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<div class="ChapSects"><a href="chap8_mj.html#X780368C083C76EDC">8 <span class="Heading">Crossed squares and Cat<span class="SimpleMath">\(^2\)</span>-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7C4AFE8D85848C8F">8.1 <span class="Heading">Definition of a crossed square
and a crossed <span class="SimpleMath">\(n\)</span>-cube of groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X820A7D30847BC828">8.2 <span class="Heading">Constructions for crossed squares</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X866A7FAC7FCB62C2">8.2-1 CrossedSquareByXMods</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7FA98BD47FF9B044">8.2-2 Size3d</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7896DAF786F46234">8.2-3 CrossedSquareByNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7FA367977B1895A7">8.2-4 CrossedSquareByNormalSubXMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X833362FE87ED3C48">8.2-5 ActorCrossedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82E4EA93824F7B26">8.2-6 CrossedSquareByAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X839E065783795CB8">8.2-7 CrossedSquareByPullback</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F5554907AF73190">8.2-8 CrossedSquareByXModSplitting</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X87FBE3CE87DC8CD5">8.2-9 CrossedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F94830681EA19BE">8.2-10 Transpose3DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7C2647CB82DDD065">8.2-11 CentralQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8645AA3686F126D5">8.2-12 IsCrossedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X828CFC5A83097189">8.2-13 Up2DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F81DA90820F4405">8.2-14 IsSymmetric3DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7AE671C7798F99FD">8.2-15 Crossed Pairing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X79A59CED7C69BF18">8.3 <span class="Heading">Substructures of Crossed Squares</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X83F16E94857407F3">8.3-1 SubCrossedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D0D7F787D438D9A">8.3-2 TrivialSub3DimensionalGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X78A79A7E85128C7B">8.4 <span class="Heading">Morphisms of crossed squares</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X83E733547A14FD61">8.4-1 CrossedSquareMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DE8173F80E07AB1">8.4-2 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8284240C7B9BB783">8.4-3 IsCrossedSquareMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8614E38E7A67690E">8.4-4 InclusionMorphismHigherDimensionalDomains</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X86D5AA247B64ED51">8.5 <span class="Heading">Definitions and constructions for cat<span class="SimpleMath">\(^2\)</span>-groups and their morphisms
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X849D845586F92444">8.5-1 Cat2Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X81BD4011837BCC2E">8.5-2 Up2DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X861BA02C7902A4F4">8.5-3 DirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X85713B1C7C34324E">8.5-4 DisplayLeadMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X864B346686AAA522">8.5-5 Transpose3DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X83616C237F4745FB">8.5-6 Cat2GroupMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D46CB517FCAA83B">8.5-7 Cat2GroupOfCrossedSquare</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82F896C7851294B7">8.5-8 Subdiagonal2DimensionalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DB082D282C52855">8.5-9 SubCat2Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X838423B47FEE09B2">8.5-10 TrivialSubCat2Group</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X80FB2B328578DE42">8.6 <span class="Heading">Enumerating cat<span class="SimpleMath">\(^2\)</span>-groups with a given source</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D421DE57B44F37A">8.6-1 AllCat2GroupsWithImagesIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X783F7FEB7B79B062">8.6-2 AllCat2GroupsWithFixedUp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X84636E047CDF2DF3">8.6-3 AllCat2GroupsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7EFCF9697E845B2C">8.6-4 AllCat2GroupsIterator</a></span>
</div></div>
</div>
<h3>8 <span class="Heading">Crossed squares and Cat<span class="SimpleMath">\(^2\)</span>-groups</span></h3>
<p>The term <em>3d-group</em> refers to a set of equivalent categories of which the most common are the categories of <em>crossed squares</em> and <em>cat<span class="SimpleMath">\(^2\)</span>-groups</em>. A <em>3d-mapping</em> is a function between two 3d-groups which preserves all the structure.</p>
<p>The material in this chapter should be considered experimental. A major overhaul took place in time for <strong class="pkg">XMod</strong> version 2.73, with the names of a number of operations being changed.</p>
<p><a id="X7C4AFE8D85848C8F" name="X7C4AFE8D85848C8F"></a></p>
<h4>8.1 <span class="Heading">Definition of a crossed square
and a crossed <span class="SimpleMath">\(n\)</span>-cube of groups</span></h4>
<p>Crossed squares were introduced by Guin-Waléry and Loday (see, for example, <a href="chapBib_mj.html#biBbrow:lod">[BL87]</a>) as fundamental crossed squares of commutative squares of spaces, but are also of purely algebraic interest. We denote by <span class="SimpleMath">\([n]\)</span> the set <span class="SimpleMath">\(\{1,2,\ldots,n\}\)</span>. We use the <span class="SimpleMath">\(n=2\)</span> version of the definition of crossed <span class="SimpleMath">\(n\)</span>-cube as given by Ellis and Steiner <a href="chapBib_mj.html#biBell:st">[ES87]</a>.</p>
<p>A <em>crossed square</em> <span class="SimpleMath">\(\calS\)</span> consists of the following:</p>
<ul>
<li><p>groups <span class="SimpleMath">\(S_J\)</span> for each of the four subsets <span class="SimpleMath">\(J \subseteq [2]\)</span> (we often find it convenient to write <span class="SimpleMath">\(L = S_{[2]},~ M = S_{\{1\}},~ N = S_{\{2\}}\)</span> and <span class="SimpleMath">\(P = S_{\emptyset}\)</span>);</p>
</li>
<li><p>a commutative diagram of group homomorphisms:</p>
<p class="center">\[
\ddot{\partial}_1 : S_{[2]} \to S_{\{2\}}, \quad
\ddot{\partial}_2 : S_{[2]} \to S_{\{1\}}, \quad
\dot{\partial}_2 : S_{\{2\}} \to S_{\emptyset}, \quad
\dot{\partial}_1 : S_{\{1\}} \to S_{\emptyset}
\]</p>
<p>(again we often write <span class="SimpleMath">\(\kappa = \ddot{\partial}_1,~ \lambda = \ddot{\partial}_2,~ \mu = \dot{\partial}_2\)</span> and <span class="SimpleMath">\(\nu = \dot{\partial}_1\)</span>);</p>
</li>
<li><p>actions of <span class="SimpleMath">\(S_{\emptyset}\)</span> on <span class="SimpleMath">\(S_{\{1\}}, S_{\{2\}}\)</span> and <span class="SimpleMath">\(S_{[2]}\)</span> which determine actions of <span class="SimpleMath">\(S_{\{1\}}\)</span> on <span class="SimpleMath">\(S_{\{2\}}\)</span> and <span class="SimpleMath">\(S_{[2]}\)</span> via <span class="SimpleMath">\(\dot{\partial}_1\)</span> and actions of <span class="SimpleMath">\(S_{\{2\}}\)</span> on <span class="SimpleMath">\(S_{\{1\}}\)</span> and <span class="SimpleMath">\(S_{[2]}\)</span> via <span class="SimpleMath">\(\dot{\partial}_2\;\)</span>;</p>
</li>
<li><p>a function <span class="SimpleMath">\(\boxtimes : S_{\{1\}} \times S_{\{2\}} \to S_{[2]}\)</span>.</p>
</li>
</ul>
<p>Here is a picture of the situation:</p>
<p class="center">\[
\vcenter{\xymatrix{
& & S_{[2]} \ar[rr]^{\ddot{\partial}_1} \ar[dd]_{\ddot{\partial}_2}
&& S_{\{2\}} \ar[dd]^{\dot{\partial}_2} &&
L \ar[rr]^{\kappa} \ar[dd]_{\lambda}
&& M \ar[dd]^{\mu} & \\
\mathcal{S} & = & && & = && \\
& & S_{\{1\}} \ar[rr]_{\dot{\partial}_1}
&& S_{\emptyset} &&
N \ar[rr]_{\nu}
&& P
}}
\]</p>
<p>The following axioms must be satisfied for all <span class="SimpleMath">\(l \in L,\; m,m_1,m_2 \in M,\; n,n_1,n_2 \in N,\; p \in P\)</span>.</p>
<ul>
<li><p>The homomorphisms <span class="SimpleMath">\(\kappa, \lambda\)</span> preserve the action of <span class="SimpleMath">\(P\;\)</span>.</p>
</li>
<li><p>Each of the upper, left-hand, right-hand and lower sides of the square,</p>
<p class="center">\[
\ddot{\calS}_1 = (\kappa : L \to M), \quad
\ddot{\calS}_2 = (\lambda : L \to N), \quad
\dot{\calS}_2 = (\mu : M \to P), \quad
\dot{\calS}_1 = (\nu : N \to P),
\]</p>
<p>and the diagonal</p>
<p class="center">\[
\calS_{12} = (\partial_{12} :=
\mu \circ \kappa = \nu \circ \lambda : L \to P)
\]</p>
<p>are crossed modules (with actions via <span class="SimpleMath">\(P\)</span>).</p>
<p>These will be called the <em>up, left, right, down</em> and <em>diagonal</em> crossed modules of <span class="SimpleMath">\(\calS\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\(\boxtimes\)</span> is a <em>crossed pairing</em>:</p>
<ul>
<li><p><span class="SimpleMath">\((n_1n_2 \boxtimes m)\;=\; {(n_1 \boxtimes m)}^{n_2}\;(n_2 \boxtimes m)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\((n \boxtimes m_1m_2) \;=\; (n \boxtimes m_2)\;{(n \boxtimes m_1)}^{m_2}\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\((n \boxtimes m)^{p} \;=\; (n^p \boxtimes m^p)\)</span>.</p>
</li>
</ul>
</li>
<li><p><span class="SimpleMath">\(\kappa(n \boxtimes m) \;=\; (m^{-1})^{n}\;m \quad \mbox{and} \quad \lambda(n \boxtimes m) \;=\; n^{-1}\;n^{m}\)</span>.</p>
</li>
<li><p><span class="SimpleMath">\((n \boxtimes \kappa l) \;=\; (l^{-1})^{n}\;l \quad \mbox{and} \quad (\lambda l \boxtimes m) \;=\; l^{-1}\;l^m\)</span>.</p>
</li>
</ul>
<p>Note that the actions of <span class="SimpleMath">\(M\)</span> on <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(N\)</span> on <span class="SimpleMath">\(M\)</span> via <span class="SimpleMath">\(P\)</span> are compatible since</p>
<p class="center">\[
{n_1}^{(m^n)} \;=\; {n_1}^{\mu(m^n)}
\;=\; {n_1}^{n^{-1}(\mu m)n}
\;=\; (({n_1}^{n^{-1}})^m)^n.
\]</p>
<p>A <em>precrossed square</em> is a similar structure which satisfies some subset of these axioms. (This theoretical notion needs to be clarified.) In this implementation <code class="code">IsPreCrossedSquare</code> only checks that the five pre-crossed modules fit together correctly and that <span class="SimpleMath">\((\kappa,\nu)\)</span> and <span class="SimpleMath">\((\lambda,\mu)\)</span> are both morphisms of pre-crossed modules.</p>
<p>Crossed squares are the <span class="SimpleMath">\(n=2\)</span> case of a crossed <span class="SimpleMath">\(n\)</span>-cube of groups, defined as follows. (This is an attempt to translate Definition 2.1 in Ronnie Brown's Computing homotopy types using crossed n-cubes of groups into right actions -- but this definition is not yet completely understood!)
<p>A <em>crossed</em> <span class="SimpleMath">\(n\)</span><em>-cube of groups</em> consists of the following:</p>
<ul>
<li><p>groups <span class="SimpleMath">\(S_A\)</span> for every subset <span class="SimpleMath">\(A \subseteq [n]\)</span>;</p>
</li>
<li><p>a commutative diagram of group homomorphisms <span class="SimpleMath">\(\partial_i : S_A \to S_{A \setminus \{i\}},\; i \in [n]\)</span>; with composites <span class="SimpleMath">\(\partial_B : S_A \to S_{A \setminus B},\; B \subseteq [n]\)</span>;</p>
</li>
<li><p>actions of <span class="SimpleMath">\(S_{\emptyset}\)</span> on each <span class="SimpleMath">\(S_A\)</span>; and hence actions of <span class="SimpleMath">\(S_B\)</span> on <span class="SimpleMath">\(S_A\)</span> via <span class="SimpleMath">\(\partial_B\)</span> for each <span class="SimpleMath">\(B \subseteq [n]\)</span>;</p>
</li>
<li><p>functions <span class="SimpleMath">\(\boxtimes_{A,B} : S_A \times S_B \to S_{A \cup B}, (A,B \subseteq [n])\)</span>.</p>
</li>
</ul>
<p>There is then a long list of axioms which must be satisfied.</p>
<p><a id="X820A7D30847BC828" name="X820A7D30847BC828"></a></p>
<h4>8.2 <span class="Heading">Constructions for crossed squares</span></h4>
<p>Analogously to the data structure used for crossed modules, crossed squares are implemented as <code class="code">3d-groups</code>. There are also experimental implementations of cat<span class="SimpleMath">\(^2\)</span>-groups, with conversion between the two types of structure. Some standard constructions of crossed squares are listed below. At present, a limited number of constructions is implemented. Morphisms of crossed squares have also been implemented, though there is still a great deal to be done.</p>
<p><a id="X866A7FAC7FCB62C2" name="X866A7FAC7FCB62C2"></a></p>
<h5>8.2-1 CrossedSquareByXMods</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByXMods</code>( <var class="Arg">up</var>, <var class="Arg">left</var>, <var class="Arg">right</var>, <var class="Arg">down</var>, <var class="Arg">diag</var>, <var class="Arg">pairing</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">‣ PreCrossedSquareByPreXMods</code>( <var class="Arg">up</var>, <var class="Arg">left</var>, <var class="Arg">right</var>, <var class="Arg">down</var>, <var class="Arg">diag</var>, <var class="Arg">pairing</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <em>up,left,right,down,diag</em> are five (pre-)crossed modules whose sources and ranges agree, as above, then we just have to add a crossed pairing to complete the data for a (pre-)crossed square.</p>
<p>The <code class="code">Display</code> function is used to print details of 3d-groups.</p>
<p>We take as our example a simple, but significant case. We start with five crossed modules formed from subgroups of <span class="SimpleMath">\(D_8\)</span> with generators <span class="SimpleMath">\([(1,2,3,4),(3,4)\)</span>. The result is a pre-crossed square which is <em>not</em> a crossed square.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b := (2,4);; c := (1,2)(3,4);; p := (1,2,3,4);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d8 := Group( b, c );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d8, "d8" );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := Subgroup( d8, [p^2] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := Subgroup( d8, [b] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := Subgroup( d8, [c] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := TrivialSubgroup( d8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">kappa := GroupHomomorphismByImages( L, M, [p^2], [b] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">lambda := GroupHomomorphismByImages( L, N, [p^2], [c] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">delta := GroupHomomorphismByImages( L, P, [p^2], [()] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mu := GroupHomomorphismByImages( M, P, [b], [()] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">nu := GroupHomomorphismByImages( N, P, [c], [()] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">up := XModByTrivialAction( kappa );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">left := XModByTrivialAction( lambda );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">diag := XModByTrivialAction( delta );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">right := XModByTrivialAction( mu );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">down := XModByTrivialAction( nu );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xp := CrossedPairingByCommutators( N, M, L );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( "xp([c,b]) = ", xp( c, b ), "\n" ); </span>
xp([c,b]) = (1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">PXS := PreCrossedSquareByPreXMods( up, left, right, down, diag, xp );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( PXS ); </span>
(pre-)crossed square with (pre-)crossed modules:
up = [Group( [ (1,3)(2,4) ] ) -> Group( [ (2,4) ] )]
left = [Group( [ (1,3)(2,4) ] ) -> Group( [ (1,2)(3,4) ] )]
right = [Group( [ (2,4) ] ) -> Group( () )]
down = [Group( [ (1,2)(3,4) ] ) -> Group( () )]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCrossedSquare( PXS ); </span>
false
</pre></div>
<p><a id="X7FA98BD47FF9B044" name="X7FA98BD47FF9B044"></a></p>
<h5>8.2-2 Size3d</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size3d</code>( <var class="Arg">XS</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Just as <code class="code">Size2d</code> was used in place of <code class="code">Size</code> for crossed modules, so <code class="code">Size3d</code> is used for crossed squares: <code class="code">Size3d( XS )</code> returns a four-element list containing the sizes of the four groups at the corners of the square.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size3d( PXS ); </span>
[ 2, 2, 2, 1 ]
</pre></div>
<p><a id="X7896DAF786F46234" name="X7896DAF786F46234"></a></p>
<h5>8.2-3 CrossedSquareByNormalSubgroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByNormalSubgroups</code>( <var class="Arg">L</var>, <var class="Arg">M</var>, <var class="Arg">N</var>, <var class="Arg">P</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">‣ CrossedPairingByCommutators</code>( <var class="Arg">N</var>, <var class="Arg">M</var>, <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <span class="SimpleMath">\(L, M, N\)</span> are normal subgroups of a group <span class="SimpleMath">\(P\)</span>, and <span class="SimpleMath">\([M,N] \leqslant L \leqslant M \cap N\)</span>, then the four inclusions <span class="SimpleMath">\(L \to M,~ L \to N,~ M \to P,~ N \to P\)</span>, together with the actions of <span class="SimpleMath">\(P\)</span> on <span class="SimpleMath">\(M, N\)</span> and <span class="SimpleMath">\(L\)</span> given by conjugation, form a crossed square with crossed pairing</p>
<p class="center">\[
\boxtimes \;:\; N \times M \to L, \quad
(n,m) \mapsto [n,m] \,=\, n^{-1}m^{-1}nm \,=\,(m^{-1})^nm \,=\, n^{-1}n^m\,.
\]</p>
<p>This construction is implemented as <code class="code">CrossedSquareByNormalSubgroups(L,M,N,P)</code> (note that the parent group comes last).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">d20 := DihedralGroup( IsPermGroup, 20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gend20 := GeneratorsOfGroup( d20 ); </span>
[ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">p1 := gend20[1];; p2 := gend20[2];; p12 := p1*p2; </span>
(1,10)(2,9)(3,8)(4,7)(5,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">d10a := Subgroup( d20, [ p1^2, p2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d10b := Subgroup( d20, [ p1^2, p12 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c5d := Subgroup( d20, [ p1^2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d20, "d20" ); SetName( d10a, "d10a" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( d10b, "d10b" ); SetName( c5d, "c5d" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSconj := CrossedSquareByNormalSubgroups( c5d, d10a, d10b, d20 );</span>
[ c5d -> d10a ]
[ | | ]
[ d10b -> d20 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xpc := CrossedPairing( XSconj );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xpc( p12, p2 );</span>
(1,3,5,7,9)(2,4,6,8,10)
</pre></div>
<p><a id="X7FA367977B1895A7" name="X7FA367977B1895A7"></a></p>
<h5>8.2-4 CrossedSquareByNormalSubXMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByNormalSubXMod</code>( <var class="Arg">X0</var>, <var class="Arg">X1</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">‣ CrossedPairingBySingleXModAction</code>( <var class="Arg">X0</var>, <var class="Arg">X1</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <span class="SimpleMath">\(\calX_1 = (\partial_1 : S_1 \to R_1)\)</span> is a normal sub-crossed module of <span class="SimpleMath">\(\calX_0 = (\partial_0 : S_0 \to R_0)\)</span> then the inclusion morphism gives a crossed square with crossed pairing</p>
<p class="center">\[
\boxtimes \;:\; R_1 \times S_0 \to S_1, \quad
(r_1,s_0) \mapsto (s_0^{-1})^{r_1} s_0.
\]</p>
<p>The example constructs the same crossed square as in the previous subsection.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">X20 := XModByNormalSubgroup( d20, d10a );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X10 := XModByNormalSubgroup( d10b, c5d );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ok := IsNormalSub2DimensionalDomain( X20, X10 ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">XS20 := CrossedSquareByNormalSubXMod( X20, X10 ); </span>
[ c5d -> d10a ]
[ | | ]
[ d10b -> d20 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xp20 := CrossedPairing( XS20 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xp20( p1^2, p2 );</span>
(1,7,3,9,5)(2,8,4,10,6)
</pre></div>
<p><a id="X833362FE87ED3C48" name="X833362FE87ED3C48"></a></p>
<h5>8.2-5 ActorCrossedSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActorCrossedSquare</code>( <var class="Arg">X0</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">‣ CrossedPairingByDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The actor <span class="SimpleMath">\(\calA(\calX_0)\)</span> of a crossed module <span class="SimpleMath">\(\calX_0\)</span> has been described in Chapter 5 (see <code class="func">ActorXMod</code> (<a href="chap6_mj.html#X790EBC7C7D320C03"><span class="RefLink">6.1-2</span></a>)). The crossed pairing is given by</p>
<p class="center">\[
\boxtimes \;:\; R \times W \,\to\, S, \quad (r,\chi) \,\mapsto\, \chi r~.
\]</p>
<p>This is implemented as <code class="code">ActorCrossedSquare(X0)</code>.</p>
<p>The example constructs <code class="code">XSact</code>, the actor crossed square of the crossed module <code class="code">X20</code>. This crossed square is converted to a cat<span class="SimpleMath">\(^2\)</span>-group <code class="code">C2act</code> in section <a href="chap8_mj.html#X7D46CB517FCAA83B"><span class="RefLink">8.5-7</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSact := ActorCrossedSquare( X20 );</span>
crossed square with crossed modules:
up = Whitehead[d10a->d20]
left = [d10a->d20]
right = Actor[d10a->d20]
down = Norrie[d10a->d20]
<span class="GAPprompt">gap></span> <span class="GAPinput">W := Range( Up2DimensionalGroup( XSact ) );</span>
Group([ (2,5)(3,4), (2,3,5,4), (1,4,2,5,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( W );</span>
"C5 : C4"
<span class="GAPprompt">gap></span> <span class="GAPinput">xpa := CrossedPairing( XSact );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xpa( p1, (2,3,5,4) );</span>
(1,7,3,9,5)(2,8,4,10,6)
</pre></div>
<p><a id="X82E4EA93824F7B26" name="X82E4EA93824F7B26"></a></p>
<h5>8.2-6 CrossedSquareByAutomorphismGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByAutomorphismGroup</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">‣ CrossedPairingByConjugators</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For <span class="SimpleMath">\(G\)</span> a group let <span class="SimpleMath">\(\Inn(G)\)</span> be its inner automorphism group and <span class="SimpleMath">\(\Aut(G)\)</span> its full automorphism group. Then there is a crossed square with groups <span class="SimpleMath">\([G,\Inn(G),\Inn(G),\Aut(G)]\)</span> where the upper and left boundaries are the maps <span class="SimpleMath">\(g \mapsto \iota_g\)</span>, where <span class="SimpleMath">\(\iota_g\)</span> is conjugation of <span class="SimpleMath">\(G\)</span> by <span class="SimpleMath">\(g\)</span>, and the right and down boundaries are inclusions. The crossed pairing is gived by <span class="SimpleMath">\(\iota_g \boxtimes \iota_h = [g,h]\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AXS20 := CrossedSquareByAutomorphismGroup( d20 );</span>
[ d20 -> Inn(d20) ]
[ | | ]
[ Inn(d20) -> Aut(d20) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( AXS20 );</span>
[ "D20", "D10", "D10", "C2 x (C5 : C4)" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">I20 := Range( Up2DimensionalGroup( AXS20 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">genI20 := GeneratorsOfGroup( I20 ); </span>
[ ^(1,2,3,4,5,6,7,8,9,10), ^(2,10)(3,9)(4,8)(5,7) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xpi := CrossedPairing( AXS20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xpi( genI20[1], genI20[2] );</span>
(1,9,7,5,3)(2,10,8,6,4)
</pre></div>
<p><a id="X839E065783795CB8" name="X839E065783795CB8"></a></p>
<h5>8.2-7 CrossedSquareByPullback</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByPullback</code>( <var class="Arg">X1</var>, <var class="Arg">X2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If crossed modules <span class="SimpleMath">\(\calX_1 = (\nu : N \to P)\)</span> and <span class="SimpleMath">\(\calX_2 = (\mu : M \to P)\)</span> have a common range <span class="SimpleMath">\(P\)</span>, let <span class="SimpleMath">\(L\)</span> be the pullback of <span class="SimpleMath">\(\{\nu,\mu\}\)</span>. Then <span class="SimpleMath">\(N\)</span> acts on <span class="SimpleMath">\(L\)</span> by <span class="SimpleMath">\((n,m)^{n'} = (n^{n'},m^{\nu n'})\), and \(M\) acts on \(L\) by \((n,m)^{m'} = (n^{\mu m'}, m^{m'})\)</span>. So <span class="SimpleMath">\((\pi_1 : L \to N)\)</span> and <span class="SimpleMath">\((\pi_2 : L \to M)\)</span> are crossed modules, where <span class="SimpleMath">\(\pi_1,\pi_2\)</span> are the two projections. The crossed pairing is given by:</p>
<p class="center">\[
\boxtimes \;:\; N \times M \to L, \quad
(n,m) \mapsto (n^{-1}n^{\mu m}, (m^{-1})^{\nu n}m) .
\]</p>
<p>The second example below uses the central extension crossed module <code class="code">X12=(D12->S3)</code> which was constructed in subsection (<code class="func">XModByCentralExtension</code> (<a href="chap2_mj.html#X7D0F6FAA7AF69844"><span class="RefLink">2.1-5</span></a>)), with pullback group <code class="code">D12xC2</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">dn := Down2DimensionalGroup( XSconj );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rt := Right2DimensionalGroup( XSconj );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSP := CrossedSquareByPullback( dn, rt ); </span>
[ (d10b x_d20 d10a) -> d10a ]
[ | | ]
[ d10b -> d20 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSP ); </span>
[ "C5", "D10", "D10", "D20" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">XS12 := CrossedSquareByPullback( X12, X12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XS12 ); </span>
[ "C2 x C2 x S3", "D12", "D12", "S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">xp12 := CrossedPairing( XS12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xp12( (1,2,3,4,5,6), (2,6)(3,5) );</span>
(1,5,3)(2,6,4)(7,11,9)(8,12,10)
</pre></div>
<p><a id="X7F5554907AF73190" name="X7F5554907AF73190"></a></p>
<h5>8.2-8 CrossedSquareByXModSplitting</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareByXModSplitting</code>( <var class="Arg">X0</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">‣ CrossedPairingByPreImages</code>( <var class="Arg">X1</var>, <var class="Arg">X2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> let <span class="SimpleMath">\(Q\)</span> be the image of <span class="SimpleMath">\(\partial\)</span>. Then <span class="SimpleMath">\(\partial = \partial' \circ \iota\) where \(\partial' : S \to Q\)</span> and <span class="SimpleMath">\(\iota\)</span> is the inclusion of <span class="SimpleMath">\(Q\)</span> in <span class="SimpleMath">\(R\)</span>. The diagonal of the square is then the initial <span class="SimpleMath">\(\calX\)</span>, and the crossed pairing is given by commutators of preimages.</p>
<p>A particular case is when <span class="SimpleMath">\(S\)</span> is an <span class="SimpleMath">\(R\)</span>-module <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(\partial\)</span> is the zero map.</p>
<p class="center">\[
\vcenter{\xymatrix{
& & S \ar[rr]^{\partial'} \ar[dd]_{\partial'} \ar[rrdd]^{\partial}
&& Q \ar[dd]^{\iota} &&
A \ar[rr]^0 \ar[dd]_0 \ar[rrdd]^0
&& 1 \ar[dd]^{\iota} & \\
& & && & && \\
& & Q \ar[rr]_{\iota}
&& R &&
1 \ar[rr]_{\iota}
&& R
}}
\]</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">k4 := Group( (1,2), (3,4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AX4 := XModByAutomorphismGroup( k4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X4 := Image( IsomorphismPermObject( AX4 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSS4 := CrossedSquareByXModSplitting( X4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSS4 );</span>
[ "C2 x C2", "1", "1", "S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">XSS20 := CrossedSquareByXModSplitting( X20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">up20 := Up2DimensionalGroup( XSS20 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Range( up20 ) = d10a; </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Range( up20 ), "d10a" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Name( XSS20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSS20;</span>
[d10a->d10a,d10a->d20]
<span class="GAPprompt">gap></span> <span class="GAPinput">xps := CrossedPairing( XSS20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xps( p1^2, p2 );</span>
(1,7,3,9,5)(2,8,4,10,6)
</pre></div>
<p><a id="X87FBE3CE87DC8CD5" name="X87FBE3CE87DC8CD5"></a></p>
<h5>8.2-9 CrossedSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquare</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The function <code class="code">CrossedSquare</code> may be used to call some of the constructions described in the previous subsections.</p>
<ul>
<li><p><code class="code">CrossedSquare(X0)</code> calls <code class="code">CrossedSquareByXModSplitting</code>.</p>
</li>
<li><p><code class="code">CrossedSquare(C0)</code> calls <code class="code">CrossedSquareOfCat2Group</code>.</p>
</li>
<li><p><code class="code">CrossedSquare(X0,X1)</code> calls <code class="code">CrossedSquareByPullback</code> when there is a common range.</p>
</li>
<li><p><code class="code">CrossedSquare(X0,X1)</code> calls <code class="code">CrossedSquareByNormalXMod</code> when <code class="code">X1</code> is normal in <code class="code">X0</code> .</p>
</li>
<li><p><code class="code">CrossedSquare(L,M,N,P)</code> calls <code class="code">CrossedSquareByNormalSubgroups</code>.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">diag := Diagonal2DimensionalGroup( AXS20 );</span>
[d20->Aut(d20)]
<span class="GAPprompt">gap></span> <span class="GAPinput">XSdiag := CrossedSquare( diag );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( XSdiag ); </span>
[ "D20", "D10", "D10", "C2 x (C5 : C4)" ]
</pre></div>
<p><a id="X7F94830681EA19BE" name="X7F94830681EA19BE"></a></p>
<h5>8.2-10 Transpose3DimensionalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Transpose3DimensionalGroup</code>( <var class="Arg">S0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>transpose</em> of a crossed square <span class="SimpleMath">\(\calS\)</span> is the crossed square <span class="SimpleMath">\(\tilde{\calS}\)</span> obtained by interchanging <span class="SimpleMath">\(M\)</span> with <span class="SimpleMath">\(N\)</span>, <span class="SimpleMath">\(\kappa\)</span> with <span class="SimpleMath">\(\lambda\)</span>, and <span class="SimpleMath">\(\nu\)</span> with <span class="SimpleMath">\(\mu\)</span>. The crossed pairing is given by</p>
<p class="center">\[
\tilde{\boxtimes} \;:\; M \times N \to L, \quad
(m,n) \;\mapsto\; m\,\tilde{\boxtimes}\,n := (n \boxtimes m)^{-1}~.
\]</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">XStrans := Transpose3DimensionalGroup( XSconj ); </span>
[ c5d -> d10b ]
[ | | ]
[ d10a -> d20 ]
</pre></div>
<p><a id="X7C2647CB82DDD065" name="X7C2647CB82DDD065"></a></p>
<h5>8.2-11 CentralQuotient</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralQuotient</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The central quotient of a crossed module <span class="SimpleMath">\(\calX = (\partial : S \to R)\)</span> is the crossed square where:</p>
<ul>
<li><p>the left crossed module is <span class="SimpleMath">\(\calX\)</span>;</p>
</li>
<li><p>the right crossed module is the quotient <span class="SimpleMath">\(\calX/Z(\calX)\)</span> (see <code class="func">CentreXMod</code> (<a href="chap4_mj.html#X7B57446086BA1BF0"><span class="RefLink">4.1-7</span></a>));</p>
</li>
<li><p>the up and down homomorphisms are the natural homomorphisms onto the quotient groups;</p>
</li>
<li><p>the crossed pairing <span class="SimpleMath">\(\boxtimes : (R \times F) \to S\)</span>, where <span class="SimpleMath">\(F = \Fix(\calX,S,R)\)</span>, is the displacement element <span class="SimpleMath">\(\boxtimes(r,Fs) = \langle r,s \rangle = (s^{-1})^rs\quad\)</span> (see <code class="func">Displacement</code> (<a href="chap4_mj.html#X7E20208279038BB8"><span class="RefLink">4.1-3</span></a>) and section <a href="chap4_mj.html#X81338C977972AD83"><span class="RefLink">4.3</span></a>).</p>
</li>
</ul>
<p>This is the special case of an intended function <code class="code">CrossedSquareByCentralExtension</code> which has not yet been implemented. In the example <code class="code">Xn7</code> <span class="SimpleMath">\(\unlhd\)</span> <code class="code">X24</code>, constructed in section <a href="chap4_mj.html#X7E373BF3836B3A9C"><span class="RefLink">4.1</span></a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pos7 := Position( ids, [ [12,2], [24,5] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Xn7 := nsx[pos7];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( Xn7 );</span>
[ [ 12, 2 ], [ 24, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( CentreXMod( Xn7 ) ); </span>
[ [ 4, 1 ], [ 4, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CQXn7 := CentralQuotient( Xn7 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( CQXn7 );</span>
[ "C12", "C3", "C4 x S3", "S3" ]
</pre></div>
<p><a id="X8645AA3686F126D5" name="X8645AA3686F126D5"></a></p>
<h5>8.2-12 IsCrossedSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCrossedSquare</code>( <var class="Arg">obj</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">‣ IsPreCrossedSquare</code>( <var class="Arg">obj</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">‣ Is3dObject</code>( <var class="Arg">obj</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">‣ IsPerm3dObject</code>( <var class="Arg">obj</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">‣ IsPc3dObject</code>( <var class="Arg">obj</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">‣ IsFp3dObject</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>These are the basic properties for 3d-groups, and crossed squares in particular.</p>
<p><a id="X828CFC5A83097189" name="X828CFC5A83097189"></a></p>
<h5>8.2-13 Up2DimensionalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Up2DimensionalGroup</code>( <var class="Arg">XS</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">‣ Left2DimensionalGroup</code>( <var class="Arg">XS</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">‣ Down2DimensionalGroup</code>( <var class="Arg">XS</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">‣ Right2DimensionalGroup</code>( <var class="Arg">XS</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">‣ CrossDiagonalActions</code>( <var class="Arg">XS</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">‣ Diagonal2DimensionalGroup</code>( <var class="Arg">XS</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">‣ Name</code>( <var class="Arg">S0</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>These are the basic attributes of a crossed square <span class="SimpleMath">\(\calS\)</span>. The six objects used in the construction of <span class="SimpleMath">\(\calS\)</span> are the four crossed modules (2d-groups) on the sides of the square (up; left; right and down); the diagonal action of <span class="SimpleMath">\(P\)</span> on <span class="SimpleMath">\(L\)</span>; and the crossed pairing <span class="SimpleMath">\(\{M,N\} \to L\)</span> (see the next subsection). The diagonal crossed module <span class="SimpleMath">\((L \to P)\)</span> is an additional attribute.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Up2DimensionalGroup( XSconj );</span>
[c5d->d10a]
<span class="GAPprompt">gap></span> <span class="GAPinput">Right2DimensionalGroup( XSact );</span>
Actor[d10a->d20]
<span class="GAPprompt">gap></span> <span class="GAPinput">Name( XSconj ); </span>
"[c5d->d10a,d10b->d20]"
<span class="GAPprompt">gap></span> <span class="GAPinput">cross1 := CrossDiagonalActions( XSconj )[1];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gensa := GeneratorsOfGroup( d10a );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gensb := GeneratorsOfGroup( d10a );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act1 := ImageElm( cross1, gensb[1] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gensa[2]; ImageElm( act1, gensa[2] );</span>
(2,10)(3,9)(4,8)(5,7)
(1,5)(2,4)(6,10)(7,9)
</pre></div>
<p><a id="X7F81DA90820F4405" name="X7F81DA90820F4405"></a></p>
<h5>8.2-14 IsSymmetric3DimensionalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSymmetric3DimensionalGroup</code>( <var class="Arg">obj</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">‣ IsAbelian3DimensionalGroup</code>( <var class="Arg">obj</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">‣ IsTrivialAction3DimensionalGroup</code>( <var class="Arg">obj</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">‣ IsNormalSub3DimensionalGroup</code>( <var class="Arg">obj</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">‣ IsCentralExtension3DimensionalGroup</code>( <var class="Arg">obj</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">‣ IsAutomorphismGroup3DimensionalGroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>These are further properties for 3d-groups, and crossed squares in particular. A 3d-group is <em>symmetric</em> if its <code class="code">Up2DimensionalGroup</code> is equal to its <code class="code">Left2DimensionalGroup</code>.</p>
<p><a id="X7AE671C7798F99FD" name="X7AE671C7798F99FD"></a></p>
<h5>8.2-15 Crossed Pairing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Crossed Pairing</code>( <var class="Arg">XS</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Crossed pairings have been implemented as functions of two variables, <span class="SimpleMath">\(\{M,N\} \to L\)</span>. (Up until version 2.92 a more complicated two-variable mapping was used.)</p>
<p>The first example shows the crossed pairing in the crossed square <code class="code">XSconj</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">xp := CrossedPairing( XSconj );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">xp( (1,6)(2,5)(3,4)(7,10)(8,9), (1,5)(2,4)(6,9)(7,8) );</span>
(1,7,8,5,3)(2,9,10,6,4)
</pre></div>
<p>The second example shows how to construct a crossed pairing.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeGroup(1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := GeneratorsOfGroup(F)[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">z := GroupHomomorphismByImages( F, F, [x], [x^0] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">id := GroupHomomorphismByImages( F, F, [x], [x] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h := function(n,m)</span>
<span class="GAPprompt">></span> <span class="GAPinput"> return x^(ExponentSumWord(n,x)*ExponentSumWord(m,x));</span>
<span class="GAPprompt">></span> <span class="GAPinput"> end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h( x^3, x^4 );</span>
f1^12
<span class="GAPprompt">gap></span> <span class="GAPinput">A := AutomorphismGroup( F );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a := GeneratorsOfGroup(A)[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">act := GroupHomomorphismByImages( F, A, [x], [a^2] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X0 := XModByBoundaryAndAction( z, act );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X1 := XModByBoundaryAndAction( id, act );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">XSF := PreCrossedSquareByPreXMods( X0, X0, X1, X1, X0, h );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCrossedSquare( XSF ); </span>
true
</pre></div>
<p><a id="X79A59CED7C69BF18" name="X79A59CED7C69BF18"></a></p>
<h4>8.3 <span class="Heading">Substructures of Crossed Squares</span></h4>
<p>To specify a sub-crossed square of <span class="SimpleMath">\(\calS\)</span> we may specify four subgroups of the four groups in <span class="SimpleMath">\(\calS\)</span>. If these define five sub-crossed modules, and these comc bine together satisfactorily, then a crossed square is produced.</p>
<p><a id="X83F16E94857407F3" name="X83F16E94857407F3"></a></p>
<h5>8.3-1 SubCrossedSquare</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubCrossedSquare</code>( <var class="Arg">XS</var>, <var class="Arg">ul</var>, <var class="Arg">ur</var>, <var class="Arg">dl</var>, <var class="Arg">dr</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">‣ IsSubCrossedSquare</code>( <var class="Arg">XS</var>, <var class="Arg">subXS</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">‣ SubPreCrossedSquare</code>( <var class="Arg">XS</var>, <var class="Arg">ul</var>, <var class="Arg">ur</var>, <var class="Arg">dl</var>, <var class="Arg">dr</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">‣ IsSubPreCrossedSquare</code>( <var class="Arg">XS</var>, <var class="Arg">subXS</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( XS20 ); </span>
[ c5d -> d10a ]
[ | | ]
[ d10b -> d20 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">p5 := p1^5*p2;; [p2,p12,p5];</span>
[ (2,10)(3,9)(4,8)(5,7), (1,10)(2,9)(3,8)(4,7)(5,6),
(1,6)(2,5)(3,4)(7,10)(8,9) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L1 := TrivialSubgroup( c5d );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M1 := Subgroup( d10a, [ p2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">N1 := Subgroup( d10b, [ p5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P1 := Subgroup( d20, [ p2, p5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sub20 := SubCrossedSquare( XS20, L1, M1, N1, P1 );</span>
crossed square with crossed modules:
up = [Group( () ) -> Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7) ] )]
left = [Group( () ) -> Group( [ ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )]
right = [Group( [ ( 2,10)( 3, 9)( 4, 8)( 5, 7) ] ) -> Group(
[ ( 2,10)( 3, 9)( 4, 8)( 5, 7), ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )]
down = [Group( [ ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] ) -> Group(
[ ( 2,10)( 3, 9)( 4, 8)( 5, 7), ( 1, 6)( 2, 5)( 3, 4)( 7,10)( 8, 9) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">sxp := CrossedPairing( sub20 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sxp( p5, p2 );</span>
()
</pre></div>
<p><a id="X7D0D7F787D438D9A" name="X7D0D7F787D438D9A"></a></p>
<h5>8.3-2 TrivialSub3DimensionalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialSub3DimensionalGroup</code>( <var class="Arg">3dgp</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">‣ TrivialSubCrossedSquare</code>( <var class="Arg">XS</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">‣ TrivialSubPreCrossedSquare</code>( <var class="Arg">XS</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>A special case of the previous operation is when all the subgroups are trivial.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TC2ab := TrivialSubCat2Group( C2ab );</span>
(pre-)cat2-group with generating (pre-)cat1-groups:
1 : [Group( () ) => Group( () )]
2 : [Group( () ) => Group( () )]
</pre></div>
<p><a id="X78A79A7E85128C7B" name="X78A79A7E85128C7B"></a></p>
<h4>8.4 <span class="Heading">Morphisms of crossed squares</span></h4>
<p>This section describes an initial implementation of morphisms of (pre-)crossed squares.</p>
<p><a id="X83E733547A14FD61" name="X83E733547A14FD61"></a></p>
<h5>8.4-1 CrossedSquareMorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedSquareMorphism</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">‣ CrossedSquareMorphismByXModMorphisms</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">mors</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">‣ CrossedSquareMorphismByGroupHomomorphisms</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">homs</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">‣ PreCrossedSquareMorphismByPreXModMorphisms</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">mors</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">‣ PreCrossedSquareMorphismByGroupHomomorphisms</code>( <var class="Arg">src</var>, <var class="Arg">rng</var>, <var class="Arg">homs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><a id="X7DE8173F80E07AB1" name="X7DE8173F80E07AB1"></a></p>
<h5>8.4-2 Source</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Source</code>( <var class="Arg">map</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">‣ Range</code>( <var class="Arg">map</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">‣ Up2DimensionalMorphism</code>( <var class="Arg">map</var> )</td><td class="tdright">( attribute )</t | | |
