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### 9 Crossed modules of groupoids

#### 9.1 Constructions for crossed modules of groupoids

A typical example of a crossed module $$\mathcal{X}$$ 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 objects of the crossed module. The source of $$\mathcal{X}$$ is a totally disconnected groupoid, with the same objects. The boundary morphism is constant on objects. For details and other references see [AW10].

##### 9.1-1 DiscreteNormalPreXModWithObjects
 ‣ DiscreteNormalPreXModWithObjects( gpd, gp ) ( operation )
 ‣ PreXModWithObjectsObj( obs, bdy, act ) ( operation )

The next stage of development of this package will be to implement constuctions of crossed modules over groupoids. This will require further developments in the Gpd package. The following example is all that can be shown at the moment. More was achieved in an earlier version, but produces errors in GAP 4.7.


gap> Ga4 := SinglePieceGroupoid( a4, [-9,-8,-7] );;
gap> Display( Ga4 );
single piece groupoid:
objects: [ -9, -8, -7 ]
group: a4 = <[ (1,2,3), (2,3,4) ]>
gap> GeneratorsOfGroup( k4 );
[ (1,2)(3,4), (1,3)(2,4) ]
gap> PXO := DiscreteNormalPreXModWithObjects( Ga4, k4 );
homogeneous, discrete groupoid with:
group: k4 = <[ (1,2)(3,4), (1,3)(2,4) ]> >
objects: [ -9, -8, -7 ]
#I  now need to be able to test:   ok := IsXMod( PM );
<semigroup>
gap> Source( PXO );
perm homogeneous, discrete groupoid: < k4, [ -9, -8, -7 ] >


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