<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>
<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="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>
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