chap8.txt

  
  8 3d-groups and 3d-mappings : crossed squares and cat^2-groups
  
  The term 3d-group refers to a set of equivalent categories of which the most
  common are the categories of crossed squares and cat^2-groups.
  
  
  8.1 Definition of a crossed square and a crossed n-cube of groups
  
  Crossed  squares were introduced by Guin-Waléry and Loday (see, for example,
  [BL87]) as fundamental crossed squares of commutative squares of spaces, but
  are also of purely algebraic interest. We denote by [n] the set {1,2,...,n}.
  We use the n=2 version of the definition of crossed n-cube as given by Ellis
  and Steiner [ES87].
  
  A crossed square mathcalS consists of the following:
  
      groups S_J for each of the four subsets J ⊆ [2];
  
      a commutative diagram of group homomorphisms:
  
  
        \ddot{\partial}_1  :  S_{[2]} \to S_{\{2\}}, \quad \ddot{\partial}_2 :
        S_{[2]}   \to   S_{\{1\}},  \quad  \dot{\partial}_1  :  S_{\{1\}}  \to
        S_{\emptyset}, \quad \dot{\partial}_2 : S_{\{2\}} \to S_{\emptyset};
  
  
  
      actions  of  S_∅  on S_{1}, S_{2} and S_[2] which determine actions of
        S_{1}  on S_{2} and S_[2] via dot∂_1 and actions of S_{2} on S_{1} and
        S_[2] via dot∂_2;
  
      a function ⊠ : S_{1} × S_{2} -> S_[2].
  
  Here is a picture of the situation:
  
  
  \vcenter{\xymatrix{      &     &     S_{[2]}     \ar[rr]^{\ddot{\partial}_1}
  \ar[dd]_{\ddot{\partial}_2}  &&  S_{\{2\}}  \ar[dd]^{\dot{\partial}_2}  & \\
  \mathcal{S}  &  =  &  &&  \\  &  &  S_{\{1\}}  \ar[rr]_{\dot{\partial}_1} &&
  S_{\emptyset} }} }}
  
  
  
  The following axioms must be satisfied for all l ∈ S_[2], m,m_1,m_2 ∈ S_{1},
  n,n_1,n_2 ∈ S_{2}, p ∈ S_∅:
  
      the homomorphisms ddot∂_1, ddot∂_2 preserve the action of S_∅;
  
      each  of  the  upper,  left-hand,  lower,  and right-hand sides of the
        square,
  
  
        \ddot{\mathcal{S}}_1  =  (\ddot{\partial}_1  : S_{[2]} \to S_{\{2\}}),
        \ddot{\mathcal{S}}_2  =  (\ddot{\partial}_2  : S_{[2]} \to S_{\{1\}}),
        \dot{\mathcal{S}}_1    =    (\dot{\partial}_1    :    S_{\{1\}}    \to
        S_{\emptyset}),  \dot{\mathcal{S}}_2  =  (\dot{\partial}_2 : S_{\{2\}}
        \to S_{\emptyset}),
  
  
  
        and the diagonal
  
  
        \mathcal{S}_{12} = (\partial_{12} := \dot{\partial}_1\ddot{\partial}_2
        = \dot{\partial}_2\ddot{\partial}_1 : S_{[2]} \to S_{\emptyset})
  
  
  
        are crossed modules (with actions via S_∅);
  
      ⊠ is a crossed pairing:
  
            (m_1m_2 ⊠ n) = (m_1 ⊠ n)^m_2 (m_2 ⊠ n),
  
            (m ⊠ n_1n_2) = (m ⊠ n_2) (m ⊠ n_1)^n_2,
  
            (m ⊠ n)^p = (m^p ⊠ n^p);
  
      ddot∂_1  (m ⊠ n) = (n^-1)^m n quad mboxand quad ddot∂_2 (m ⊠ n) = m^-1
        m^n,
  
      (m  ⊠ ddot∂_1 l) = (l^-1)^m l quad mboxand quad (ddot∂_2 l ⊠ n) = l^-1
        l^n.
  
  Note  that  the  actions  of  S_{1}  on S_{2} and S_{2} on S_{1} via S_∅ are
  compatible since
  
  
  {m_1}^{(n^m)}        \;=\;        {m_1}^{\dot{\partial}_2(n^m)}        \;=\;
  {m_1}^{m^{-1}(\dot{\partial}_2 n)m} \;=\; (({m_1}^{m^{-1}})^n)^m.
  
  
  
  (A  precrossed  square is a similar structure which satisfies some subset of
  these axioms. [More needed here.])
  
  In  what  follows we shall generally use the following notation for the S_J,
  namely L = S_[2];~ M = S_{1};~ N = S_{2} and P = S_∅.
  
  Crossed  squares  are the n=2 case of a crossed n-cube of groups, defined as
  follows.  (This  is  an  attempt  to  translate  Definition  2.1 in Ronnie's
  Computing  homotopy types using crossed n-cubes of groups into right actions
  -- but this definition is not yet completely understood!)
  
  A crossed n-cube of groups consists of the following:
  
      groups S_A for every subset A ⊆ [n];
  
      a commutative diagram of group homomorphisms ∂_i : S_A -> S_A ∖ {i}, i
        ∈ [n]; with composites ∂_B : S_A -> S_A ∖ B, B ⊆ [n];
  
      actions  of  S_∅  on each S_A; and hence actions of S_B on S_A via ∂_B
        for each B ⊆ [n];
  
      functions ⊠_A,B : S_A × S_B -> S_A ∪ B, (A,B ⊆ [n]).
  
  The following axioms must be satisfied (long list to be added).
  
  
  8.2 Constructions for crossed squares
  
  Analogously  to the data structure used for crossed modules, crossed squares
  are  implemented  as 3d-groups. When times allows, cat^2-groups will also be
  implemented,  with  conversion  between  the  two  types  of structure. Some
  standard  constructions  of  crossed squares are listed below. At present, a
  limited  number  of  constructions  are  implemented.  Morphisms  of crossed
  squares have also been implemented, though there is a lot still to be done.
  
  8.2-1 CrossedSquare
  
  CrossedSquare( args )  function
  CrossedSquareByNormalSubgroups( P, N, M, L )  operation
  ActorCrossedSquare( X0 )  operation
  Transpose3dGroup( S0 )  attribute
  Name( S0 )  attribute
  
  Here are some standard examples of crossed squares.
  
      If  M,  N  are  normal subgroups of a group P, and L = M ∩ N, then the
        four  inclusions, L -> N,~ L -> M,~ M -> P,~ N -> P, together with the
        actions of P on M, N and L given by conjugation, form a crossed square
        with crossed pairing
  
  
        \boxtimes  \;:\;  M  \times  N  \to M\cap N, \quad (m,n) \mapsto [m,n]
        \,=\, m^{-1}n^{-1}mn \,=\,(n^{-1})^mn \,=\, m^{-1}m^n\,.
  
  
  
        This          construction          is          implemented         as
        CrossedSquareByNormalSubgroups(P,N,M,L);.
  
      The actor mathcalA(mathcalX_0) of a crossed module mathcalX_0 has been
        described in Chapter 5. The crossed pairing is given by
  
  
        \boxtimes  \;:\; R \times W \,\to\, S, \quad (r,\chi) \,\mapsto\, \chi
        r~.
  
  
  
        This is implemented as ActorCrossedSquare( X0 );.
  
      The  transpose  of  mathcalS  is  the  crossed  square  tildemathcalS}
        obtained  by interchanging S_{1} with S_{2}, ddot∂_1 with ddot∂_2, and
        dot∂_1 with dot∂_2. The crossed pairing is given by
  
  
        \tilde{\boxtimes}  \;:\; S_{\{2\}} \times S_{\{1\}} \to S_{[2]}, \quad
        (n,m) \;\mapsto\; n\,\tilde{\boxtimes}\,m := (m \boxtimes n)^{-1}~.
  
  
  
    Example  
    
    gap> d20 := DihedralGroup( IsPermGroup, 20 );;
    gap> gend20 := GeneratorsOfGroup( d20 ); 
    [ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ]
    gap> p1 := gend20[1];;  p2 := gend20[2];;  p12 := p1*p2; 
    (1,10)(2,9)(3,8)(4,7)(5,6)
    gap> d10a := Subgroup( d20, [ p1^2, p2 ] );;
    gap> d10b := Subgroup( d20, [ p1^2, p12 ] );;
    gap> c5d := Subgroup( d20, [ p1^2 ] );;
    gap> SetName( d20, "d20" );  SetName( d10a, "d10a" ); 
    gap> SetName( d10b, "d10b" );  SetName( c5d, "c5d" ); 
    gap> XSconj := CrossedSquareByNormalSubgroups( d20, d10b, d10a, c5d );
    [  c5d -> d10b ]
    [   |      |   ]
    [ d10a -> d20  ]
    
    gap> Name( XSconj );
    "[c5d->d10b,d10a->d20]"
    gap> XStrans := Transpose3dGroup( XSconj ); 
    [  c5d -> d10a ]
    [   |      |   ]
    [ d10b -> d20  ]
    
    gap> X20 := XModByNormalSubgroup( d20, d10a );
    [d10a->d20]
    gap> XSact := ActorCrossedSquare( X20 );
    crossed square with:
          up = Whitehead[d10a->d20]
        left = [d10a->d20]
        down = Norrie[d10a->d20]
       right = Actor[d10a->d20]
    
  
  
  8.2-2 CentralQuotient
  
  CentralQuotient( X0 )  attribute
  
  The  central  quotient  of  a  crossed module mathcalX = (∂ : S -> R) is the
  crossed square where:
  
      the left crossed module is mathcalX;
  
      the  right  crossed  module  is the quotient mathcalX/Z(mathcalX) (see
        CentreXMod (4.1-7));
  
      the  top  and  bottom homomorphisms are the natural homomorphisms onto
        the quotient groups;
  
      the  crossed pairing ⊠ : (R × F) -> S, where F = Fix(mathcalX,S,R), is
        the  displacement  element  ⊠(r,Fs)  =  ⟨  r,s  ⟩ = (s^-1)^rsquad (see
        Displacement (4.1-3) and section 4.3).
  
  This     is     the     special     case    of    an    intended    function
  CrossedSquareByCentralExtension  which  has not yet been implemented. In the
  example Xn7 ⊴ X24, constructed in section 4.1.
  
    Example  
    
    gap> pos7 := Position( ids, [ [12,2], [24,5] ] );;
    gap> Xn7 := nsx[pos7]; 
    [Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )]
    gap> IdGroup( CentreXMod(Xn7) );  
    [ [ 4, 1 ], [ 4, 1 ] ]
    gap> CQXn7 := CentralQuotient( Xn7 );
    crossed square with:
          up = [Group( [ f2, f3, f4 ] )->Group( [ f1 ] )]
        left = [Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )]
        down = [Group( [ f1, f2, f4, f5 ] )->Group( [ f1, f2 ] )]
       right = [Group( [ f1 ] )->Group( [ f1, f2 ] )]
    
    gap> IdGroup( CQXn7 );
    [ [ [ 12, 2 ], [ 3, 1 ] ], [ [ 24, 5 ], [ 6, 1 ] ] ]
    
  
  
  8.2-3 IsCrossedSquare
  
  IsCrossedSquare( obj )  property
  Is3dObject( obj )  property
  IsPerm3dObject( obj )  property
  IsPc3dObject( obj )  property
  IsFp3dObject( obj )  property
  IsPreCrossedSquare( obj )  property
  
  These  are  the  basic  properties  for  3d-groups,  and  crossed squares in
  particular.
  
  8.2-4 Up2DimensionalGroup
  
  Up2DimensionalGroup( XS )  attribute
  Left2DimensionalGroup( XS )  attribute
  Down2DimensionalGroup( XS )  attribute
  Right2DimensionalGroup( XS )  attribute
  DiagonalAction( XS )  attribute
  CrossedPairing( XS )  attribute
  ImageElmCrossedPairing( XS, pair )  operation
  
  In  this implementation the attributes used in the construction of a crossed
  square  XS  are  the  four  crossed  modules (2d-groups) on the sides of the
  square  (up;  down, left; and right); the diagonal action of P on L; and the
  crossed pairing.
  
  The  GAP  development  team  have  suggested that crossed pairings should be
  implemented  as  a  special case of BinaryMappings -- a structure which does
  not  yet  exist  in  GAP. As a temporary measure, crossed pairings have been
  implemented using Mapping2ArgumentsByFunction.
  
    Example  
    
    gap> Up2DimensionalGroup( XSconj );
    [c5d->d10b]
    gap> Right2DimensionalGroup( XSact );
    Actor[d10a->d20]
    gap> xpconj := CrossedPairing( XSconj );;
    gap> ImageElmCrossedPairing( xpconj, [ p2, p12 ] );
    (1,9,7,5,3)(2,10,8,6,4)
    gap> diag := DiagonalAction( XSact );
    [ (1,3,5,2,4)(6,10,14,8,12)(7,11,15,9,13), (1,2,5,4)(6,8,14,12)(7,11,13,9) 
     ] -> 
    [ (1,3,5,2,4)(6,10,14,8,12)(7,11,15,9,13), (1,2,5,4)(6,8,14,12)(7,11,13,9) 
     ] -> [ ^(1,3,5,7,9)(2,4,6,8,10), ^(1,2,5,4)(3,8)(6,7,10,9) ]
    
  
  
  
  8.3 Morphisms of crossed squares
  
  This   section   describes   an   initial  implementation  of  morphisms  of
  (pre-)crossed squares.
  
  8.3-1 Source
  
  Source( map )  attribute
  Range( map )  attribute
  Up2DimensionalMorphism( map )  attribute
  Left2DimensionalMorphism( map )  attribute
  Down2DimensionalMorphism( map )  attribute
  Right2DimensionalMorphism( map )  attribute
  
  Morphisms  of  3dObjects are implemented as 3dMappings. These have a pair of
  3d-groups  as  source  and  range,  together  with four 2d-morphisms mapping
  between  the four pairs of crossed modules on the four sides of the squares.
  These functions return fail when invalid data is supplied.
  
  8.3-2 IsCrossedSquareMorphism
  
  IsCrossedSquareMorphism( map )  property
  IsPreCrossedSquareMorphism( map )  property
  IsBijective( mor )  property
  IsEndomorphism3dObject( mor )  property
  IsAutomorphism3dObject( mor )  property
  
  A  morphism  mor  between  two pre-crossed squares mathcalS_1 and mathcalS_2
  consists  of  four  crossed  module  morphisms  Up2DimensionalMorphism(mor),
  mapping  the  Up2DimensionalGroup  of  mathcalS_1  to  that  of  mathcalS_2,
  Left2DimensionalMorphism(mor),       Down2DimensionalMorphism(mor)       and
  Right2DimensionalMorphism(mor). These four morphisms are required to commute
  with  the  four boundary maps and to preserve the rest of the structure. The
  current version of IsCrossedSquareMorphism does not perform all the required
  checks.
  
    Example  
    
    gap> ad20 := GroupHomomorphismByImages( d20, d20, [p1,p2], [p1,p2^p1] );;
    gap> ad10a := GroupHomomorphismByImages( d10a, d10a, [p1^2,p2], [p1^2,p2^p1] );;
    gap> ad10b := GroupHomomorphismByImages( d10b, d10b, [p1^2,p12], [p1^2,p12^p1] );;
    gap> idc5d := IdentityMapping( c5d );;
    gap> upconj := Up2DimensionalGroup( XSconj );;
    gap> leftconj := Left2DimensionalGroup( XSconj );; 
    gap> downconj := Down2DimensionalGroup( XSconj );; 
    gap> rightconj := Right2DimensionalGroup( XSconj );; 
    gap> up := XModMorphismByHoms( upconj, upconj, idc5d, ad10b );
    [[c5d->d10b] => [c5d->d10b]]
    gap> left := XModMorphismByHoms( leftconj, leftconj, idc5d, ad10a );
    [[c5d->d10a] => [c5d->d10a]]
    gap> down := XModMorphismByHoms( downconj, downconj, ad10a, ad20 );
    [[d10a->d20] => [d10a->d20]]
    gap> right := XModMorphismByHoms( rightconj, rightconj, ad10b, ad20 );
    [[d10b->d20] => [d10b->d20]]
    gap> autoconj := CrossedSquareMorphism( XSconj, XSconj, up, left, right, down );; 
    gap> ord := Order( autoconj );;
    gap> Display( autoconj );
    Morphism of crossed squares :- 
    :    Source = [c5d->d10b,d10a->d20]
    :     Range = [c5d->d10b,d10a->d20]
    :     order = 5
    :    up-left: [ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10) ], 
      [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10) ] ]
    :   up-right: 
    [ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6) ], 
      [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7) ] ]
    :  down-left: 
    [ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 2,10)( 3, 9)( 4, 8)( 5, 7) ], 
      [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1, 3)( 4,10)( 5, 9)( 6, 8) ] ]
    : down-right: 
    [ [ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10), ( 2,10)( 3, 9)( 4, 8)( 5, 7) ], 
      [ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10), ( 1, 3)( 4,10)( 5, 9)( 6, 8) ] ]
    gap> IsAutomorphismHigherDimensionalDomain( autoconj );
    true
    gap> KnownPropertiesOfObject( autoconj );
    [ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", 
      "IsSingleValued", "IsInjective", "IsSurjective", 
      "IsPreCrossedSquareMorphism", "IsCrossedSquareMorphism", 
      "IsEndomorphismHigherDimensionalDomain", 
      "IsAutomorphismHigherDimensionalDomain" ]
    
  
  
  
  8.4 Definitions and constructions for cat^2-groups and their morphisms
  
  We  shall  give  three  definitions  of  cat^2-groups and show that they are
  equivalent.  When we come to define cat^n-groups we shall give a similar set
  of three definitions.
  
  Firstly, we take the definition of a cat^2-group from Section 5 of Brown and
  Loday    [BL87],    suitably    modified.    A    cat^2-group   mathcalC   =
  (C_[2],C_{2},C_{1},C_∅)  comprises  four groups (one for each of the subsets
  of [2]) and 15 homomorphisms, as shown in the following diagram:
  
  
  \vcenter{\xymatrix{     &     C_{[2]}     \ar[ddd]     <-1.2ex>     \ar[ddd]
  <-2.0ex>_{\ddot{t}_2,\ddot{h}_2}       \ar[rrr]       <+1.2ex>      \ar[rrr]
  <+2.0ex>^{\ddot{t}_1,\ddot{h}_1}     \ar[dddrrr]     <-0.2ex>    \ar[dddrrr]
  <-1.0ex>_(0.55){t_{[2]},h_{[2]}}    &&&    C_{\{2\}}   \ar[lll]^{\ddot{e}_1}
  \ar[ddd]<+1.2ex>   \ar[ddd]  <+2.0ex>^{\dot{t}_2,\dot{h}_2}  \\  \mathcal{C}
  \quad  =  \quad & &&& \\ & &&& \\ & C_{\{1\}} \ar[uuu]_{\ddot{e}_2} \ar[rrr]
  <-1.2ex>    \ar[rrr]    <-2.0ex>_{\dot{t}_1,\dot{h}_1}   &&&   C_{\emptyset}
  \ar[uuu]^{\dot{e}_2}  \ar[lll]_{\dot{e}_1} \ar[uuulll] <-1.0ex>_{e_{[2]}} \\
  }}
  
  
  
  The following axioms are satisfied by these homomorphisms:
  
      the   four   sides   of   the   square   are   cat^1-groups,   denoted
        ddotmathcalC}_1, ddotmathcalC}_2, dotmathcalC}_1, dotmathcalC}_2,
  
      dott_1∘ddoth_2  = doth_2∘ddott_1, ~ dott_2∘ddoth_1 = doth_1∘ddott_2, ~
        dote_1∘dott_2  = ddott_2∘ddote_1, ~ dote_2∘dott_1 = ddott_1∘ddote_2, ~
        dote_1∘doth_2 = ddoth_2∘ddote_1, ~ dote_2∘doth_1 = ddoth_1∘ddote_2,
  
      dott_1∘ddott_2   =   dott_2∘ddott_1   =   t_[2],  ~  doth_1∘ddoth_2  =
        doth_2∘ddoth_1  =  h_[2],  ~  dote_1∘ddote_2 = dote_2∘ddote_1 = e_[2],
        making  the  diagonal  a  cat^1-group  (e_[2]; t_[2], h_[2] : C_[2] ->
        C_∅).
  
  It  follows from these identities that (ddott_1,dott_1),(ddoth_1,doth_1) and
  (ddote_1,dote_1) are morphisms of cat^1-groups.
  
  Secondly,  we  give  the  simplest  of  the  three definitions, adapted from
  Ellis-Steiner  [ES87].  A cat^2-group mathcalC consists of groups G, R_1,R_2
  and  six  homomorphisms t_1,h_1 : G -> R_2,~ e_1 : R_2 -> G,~ t_2,h_2 : G ->
  R_1,~  e_2  : R_1 -> G, satisfying the following axioms for all 1 leqslant i
  leqslant 2,
  
      (t_i  ∘  e_i)r  =  r,~ (h_i ∘ e_i)r = r,~ ∀ r ∈ R_[2] ∖ {i}, quad [ker
        t_i, ker h_i] = 1,
  
      (e_1  ∘  t_1)  ∘  (e_2 ∘ t_2) = (e_2 ∘ t_2) ∘ (e_1 ∘ t_1), quad (e_1 ∘
        h_1) ∘ (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ h_1),
  
      (e_1  ∘  t_1)  ∘  (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ t_1), quad (e_2 ∘
        t_2) ∘ (e_1 ∘ h_1) = (e_1 ∘ h_1) ∘ (e_2 ∘ t_2).
  
  Our   third   definition   defines   a  cat^2-group  as  a  "cat^1-group  of
  cat^1-groups".   A   cat^2-group   mathcalC  consists  of  two  cat^1-groups
  mathcalC_1  = (e_1;t_1,h_1 : G_1 -> R_1) and mathcalC_2 = (e_2;t_2,h_2 : G_2
  ->  R_2) and cat^1-morphisms t = (ddott,dott), h = (ddoth,doth) : mathcalC_1
  ->  mathcalC_2,  e = (ddote,dote) : mathcalC_2 -> mathcalC_1, subject to the
  following conditions:
  
  
  (t  \circ  e)  ~\mbox{and}~ (h \circ e) ~\mbox{are the identity mapping on}~
  \mathcal{C}_2, \qquad [\ker t, \ker h] = \{ 1_{\mathcal{C}_1} \},
  
  
  
  where ker t = (ker ddott, ker dott), and similarly for ker h.
  
  8.4-1 Cat2Group
  
  Cat2Group( args )  function
  PreCat2Group( args )  function
  PreCat2GroupByPreCat1Groups( L )  operation
  
  The  global  functions Cat2Group and PreCat2Group are normally called with a
  single argument, a list of cat1-groups.
  
    Example  
    
    gap> CC6 := Cat2Group( Cat1Group(6,2,2), Cat1Group(6,2,3) );
    generating (pre-)cat1-groups:
    1 : [C6=>Group( [ f1 ] )]
    2 : [C6=>Group( [ f2 ] )]
    
    gap> IsCat2Group( CC6 );
    true
    
  
  
  8.4-2 Cat2GroupOfCrossedSquare
  
  Cat2GroupOfCrossedSquare( xsq )  attribute
  CrossedSquareOfCat2Group( CC )  attribute
  
  These functions are very experimental, and should not be relied on!
  
  These  functions  provide  the conversion from crossed square to cat2-group,
  and  conversely.  (They are the 3-dimensional equivalents of Cat1GroupOfXMod
  and XModOfCat1Group.)
  
    Example  
    
    gap> xsCC6 := CrossedSquareOfCat2Group( CC6 );
    crossed square with:
          up = [Group( () )->Group( [ (1,2) ] )]
        left = [Group( () )->Group( [ (), (3,4,5) ] )]
        down = [Group( [ (), (3,4,5) ] ) -> Group( () )]
       right = [Group( [ (1,2) ] ) -> Group( () )]
    gap> Cat2GroupOfCrossedSquare( XSact );
    Warning: these conversion functions are still under development
    fail
    
  
  
  
  8.5 Definition and constructions for cat^n-groups and their morphisms
  
  In  this  chapter we are interested in cat^2-groups, but it is convenient in
  this section to give the more general definition. There are three equivalent
  description of a cat^n-group.
  
  A cat^n-group consists of the following.
  
      2^n  groups  G_A,  one  for  each  subset A of [n], the vertices of an
        n-cube.
  
      Group homomorphisms forming n2^n-1 commuting cat^1-groups,
  
  
        \mathcal{C}_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\ G_A \to G_{A
        \setminus  \{i\}}),  \quad\mbox{for all} \quad A \subseteq [n],~ i \in
        A,
  
  
  
        the edges of the cube.
  
      These  cat^1-groups  combine  (in  sets  of  4)  to  form  n(n-1)2^n-3
        cat^2-groups  mathcalC_A,{i,j}  for  all  {i,j} ⊆ A ⊆ [n],~ i ≠ j, the
        faces of the cube.
  
  Note that, since the t_A,i, h_A,i and e_A,i commute, composite homomorphisms
  t_A,B,  h_A,B  :  G_A -> G_A ∖ B and e_A,B : G_A ∖ B -> G_A are well defined
  for all B ⊆ A ⊆ [n].
  
  Secondly, we give the simplest of the three descriptions, again adapted from
  Ellis-Steiner [ES87].
  
  A  cat^n-group mathcalC consists of 2^n groups G_A, one for each subset A of
  [n], and 3n homomorphisms
  
  
  t_{[n],i},  h_{[n],i}  :  G_{[n]}  \to G_{[n] \setminus \{i\}},~ e_{[n],i} :
  G_{[n] \setminus \{i\}} \to G_{[n]},
  
  
  
  satisfying the following axioms for all 1 leqslant i leqslant n,}
  
      the  mathcalC_[n],i  ~=~ (e_[n],i; t_[n],i, h_[n],i : G_[n] -> G_[n] ∖
        {i})~ are commuting cat^1-groups, so that:
  
      (e_1  ∘  t_1)  ∘  (e_2 ∘ t_2) = (e_2 ∘ t_2) ∘ (e_1 ∘ t_1), quad (e_1 ∘
        h_1) ∘ (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ h_1),
  
      (e_1  ∘  t_1)  ∘  (e_2 ∘ h_2) = (e_2 ∘ h_2) ∘ (e_1 ∘ t_1), quad (e_2 ∘
        t_2) ∘ (e_1 ∘ h_1) = (e_1 ∘ h_1) ∘ (e_2 ∘ t_2).
  
  Our   third   description   defines  a  cat^n-group  as  a  "cat^1-group  of
  cat^(n-1)-groups".
  
  A cat^n-group mathcalC consists of two cat^(n-1)-groups:
  
      mathcalA  with  groups  G_A,  A  ⊆ [n-1], and homomorphisms ddott_A,i,
        ddoth_A,i, ddote_A,i,
  
      mathcalB  with  groups  H_B,  B  ⊆  [n-1], and homomorphisms dott_B,i,
        doth_B,i, dote_B,i, and
  
      cat^(n-1)-morphisms  t,h  :  mathcalA  -> mathcalB and e : mathcalB ->
        mathcalA subject to the following conditions:
  
  
        (t  \circ  e) ~\mbox{and}~ (h \circ e) ~\mbox{are the identity mapping
        on}~ \mathcal{B}, \qquad [\ker t, \ker h] = \{ 1_{\mathcal{A}} \}.