chap9.txt

  
  9 Crossed modules of groupoids
  
  The material documented in this chapter is experimental, and is likely to be
  changed very soon.
  
  
  9.1 Constructions for crossed modules of groupoids
  
  A  typical  example of a crossed module mathcalX 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 mathcalX 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 [AW10].
  
  9.1-1 SinglePiecePreXModWithObjects
  
  SinglePiecePreXModWithObjects( pxmod, obs, isdisc )  operation
  
  At  present the experimental operation SinglePiecePreXModWithObjects accepts
  a  precrossed module pxmod, a set of objects obs, and a boolean isdisc which
  is  true when the source groupoid is homogeneous and discrete and false when
  the  source  groupoid  is  connected. Other operations will be added as time
  permits.
  
  In the example the crossed module DX4 has discrete source, and is a groupoid
  equivalent of XModByNormalSubgroup.
  
    Example  
    
    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> DX4 := SinglePiecePreXModWithObjects( X4, [-9,-8,-7], false );
    precrossed module with source groupoid:
    single piece groupoid: < a4, [ -9, -8, -7 ] >
    and range groupoid:
    single piece groupoid: < s4, [ -9, -8, -7 ] >
    gap> Ga4 := Source( DX4 );; 
    gap> Gs4 := Range( DX4 );;
    
  
  
  9.1-2 IsXModWithObjects
  
  IsXModWithObjects( pxmod )  property
  IsPreXModWithObjects( pxmod )  property
  IsDirectProductWithCompleteDigraphDomain( pxmod )  property
  
  The     precrossed     module     DX4     belongs     to     the    category
  Is2DimensionalGroupWithObjects and is, of course, a crossed module.
  
    Example  
    
    gap> IsXModWithObjects( DX4 ); 
    true
    gap> KnownPropertiesOfObject( DX4 ); 
    [ "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", 
      "IsGeneratorsOfSemigroup", "IsSinglePieceDomain", 
      "IsDirectProductWithCompleteDigraphDomain", "IsPreXModWithObjects", 
      "IsXModWithObjects" ]
    
  
  
  9.1-3 IsPermPreXModWithObjects
  
  IsPermPreXModWithObjects( pxmod )  property
  IsPcPreXModWithObjects( pxmod )  property
  IsFpPreXModWithObjects( pxmod )  property
  
  To test these properties we test the precrossed modules from which they were
  constructed.
  
    Example  
    
    gap> IsPermPreXModWithObjects( DX4 );
    true
    gap> IsPcPreXModWithObjects( DX4 );  
    false
    gap> IsFpPreXModWithObjects( DX4 );
    false
    
  
  
  9.1-4 Root2dGroup
  
  Root2dGroup( pxmod )  attribute
  XModAction( pxmod )  attribute
  
  The  attributes  of  a  precrossed  module with objects include the standard
  Source;  Range;  Boundary;  and  XModAction  as  with  precrossed modules of
  groups.  There is also ObjectList, as in the groupoids package. Additionally
  there  is  Root2dGroup which is the underlying precrossed module used in the
  construction.
  
  Note that XModAction is now a groupoid homomorphism from the source groupoid
  to a one-object groupoid (with object 0) where the group is the automorphism
  group of the range groupoid.
  
    Example  
    
    gap> KnownAttributesOfObject(DX4);
    [ "Range", "Source", "Boundary", "ObjectList", "XModAction", "Root2dGroup" ]
    gap> Root2dGroup( DX4 ); 
    [a4->s4]
    gap> act := XModAction( DX4 );; 
    gap> r := Arrow( Gs4, (1,2,3,4), -7, -8 );; 
    gap> ImageElm( act, r );            
    [groupoid homomorphism : 
    [ [ [(1,2,3) : -9 -> -9], [(2,3,4) : -9 -> -9], [() : -9 -> -8], 
          [() : -9 -> -7] ], 
      [ [(2,3,4) : -9 -> -9], [(1,3,4) : -9 -> -9], [() : -9 -> -7], 
          [() : -9 -> -8] ] ] : 0 -> 0]
    gap> s := Arrow( Ga4, (1,2,4), -8, -8 );;
    gap> ##  calculate s^r 
    gap> ims := ImageElmXModAction( DX4, s, r );
    [(1,2,3) : -7 -> -7]
    
  
  
  There is much more to be done with these constructions.