  
  [1X5. Actors of 2d-objects[0X
  
  
  [1X5.1 Actor of a crossed module[0X
  
  The [13Xactor[0X of cal X is a crossed module (Delta ~:~ cal W(cal X) -> rm Aut(cal
  X))  which  was  shown by Lue and Norrie, in \cite{N2} and \cite{N1} to give
  the  automorphism  object of a crossed module cal X. In this implementation,
  the  source  of the actor is a permutation representation W of the Whitehead
  group  of regular derivations, and the range is a permutation representation
  A of the automorphism group rm Aut(cal X) of cal X.
  
  [1X5.1-1 WhiteheadXMod[0X
  
  [2X> WhiteheadXMod( [0X[3Xxmod[0X[2X ) ___________________________________________[0Xattribute
  [2X> LueXMod( [0X[3Xxmod[0X[2X ) _________________________________________________[0Xattribute
  [2X> NorrieXMod( [0X[3Xxmod[0X[2X ) ______________________________________________[0Xattribute
  [2X> ActorXMod( [0X[3Xxmod[0X[2X ) _______________________________________________[0Xattribute
  [2X> AutomorphismPermGroup( [0X[3Xxmod[0X[2X ) ___________________________________[0Xattribute
  
  An  automorphism  (  sigma,  rho  )  of  [10XX[0X  acts  on the Whitehead monoid by
  chi^(sigma,rho) = sigma circ chi circ rho^-1, and this action determines the
  action  for the actor. In fact the four groups R, S, W, A, the homomorphisms
  between  them,  and the various actions, give five crossed modules forming a
  [13Xcrossed square[0X:
  
  --    cal X = (partial : S -> R),~ the initial crossed module, on the left,
  
  --    cal  W(X) = (eta : S -> W),~ the Whitehead crossed module of cal X, at
        the top,
  
  --    cal  L(X)  =  (Deltacirceta  =  alphacircpartial  :  S -> A),~ the Lue
        crossed module of cal X, along the top-left to bottom-right diagonal,
  
  --    cal  N(X)  = (alpha : R -> A),~ the Norrie crossed module of cal X, at
        the bottom, and
  
  --    rm Act(cal X) = ( Delta : W -> A),~ the actor crossed module of cal X,
        on the right.
  
  [1X5.1-2 Centre[0X
  
  [2X> Centre( [0X[3Xxmod[0X[2X ) __________________________________________________[0Xattribute
  [2X> InnerActor( [0X[3Xxmod[0X[2X ) ______________________________________________[0Xattribute
  [2X> InnerMorphism( [0X[3Xxmod[0X[2X ) ___________________________________________[0Xattribute
  
  Pairs  of  boundaries  or identity mappings provide six morphisms of crossed
  modules.   In   particular,   the   boundaries   of  mathcalW(mathcalX)  and
  mathcalN(mathcalX)  form  the  [13Xinner  morphism[0X  of  mathcalX, mapping source
  elements to principal derivations and range elements to inner automorphisms.
  The  image  of  mathcalX under this morphism is the [13Xinner actor[0X of mathcalX,
  while  the  kernel  is the [13Xcentre[0X of mathcalX. In the example which follows,
  using the crossed module [10X(X3 : c3 -> s3)[0X from Chapter [14X4.[0X, the inner morphism
  is an inclusion of crossed modules.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> X3;[0X
    [4X[c3->s3]][0X
    [4Xgap> WGX3 := WhiteheadPermGroup( X3 );[0X
    [4XGroup( [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ] )[0X
    [4Xgap> APX3 := AutomorphismPermGroup( X3 );[0X
    [4XGroup( [ (3,4,5), (1,2)(4,5) ] )[0X
    [4Xgap> WX3 := WhiteheadXMod( X3 );;  Display( WX3 );[0X
    [4XCrossed module Whitehead[c3->s3] :-[0X
    [4X: Source group has generators:[0X
    [4X  [ ( 1, 2, 3)( 4, 6, 5) ][0X
    [4X: Range group has generators:[0X
    [4X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][0X
    [4X: Boundary homomorphism maps source generators to:[0X
    [4X  [ (1,3,2)(4,6,5) ][0X
    [4X: Action homomorphism maps range generators to automorphisms:[0X
    [4X  (1,2,3)(4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }[0X
    [4X  (1,4)(2,6)(3,5) --> { source gens --> [ (1,3,2)(4,5,6) ] }[0X
    [4X  These 2 automorphisms generate the group of automorphisms.[0X
    [4Xgap> LX3 := LueXMod( X3 );[0X
    [4XLue[c3->s3][0X
    [4Xgap> NX3 := NorrieXMod( X3 );[0X
    [4XNorrie[c3->s3][0X
    [4Xgap> AX3 := ActorXMod( X3 );;  Display( AX3);[0X
    [4XCrossed module Actor[c3->s3] :-[0X
    [4X: Source group has generators:[0X
    [4X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][0X
    [4X: Range group has generators:[0X
    [4X  [ (3,4,5), (1,2)(4,5) ][0X
    [4X: Boundary homomorphism maps source generators to:[0X
    [4X  [ (3,5,4), (1,2)(4,5) ][0X
    [4X: Action homomorphism maps range generators to automorphisms:[0X
    [4X  (3,4,5) --> { source gens --> [ (1,2,3)(4,5,6), (1,5)(2,4)(3,6) ] }[0X
    [4X  (1,2)(4,5) --> { source gens --> [ (1,3,2)(4,6,5), (1,4)(2,6)(3,5) ] }[0X
    [4X  These 2 automorphisms generate the group of automorphisms.[0X
    [4Xgap> IAX3 := InnerActorXMod( X3 );;  Display( IAX3 );[0X
    [4XCrossed module InnerActor[c3->s3] :-[0X
    [4X: Source group has generators:[0X
    [4X  [ (1,3,2)(4,6,5) ][0X
    [4X: Range group has generators:[0X
    [4X  [ (3,5,4), (1,2)(4,5) ][0X
    [4X: Boundary homomorphism maps source generators to:[0X
    [4X  [ (3,4,5) ][0X
    [4X: Action homomorphism maps range generators to automorphisms:[0X
    [4X  (3,5,4) --> { source gens --> [ (1,3,2)(4,6,5) ] }[0X
    [4X  (1,2)(4,5) --> { source gens --> [ (1,2,3)(4,5,6) ] }[0X
    [4X  These 2 automorphisms generate the group of automorphisms.[0X
    [4Xgap> IMX3 := InnerMorphism( X3 );; Display( IMX3 );[0X
    [4XMorphism of crossed modules :-[0X
    [4X: Source = [c3->s3] with generating sets:[0X
    [4X  [ ( 1, 2, 3)( 4, 6, 5) ][0X
    [4X  [ (4,5,6), (2,3)(5,6) ][0X
    [4X:  Range = Actor[c3->s3] with generating sets:[0X
    [4X  [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ][0X
    [4X  [ (3,4,5), (1,2)(4,5) ][0X
    [4X: Source Homomorphism maps source generators to:[0X
    [4X  [ (1,3,2)(4,6,5) ][0X
    [4X: Range Homomorphism maps range generators to:[0X
    [4X  [ (3,5,4), (1,2)(4,5) ][0X
    [4Xgap> Centre( X3 );[0X
    [4X[Group( () )->Group( () )][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
