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<div class="ChapSects"><a href="chap1_mj.html#X7BD6D8987DF55657">1 <span class="Heading">Generalized Morphism Category</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7D03633A7D98026B">1.1 <span class="Heading">GAP Categories</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X79F2D53F7B79450C">1.1-1 IsGeneralizedMorphismCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BEE0C4D8324D332">1.1-2 IsGeneralizedMorphismCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X805BE326803873AC">1.1-3 IsGeneralizedMorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7C701DBF7BAE649A">1.2 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X78C35DEB7A844B1D">1.2-1 UnderlyingHonestObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E72DC48842AFA58">1.2-2 DomainEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X86B43E60846202CA">1.2-3 GeneralizedImageEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7EF102897D38A0CE">1.2-4 DefectEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X834716A87EADE83E">1.2-5 GeneralizedKernelEmbedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X867950C684A50A6E">1.2-6 CodomainProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X80E5FD8C84054135">1.2-7 GeneralizedCokernelProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E323705780A7472">1.2-8 CodefectProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E0DAE747945216A">1.2-9 GeneralizedCoimageProjection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8056D2E886060EB7">1.2-10 AssociatedMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7CF4DA3478101045">1.2-11 DomainAssociatedMorphismCodomainTriple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8129D15C87A5642B">1.2-12 HonestRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8180D14B85D8DDFD">1.2-13 GeneralizedInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87BA1A897A3019B0">1.2-14 IdempotentDefinedBySubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F3F851D7FD6BE60">1.2-15 IdempotentDefinedByFactorobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8387D5C484A26089">1.2-16 UnderlyingHonestCategory</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7DE8E16C7C2D387B">1.3 <span class="Heading">Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X82FE8F3E84432929">1.3-1 GeneralizedMorphismFromFactorToSubobject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B8E3FBC82429136">1.3-2 CommonRestriction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X83CB98F3783972DE">1.3-3 ConcatenationProduct</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X871597447BB998A1">1.4 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X78B6216F7869E215">1.4-1 IsHonest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X806C879182E335B6">1.4-2 HasFullDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7A5A03577DF86A1F">1.4-3 HasFullCodomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B025572785DACBA">1.4-4 IsSingleValued</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85E05A137F6C759A">1.4-5 IsTotal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7B40ED8B78D067A5">1.5 <span class="Heading">Convenience methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8276ADE97D53B8F0">1.5-1 GeneralizedMorphismCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X846BEA5784436575">1.5-2 GeneralizedMorphismObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F89465C80D5CB2F">1.5-3 AsGeneralizedMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BCAEAEC78CE8093">1.5-4 GeneralizedMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87C5A2AC86D384BC">1.5-5 GeneralizedMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7DC1AF448411DFCE">1.5-6 GeneralizedMorphismWithRangeAid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D32834C7EA50379">1.5-7 GeneralizedMorphismWithSourceAid</a></span>
</div></div>
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<h3>1 <span class="Heading">Generalized Morphism Category</span></h3>

<p>Let <span class="SimpleMath">\(\mathbf{A}\)</span> be an abelian category. We denote its generalized morphism category by <span class="SimpleMath">\(\mathbf{G(A)}\)</span>.</p>

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

<h4>1.1 <span class="Heading">GAP Categories</span></h4>

<p><a id="X79F2D53F7B79450C" name="X79F2D53F7B79450C"></a></p>

<h5>1.1-1 IsGeneralizedMorphismCategory</h5>

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

<p><a id="X7BEE0C4D8324D332" name="X7BEE0C4D8324D332"></a></p>

<h5>1.1-2 IsGeneralizedMorphismCategoryObject</h5>

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

<p><a id="X805BE326803873AC" name="X805BE326803873AC"></a></p>

<h5>1.1-3 IsGeneralizedMorphism</h5>

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

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

<h4>1.2 <span class="Heading">Attributes</span></h4>

<p><a id="X78C35DEB7A844B1D" name="X78C35DEB7A844B1D"></a></p>

<h5>1.2-1 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="SimpleMath">\(\mathbf{A}\)</span></p>

<p>The argument is an object <span class="SimpleMath">\(a\)</span> in the generalized morphism category. The output is its underlying honest object</p>

<p><a id="X7E72DC48842AFA58" name="X7E72DC48842AFA58"></a></p>

<h5>1.2-2 DomainEmbedding</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DomainEmbedding</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( d, a )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its domain <span class="SimpleMath">\(d \hookrightarrow a \in \mathbf{A}\)</span>.</p>

<p><a id="X86B43E60846202CA" name="X86B43E60846202CA"></a></p>

<h5>1.2-3 GeneralizedImageEmbedding</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedImageEmbedding</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( i, b )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its generalized image <span class="SimpleMath">\(i \hookrightarrow b \in \mathbf{A}\)</span>.</p>

<p><a id="X7EF102897D38A0CE" name="X7EF102897D38A0CE"></a></p>

<h5>1.2-4 DefectEmbedding</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefectEmbedding</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( d, b )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its defect <span class="SimpleMath">\(d \hookrightarrow b \in \mathbf{A}\)</span>.</p>

<p><a id="X834716A87EADE83E" name="X834716A87EADE83E"></a></p>

<h5>1.2-5 GeneralizedKernelEmbedding</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedKernelEmbedding</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( k, a )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its generalized kernel <span class="SimpleMath">\(k \hookrightarrow a \in \mathbf{A}\)</span>.</p>

<p><a id="X867950C684A50A6E" name="X867950C684A50A6E"></a></p>

<h5>1.2-6 CodomainProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CodomainProjection</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( b, c )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its codomain <span class="SimpleMath">\(b \twoheadrightarrow c \in \mathbf{A}\)</span>.</p>

<p><a id="X80E5FD8C84054135" name="X80E5FD8C84054135"></a></p>

<h5>1.2-7 GeneralizedCokernelProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedCokernelProjection</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( b, c )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its generalized cokernel <span class="SimpleMath">\(b \twoheadrightarrow c \in \mathbf{A}\)</span>.</p>

<p><a id="X7E323705780A7472" name="X7E323705780A7472"></a></p>

<h5>1.2-8 CodefectProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CodefectProjection</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( a, c )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its codefect <span class="SimpleMath">\(a \twoheadrightarrow c \in \mathbf{A}\)</span>.</p>

<p><a id="X7E0DAE747945216A" name="X7E0DAE747945216A"></a></p>

<h5>1.2-9 GeneralizedCoimageProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedCoimageProjection</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( a, c )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its generalized coimage <span class="SimpleMath">\(a \twoheadrightarrow c \in \mathbf{A}\)</span>.</p>

<p><a id="X8056D2E886060EB7" name="X8056D2E886060EB7"></a></p>

<h5>1.2-10 AssociatedMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedMorphism</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( d, c )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is its associated morphism <span class="SimpleMath">\(d \rightarrow c \in \mathbf{A}\)</span>.</p>

<p><a id="X7CF4DA3478101045" name="X7CF4DA3478101045"></a></p>

<h5>1.2-11 DomainAssociatedMorphismCodomainTriple</h5>

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

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is a triple <span class="SimpleMath">\(( d \hookrightarrow a, d \rightarrow c, b \twoheadrightarrow c )\)</span> consisting of its domain, associated morphism, and codomain.</p>

<p><a id="X8129D15C87A5642B" name="X8129D15C87A5642B"></a></p>

<h5>1.2-12 HonestRepresentative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HonestRepresentative</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{A}}( a, b )\)</span></p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha: a \rightarrow b\)</span>. The output is the honest representative in <span class="SimpleMath">\(\mathbf{A}\)</span> of <span class="SimpleMath">\(\alpha\)</span>, if it exists, otherwise an error is thrown.</p>

<p><a id="X8180D14B85D8DDFD" name="X8180D14B85D8DDFD"></a></p>

<h5>1.2-13 GeneralizedInverse</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedInverse</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{G(A)}}(b,a)\)</span></p>

<p>The argument is a morphism <span class="SimpleMath">\(\alpha: a \rightarrow b \in \mathbf{A}\)</span>. The output is its generalized inverse <span class="SimpleMath">\(b \rightarrow a\)</span>.</p>

<p><a id="X87BA1A897A3019B0" name="X87BA1A897A3019B0"></a></p>

<h5>1.2-14 IdempotentDefinedBySubobject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentDefinedBySubobject</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)</span></p>

<p>The argument is a subobject <span class="SimpleMath">\(\alpha: a \hookrightarrow b \in \mathbf{A}\)</span>. The output is the idempotent <span class="SimpleMath">\(b \rightarrow b \in \mathbf{G(A)}\)</span> defined by <span class="SimpleMath">\(\alpha\)</span>.</p>

<p><a id="X7F3F851D7FD6BE60" name="X7F3F851D7FD6BE60"></a></p>

<h5>1.2-15 IdempotentDefinedByFactorobject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdempotentDefinedByFactorobject</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a morphism in <span class="SimpleMath">\(\mathrm{Hom}_{\mathbf{G(A)}}(b,b)\)</span></p>

<p>The argument is a factorobject <span class="SimpleMath">\(\alpha: b \twoheadrightarrow a \in \mathbf{A}\)</span>. The output is the idempotent <span class="SimpleMath">\(b \rightarrow b \in \mathbf{G(A)}\)</span> defined by <span class="SimpleMath">\(\alpha\)</span>.</p>

<p><a id="X8387D5C484A26089" name="X8387D5C484A26089"></a></p>

<h5>1.2-16 UnderlyingHonestCategory</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingHonestCategory</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a category</p>

<p>The argument is a generalized morphism category <span class="SimpleMath">\(C = \mathbf{G(A)}\)</span>. The output is <span class="SimpleMath">\(\mathbf{A}\)</span>.</p>

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

<h4>1.3 <span class="Heading">Operations</span></h4>

<p><a id="X82FE8F3E84432929" name="X82FE8F3E84432929"></a></p>

<h5>1.3-1 GeneralizedMorphismFromFactorToSubobject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismFromFactorToSubobject</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="SimpleMath">\(\mathrm{Hom}_{\mathbf{G(A)}}(c,a)\)</span></p>

<p>The arguments are a a factorobject <span class="SimpleMath">\(\beta: b \twoheadrightarrow c\)</span>, and a subobject <span class="SimpleMath">\(\alpha: a \hookrightarrow b\)</span>. The output is the generalized morphism from the factorobject to the subobject.</p>

<p><a id="X7B8E3FBC82429136" name="X7B8E3FBC82429136"></a></p>

<h5>1.3-2 CommonRestriction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommonRestriction</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="SimpleMath">\(L\)</span> of generalized morphisms by three arrows having the same source. The output is a list of generalized morphisms by three arrows which is the comman restriction of <span class="SimpleMath">\(L\)</span>.</p>

<p><a id="X83CB98F3783972DE" name="X83CB98F3783972DE"></a></p>

<h5>1.3-3 ConcatenationProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConcatenationProduct</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalized moprhism</p>

<p>The argument is a list <span class="SimpleMath">\(L = ( \alpha_1, \dots, \alpha_n )\)</span> of generalized morphisms (with same data structures). The output is their concatenation product, i.e., a generalized morphism <span class="SimpleMath">\(\alpha\)</span> with <span class="SimpleMath">\(\mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Source}( \alpha_i ) )\)</span>, and <span class="SimpleMath">\(\mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha ) ) = \bigoplus_{i=1}^n \mathrm{UnderlyingHonestObject}( \mathrm{Range}( \alpha_i ) )\)</span>, and with morphisms in the representation of <span class="SimpleMath">\(\alpha\)</span> given as the direct sums of the corresponding morphisms of the <span class="SimpleMath">\(\alpha_i\)</span>.</p>

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

<h4>1.4 <span class="Heading">Properties</span></h4>

<p><a id="X78B6216F7869E215" name="X78B6216F7869E215"></a></p>

<h5>1.4-1 IsHonest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHonest</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha\)</span>. The output is <code class="code">true</code> if <span class="SimpleMath">\(\alpha\)</span> admits an honest representative, otherwise <code class="code">false</code>.</p>

<p><a id="X806C879182E335B6" name="X806C879182E335B6"></a></p>

<h5>1.4-2 HasFullDomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasFullDomain</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha\)</span>. The output is <code class="code">true</code> if the domain of <span class="SimpleMath">\(\alpha\)</span> is an isomorphism, otherwise <code class="code">false</code>.</p>

<p><a id="X7A5A03577DF86A1F" name="X7A5A03577DF86A1F"></a></p>

<h5>1.4-3 HasFullCodomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasFullCodomain</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha\)</span>. The output is <code class="code">true</code> if the codomain of <span class="SimpleMath">\(\alpha\)</span> is an isomorphism, otherwise <code class="code">false</code>.</p>

<p><a id="X7B025572785DACBA" name="X7B025572785DACBA"></a></p>

<h5>1.4-4 IsSingleValued</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSingleValued</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha\)</span>. The output is <code class="code">true</code> if the codomain of <span class="SimpleMath">\(\alpha\)</span> is an isomorphism, otherwise <code class="code">false</code>.</p>

<p><a id="X85E05A137F6C759A" name="X85E05A137F6C759A"></a></p>

<h5>1.4-5 IsTotal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTotal</code>( <var class="Arg">alpha</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>The argument is a generalized morphism <span class="SimpleMath">\(\alpha\)</span>. The output is <code class="code">true</code> if the domain of <span class="SimpleMath">\(\alpha\)</span> is an isomorphism, otherwise <code class="code">false</code>.</p>

<p><a id="X7B40ED8B78D067A5" name="X7B40ED8B78D067A5"></a></p>

<h4>1.5 <span class="Heading">Convenience methods</span></h4>

<p>This section contains operations which, depending on the current generalized morphism standard of the system and the category, might point to other Operations. Please use them only as convenience and never in serious code.</p>

<p><a id="X8276ADE97D53B8F0" name="X8276ADE97D53B8F0"></a></p>

<h5>1.5-1 GeneralizedMorphismCategory</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismCategory</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a category</p>

<p>Creates a new category of generalized morphisms. Might point to GeneralizedMorphismCategoryByThreeArrows, GeneralizedMorphismCategoryByCospans, or GeneralizedMorphismCategoryBySpans</p>

<p><a id="X846BEA5784436575" name="X846BEA5784436575"></a></p>

<h5>1.5-2 GeneralizedMorphismObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismObject</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an object in the generalized morphism category</p>

<p>Creates an object in the current generalized morphism category, depending on the standard</p>

<p><a id="X7F89465C80D5CB2F" name="X7F89465C80D5CB2F"></a></p>

<h5>1.5-3 AsGeneralizedMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsGeneralizedMorphism</code>( <var class="Arg">phi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalized morphism</p>

<p>Returns the corresponding morphism to phi in the current generalized morphism category.</p>

<p><a id="X7BCAEAEC78CE8093" name="X7BCAEAEC78CE8093"></a></p>

<h5>1.5-4 GeneralizedMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphism</code>( <var class="Arg">phi</var>, <var class="Arg">psi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalized morphism</p>

<p>Returns the corresponding morphism to phi and psi in the current generalized morphism category.</p>

<p><a id="X87C5A2AC86D384BC" name="X87C5A2AC86D384BC"></a></p>

<h5>1.5-5 GeneralizedMorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphism</code>( <var class="Arg">iota</var>, <var class="Arg">phi</var>, <var class="Arg">pi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalized morphism</p>

<p>Returns the corresponding morphism to iota, phi and psi in the current generalized morphism category.</p>

<p><a id="X7DC1AF448411DFCE" name="X7DC1AF448411DFCE"></a></p>

<h5>1.5-6 GeneralizedMorphismWithRangeAid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismWithRangeAid</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns a generalized morphism with range aid by three arrows or by span, or a generalized morphism by cospan, depending on the standard.</p>

<p><a id="X7D32834C7EA50379" name="X7D32834C7EA50379"></a></p>

<h5>1.5-7 GeneralizedMorphismWithSourceAid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedMorphismWithSourceAid</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns a generalized morphism with source aid by three arrows or by cospan, or a generalized morphism by span, depending on the standard.</p>


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