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<title>GAP (GeneralizedMorphismsForCAP) - Chapter 4: Generalized Morphism Category by Three Arrows</title>
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<p><a id="X7F39133179F83B50" name="X7F39133179F83B50"></a></p>
<div class="ChapSects"><a href="chap4.html#X7F39133179F83B50">4 <span class="Heading">Generalized Morphism Category by Three Arrows</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D03633A7D98026B">4.1 <span class="Heading">GAP Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X869A7C2A85C2743A">4.1-1 IsGeneralizedMorphismCategoryByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E861596801CA565">4.1-2 IsGeneralizedMorphismCategoryByThreeArrowsObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87934BAA84C6E654">4.1-3 IsGeneralizedMorphismByThreeArrows</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X871597447BB998A1">4.2 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8247F41681D15233">4.2-1 HasIdentitiesAsReversedArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8347A40179DBD909">4.2-2 HasIdentityAsSourceAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81F0F4FA78F9F361">4.2-3 HasIdentityAsRangeAid</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C701DBF7BAE649A">4.3 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8028877F819A0DBF">4.3-1 UnderlyingHonestObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B17B7587A449B8E">4.3-2 SourceAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87B0A0AF84F8AF7D">4.3-3 RangeAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CCBCD5385E55586">4.3-4 Arrow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X784C006B8525E205">4.3-5 PseudoInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78A8A55883C06AB1">4.3-6 GeneralizedInverseByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80E329707BAA3B29">4.3-7 IdempotentDefinedBySubobjectByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8217014B870B3EC2">4.3-8 IdempotentDefinedByFactorobjectByThreeArrows</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DE8E16C7C2D387B">4.4 <span class="Heading">Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78C1A22D86943E97">4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A50AC43855B33B1">4.4-2 CommonCoastriction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X86EC0F0A78ECBC10">4.5 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8477D6E679E15474">4.5-1 GeneralizedMorphismByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D60D9F97A2F7D5F">4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F9E6B188755170A">4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E0CF2A87AC08A8B">4.5-4 AsGeneralizedMorphismByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8296606484DDDF6C">4.5-5 GeneralizedMorphismCategoryByThreeArrows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8165802A8625467E">4.5-6 GeneralizedMorphismByThreeArrowsObject</a></span>
</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">‣ IsGeneralizedMorphismCategoryByThreeArrows</code>( <var class="Arg">object</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<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">‣ IsGeneralizedMorphismCategoryByThreeArrowsObject</code>( <var class="Arg">object</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<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">‣ IsGeneralizedMorphismByThreeArrows</code>( <var class="Arg">object</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<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">‣ HasIdentitiesAsReversedArrows</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is <code class="code">true</code> if <span class="Math">a \leftarrow b</span> and <span class="Math">c \leftarrow d</span> are congruent to identity morphisms, <code class="code">false</code> otherwise.</p>

<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">‣ HasIdentityAsSourceAid</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is <code class="code">true</code> if <span class="Math">a \leftarrow b</span> is congruent to an identity morphism, <code class="code">false</code> otherwise.</p>

<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">‣ HasIdentityAsRangeAid</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is <code class="code">true</code> if <span class="Math">c \leftarrow d</span> is congruent to an identity morphism, <code class="code">false</code> otherwise.</p>

<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">‣ UnderlyingHonestObject</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 three arrows. The output is its underlying honest object.</p>

<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">‣ SourceAid</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is its source aid <span class="Math">a \leftarrow b</span>.</p>

<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">‣ RangeAid</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is its range aid <span class="Math">c \leftarrow d</span>.</p>

<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">‣ Arrow</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 class="Math">a \leftarrow b \rightarrow c \leftarrow d</span>. The output is its range aid <span class="Math">b \rightarrow c</span>.</p>

<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">‣ PseudoInverse</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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 three arrows. The output is its pseudo inverse <span class="Math">b \rightarrow a</span>.</p>

<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">‣ GeneralizedInverseByThreeArrows</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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>. The output is its generalized inverse <span class="Math">b \rightarrow a</span> by three arrows.</p>

<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">‣ IdempotentDefinedBySubobjectByThreeArrows</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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}</span>. The output is the idempotent <span class="Math">b \rightarrow b \in \mathbf{G(A)}</span> by three arrows defined by <span class="Math">\alpha</span>.</p>

<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">‣ IdempotentDefinedByFactorobjectByThreeArrows</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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{A}</span>. The output is the idempotent <span class="Math">b \rightarrow b \in \mathbf{G(A)}</span> by three arrows defined by <span class="Math">\alpha</span>.</p>

<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">‣ GeneralizedMorphismFromFactorToSubobjectByThreeArrows</code>( <var class="Arg">beta</var>, <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<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>, and a subobject <span class="Math">\alpha: a \hookrightarrow b</span>. The output is the generalized morphism by three arrows from the factorobject to the subobject.</p>

<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">‣ CommonCoastriction</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<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 having the same range. The output is a list of generalized morphisms by three arrows which is the comman coastriction of <span class="Math">L</span>.</p>

<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">‣ GeneralizedMorphismByThreeArrows</code>( <var class="Arg">alpha</var>, <var class="Arg">beta</var>, <var class="Arg">gamma</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<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="Math">\beta: b \rightarrow c</span>, and <span class="Math">\gamma: c \leftarrow d</span> in <span class="Math">\mathbf{A}</span>. The output is a generalized morphism by three arrows with source aid <span class="Math">\alpha</span>, arrow <span class="Math">\beta</span>, and range aid <span class="Math">\gamma</span>.</p>

<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">‣ GeneralizedMorphismByThreeArrowsWithSourceAid</code>( <var class="Arg">alpha</var>, <var class="Arg">beta</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<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="Math">\beta: b \rightarrow c</span> in <span class="Math">\mathbf{A}</span>. The output is a generalized morphism by three arrows defined by the composition of the given two arrows regarded as generalized morphisms.</p>

<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">‣ GeneralizedMorphismByThreeArrowsWithRangeAid</code>( <var class="Arg">beta</var>, <var class="Arg">gamma</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<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="Math">\gamma: c \leftarrow d</span> in <span class="Math">\mathbf{A}</span>. The output is a generalized morphism by three arrows defined by the composition of the given two arrows regarded as generalized morphisms.</p>

<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">‣ AsGeneralizedMorphismByThreeArrows</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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="Math">\mathbf{A}</span>. The output is the honest generalized morphism by three arrows defined by <span class="Math">\alpha</span>.</p>

<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">‣ GeneralizedMorphismCategoryByThreeArrows</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a category</p>

<p>The argument is an abelian category <span class="Math">\mathbf{A}</span>. The output is its generalized morphism category <span class="Math">\mathbf{G(A)}</span> by three arrows.</p>

<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">‣ GeneralizedMorphismByThreeArrowsObject</code>( <var class="Arg">a</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<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">\mathbf{A}</span>. The output is the object in the generalized morphism category by three arrows whose underlying honest object is <span class="Math">a</span>.</p>


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