<!-- ------------------------------------------------------------------- --> <!-- --> <!-- gpd2obj.xml XMod documentation Chris Wensley --> <!-- --> <!-- Copyright (C) 2001-2025, Chris Wensley et al, --> <!-- --> <!-- ------------------------------------------------------------------- -->
<?xmlversion="1.0"encoding="UTF-8"?>
<Chapter Label="chap-gpd2o">
<Heading>Crossed modules of groupoids</Heading>
<Index>crossed module of groupoids</Index>
The material documented in this chapter is experimental,
and is likely to be changed in due course.
<Section>
<Heading>Constructions for crossed modules of groupoids</Heading>
<Index>crossed module over a groupoid</Index>
<Index>2dimensional-domain with objects</Index>
A typical example of a crossed module <M>\calX</M>
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 <E>objects</E>
of the crossed module.
The source of <M>\calX</M> 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 <Cite Key="AW2" />.
<ManSection>
<Oper Name="PreXModWithObjectsByBoundaryAndAction"
Arg="bdy act" />
<Description>
This is the groupoid generalisation of the operation
<C>PreXModByBoundaryAndAction</C>.
</Description>
</ManSection>
<ManSection>
<Oper Name="SinglePiecePreXModWithObjects"
Arg="pxmod obs isdisc" />
<Description>
At present the experimental operation <C>SinglePiecePreXModWithObjects</C>
accepts a precrossed module <C>pxmod</C>, a set of objects <C>obs</C>,
and a boolean <C>isdisc</C> which is <K>true</K> when the source groupoid
is homogeneous and discrete and <K>false</K> when the source groupoid
is connected.
Other operations will be added as time permits.
<P/>
In the example the crossed module <C>DX4</C> has discrete source,
while the crossed module <C>CX4</C> has connected source.
(Calculations with <C>DX4</C> temporarily removed while this function
is being developed.)
These are groupoid generalisations of <Ref Oper="XModByNormalSubgroup"/>
and the example <C>X4</C> in <Ref Oper="NormalSubXMods"/>.
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> s4 := Group( (1,2,3,4), (3,4) );;
gap> SetName( s4, "s4" );
gap> a4 := Subgroup( s4, [ (1,2,3), (2,3,4) ] );;
gap> SetName( a4, "a4" );
gap> X4 := XModByNormalSubgroup( s4, a4 );;
gap> CX4 := SinglePiecePreXModWithObjects( X4, [-6,-5,-4], false );
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 ] >
gap> SetName( CX4, "CX4" );
gap> Ca4 := Source( CX4 );; SetName( Ca4, "Ca4" );
gap> Cs4 := Range( CX4 );; SetName( Cs4, "Cs4" );
]]>
</Example>
<ManSection>
<Prop Name="IsXModWithObjects"
Arg="pxmod" />
<Prop Name="IsPreXModWithObjects"
Arg="pxmod" />
<Prop Name="IsDirectProductWithCompleteDigraphDomain"
Arg="pxmod" />
<Description>
<Index>Is2DimensionalGroupWithObjects</Index>
The precrossed module <C>DX4</C> belongs to the category
<C>Is2DimensionalGroupWithObjects</C> and is, of course, a crossed module.
<P/>
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> Set( KnownPropertiesOfObject( CX4 ) );
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsAssociative", "IsDuplicateFree", "IsGeneratorsOfSemigroup", "IsPreXModWithObjects", "IsSinglePieceDomain", "IsXModWithObjects" ]
]]>
</Example>
<ManSection>
<Prop Name="IsPermPreXModWithObjects"
Arg="pxmod" />
<Prop Name="IsPcPreXModWithObjects"
Arg="pxmod" />
<Prop Name="IsFpPreXModWithObjects"
Arg="pxmod" />
<Description>
To test these properties we test the precrossed modules from which
they were constructed.
<P/>
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> IsPermPreXModWithObjects( CX4 );
true
gap> IsPcPreXModWithObjects( CX4 );
false
gap> IsFpPreXModWithObjects( CX4 );
false
]]>
</Example>
<ManSection>
<Attr Name="Root2dGroup"
Arg="pxmod" />
<Attr Name="XModAction"
Arg="pxmod" Label="for crossed modules of groupoids" />
<Description>
<Index>ObjectList</Index>
The attributes of a precrossed module with objects include the standard
<C>Source</C>; <C>Range</C>;
<Ref Attr="Boundary" Label="for crossed modules"/>;
and <Ref Meth="XModAction" Label="for crossed modules of groups"/>
as with precrossed modules of groups.
There is also <C>ObjectList</C>,
as in the <Package>groupoids</Package> package.
Additionally there is <C>Root2dGroup</C> which is the underlying
precrossed module used in the construction.
<P/>
Note that <C>XModAction</C> is now a groupoid homomorphism
from the source groupoid to a one-object groupoid (with object <C>0</C>)
where the group is the automorphism group of the range groupoid.
<P/>
</Description>
</ManSection>
<P/>
<Example>
<![CDATA[
gap> Root2dGroup( CX4 );
[a4->s4]
gap> actC := XModAction( CX4 );;
gap> Size( Range( actC ) );
20736
gap> r1 := Arrow( Cs4, (1,2,3,4), -4, -5 );;
gap> ImageElm( actC, r1 );
[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]
gap> s1 := Arrow( Ca4, (1,2,4), -5, -5 );;
gap> ## calculate s1^r1
gap> ims1 := ImageElmXModAction( CX4, s1, r1 );
[(1,2,3) : -6 -> -6]
]]>
</Example>
There is much more to be done with these constructions.
</Section>
</Chapter>
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