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<title>GAP (GeneralizedMorphismsForCAP) - Chapter 2: Generalized Morphism Category by Cospans</title>
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<p><a id="X87EB52217E1FF49C" name="X87EB52217E1FF49C"></a></p>
<div class="ChapSects"><a href="chap2.html#X87EB52217E1FF49C">2 <span class="Heading">Generalized Morphism Category by Cospans</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D03633A7D98026B">2.1 <span class="Heading">GAP Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8187BB8479575083">2.1-1 IsGeneralizedMorphismCategoryByCospans</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86C1877D7EC9F075">2.1-2 IsGeneralizedMorphismCategoryByCospansObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82D8F9C38330F81E">2.1-3 IsGeneralizedMorphismByCospan</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X871597447BB998A1">2.2 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X815D6FE986CC3F99">2.2-1 HasIdentityAsReversedArrow</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7C701DBF7BAE649A">2.3 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D7E0AB87D8C6BE4">2.3-1 UnderlyingHonestObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7863825F85197D30">2.3-2 Arrow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X871D3BBC7E6B544B">2.3-3 ReversedArrow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7DCA5A8278F4C569">2.3-4 NormalizedCospanTuple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7F4AC0FE847140A4">2.3-5 PseudoInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87E4921884F9F1FF">2.3-6 GeneralizedInverseByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CFE9A12793AF805">2.3-7 IdempotentDefinedBySubobjectByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8330AB9480BC933A">2.3-8 IdempotentDefinedByFactorobjectByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B636E0D7A9FA235">2.3-9 NormalizedCospan</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7DE8E16C7C2D387B">2.4 <span class="Heading">Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7988FD538374DC5C">2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86EC0F0A78ECBC10">2.5 <span class="Heading">Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8313697F833CA064">2.5-1 GeneralizedMorphismByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X865028F47CB5A9FA">2.5-2 GeneralizedMorphismByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B744FE28475F25B">2.5-3 GeneralizedMorphismByCospanWithSourceAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79F31F9E80A5BFAD">2.5-4 AsGeneralizedMorphismByCospan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X879C3A4F82EC9C13">2.5-5 GeneralizedMorphismCategoryByCospans</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X81ADB9C17D822C05">2.5-6 GeneralizedMorphismByCospansObject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X83F3C1E6877018D8">2.6 <span class="Heading">Constructors of lifts of exact functors and natrual (iso)morphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A094900868E02F1">2.6-1 AsGeneralizedMorphismByCospan</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Generalized Morphism Category by Cospans</span></h3>

<p><a id="X7D03633A7D98026B" name="X7D03633A7D98026B"></a></p>

<h4>2.1 <span class="Heading">GAP Categories</span></h4>

<p><a id="X8187BB8479575083" name="X8187BB8479575083"></a></p>

<h5>2.1-1 IsGeneralizedMorphismCategoryByCospans</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedMorphismCategoryByCospans</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 cospans.</p>

<p><a id="X86C1877D7EC9F075" name="X86C1877D7EC9F075"></a></p>

<h5>2.1-2 IsGeneralizedMorphismCategoryByCospansObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedMorphismCategoryByCospansObject</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 cospans.</p>

<p><a id="X82D8F9C38330F81E" name="X82D8F9C38330F81E"></a></p>

<h5>2.1-3 IsGeneralizedMorphismByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralizedMorphismByCospan</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 cospans.</p>

<p><a id="X871597447BB998A1" name="X871597447BB998A1"></a></p>

<h4>2.2 <span class="Heading">Properties</span></h4>

<p><a id="X815D6FE986CC3F99" name="X815D6FE986CC3F99"></a></p>

<h5>2.2-1 HasIdentityAsReversedArrow</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasIdentityAsReversedArrow</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 a cospan <span class="Math">a \rightarrow b \leftarrow c</span>. The output is <code class="code">true</code> if <span class="Math">b \leftarrow c</span> is congruent to an identity morphism, <code class="code">false</code> otherwise.</p>

<p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p>

<h4>2.3 <span class="Heading">Attributes</span></h4>

<p><a id="X7D7E0AB87D8C6BE4" name="X7D7E0AB87D8C6BE4"></a></p>

<h5>2.3-1 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 cospans. The output is its underlying honest object.</p>

<p><a id="X7863825F85197D30" name="X7863825F85197D30"></a></p>

<h5>2.3-2 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}}(a,c)</span></p>

<p>The argument is a generalized morphism <span class="Math">\alpha</span> by a cospan <span class="Math">a \rightarrow b \leftarrow c</span>. The output is its arrow <span class="Math">a \rightarrow b</span>.</p>

<p><a id="X871D3BBC7E6B544B" name="X871D3BBC7E6B544B"></a></p>

<h5>2.3-3 ReversedArrow</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReversedArrow</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}}(c,b)</span></p>

<p>The argument is a generalized morphism <span class="Math">\alpha</span> by a cospan <span class="Math">a \rightarrow b \leftarrow c</span>. The output is its reversed arrow <span class="Math">b \leftarrow c</span>.</p>

<p><a id="X7DCA5A8278F4C569" name="X7DCA5A8278F4C569"></a></p>

<h5>2.3-4 NormalizedCospanTuple</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizedCospanTuple</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a pair of morphisms in <span class="Math">\mathbf{A}</span>.</p>

<p>The argument is a generalized morphism <span class="Math">\alpha: a \rightarrow b</span> by a cospan. The output is its normalized cospan pair <span class="Math">(a \rightarrow d, d \leftarrow b)</span>.</p>

<p><a id="X7F4AC0FE847140A4" name="X7F4AC0FE847140A4"></a></p>

<h5>2.3-5 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 a cospan. The output is its pseudo inverse <span class="Math">b \rightarrow a</span>.</p>

<p><a id="X87E4921884F9F1FF" name="X87E4921884F9F1FF"></a></p>

<h5>2.3-6 GeneralizedInverseByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedInverseByCospan</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 cospan.</p>

<p><a id="X7CFE9A12793AF805" name="X7CFE9A12793AF805"></a></p>

<h5>2.3-7 IdempotentDefinedBySubobjectByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentDefinedBySubobjectByCospan</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 cospan defined by <span class="Math">\alpha</span>.</p>

<p><a id="X8330AB9480BC933A" name="X8330AB9480BC933A"></a></p>

<h5>2.3-8 IdempotentDefinedByFactorobjectByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentDefinedByFactorobjectByCospan</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 cospan defined by <span class="Math">\alpha</span>.</p>

<p><a id="X7B636E0D7A9FA235" name="X7B636E0D7A9FA235"></a></p>

<h5>2.3-9 NormalizedCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizedCospan</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 generalized morphism <span class="Math">\alpha: a \rightarrow b</span> by a cospan. The output is its normalization by cospan.</p>

<p><a id="X7DE8E16C7C2D387B" name="X7DE8E16C7C2D387B"></a></p>

<h4>2.4 <span class="Heading">Operations</span></h4>

<p><a id="X7988FD538374DC5C" name="X7988FD538374DC5C"></a></p>

<h5>2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismFromFactorToSubobjectByCospan</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 cospan from the factorobject to the subobject.</p>

<p><a id="X86EC0F0A78ECBC10" name="X86EC0F0A78ECBC10"></a></p>

<h4>2.5 <span class="Heading">Constructors</span></h4>

<p><a id="X8313697F833CA064" name="X8313697F833CA064"></a></p>

<h5>2.5-1 GeneralizedMorphismByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismByCospan</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 \rightarrow b</span> and <span class="Math">\beta: c \rightarrow b</span> in <span class="Math">\mathbf{A}</span>. The output is a generalized morphism by cospan with arrow <span class="Math">\alpha</span> and reversed arrow <span class="Math">\beta</span>.</p>

<p><a id="X865028F47CB5A9FA" name="X865028F47CB5A9FA"></a></p>

<h5>2.5-2 GeneralizedMorphismByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismByCospan</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 cospan defined by the composition of the given three arrows regarded as generalized morphisms.</p>

<p><a id="X7B744FE28475F25B" name="X7B744FE28475F25B"></a></p>

<h5>2.5-3 GeneralizedMorphismByCospanWithSourceAid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismByCospanWithSourceAid</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 cospan defined by the composition of the given two arrows regarded as generalized morphisms.</p>

<p><a id="X79F31F9E80A5BFAD" name="X79F31F9E80A5BFAD"></a></p>

<h5>2.5-4 AsGeneralizedMorphismByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGeneralizedMorphismByCospan</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 cospan defined by <span class="Math">\alpha</span>.</p>

<p><a id="X879C3A4F82EC9C13" name="X879C3A4F82EC9C13"></a></p>

<h5>2.5-5 GeneralizedMorphismCategoryByCospans</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismCategoryByCospans</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 cospans.</p>

<p><a id="X81ADB9C17D822C05" name="X81ADB9C17D822C05"></a></p>

<h5>2.5-6 GeneralizedMorphismByCospansObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismByCospansObject</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 cospans whose underlying honest object is <span class="Math">a</span>.</p>

<p><a id="X83F3C1E6877018D8" name="X83F3C1E6877018D8"></a></p>

<h4>2.6 <span class="Heading">Constructors of lifts of exact functors and natrual (iso)morphisms</span></h4>

<p><a id="X7A094900868E02F1" name="X7A094900868E02F1"></a></p>

<h5>2.6-1 AsGeneralizedMorphismByCospan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGeneralizedMorphismByCospan</code>( <var class="Arg">F</var>, <var class="Arg">name</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Lift the <em>exact</em> functor <var class="Arg">F</var> to a functor <span class="SimpleMath">A -> B</span>, where <span class="SimpleMath">A :=</span> <code class="code">GeneralizedMorphismCategoryByCospans( SourceOfFunctor( </code><var class="Arg">F</var><code class="code"> ) )</code> and <span class="SimpleMath">B :=</span> <code class="code">GeneralizedMorphismCategoryByCospans( RangeOfFunctor( </code><var class="Arg">F</var><code class="code"> ) )</code>.</p>


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