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<div class="ChapSects"><a href="chap34_mj.html#X83AAC8367CC7686F">34 <span class="Heading"> Commutative diagrams and abstract categories</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34_mj.html#X7CFDEEC07F15CF82">34.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7B9157D578F3A25A">34.1-1 HomomorphismChainToCommutativeDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X80840B51839EDEF3">34.1-2 NormalSeriesToQuotientDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7F433F027A1093E6">34.1-3 NerveOfCommutativeDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X85278C6C7D2495A5">34.1-4 GroupHomologyOfCommutativeDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7E687DBD787A68BD">34.1-5 PersistentHomologyOfCommutativeDiagramOfPGroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34_mj.html#X7CFDEEC07F15CF82">34.2 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X806E560D7883B995">34.2-1 CategoricalEnrichment</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X78C3228682279032">34.2-2 IdentityArrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7E5C1D4D785E20B6">34.2-3 InitialArrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7B9F14497D770AB0">34.2-4 TerminalArrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X864688708741BADF">34.2-5 HasInitialObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X78ED3C0778BB65FE">34.2-6 HasTerminalObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7DE8173F80E07AB1">34.2-7 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X7C76423782BA2868">34.2-8 Target</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X81C3CA4183D53AD5">34.2-9 CategoryName</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X80825C317FCA49E3">34.2-10 CompositionEqualityAdditionMinus</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X829A6A767BD96D34">34.2-11 Object</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X87F6B07083307724">34.2-12 Mapping</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X78E3186D83B7B92B">34.2-13 IsCategoryObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap34_mj.html#X792400427CBA758A">34.2-14 IsCategoryArrow</a></span>
</div></div>
</div>
<h3>34 <span class="Heading"> Commutative diagrams and abstract categories</span></h3>
<p><strong class="button">COMMUTATIVE DIAGRAMS</strong> <br /></p>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>34.1 <span class="Heading"> </span></h4>
<p><a id="X7B9157D578F3A25A" name="X7B9157D578F3A25A"></a></p>
<h5>34.1-1 HomomorphismChainToCommutativeDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomomorphismChainToCommutativeDiagram</code>( <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(H=[h_1,h_2,...,h_n]\)</span> of mappings such that the composite <span class="SimpleMath">\(h_1h_2...h_n\)</span> is defined. It returns the list of composable homomorphism as a commutative diagram.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X80840B51839EDEF3" name="X80840B51839EDEF3"></a></p>
<h5>34.1-2 NormalSeriesToQuotientDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NormalSeriesToQuotientDiagram</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NormalSeriesToQuotientDiagram</code>( <var class="Arg">L</var>, <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an increasing (or decreasing) list <span class="SimpleMath">\(L=[L_1,L_2,...,L_n]\)</span> of normal subgroups of a group <span class="SimpleMath">\(G\)</span> with <span class="SimpleMath">\(G=L_n\)</span>. It returns the chain of quotient homomorphisms <span class="SimpleMath">\(G/L_i \rightarrow G/L_{i+1}\)</span> as a commutative diagram.</p>
<p>Optionally a subseries <span class="SimpleMath">\(M\)</span> of <span class="SimpleMath">\(L\)</span> can be entered as a second variable. Then the resulting diagram of quotient groups has two rows of horizontal arrows and one row of vertical arrows.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7F433F027A1093E6" name="X7F433F027A1093E6"></a></p>
<h5>34.1-3 NerveOfCommutativeDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NerveOfCommutativeDiagram</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a commutative diagram <span class="SimpleMath">\(D\)</span> and returns the commutative diagram <span class="SimpleMath">\(ND\)</span> consisting of all possible composites of the arrows in <span class="SimpleMath">\(D\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X85278C6C7D2495A5" name="X85278C6C7D2495A5"></a></p>
<h5>34.1-4 GroupHomologyOfCommutativeDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupHomologyOfCommutativeDiagram</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupHomologyOfCommutativeDiagram</code>( <var class="Arg">D</var>, <var class="Arg">n</var>, <var class="Arg">prime</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupHomologyOfCommutativeDiagram</code>( <var class="Arg">D</var>, <var class="Arg">n</var>, <var class="Arg">prime</var>, <var class="Arg">Resolution_Algorithm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a commutative diagram <span class="SimpleMath">\(D\)</span> of <span class="SimpleMath">\(p\)</span>-groups and positive integer <span class="SimpleMath">\(n\)</span>. It returns the commutative diagram of vector spaces obtained by applying mod p homology.</p>
<p>Non-prime power groups can also be handled if a prime <span class="SimpleMath">\(p\)</span> is entered as the third argument. Integral homology can be obtained by setting <span class="SimpleMath">\(p=0\)</span>. For <span class="SimpleMath">\(p=0\)</span> the result is a diagram of groups.</p>
<p>A particular resolution algorithm, such as ResolutionNilpotentGroup, can be entered as a fourth argument. For positive <span class="SimpleMath">\(p\)</span> the default is ResolutionPrimePowerGroup. For <span class="SimpleMath">\(p=0\)</span> the default is ResolutionFiniteGroup.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7E687DBD787A68BD" name="X7E687DBD787A68BD"></a></p>
<h5>34.1-5 PersistentHomologyOfCommutativeDiagramOfPGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PersistentHomologyOfCommutativeDiagramOfPGroups</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a commutative diagram <span class="SimpleMath">\(D\)</span> of finite <span class="SimpleMath">\(p\)</span>-groups and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a list containing, for each homomorphism in the nerve of <span class="SimpleMath">\(D\)</span>, a triple <span class="SimpleMath">\([k,l,m]\)</span> where <span class="SimpleMath">\(k\)</span> is the dimension of the source of the induced mod <span class="SimpleMath">\(p\)</span> homology map in degree <span class="SimpleMath">\(n\)</span>, <span class="SimpleMath">\(l\)</span> is the dimension of the image, and <span class="SimpleMath">\(m\)</span> is the dimension of the cokernel.</p>
<p><strong class="button">Examples:</strong></p>
<p><strong class="button">ABSTRACT CATEGORIES</strong> <br /> <br /></p>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>34.2 <span class="Heading"> </span></h4>
<p><a id="X806E560D7883B995" name="X806E560D7883B995"></a></p>
<h5>34.2-1 CategoricalEnrichment</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CategoricalEnrichment</code>( <var class="Arg">X</var>, <var class="Arg">Name</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a structure <span class="SimpleMath">\(X\)</span> such as a group or group homomorphism, together with the name of some existing category such as Name:=Category_of_Groups or Category_of_Abelian_Groups. It returns, as appropriate, an object or arrow in the named category.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X78C3228682279032" name="X78C3228682279032"></a></p>
<h5>34.2-2 IdentityArrow</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IdentityArrow</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">\(X\)</span> in some category, and returns the identity arrow on the object <span class="SimpleMath">\(X\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X7E5C1D4D785E20B6" name="X7E5C1D4D785E20B6"></a></p>
<h5>34.2-3 InitialArrow</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; InitialArrow</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">\(X\)</span> in some category, and returns the arrow from the initial object in the category to <span class="SimpleMath">\(X\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X7B9F14497D770AB0" name="X7B9F14497D770AB0"></a></p>
<h5>34.2-4 TerminalArrow</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TerminalArrow</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">\(X\)</span> in some category, and returns the arrow from <span class="SimpleMath">\(X\)</span> to the terminal object in the category.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X864688708741BADF" name="X864688708741BADF"></a></p>
<h5>34.2-5 HasInitialObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HasInitialObject</code>( <var class="Arg">Name</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs the name of a category and returns true or false depending on whether the category has an initial object.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X78ED3C0778BB65FE" name="X78ED3C0778BB65FE"></a></p>
<h5>34.2-6 HasTerminalObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HasTerminalObject</code>( <var class="Arg">Name</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs the name of a category and returns true or false depending on whether the category has a terminal object.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7DE8173F80E07AB1" name="X7DE8173F80E07AB1"></a></p>
<h5>34.2-7 Source</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Source</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an arrow <span class="SimpleMath">\(f\)</span> in some category, and returns its source.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap2.html">1</a></span> , <span class="URL"><a href="../tutorial/chap4.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap8.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutNonabelian.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoefficientSequence.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutFunctorial.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">11</a></span> </p>
<p><a id="X7C76423782BA2868" name="X7C76423782BA2868"></a></p>
<h5>34.2-8 Target</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Target</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an arrow <span class="SimpleMath">\(f\)</span> in some category, and returns its target.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap2.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap8.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoefficientSequence.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">8</a></span> </p>
<p><a id="X81C3CA4183D53AD5" name="X81C3CA4183D53AD5"></a></p>
<h5>34.2-9 CategoryName</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CategoryName</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object or arrow <span class="SimpleMath">\(X\)</span> in some category, and returns the name of the category.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">1</a></span> </p>
<p><a id="X80825C317FCA49E3" name="X80825C317FCA49E3"></a></p>
<h5>34.2-10 CompositionEqualityAdditionMinus</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CompositionEqualityAdditionMinus</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Composition of suitable arrows <span class="SimpleMath">\(f,g\)</span> is given by <span class="SimpleMath">\(f*g\)</span> when the source of <span class="SimpleMath">\(f\)</span> equals the target of <span class="SimpleMath">\(g\)</span>. (Warning: this differes to the standard GAP convention.)</p>
<p>Equality is tested using <span class="SimpleMath">\(f=g\)</span>.</p>
<p>In an additive category the sum and difference of suitable arrows is given by <span class="SimpleMath">\(f+g\)</span> and <span class="SimpleMath">\(f-g\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X829A6A767BD96D34" name="X829A6A767BD96D34"></a></p>
<h5>34.2-11 Object</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Object</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an object <span class="SimpleMath">\(X\)</span> in some category, and returns the GAP structure <span class="SimpleMath">\(Y\)</span> such that <span class="SimpleMath">\(X=CategoricalEnrichment(Y,CategoryName(X))\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap10.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">2</a></span> </p>
<p><a id="X87F6B07083307724" name="X87F6B07083307724"></a></p>
<h5>34.2-12 Mapping</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Mapping</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs an arrow <span class="SimpleMath">\(f\)</span> in some category, and returns the GAP structure <span class="SimpleMath">\(Y\)</span> such that <span class="SimpleMath">\(f=CategoricalEnrichment(Y,CategoryName(X))\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../tutorial/chap13.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAbelianCategories.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoefficientSequence.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">6</a></span> </p>
<p><a id="X78E3186D83B7B92B" name="X78E3186D83B7B92B"></a></p>
<h5>34.2-13 IsCategoryObject</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCategoryObject</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs <span class="SimpleMath">\(X\)</span> and returns true if <span class="SimpleMath">\(X\)</span> is an object in some category.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X792400427CBA758A" name="X792400427CBA758A"></a></p>
<h5>34.2-14 IsCategoryArrow</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsCategoryArrow</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs <span class="SimpleMath">\(X\)</span> and returns true if <span class="SimpleMath">\(X\)</span> is an arrow in some category.</p>
<p><strong class="button">Examples:</strong></p>
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