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<p id="mathjaxlink" class="pcenter"><a href="chap10_mj.html">[MathJax on]</a></p>
<p><a id="X80B3A81B7E5CA3A9" name="X80B3A81B7E5CA3A9"></a></p>
<div class="ChapSects"><a href="chap10.html#X80B3A81B7E5CA3A9">10 <span class="Heading">Crossed modules of groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X847F4ED77F50528C">10.1 <span class="Heading">Constructions for crossed modules of groupoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X78F89CAB7A281B8F">10.1-1 PreXModWithObjectsByBoundaryAndAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X86CD034F82F5F029">10.1-2 SinglePiecePreXModWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B76F2BF82E075FF">10.1-3 IsXModWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X858EB4F97D04D012">10.1-4 IsPermPreXModWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X797B1CD07C3682EE">10.1-5 Root2dGroup</a></span>
</div></div>
</div>

<h3>10 <span class="Heading">Crossed modules of groupoids</span></h3>

<p>The material documented in this chapter is experimental, and is likely to be changed in due course.</p>

<p><a id="X847F4ED77F50528C" name="X847F4ED77F50528C"></a></p>

<h4>10.1 <span class="Heading">Constructions for crossed modules of groupoids</span></h4>

<p>A typical example of a crossed module <span class="SimpleMath">calX</span> over a groupoid has for its range a connected groupoid. This is a direct product of a group with a complete graph, and we call the vertices of the graph the <em>objects</em> of the crossed module. The source of <span class="SimpleMath">calX</span> is a groupoid, with the same objects, which is either discrete or connected. The boundary morphism is constant on objects. For details and other references see <a href="chapBib.html#biBAW2">[AW10]</a>.</p>

<p><a id="X78F89CAB7A281B8F" name="X78F89CAB7A281B8F"></a></p>

<h5>10.1-1 PreXModWithObjectsByBoundaryAndAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreXModWithObjectsByBoundaryAndAction</code>( <var class="Arg">bdy</var>, <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This is the groupoid generalisation of the operation <code class="code">PreXModByBoundaryAndAction</code>.</p>

<p><a id="X86CD034F82F5F029" name="X86CD034F82F5F029"></a></p>

<h5>10.1-2 SinglePiecePreXModWithObjects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SinglePiecePreXModWithObjects</code>( <var class="Arg">pxmod</var>, <var class="Arg">obs</var>, <var class="Arg">isdisc</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>At present the experimental operation <code class="code">SinglePiecePreXModWithObjects</code> accepts a precrossed module <code class="code">pxmod</code>, a set of objects <code class="code">obs</code>, and a boolean <code class="code">isdisc</code> which is <code class="keyw">true</code> when the source groupoid is homogeneous and discrete and <code class="keyw">false</code> when the source groupoid is connected. Other operations will be added as time permits.</p>

<p>In the example the crossed module <code class="code">DX4</code> has discrete source, while the crossed module <code class="code">CX4</code> has connected source. (Calculations with <code class="code">DX4</code> temporarily removed while this function is being developed.) These are groupoid generalisations of <code class="func">XModByNormalSubgroup</code> (<a href="chap2.html#X83050ED686776933"><span class="RefLink">2.1-2</span></a>) and the example <code class="code">X4</code> in <code class="func">NormalSubXMods</code> (<a href="chap2.html#X7884284383284A87"><span class="RefLink">2.2-2</span></a>).</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">s4 := Group( (1,2,3,4), (3,4) );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( s4, "s4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a4 := Subgroup( s4, [ (1,2,3), (2,3,4) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( a4, "a4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X4 := XModByNormalSubgroup( s4, a4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CX4 := SinglePiecePreXModWithObjects( X4, [-6,-5,-4], false );</span>
single piece precrossed module with objects
  source groupoid:
    single piece groupoid: < a4, [ -6, -5, -4 ] >
  and range groupoid:
    single piece groupoid: < s4, [ -6, -5, -4 ] >
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( CX4, "CX4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ca4 := Source( CX4 );;  SetName( Ca4, "Ca4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cs4 := Range( CX4 );;  SetName( Cs4, "Cs4" );</span>

</pre></div>

<p><a id="X7B76F2BF82E075FF" name="X7B76F2BF82E075FF"></a></p>

<h5>10.1-3 IsXModWithObjects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsXModWithObjects</code>( <var class="Arg">pxmod</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">‣ IsPreXModWithObjects</code>( <var class="Arg">pxmod</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">‣ IsDirectProductWithCompleteDigraphDomain</code>( <var class="Arg">pxmod</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>The precrossed module <code class="code">DX4</code> belongs to the category <code class="code">Is2DimensionalGroupWithObjects</code> and is, of course, a crossed module.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Set( KnownPropertiesOfObject( CX4 ) ); </span>
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsAssociative", 
  "IsDuplicateFree", "IsGeneratorsOfSemigroup", "IsPreXModWithObjects", 
  "IsSinglePieceDomain", "IsXModWithObjects" ]

</pre></div>

<p><a id="X858EB4F97D04D012" name="X858EB4F97D04D012"></a></p>

<h5>10.1-4 IsPermPreXModWithObjects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermPreXModWithObjects</code>( <var class="Arg">pxmod</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">‣ IsPcPreXModWithObjects</code>( <var class="Arg">pxmod</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">‣ IsFpPreXModWithObjects</code>( <var class="Arg">pxmod</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>To test these properties we test the precrossed modules from which they were constructed.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermPreXModWithObjects( CX4 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPcPreXModWithObjects( CX4 );  </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFpPreXModWithObjects( CX4 );</span>
false

</pre></div>

<p><a id="X797B1CD07C3682EE" name="X797B1CD07C3682EE"></a></p>

<h5>10.1-5 Root2dGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Root2dGroup</code>( <var class="Arg">pxmod</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">‣ XModAction</code>( <var class="Arg">pxmod</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The attributes of a precrossed module with objects include the standard <code class="code">Source</code>; <code class="code">Range</code>; <code class="func">Boundary</code> (<a href="chap2.html#X790248A67CB9C33A"><span class="RefLink">2.1-9</span></a>); and <code class="func">XModAction</code> (<a href="chap2.html#X790248A67CB9C33A"><span class="RefLink">2.1-9</span></a>) as with precrossed modules of groups. There is also <code class="code">ObjectList</code>, as in the <strong class="pkg">groupoids</strong> package. Additionally there is <code class="code">Root2dGroup</code> which is the underlying precrossed module used in the construction.</p>

<p>Note that <code class="code">XModAction</code> is now a groupoid homomorphism from the source groupoid to a one-object groupoid (with object <code class="code">0</code>) where the group is the automorphism group of the range groupoid.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Root2dGroup( CX4 ); </span>
[a4->s4]
<span class="GAPprompt">gap></span> <span class="GAPinput">actC := XModAction( CX4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Range( actC ) ); </span>
20736
<span class="GAPprompt">gap></span> <span class="GAPinput">r1 := Arrow( Cs4, (1,2,3,4), -4, -5 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( actC, r1 );            </span>
[groupoid homomorphism : Ca4 -> Ca4
[ [ [(1,2,3) : -6 -> -6], [(2,3,4) : -6 -> -6], [()  : -6 -> -5], 
      [() : -6 -> -4] ], 
  [ [(2,3,4) : -4 -> -4], [(1,3,4) : -4 -> -4], [() : -4 -> -6], 
      [() : -4 -> -5] ] ] : 0 -> 0]
<span class="GAPprompt">gap></span> <span class="GAPinput">s1 := Arrow( Ca4, (1,2,4), -5, -5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">##  calculate s1^r1 </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ims1 := ImageElmXModAction( CX4, s1, r1 );</span>
[(1,2,3) : -6 -> -6]

</pre></div>

<p>There is much more to be done with these constructions.</p>


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