  
  [1X8 [33X[0;0Y3d-groups and 3d-mappings : crossed squares and cat[22X^2[122X[101X[1X-groups[133X[101X
  
  [33X[0;0YThe term [13X3d-group[113X refers to a set of equivalent categories of which the most
  common are the categories of [13Xcrossed squares[113X and [13Xcat[22X^2[122X-groups[113X.[133X
  
  
  [1X8.1 [33X[0;0YDefinition of a crossed square and a crossed [22Xn[122X[101X[1X-cube of groups[133X[101X
  
  [33X[0;0YCrossed  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 [22X[n][122X the set [22X{1,2,...,n}[122X.
  We use the [22Xn=2[122X version of the definition of crossed [22Xn[122X-cube as given by Ellis
  and Steiner [ES87].[133X
  
  [33X[0;0YA [13Xcrossed square[113X [22XmathcalS[122X consists of the following:[133X
  
  [30X    [33X[0;6Ygroups [22XS_J[122X for each of the four subsets [22XJ ⊆ [2][122X;[133X
  
  [30X    [33X[0;6Ya commutative diagram of group homomorphisms:[133X
  
  
  [24X      [33X[0;6Y\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};[133X
  
  [124X
  
  [30X    [33X[0;6Yactions  of  [22XS_∅[122X  on [22XS_{1}, S_{2}[122X and [22XS_[2][122X which determine actions of
        [22XS_{1}[122X  on [22XS_{2}[122X and [22XS_[2][122X via [22Xdot∂_1[122X and actions of [22XS_{2}[122X on [22XS_{1}[122X and
        [22XS_[2][122X via [22Xdot∂_2[122X;[133X
  
  [30X    [33X[0;6Ya function [22X⊠ : S_{1} × S_{2} -> S_[2][122X.[133X
  
  [33X[0;0YHere is a picture of the situation:[133X
  
  
  [24X[33X[0;6Y\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} }} }}[133X
  
  [124X
  
  [33X[0;0YThe following axioms must be satisfied for all [22Xl ∈ S_[2], m,m_1,m_2 ∈ S_{1},
  n,n_1,n_2 ∈ S_{2}, p ∈ S_∅[122X:[133X
  
  [30X    [33X[0;6Ythe homomorphisms [22Xddot∂_1, ddot∂_2[122X preserve the action of [22XS_∅[122X;[133X
  
  [30X    [33X[0;6Yeach  of  the  upper,  left-hand,  lower,  and right-hand sides of the
        square,[133X
  
  
  [24X      [33X[0;6Y\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}),[133X
  
  [124X
  
        [33X[0;6Yand the diagonal[133X
  
  
  [24X      [33X[0;6Y\mathcal{S}_{12} = (\partial_{12} := \dot{\partial}_1\ddot{\partial}_2
        = \dot{\partial}_2\ddot{\partial}_1 : S_{[2]} \to S_{\emptyset})[133X
  
  [124X
  
        [33X[0;6Yare crossed modules (with actions via [22XS_∅[122X);[133X
  
  [30X    [33X[0;6Y[22X⊠[122X is a [13Xcrossed pairing[113X:[133X
  
        [30X    [33X[0;12Y[22X(m_1m_2 ⊠ n) = (m_1 ⊠ n)^m_2 (m_2 ⊠ n)[122X,[133X
  
        [30X    [33X[0;12Y[22X(m ⊠ n_1n_2) = (m ⊠ n_2) (m ⊠ n_1)^n_2[122X,[133X
  
        [30X    [33X[0;12Y[22X(m ⊠ n)^p = (m^p ⊠ n^p)[122X;[133X
  
  [30X    [33X[0;6Y[22Xddot∂_1  (m ⊠ n) = (n^-1)^m n quad mboxand quad ddot∂_2 (m ⊠ n) = m^-1
        m^n[122X,[133X
  
  [30X    [33X[0;6Y[22X(m  ⊠ ddot∂_1 l) = (l^-1)^m l quad mboxand quad (ddot∂_2 l ⊠ n) = l^-1
        l^n[122X.[133X
  
  [33X[0;0YNote  that  the  actions  of  [22XS_{1}[122X  on [22XS_{2}[122X and [22XS_{2}[122X on [22XS_{1}[122X via [22XS_∅[122X are
  compatible since[133X
  
  
  [24X[33X[0;6Y{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.[133X
  
  [124X
  
  [33X[0;0Y(A  [13Xprecrossed  square[113X is a similar structure which satisfies some subset of
  these axioms. [More needed here.])[133X
  
  [33X[0;0YIn  what  follows we shall generally use the following notation for the [22XS_J[122X,
  namely [22XL = S_[2];~ M = S_{1};~ N = S_{2}[122X and [22XP = S_∅[122X.[133X
  
  [33X[0;0YCrossed  squares  are the [22Xn=2[122X case of a crossed [22Xn[122X-cube of groups, defined as
  follows.  (This  is  an  attempt  to  translate  Definition  2.1 in Ronnie's
  [13XComputing  homotopy types using crossed n-cubes of groups[113X into right actions
  -- but this definition is not yet completely understood!)[133X
  
  [33X[0;0YA [13Xcrossed[113X [22Xn[122X[13X-cube of groups[113X consists of the following:[133X
  
  [30X    [33X[0;6Ygroups [22XS_A[122X for every subset [22XA ⊆ [n][122X;[133X
  
  [30X    [33X[0;6Ya commutative diagram of group homomorphisms [22X∂_i : S_A -> S_A ∖ {i}, i
        ∈ [n][122X; with composites [22X∂_B : S_A -> S_A ∖ B, B ⊆ [n][122X;[133X
  
  [30X    [33X[0;6Yactions  of  [22XS_∅[122X  on each [22XS_A[122X; and hence actions of [22XS_B[122X on [22XS_A[122X via [22X∂_B[122X
        for each [22XB ⊆ [n][122X;[133X
  
  [30X    [33X[0;6Yfunctions [22X⊠_A,B : S_A × S_B -> S_A ∪ B, (A,B ⊆ [n])[122X.[133X
  
  [33X[0;0YThe following axioms must be satisfied (long list to be added).[133X
  
  
  [1X8.2 [33X[0;0YConstructions for crossed squares[133X[101X
  
  [33X[0;0YAnalogously  to the data structure used for crossed modules, crossed squares
  are  implemented  as [10X3d-groups[110X. When times allows, cat[22X^2[122X-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.[133X
  
  [1X8.2-1 CrossedSquare[101X
  
  [29X[2XCrossedSquare[102X( [3Xargs[103X ) [32X function
  [29X[2XCrossedSquareByNormalSubgroups[102X( [3XP[103X, [3XN[103X, [3XM[103X, [3XL[103X ) [32X operation
  [29X[2XActorCrossedSquare[102X( [3XX0[103X ) [32X operation
  [29X[2XTranspose3dGroup[102X( [3XS0[103X ) [32X attribute
  [29X[2XName[102X( [3XS0[103X ) [32X attribute
  
  [33X[0;0YHere are some standard examples of crossed squares.[133X
  
  [30X    [33X[0;6YIf  [22XM,  N[122X  are  normal subgroups of a group [22XP[122X, and [22XL = M ∩ N[122X, then the
        four  inclusions, [22XL -> N,~ L -> M,~ M -> P,~ N -> P[122X, together with the
        actions of [22XP[122X on [22XM, N[122X and [22XL[122X given by conjugation, form a crossed square
        with crossed pairing[133X
  
  
  [24X      [33X[0;6Y\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\,.[133X
  
  [124X
  
        [33X[0;6YThis          construction          is          implemented         as
        [10XCrossedSquareByNormalSubgroups(P,N,M,L);[110X.[133X
  
  [30X    [33X[0;6YThe actor [22XmathcalA(mathcalX_0)[122X of a crossed module [22XmathcalX_0[122X has been
        described in Chapter 5. The crossed pairing is given by[133X
  
  
  [24X      [33X[0;6Y\boxtimes  \;:\; R \times W \,\to\, S, \quad (r,\chi) \,\mapsto\, \chi
        r~.[133X
  
  [124X
  
        [33X[0;6YThis is implemented as [10XActorCrossedSquare( X0 );[110X.[133X
  
  [30X    [33X[0;6YThe  [13Xtranspose[113X  of  [22XmathcalS[122X  is  the  crossed  square  [22XtildemathcalS}[122X
        obtained  by interchanging [22XS_{1}[122X with [22XS_{2}[122X, [22Xddot∂_1[122X with [22Xddot∂_2[122X, and
        [22Xdot∂_1[122X with [22Xdot∂_2[122X. The crossed pairing is given by[133X
  
  
  [24X      [33X[0;6Y\tilde{\boxtimes}  \;:\; S_{\{2\}} \times S_{\{1\}} \to S_{[2]}, \quad
        (n,m) \;\mapsto\; n\,\tilde{\boxtimes}\,m := (m \boxtimes n)^{-1}~.[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xd20 := DihedralGroup( IsPermGroup, 20 );;[127X[104X
    [4X[25Xgap>[125X [27Xgend20 := GeneratorsOfGroup( d20 ); [127X[104X
    [4X[28X[ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ][128X[104X
    [4X[25Xgap>[125X [27Xp1 := gend20[1];;  p2 := gend20[2];;  p12 := p1*p2; [127X[104X
    [4X[28X(1,10)(2,9)(3,8)(4,7)(5,6)[128X[104X
    [4X[25Xgap>[125X [27Xd10a := Subgroup( d20, [ p1^2, p2 ] );;[127X[104X
    [4X[25Xgap>[125X [27Xd10b := Subgroup( d20, [ p1^2, p12 ] );;[127X[104X
    [4X[25Xgap>[125X [27Xc5d := Subgroup( d20, [ p1^2 ] );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( d20, "d20" );  SetName( d10a, "d10a" ); [127X[104X
    [4X[25Xgap>[125X [27XSetName( d10b, "d10b" );  SetName( c5d, "c5d" ); [127X[104X
    [4X[25Xgap>[125X [27XXSconj := CrossedSquareByNormalSubgroups( d20, d10b, d10a, c5d );[127X[104X
    [4X[28X[  c5d -> d10b ][128X[104X
    [4X[28X[   |      |   ][128X[104X
    [4X[28X[ d10a -> d20  ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XName( XSconj );[127X[104X
    [4X[28X"[c5d->d10b,d10a->d20]"[128X[104X
    [4X[25Xgap>[125X [27XXStrans := Transpose3dGroup( XSconj ); [127X[104X
    [4X[28X[  c5d -> d10a ][128X[104X
    [4X[28X[   |      |   ][128X[104X
    [4X[28X[ d10b -> d20  ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XX20 := XModByNormalSubgroup( d20, d10a );[127X[104X
    [4X[28X[d10a->d20][128X[104X
    [4X[25Xgap>[125X [27XXSact := ActorCrossedSquare( X20 );[127X[104X
    [4X[28Xcrossed square with:[128X[104X
    [4X[28X      up = Whitehead[d10a->d20][128X[104X
    [4X[28X    left = [d10a->d20][128X[104X
    [4X[28X    down = Norrie[d10a->d20][128X[104X
    [4X[28X   right = Actor[d10a->d20][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 CentralQuotient[101X
  
  [29X[2XCentralQuotient[102X( [3XX0[103X ) [32X attribute
  
  [33X[0;0YThe  central  quotient  of  a  crossed module [22XmathcalX = (∂ : S -> R)[122X is the
  crossed square where:[133X
  
  [30X    [33X[0;6Ythe left crossed module is [22XmathcalX[122X;[133X
  
  [30X    [33X[0;6Ythe  right  crossed  module  is the quotient [22XmathcalX/Z(mathcalX)[122X (see
        [2XCentreXMod[102X ([14X4.1-7[114X));[133X
  
  [30X    [33X[0;6Ythe  top  and  bottom homomorphisms are the natural homomorphisms onto
        the quotient groups;[133X
  
  [30X    [33X[0;6Ythe  crossed pairing [22X⊠ : (R × F) -> S[122X, where [22XF = Fix(mathcalX,S,R)[122X, is
        the  displacement  element  [22X⊠(r,Fs)  =  ⟨  r,s  ⟩ = (s^-1)^rsquad[122X (see
        [2XDisplacement[102X ([14X4.1-3[114X) and section [14X4.3[114X).[133X
  
  [33X[0;0YThis     is     the     special     case    of    an    intended    function
  [10XCrossedSquareByCentralExtension[110X  which  has not yet been implemented. In the
  example [10XXn7[110X [22X⊴[122X [10XX24[110X, constructed in section [14X4.1[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xpos7 := Position( ids, [ [12,2], [24,5] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn7 := nsx[pos7]; [127X[104X
    [4X[28X[Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27XIdGroup( CentreXMod(Xn7) );  [127X[104X
    [4X[28X[ [ 4, 1 ], [ 4, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XCQXn7 := CentralQuotient( Xn7 );[127X[104X
    [4X[28Xcrossed square with:[128X[104X
    [4X[28X      up = [Group( [ f2, f3, f4 ] )->Group( [ f1 ] )][128X[104X
    [4X[28X    left = [Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )][128X[104X
    [4X[28X    down = [Group( [ f1, f2, f4, f5 ] )->Group( [ f1, f2 ] )][128X[104X
    [4X[28X   right = [Group( [ f1 ] )->Group( [ f1, f2 ] )][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIdGroup( CQXn7 );[127X[104X
    [4X[28X[ [ [ 12, 2 ], [ 3, 1 ] ], [ [ 24, 5 ], [ 6, 1 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-3 IsCrossedSquare[101X
  
  [29X[2XIsCrossedSquare[102X( [3Xobj[103X ) [32X property
  [29X[2XIs3dObject[102X( [3Xobj[103X ) [32X property
  [29X[2XIsPerm3dObject[102X( [3Xobj[103X ) [32X property
  [29X[2XIsPc3dObject[102X( [3Xobj[103X ) [32X property
  [29X[2XIsFp3dObject[102X( [3Xobj[103X ) [32X property
  [29X[2XIsPreCrossedSquare[102X( [3Xobj[103X ) [32X property
  
  [33X[0;0YThese  are  the  basic  properties  for  3d-groups,  and  crossed squares in
  particular.[133X
  
  [1X8.2-4 Up2DimensionalGroup[101X
  
  [29X[2XUp2DimensionalGroup[102X( [3XXS[103X ) [32X attribute
  [29X[2XLeft2DimensionalGroup[102X( [3XXS[103X ) [32X attribute
  [29X[2XDown2DimensionalGroup[102X( [3XXS[103X ) [32X attribute
  [29X[2XRight2DimensionalGroup[102X( [3XXS[103X ) [32X attribute
  [29X[2XDiagonalAction[102X( [3XXS[103X ) [32X attribute
  [29X[2XCrossedPairing[102X( [3XXS[103X ) [32X attribute
  [29X[2XImageElmCrossedPairing[102X( [3XXS[103X, [3Xpair[103X ) [32X operation
  
  [33X[0;0YIn  this implementation the attributes used in the construction of a crossed
  square  [10XXS[110X  are  the  four  crossed  modules (2d-groups) on the sides of the
  square  (up;  down, left; and right); the diagonal action of [22XP[122X on [22XL[122X; and the
  crossed pairing.[133X
  
  [33X[0;0YThe  [5XGAP[105X  development  team  have  suggested that crossed pairings should be
  implemented  as  a  special case of [10XBinaryMappings[110X -- a structure which does
  not  yet  exist  in  [5XGAP[105X. As a temporary measure, crossed pairings have been
  implemented using [10XMapping2ArgumentsByFunction[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XUp2DimensionalGroup( XSconj );[127X[104X
    [4X[28X[c5d->d10b][128X[104X
    [4X[25Xgap>[125X [27XRight2DimensionalGroup( XSact );[127X[104X
    [4X[28XActor[d10a->d20][128X[104X
    [4X[25Xgap>[125X [27Xxpconj := CrossedPairing( XSconj );;[127X[104X
    [4X[25Xgap>[125X [27XImageElmCrossedPairing( xpconj, [ p2, p12 ] );[127X[104X
    [4X[28X(1,9,7,5,3)(2,10,8,6,4)[128X[104X
    [4X[25Xgap>[125X [27Xdiag := DiagonalAction( XSact );[127X[104X
    [4X[28X[ (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) [128X[104X
    [4X[28X ] -> [128X[104X
    [4X[28X[ (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) [128X[104X
    [4X[28X ] -> [ ^(1,3,5,7,9)(2,4,6,8,10), ^(1,2,5,4)(3,8)(6,7,10,9) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YMorphisms of crossed squares[133X[101X
  
  [33X[0;0YThis   section   describes   an   initial  implementation  of  morphisms  of
  (pre-)crossed squares.[133X
  
  [1X8.3-1 Source[101X
  
  [29X[2XSource[102X( [3Xmap[103X ) [32X attribute
  [29X[2XRange[102X( [3Xmap[103X ) [32X attribute
  [29X[2XUp2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute
  [29X[2XLeft2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute
  [29X[2XDown2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute
  [29X[2XRight2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute
  
  [33X[0;0YMorphisms  of  [10X3dObjects[110X are implemented as [10X3dMappings[110X. 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 [10Xfail[110X when invalid data is supplied.[133X
  
  [1X8.3-2 IsCrossedSquareMorphism[101X
  
  [29X[2XIsCrossedSquareMorphism[102X( [3Xmap[103X ) [32X property
  [29X[2XIsPreCrossedSquareMorphism[102X( [3Xmap[103X ) [32X property
  [29X[2XIsBijective[102X( [3Xmor[103X ) [32X property
  [29X[2XIsEndomorphism3dObject[102X( [3Xmor[103X ) [32X property
  [29X[2XIsAutomorphism3dObject[102X( [3Xmor[103X ) [32X property
  
  [33X[0;0YA  morphism  [10Xmor[110X  between  two pre-crossed squares [22XmathcalS_1[122X and [22XmathcalS_2[122X
  consists  of  four  crossed  module  morphisms  [10XUp2DimensionalMorphism(mor)[110X,
  mapping  the  [10XUp2DimensionalGroup[110X  of  [22XmathcalS_1[122X  to  that  of  [22XmathcalS_2[122X,
  [10XLeft2DimensionalMorphism(mor)[110X,       [10XDown2DimensionalMorphism(mor)[110X       and
  [10XRight2DimensionalMorphism(mor)[110X. These four morphisms are required to commute
  with  the  four boundary maps and to preserve the rest of the structure. The
  current version of [10XIsCrossedSquareMorphism[110X does not perform all the required
  checks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xad20 := GroupHomomorphismByImages( d20, d20, [p1,p2], [p1,p2^p1] );;[127X[104X
    [4X[25Xgap>[125X [27Xad10a := GroupHomomorphismByImages( d10a, d10a, [p1^2,p2], [p1^2,p2^p1] );;[127X[104X
    [4X[25Xgap>[125X [27Xad10b := GroupHomomorphismByImages( d10b, d10b, [p1^2,p12], [p1^2,p12^p1] );;[127X[104X
    [4X[25Xgap>[125X [27Xidc5d := IdentityMapping( c5d );;[127X[104X
    [4X[25Xgap>[125X [27Xupconj := Up2DimensionalGroup( XSconj );;[127X[104X
    [4X[25Xgap>[125X [27Xleftconj := Left2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xdownconj := Down2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xrightconj := Right2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xup := XModMorphismByHoms( upconj, upconj, idc5d, ad10b );[127X[104X
    [4X[28X[[c5d->d10b] => [c5d->d10b]][128X[104X
    [4X[25Xgap>[125X [27Xleft := XModMorphismByHoms( leftconj, leftconj, idc5d, ad10a );[127X[104X
    [4X[28X[[c5d->d10a] => [c5d->d10a]][128X[104X
    [4X[25Xgap>[125X [27Xdown := XModMorphismByHoms( downconj, downconj, ad10a, ad20 );[127X[104X
    [4X[28X[[d10a->d20] => [d10a->d20]][128X[104X
    [4X[25Xgap>[125X [27Xright := XModMorphismByHoms( rightconj, rightconj, ad10b, ad20 );[127X[104X
    [4X[28X[[d10b->d20] => [d10b->d20]][128X[104X
    [4X[25Xgap>[125X [27Xautoconj := CrossedSquareMorphism( XSconj, XSconj, up, left, right, down );; [127X[104X
    [4X[25Xgap>[125X [27Xord := Order( autoconj );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( autoconj );[127X[104X
    [4X[28XMorphism of crossed squares :- [128X[104X
    [4X[28X:    Source = [c5d->d10b,d10a->d20][128X[104X
    [4X[28X:     Range = [c5d->d10b,d10a->d20][128X[104X
    [4X[28X:     order = 5[128X[104X
    [4X[28X:    up-left: [ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10) ], [128X[104X
    [4X[28X  [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10) ] ][128X[104X
    [4X[28X:   up-right: [128X[104X
    [4X[28X[ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6) ], [128X[104X
    [4X[28X  [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7) ] ][128X[104X
    [4X[28X:  down-left: [128X[104X
    [4X[28X[ [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 2,10)( 3, 9)( 4, 8)( 5, 7) ], [128X[104X
    [4X[28X  [ ( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10), ( 1, 3)( 4,10)( 5, 9)( 6, 8) ] ][128X[104X
    [4X[28X: down-right: [128X[104X
    [4X[28X[ [ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10), ( 2,10)( 3, 9)( 4, 8)( 5, 7) ], [128X[104X
    [4X[28X  [ ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10), ( 1, 3)( 4,10)( 5, 9)( 6, 8) ] ][128X[104X
    [4X[25Xgap>[125X [27XIsAutomorphismHigherDimensionalDomain( autoconj );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XKnownPropertiesOfObject( autoconj );[127X[104X
    [4X[28X[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", [128X[104X
    [4X[28X  "IsSingleValued", "IsInjective", "IsSurjective", [128X[104X
    [4X[28X  "IsPreCrossedSquareMorphism", "IsCrossedSquareMorphism", [128X[104X
    [4X[28X  "IsEndomorphismHigherDimensionalDomain", [128X[104X
    [4X[28X  "IsAutomorphismHigherDimensionalDomain" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X8.4 [33X[0;0YDefinitions and constructions for cat[22X^2[122X[101X[1X-groups and their morphisms[133X[101X
  
  [33X[0;0YWe  shall  give  three  definitions  of  cat[22X^2[122X-groups and show that they are
  equivalent.  When we come to define cat[22X^n[122X-groups we shall give a similar set
  of three definitions.[133X
  
  [33X[0;0YFirstly, we take the definition of a cat[22X^2[122X-group from Section 5 of Brown and
  Loday    [BL87],    suitably    modified.    A    cat[22X^2[122X-group   [22XmathcalC   =
  (C_[2],C_{2},C_{1},C_∅)[122X  comprises  four groups (one for each of the subsets
  of [22X[2][122X) and [22X15[122X homomorphisms, as shown in the following diagram:[133X
  
  
  [24X[33X[0;6Y\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]}} \\
  }}[133X
  
  [124X
  
  [33X[0;0YThe following axioms are satisfied by these homomorphisms:[133X
  
  [30X    [33X[0;6Ythe   four   sides   of   the   square   are   cat[22X^1[122X-groups,   denoted
        [22XddotmathcalC}_1, ddotmathcalC}_2, dotmathcalC}_1, dotmathcalC}_2[122X,[133X
  
  [30X    [33X[0;6Y[22Xdott_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,[122X[133X
  
  [30X    [33X[0;6Y[22Xdott_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],[122X
        making  the  diagonal  a  cat[22X^1[122X-group  [22X(e_[2]; t_[2], h_[2] : C_[2] ->
        C_∅)[122X.[133X
  
  [33X[0;0YIt  follows from these identities that [22X(ddott_1,dott_1),(ddoth_1,doth_1)[122X and
  [22X(ddote_1,dote_1)[122X are morphisms of cat[22X^1[122X-groups.[133X
  
  [33X[0;0YSecondly,  we  give  the  simplest  of  the  three definitions, adapted from
  Ellis-Steiner  [ES87].  A cat[22X^2[122X-group [22XmathcalC[122X consists of groups [22XG, R_1,R_2[122X
  and  six  homomorphisms [22Xt_1,h_1 : G -> R_2,~ e_1 : R_2 -> G,~ t_2,h_2 : G ->
  R_1,~  e_2  : R_1 -> G[122X, satisfying the following axioms for all [22X1 leqslant i
  leqslant 2[122X,[133X
  
  [30X    [33X[0;6Y[22X(t_i  ∘  e_i)r  =  r,~ (h_i ∘ e_i)r = r,~ ∀ r ∈ R_[2] ∖ {i}, quad [ker
        t_i, ker h_i] = 1,[122X[133X
  
  [30X    [33X[0;6Y[22X(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),[122X[133X
  
  [30X    [33X[0;6Y[22X(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).[122X[133X
  
  [33X[0;0YOur   third   definition   defines   a  cat[22X^2[122X-group  as  a  "cat[22X^1[122X-group  of
  cat[22X^1[122X-groups".   A   cat[22X^2[122X-group   [22XmathcalC[122X  consists  of  two  cat[22X^1[122X-groups
  [22XmathcalC_1  = (e_1;t_1,h_1 : G_1 -> R_1)[122X and [22XmathcalC_2 = (e_2;t_2,h_2 : G_2
  ->  R_2)[122X and cat[22X^1[122X-morphisms [22Xt = (ddott,dott), h = (ddoth,doth) : mathcalC_1
  ->  mathcalC_2,  e = (ddote,dote) : mathcalC_2 -> mathcalC_1[122X, subject to the
  following conditions:[133X
  
  
  [24X[33X[0;6Y(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} \},[133X
  
  [124X
  
  [33X[0;0Ywhere [22Xker t = (ker ddott, ker dott)[122X, and similarly for [22Xker h[122X.[133X
  
  [1X8.4-1 Cat2Group[101X
  
  [29X[2XCat2Group[102X( [3Xargs[103X ) [32X function
  [29X[2XPreCat2Group[102X( [3Xargs[103X ) [32X function
  [29X[2XPreCat2GroupByPreCat1Groups[102X( [3XL[103X ) [32X operation
  
  [33X[0;0YThe  global  functions [10XCat2Group[110X and [10XPreCat2Group[110X are normally called with a
  single argument, a list of cat1-groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCC6 := Cat2Group( Cat1Group(6,2,2), Cat1Group(6,2,3) );[127X[104X
    [4X[28Xgenerating (pre-)cat1-groups:[128X[104X
    [4X[28X1 : [C6=>Group( [ f1 ] )][128X[104X
    [4X[28X2 : [C6=>Group( [ f2 ] )][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsCat2Group( CC6 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.4-2 Cat2GroupOfCrossedSquare[101X
  
  [29X[2XCat2GroupOfCrossedSquare[102X( [3Xxsq[103X ) [32X attribute
  [29X[2XCrossedSquareOfCat2Group[102X( [3XCC[103X ) [32X attribute
  
  [33X[0;0Y[13XThese functions are very experimental, and should not be relied on![113X[133X
  
  [33X[0;0YThese  functions  provide  the conversion from crossed square to cat2-group,
  and  conversely.  (They are the 3-dimensional equivalents of [10XCat1GroupOfXMod[110X
  and [10XXModOfCat1Group[110X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XxsCC6 := CrossedSquareOfCat2Group( CC6 );[127X[104X
    [4X[28Xcrossed square with:[128X[104X
    [4X[28X      up = [Group( () )->Group( [ (1,2) ] )][128X[104X
    [4X[28X    left = [Group( () )->Group( [ (), (3,4,5) ] )][128X[104X
    [4X[28X    down = [Group( [ (), (3,4,5) ] ) -> Group( () )][128X[104X
    [4X[28X   right = [Group( [ (1,2) ] ) -> Group( () )][128X[104X
    [4X[25Xgap>[125X [27XCat2GroupOfCrossedSquare( XSact );[127X[104X
    [4X[28XWarning: these conversion functions are still under development[128X[104X
    [4X[28Xfail[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X8.5 [33X[0;0YDefinition and constructions for cat[22X^n[122X[101X[1X-groups and their morphisms[133X[101X
  
  [33X[0;0YIn  this  chapter we are interested in cat[22X^2[122X-groups, but it is convenient in
  this section to give the more general definition. There are three equivalent
  description of a cat[22X^n[122X-group.[133X
  
  [33X[0;0YA [13Xcat[22X^n[122X-group[113X consists of the following.[133X
  
  [30X    [33X[0;6Y[22X2^n[122X  groups  [22XG_A[122X,  one  for  each  subset [22XA[122X of [22X[n][122X, the [13Xvertices[113X of an
        [22Xn[122X-cube.[133X
  
  [30X    [33X[0;6YGroup homomorphisms forming [22Xn2^n-1[122X commuting cat[22X^1[122X-groups,[133X
  
  
  [24X      [33X[0;6Y\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,[133X
  
  [124X
  
        [33X[0;6Ythe [13Xedges[113X of the cube.[133X
  
  [30X    [33X[0;6YThese  cat[22X^1[122X-groups  combine  (in  sets  of  [22X4[122X)  to  form  [22Xn(n-1)2^n-3[122X
        cat[22X^2[122X-groups  [22XmathcalC_A,{i,j}[122X  for  all  [22X{i,j} ⊆ A ⊆ [n],~ i ≠ j[122X, the
        [13Xfaces[113X of the cube.[133X
  
  [33X[0;0YNote that, since the [22Xt_A,i, h_A,i[122X and [22Xe_A,i[122X commute, composite homomorphisms
  [22Xt_A,B,  h_A,B  :  G_A -> G_A ∖ B[122X and [22Xe_A,B : G_A ∖ B -> G_A[122X are well defined
  for all [22XB ⊆ A ⊆ [n][122X.[133X
  
  [33X[0;0YSecondly, we give the simplest of the three descriptions, again adapted from
  Ellis-Steiner [ES87].[133X
  
  [33X[0;0YA  cat[22X^n[122X-group [22XmathcalC[122X consists of [22X2^n[122X groups [22XG_A[122X, one for each subset [22XA[122X of
  [22X[n][122X, and [22X3n[122X homomorphisms[133X
  
  
  [24X[33X[0;6Yt_{[n],i},  h_{[n],i}  :  G_{[n]}  \to G_{[n] \setminus \{i\}},~ e_{[n],i} :
  G_{[n] \setminus \{i\}} \to G_{[n]},[133X
  
  [124X
  
  [33X[0;0Ysatisfying the following axioms for all [22X1 leqslant i leqslant n[122X,}[133X
  
  [30X    [33X[0;6Ythe  [22XmathcalC_[n],i  ~=~ (e_[n],i; t_[n],i, h_[n],i : G_[n] -> G_[n] ∖
        {i})~[122X are [13Xcommuting[113X cat[22X^1[122X-groups, so that:[133X
  
  [30X    [33X[0;6Y[22X(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),[122X[133X
  
  [30X    [33X[0;6Y[22X(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).[122X[133X
  
  [33X[0;0YOur   third   description   defines  a  cat[22X^n[122X-group  as  a  "cat[22X^1[122X-group  of
  cat[22X^(n-1)[122X-groups".[133X
  
  [33X[0;0YA [13Xcat[22X^n[122X-group[113X [22XmathcalC[122X consists of two cat[22X^(n-1)[122X-groups:[133X
  
  [30X    [33X[0;6Y[22XmathcalA[122X  with  groups  [22XG_A,  A  ⊆ [n-1][122X, and homomorphisms [22Xddott_A,i,
        ddoth_A,i, ddote_A,i[122X,[133X
  
  [30X    [33X[0;6Y[22XmathcalB[122X  with  groups  [22XH_B,  B  ⊆  [n-1][122X, and homomorphisms [22Xdott_B,i,
        doth_B,i, dote_B,i[122X, and[133X
  
  [30X    [33X[0;6Ycat[22X^(n-1)[122X-morphisms  [22Xt,h  :  mathcalA  -> mathcalB[122X and [22Xe : mathcalB ->
        mathcalA[122X subject to the following conditions:[133X
  
  
  [24X      [33X[0;6Y(t  \circ  e) ~\mbox{and}~ (h \circ e) ~\mbox{are the identity mapping
        on}~ \mathcal{B}, \qquad [\ker t, \ker h] = \{ 1_{\mathcal{A}} \}.[133X
  
  [124X
  
