  
  [1X9 [33X[0;0YCrossed modules of groupoids[133X[101X
  
  
  [1X9.1 [33X[0;0YConstructions for crossed modules of groupoids[133X[101X
  
  [33X[0;0YA  typical  example of a crossed module [22XmathcalX[122X 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 [13Xobjects[113X of the
  crossed  module.  The source of [22XmathcalX[122X is a totally disconnected groupoid,
  with  the  same  objects.  The boundary morphism is constant on objects. For
  details and other references see [AW10].[133X
  
  [1X9.1-1 DiscreteNormalPreXModWithObjects[101X
  
  [29X[2XDiscreteNormalPreXModWithObjects[102X( [3Xgpd[103X, [3Xgp[103X ) [32X operation
  [29X[2XPreXModWithObjectsObj[102X( [3Xobs[103X, [3Xbdy[103X, [3Xact[103X ) [32X operation
  
  [33X[0;0YThe  next  stage  of  development  of  this  package  will  be  to implement
  constuctions  of  crossed  modules over groupoids. This will require further
  developments  in  the  [5XGpd[105X package. The following example is all that can be
  shown  at  the moment. More was achieved in an earlier version, but produces
  errors in [5XGAP[105X 4.7.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XGa4 := SinglePieceGroupoid( a4, [-9,-8,-7] );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( Ga4 );[127X[104X
    [4X[28Xsingle piece groupoid: [128X[104X
    [4X[28X  objects: [ -9, -8, -7 ][128X[104X
    [4X[28X    group: a4 = <[ (1,2,3), (2,3,4) ]>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( k4 );[127X[104X
    [4X[28X[ (1,2)(3,4), (1,3)(2,4) ][128X[104X
    [4X[25Xgap>[125X [27XPXO := DiscreteNormalPreXModWithObjects( Ga4, k4 );[127X[104X
    [4X[28Xhomogeneous, discrete groupoid with:[128X[104X
    [4X[28X  group: k4 = <[ (1,2)(3,4), (1,3)(2,4) ]> >[128X[104X
    [4X[28Xobjects: [ -9, -8, -7 ][128X[104X
    [4X[28X#I  now need to be able to test:   ok := IsXMod( PM );[128X[104X
    [4X[28X<semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSource( PXO ); [127X[104X
    [4X[28Xperm homogeneous, discrete groupoid: < k4, [ -9, -8, -7 ] >[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
