p> <h5>4.3-1 UnderlyingHonestObject</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: an object in <span class="Math">\mathbf{A}</span></p> <p>The argument is an object <span class="Math">a</span> in the generalized morphism category by th <p><a id="X7B17B7587A449B8E" name="X7B17B7587A449B8E"></a></p> <h5>4.3-2 SourceAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{A}}(b,a)</span></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X87B0A0AF84F8AF7D" name="X87B0A0AF84F8AF7D"></a></p> <h5>4.3-3 RangeAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{A}}(d,c)</span></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X7CCBCD5385E55586" name="X7CCBCD5385E55586"></a></p> <h5>4.3-4 Arrow</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{A}}(b,c)</span></p> <p>The argument is a generalized morphism <span class="Math">\alpha</span> by three arrows <span cl <p><a id="X784C006B8525E205" name="X784C006B8525E205"></a></p> <h5>4.3-5 PseudoInverse</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(b,a)</span></p> <p>The argument is a generalized morphism <span class="Math">\alpha: a \rightarrow b</span> by thr <p><a id="X78A8A55883C06AB1" name="X78A8A55883C06AB1"></a></p> <h5>4.3-6 GeneralizedInverseByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(b,a)</span></p> <p>The argument is a morphism <span class="Math">\alpha: a \rightarrow b \in \mathbf{A}</span>. Th <p><a id="X80E329707BAA3B29" name="X80E329707BAA3B29"></a></p> <h5>4.3-7 IdempotentDefinedBySubobjectByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(b,b)</span></p> <p>The argument is a subobject <span class="Math">\alpha: a \hookrightarrow b \in \mathbf{A}</sp <p><a id="X8217014B870B3EC2" name="X8217014B870B3EC2"></a></p> <h5>4.3-8 IdempotentDefinedByFactorobjectByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(b,b)</span></p> <p>The argument is a factorobject <span class="Math">\alpha: b \twoheadrightarrow a \in \mathbf <p><a id="X7DE8E16C7C2D387B" name="X7DE8E16C7C2D387B"></a></p> <h4>4.4 <span class="Heading">Operations</span></h4> <p><a id="X78C1A22D86943E97" name="X78C1A22D86943E97"></a></p> <h5>4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(c,a)</span></p> <p>The arguments are a a factorobject <span class="Math">\beta: b \twoheadrightarrow c</span>, an <p><a id="X7A50AC43855B33B1" name="X7A50AC43855B33B1"></a></p> <h5>4.4-2 CommonCoastriction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a list of generalized morphisms</p> <p>The argument is a list <span class="Math">L</span> of generalized morphisms by three arrows havi <p><a id="X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10"></a></p> <h4>4.5 <span class="Heading">Constructors</span></h4> <p><a id="X8477D6E679E15474" name="X8477D6E679E15474"></a></p> <h5>4.5-1 GeneralizedMorphismByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(a,d)</span></p> <p>The arguments are morphisms <span class="Math">\alpha: a \leftarrow b</span>, <span class="Mat <p><a id="X7D60D9F97A2F7D5F" name="X7D60D9F97A2F7D5F"></a></p> <h5>4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(a,c)</span></p> <p>The arguments are morphisms <span class="Math">\alpha: a \leftarrow b</span>, and <span class=" <p><a id="X7F9E6B188755170A" name="X7F9E6B188755170A"></a></p> <h5>4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(b,d)</span></p> <p>The arguments are morphisms <span class="Math">\beta: b \rightarrow c</span>, and <span class=" <p><a id="X7E0CF2A87AC08A8B" name="X7E0CF2A87AC08A8B"></a></p> <h5>4.5-4 AsGeneralizedMorphismByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a morphism in <span class="Math">\mathrm{Hom}_{\mathbf{G(A)}}(a,b)</span></p> <p>The argument is a morphism <span class="Math">\alpha: a \rightarrow b</span> in <span class="Mat <p><a id="X8296606484DDDF6C" name="X8296606484DDDF6C"></a></p> <h5>4.5-5 GeneralizedMorphismCategoryByThreeArrows</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: a category</p> <p>The argument is an abelian category <span class="Math">\mathbf{A}</span>. The output is its gen <p><a id="X8165802A8625467E" name="X8165802A8625467E"></a></p> <h5>4.5-6 GeneralizedMorphismByThreeArrowsObject</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Returns: an object in <span class="Math">\mathbf{G(A)}</span></p> <p>The argument is an object <span class="Math">a</span> in an abelian category <span class="Math">\ <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
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