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<p><a id="X7DBA3A7E81C71A64" name="X7DBA3A7E81C71A64"></a></p>
<div class="ChapSects"><a href="chap9.html#X7DBA3A7E81C71A64">9 <span class="Heading">Cat<span class="SimpleMath">^3</span>-groups and Crossed cubes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7CC52AF4840F478E">9.1 <span class="Heading">Functions for (pre-)cat<span class="SimpleMath">^3</span>-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7828496F7D72E232">9.1-1 Cat3Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X85A5A6967D942463">9.1-2 Front3DimensionalGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X80E074E37D02B2F6">9.2 <span class="Heading">Enumerating cat<span class="SimpleMath">^3</span>-groups with a given source</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X83E4A7367DD13598">9.2-1 AllCat3GroupTriples</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7F8538B580847268">9.3 <span class="Heading">
Definition and constructions for cat<span class="SimpleMath">^n</span>-groups and their morphisms
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X81C3D39A81B20D76">9.3-1 PreCatnGroup</a></span>
</div></div>
</div>
<h3>9 <span class="Heading">Cat<span class="SimpleMath">^3</span>-groups and Crossed cubes</span></h3>
<p>The term <em>4d-group</em> refers to a set of equivalent categories of which the ones we are most interested are the categories of <em>crossed cubes</em> and <em>cat<span class="SimpleMath">^3</span>-groups</em>. A <em>4d-mapping</em> is a function between two 4d-groups which preserves all the structure.</p>
<p>The material in this chapter should be considered very experimental. As yet there are no functions for crossed cubes.</p>
<p><a id="X7CC52AF4840F478E" name="X7CC52AF4840F478E"></a></p>
<h4>9.1 <span class="Heading">Functions for (pre-)cat<span class="SimpleMath">^3</span>-groups</span></h4>
<p>We shall use the following standard orientation of a cat<span class="SimpleMath">^3</span>-group <span class="SimpleMath">calE</span> on a group <span class="SimpleMath">G</span>. <span class="SimpleMath">calE</span> contains <span class="SimpleMath">8</span> groups; <span class="SimpleMath">12</span> cat<span class="SimpleMath">^1</span>-groups and <span class="SimpleMath">6</span> cat<span class="SimpleMath">^2</span>-groups forming the vertices; edges and faces of a cube, as shown in the following diagram.</p>
<p class="pcenter">
\vcenter{\xymatrix{
& H \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex>
\ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex>
\ar[dr] <+0.6ex>
& & & & N \ar[llll] <+0.6ex>
\ar[dddd] <+0.6ex> \ar[dddd] <+0.0ex>
\ar[dr] <+0.6ex>
& \\
& & G \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex>
\ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex>
\ar[ul] <+0.5ex> \ar[ul] <+0.0ex>
& & & & R \ar[llll] <+0.6ex>
\ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex>
\ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\
& & & & & & \\
& & & & & & \\
& M \ar[uuuu] <+0.6ex>
\ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex>
\ar[dr] <+0.6ex>
& & & & L \ar[uuuu] <+0.6ex>
\ar[llll] <+0.6ex>
\ar[dr] <+0.6ex>
& \\
& & Q \ar[uuuu] <+0.6ex>
\ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex>
\ar[ul] <+0.5ex> \ar[ul] <+0.0ex>
& & & & P \ar[uuuu] <+0.6ex>
\ar[llll] <+0.6ex>
\ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\
}}
</p>
<p>By definition, <span class="SimpleMath">calE</span> is generated by three commuting cat<span class="SimpleMath">^1</span>-groups <span class="SimpleMath">(G ⇒ R), (G ⇒ Q)</span> and <span class="SimpleMath">(G ⇒ H)</span>, but it is more convenient to think of <span class="SimpleMath">calE</span> as generated by two cat<span class="SimpleMath">^2</span>-groups</p>
<ul>
<li><p><em>front</em><span class="SimpleMath">(calE)</span>, generated by <span class="SimpleMath">(G ⇒ R)</span> and <span class="SimpleMath">(G ⇒ Q)</span>;</p>
</li>
<li><p><em>left</em><span class="SimpleMath">(calE)</span>, generated by <span class="SimpleMath">(G ⇒ Q)</span> and <span class="SimpleMath">(G ⇒ H)</span>.</p>
</li>
</ul>
<p>Because the tail, head and embedding maps all commute, it follows that <em>up</em><span class="SimpleMath">(calE)</span>, generated by <span class="SimpleMath">(G ⇒ H)</span> and <span class="SimpleMath">(G ⇒ R)</span>, is a third cat<span class="SimpleMath">^2</span>-group. The three remaining faces (cat<span class="SimpleMath">^2</span>-groups) <em>right</em><span class="SimpleMath">(calE)</span>, <em>down</em><span class="SimpleMath">(calE)</span> and <em>back</em><span class="SimpleMath">(calE)</span> are then easily constructed. We shall always use the order [<em>front, left, up, right, down, back</em>] for the six faces.</p>
<p><a id="X7828496F7D72E232" name="X7828496F7D72E232"></a></p>
<h5>9.1-1 Cat3Group</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Cat3Group</code>( <var class="Arg">args</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; PreCat3Group</code>( <var class="Arg">args</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; IsCat3Group</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PreCat3GroupByPreCat2Groups</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The global functions <code class="code">Cat3Group</code> and <code class="code">PreCat3Group</code> normally take as arguments a pair of cat<span class="SimpleMath">^2</span>-groups [<em>front, left</em>], or a trio of cat<span class="SimpleMath">^1</span>-groups [<em>front-up, front-left = left-up, left-left</em>].</p>
<p>In subsection <code class="func">AllCat2GroupsIterator</code> (<a href="chap8.html#X7EFCF9697E845B2C"><span class="RefLink">8.6-4</span></a>) the list of pairs <code class="code">CatnGroupLists(d12).pairs</code> contains the three entries <code class="code">[6,8],[8,11]</code> and <code class="code">[6,11]</code>. It follows that the sixth, eighth and eleventh cat<span class="SimpleMath">^1</span>-groups for <code class="code">d12</code> generate a cat<span class="SimpleMath">^3</span>-group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">alld12 := AllCat1Groups( d12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C68 := Cat2Group( alld12[6], alld12[8] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C811 := Cat2Group( alld12[8], alld12[11] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga := Cat3Group( C68, C811 );</span>
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Gb := Cat3Group( alld12[6], alld12[8], alld12[11] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga = C3Gb;</span>
true
</pre></div>
<p><a id="X85A5A6967D942463" name="X85A5A6967D942463"></a></p>
<h5>9.1-2 Front3DimensionalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Front3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Left3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Up3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Right3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Down3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Back3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The six faces of a cat<span class="SimpleMath">^3</span>-group are stored as these attributes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C116 := Cat2Group( alld12[11], alld12[6] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Up3DimensionalGroup( C3Ga ) = C116;</span>
true
</pre></div>
<p><a id="X80E074E37D02B2F6" name="X80E074E37D02B2F6"></a></p>
<h4>9.2 <span class="Heading">Enumerating cat<span class="SimpleMath">^3</span>-groups with a given source</span></h4>
<p>Once the list <code class="code">CatnGroupLists(G).pairs</code> has been obtained we may seek all triples <span class="SimpleMath">[i,j],[j,k]</span> and <span class="SimpleMath">[k,i]</span> or <span class="SimpleMath">[i,k]</span> of pairs in this list and then, for each such triple, construct a cat<span class="SimpleMath">^3</span>-group generated by the <span class="SimpleMath">i</span>-th, <span class="SimpleMath">j</span>-th and <span class="SimpleMath">k</span>-th cat<span class="SimpleMath">^1</span>-group on <span class="SimpleMath">G</span>.</p>
<p><a id="X83E4A7367DD13598" name="X83E4A7367DD13598"></a></p>
<h5>9.2-1 AllCat3GroupTriples</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllCat3GroupTriples</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllCat3GroupsNumber</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllCat3Groups</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The list of triples returned by the operation <code class="code">AllCat3GroupTriples</code> is saved as <code class="code">CatnGroupLists(G).cat3triples</code>. The length of this list is the number of cat<span class="SimpleMath">^3</span>-groups on <span class="SimpleMath">G</span>, and is saved as <code class="code">CatnGroupNumbers(G).cat3</code>.</p>
<p>As yet there is no operation <code class="code">AllCat3GroupsUpToIsomorphism(G)</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">triples := AllCat3GroupTriples( d12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CatnGroupNumbers( d12 ).cat3; </span>
94
<span class="GAPprompt">gap></span> <span class="GAPinput">triples[46];</span>
[ 5, 7, 11 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">alld12 := AllCat1Groups( d12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cat3Group( alld12[5], alld12[7], alld12[11] );</span>
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
</pre></div>
<p><a id="X7F8538B580847268" name="X7F8538B580847268"></a></p>
<h4>9.3 <span class="Heading">
Definition and constructions for cat<span class="SimpleMath">^n</span>-groups and their morphisms
</span></h4>
<p>In this chapter and the previous one we are interested in cat<span class="SimpleMath">^2</span>-groups and cat<span class="SimpleMath">^3</span>-groups, and it is convenient in this section to give the more general definition. There are three equivalent descriptions of a cat<span class="SimpleMath">^n</span>-group.</p>
<p>A <em>cat<span class="SimpleMath">^n</span>-group</em> consists of the following.</p>
<ul>
<li><p><span class="SimpleMath">2^n</span> groups <span class="SimpleMath">G_A</span>, one for each subset <span class="SimpleMath">A</span> of <span class="SimpleMath">[n]</span>, the <em>vertices</em> of an <span class="SimpleMath">n</span>-cube.</p>
</li>
<li><p>Group homomorphisms forming <span class="SimpleMath">n2^n-1</span> commuting cat<span class="SimpleMath">^1</span>-groups,</p>
<p class="pcenter">
\calC_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\
G_A \to G_{A \setminus \{i\}}),
\quad\mbox{for all} \quad A \subseteq [n],~ i \in A,
</p>
<p>the <em>edges</em> of the cube.</p>
</li>
<li><p>These cat<span class="SimpleMath">^1</span>-groups combine (in sets of <span class="SimpleMath">4</span>) to form <span class="SimpleMath">n(n-1)2^n-3</span> cat<span class="SimpleMath">^2</span>-groups <span class="SimpleMath">calC_A,{i,j}</span> for all <span class="SimpleMath">{i,j} ⊆ A ⊆ [n],~ i ≠ j</span>, the <em>faces</em> of the cube.</p>
</li>
</ul>
<p>Note that, since the <span class="SimpleMath">t_A,i, h_A,i</span> and <span class="SimpleMath">e_A,i</span> commute, composite homomorphisms <span class="SimpleMath">t_A,B, h_A,B : G_A -> G_A ∖ B</span> and <span class="SimpleMath">e_A,B : G_A ∖ B -> G_A</span> are well defined for all <span class="SimpleMath">B ⊆ A ⊆ [n]</span>.</p>
<p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <a href="chapBib.html#biBell:st">[ES87]</a>.</p>
<p>A cat<span class="SimpleMath">^n</span>-group <span class="SimpleMath">calC</span> consists of <span class="SimpleMath">2^n</span> groups <span class="SimpleMath">G_A</span>, one for each subset <span class="SimpleMath">A</span> of <span class="SimpleMath">[n]</span>, and <span class="SimpleMath">3n</span> homomorphisms</p>
<p class="pcenter">
t_{[n],i}, h_{[n],i} : G_{[n]} \to G_{[n] \setminus \{i\}},~
e_{[n],i} : G_{[n] \setminus \{i\}} \to G_{[n]},
</p>
<p>satisfying the following axioms for all <span class="SimpleMath">1 leqslant i leqslant n</span>,}</p>
<ul>
<li><p>the <span class="SimpleMath">calC_[n],i ~=~ (e_[n],i; t_[n],i, h_[n],i : G_[n] -> G_[n] ∖ {i})~</span> are <em>commuting</em> cat<span class="SimpleMath">^1</span>-groups, so that:</p>
</li>
<li><p><span class="SimpleMath">(e_1 ∘ t_1) ∘ (e_2 ∘ t_2) = (e_2 ∘ t_2) ∘ (e_1 ∘ t_1), quad (e_1 ∘ h_1) ∘ (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ h_1),</span></p>
</li>
<li><p><span class="SimpleMath">(e_1 ∘ t_1) ∘ (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ t_1), quad (e_2 ∘ t_2) ∘ (e_1 ∘ h_1) = (e_1 ∘ h_1) ∘ (e_2 ∘ t_2).</span></p>
</li>
</ul>
<p>Our third description defines a cat<span class="SimpleMath">^n</span>-group as a "cat<span class="SimpleMath">^1</span>-group of cat<span class="SimpleMath">^(n-1)</span>-groups".</p>
<p>A <em>cat<span class="SimpleMath">^n</span>-group</em> <span class="SimpleMath">calC</span> consists of two cat<span class="SimpleMath">^(n-1)</span>-groups:</p>
<ul>
<li><p><span class="SimpleMath">calA</span> with groups <span class="SimpleMath">G_A, A ⊆ [n-1]</span>, and homomorphisms <span class="SimpleMath">ddott_A,i, ddoth_A,i, ddote_A,i</span>,</p>
</li>
<li><p><span class="SimpleMath">calB</span> with groups <span class="SimpleMath">H_B, B ⊆ [n-1]</span>, and homomorphisms <span class="SimpleMath">dott_B,i, doth_B,i, dote_B,i</span>, and</p>
</li>
<li><p>cat<span class="SimpleMath">^(n-1)</span>-morphisms <span class="SimpleMath">t,h : calA -> calB</span> and <span class="SimpleMath">e : calB -> calA</span> subject to the following conditions:</p>
<p class="pcenter">
(t \circ e) ~\mbox{and}~ (h \circ e)
~\mbox{are the identity mapping on}~ \calB, \qquad
[\ker t, \ker h] = \{ 1_{\calA} \}.
</p>
</li>
</ul>
<p><a id="X81C3D39A81B20D76" name="X81C3D39A81B20D76"></a></p>
<h5>9.3-1 PreCatnGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PreCatnGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CatnGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The operation <code class="code">(Pre)CatnGroup</code> expects as input a list of cat<span class="SimpleMath">^1</span>-groups. For our group <code class="code">d12</code> we may construct various cat<span class="SimpleMath">^4</span>-groups, and here is one of them.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PC4 := PreCatnGroup( [ alld12[5], alld12[7], alld12[11], alld12[12] ] );</span>
(pre-)cat4-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
4 : [d12 => Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCatnGroup( PC4 ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC4 );</span>
5
</pre></div>
<p>For a cat<span class="SimpleMath">^5</span>-group we may start with the cyclic group whose order is the product of the first five primes. With this group we may form <span class="SimpleMath">32</span> cat<span class="SimpleMath">^1</span>-groups and <span class="SimpleMath">528</span> cat<span class="SimpleMath">^2</span>-groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( (1,2), (3,4,5), (6,7,8,9,10), (11,12,13,14,15,16,17),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (20,21,22,23,24,25,26,27,28,29,30) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( G, "C2310" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">all1 := AllCat1Groups( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( "G has ", CatnGroupNumbers( G ).cat1, " cat1-groups\n" );</span>
G has 32 cat1-groups
<span class="GAPprompt">gap></span> <span class="GAPinput">PC5 := PreCatnGroup( [ all1[2], all1[5], all1[13], all1[25], all1[32] ] );</span>
(pre-)cat5-group with generating (pre-)cat1-groups:
1 : [C2310 => Group( [ (), (), (), (), (1,2) ] )]
2 : [C2310 => Group( [ (), (), (), (3,4,5), (1,2) ] )]
3 : [C2310 => Group( [ (), (), ( 6, 7, 8, 9,10), (3,4,5), (1,2) ] )]
4 : [C2310 => Group( [ (), (11,12,13,14,15,16,17), ( 6, 7, 8, 9,10), (3,4,5),
(1,2) ] )]
5 : [C2310 => Group( [ (20,21,22,23,24,25,26,27,28,29,30),
(11,12,13,14,15,16,17), ( 6, 7, 8, 9,10), (3,4,5), (1,2) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC5 );</span>
6
</pre></div>
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