  
  [1X8 [33X[0;0YCrossed 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. A [13X3d-mapping[113X
  is a function between two 3d-groups which preserves all the structure.[133X
  
  [33X[0;0YThe  material  in  this  chapter  should be considered experimental. A major
  overhaul  took  place  in  time  for  [5XXMod[105X version 2.73, with the names of a
  number of operations being changed.[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 (we often find it
        convenient to write [22XL = S_[2],~ M = S_{1},~ N = S_{2}[122X and [22XP = S_∅[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}_2  :  S_{\{2\}}  \to
        S_{\emptyset}, \quad \dot{\partial}_1 : S_{\{1\}} \to S_{\emptyset}[133X
  
  [124X
  
        [33X[0;6Y(again  we  often write [22Xκ = ddot∂_1,~ λ = ddot∂_2,~ μ = dot∂_2[122X and [22Xν =
        dot∂_1[122X);[133X
  
  [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}  && L
  \ar[rr]^{\kappa} \ar[dd]_{\lambda} && M \ar[dd]^{\mu} & \\ \mathcal{S} & = &
  &&  & = && \\ & & S_{\{1\}} \ar[rr]_{\dot{\partial}_1} && S_{\emptyset} && N
  \ar[rr]_{\nu} && P }}[133X
  
  [124X
  
  [33X[0;0YThe  following  axioms  must  be  satisfied  for  all  [22Xl ∈ L, m,m_1,m_2 ∈ M,
  n,n_1,n_2 ∈ N, p ∈ P[122X.[133X
  
  [30X    [33X[0;6YThe homomorphisms [22Xκ, λ[122X preserve the action of [22XP[122X.[133X
  
  [30X    [33X[0;6YEach  of  the  upper,  left-hand,  right-hand  and  lower sides of the
        square,[133X
  
  
  [24X      [33X[0;6Y\ddot{\mathcal{S}}_1  = (\kappa : L \to M), \quad \ddot{\mathcal{S}}_2
        =  (\lambda  :  L \to N), \quad \dot{\mathcal{S}}_2 = (\mu : M \to P),
        \quad \dot{\mathcal{S}}_1 = (\nu : N \to P),[133X
  
  [124X
  
        [33X[0;6Yand the diagonal[133X
  
  
  [24X      [33X[0;6Y\mathcal{S}_{12}  =  (\partial_{12}  :=  \mu  \circ \kappa = \nu \circ
        \lambda : L \to P)[133X
  
  [124X
  
        [33X[0;6Yare crossed modules (with actions via [22XP[122X).[133X
  
        [33X[0;6YThe  first  four  of  these  are  called  the [13Xup, left, right[113X and [13Xdown[113X
        crossed modules of [22XmathcalS[122X.[133X
  
  [30X    [33X[0;6Y[22X⊠[122X is a [13Xcrossed pairing[113X:[133X
  
        [30X    [33X[0;12Y[22X(n_1n_2 ⊠ m) = (n_1 ⊠ m)^n_2 (n_2 ⊠ m)[122X,[133X
  
        [30X    [33X[0;12Y[22X(n ⊠ m_1m_2) = (n ⊠ m_2) (n ⊠ m_1)^m_2[122X,[133X
  
        [30X    [33X[0;12Y[22X(n ⊠ m)^p = (n^p ⊠ m^p)[122X.[133X
  
  [30X    [33X[0;6Y[22Xddot∂_1  (n ⊠ m) = (m^-1)^n m quad mboxand quad ddot∂_2 (n ⊠ m) = n^-1
        n^m[122X.[133X
  
  [30X    [33X[0;6Y[22X(n  ⊠ ddot∂_1 l) = (l^-1)^n l quad mboxand quad (ddot∂_2 l ⊠ m) = l^-1
        l^m[122X.[133X
  
  [33X[0;0YNote that the actions of [22XM[122X on [22XN[122X and [22XN[122X on [22XM[122X via [22XP[122X are compatible since[133X
  
  
  [24X[33X[0;6Y{n_1}^{(m^n)}        \;=\;        {n_1}^{\dot{\partial}_2(m^n)}        \;=\;
  {n_1}^{n^{-1}(\dot{\partial}_2 m)n} \;=\; (({n_1}^{n^{-1}})^m)^n.[133X
  
  [124X
  
  [33X[0;0Y(A  [13Xprecrossed  square[113X is a similar structure which satisfies some subset of
  these axioms. This notion needs to be clarified.)[133X
  
  [33X[0;0YCrossed  squares  are the [22Xk=2[122X case of a crossed [22Xk[122X-cube of groups, defined as
  follows.  (This  is an attempt to translate Definition 2.1 in Ronnie Brown'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 [22Xk[122X[13X-cube of groups[113X consists of the following:[133X
  
  [30X    [33X[0;6Ygroups [22XS_A[122X for every subset [22XA ⊆ [k][122X;[133X
  
  [30X    [33X[0;6Ya commutative diagram of group homomorphisms [22X∂_i : S_A -> S_A ∖ {i}, i
        ∈ [k][122X; with composites [22X∂_B : S_A -> S_A ∖ B, B ⊆ [k][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 ⊆ [k][122X;[133X
  
  [30X    [33X[0;6Yfunctions [22X⊠_A,B : S_A × S_B -> S_A ∪ B, (A,B ⊆ [k])[122X.[133X
  
  [33X[0;0YThere is then a long list of axioms which must be satisfied.[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. There are also experimental implementations of
  cat[22X^2[122X-groups,  with  conversion  between  the  two  types of structure. Some
  standard  constructions  of  crossed squares are listed below. At present, a
  limited number of constructions is implemented. Morphisms of crossed squares
  have also been implemented, though there is still a lot to be done.[133X
  
  [1X8.2-1 CrossedSquareByNormalSubgroups[101X
  
  [33X[1;0Y[29X[2XCrossedSquareByNormalSubgroups[102X( [3XL[103X, [3XM[103X, [3XN[103X, [3XP[103X ) [32X operation[133X
  
  [33X[0;0YIf  [22XL, M, N[122X are normal subgroups of a group [22XP[122X, and [22X[M,N] leqslant L leqslant
  M  ∩ N[122X, then the four inclusions [22XL -> M,~ L -> N,~ 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  \;:\;  N  \times  M  \to  L,  \quad  (n,m)  \mapsto  [n,m]  \,=\,
  n^{-1}m^{-1}nm \,=\,(m^{-1})^nm \,=\, n^{-1}n^m\,.[133X
  
  [124X
  
  [33X[0;0YThis construction is implemented as [10XCrossedSquareByNormalSubgroups(L,M,N,P);[110X
  (note that the parent group come last).[133X
  
  [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( c5d, d10a, d10b, d20 );[127X[104X
    [4X[28X[  c5d -> d10a ][128X[104X
    [4X[28X[   |      |   ][128X[104X
    [4X[28X[ d10b -> d20  ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-2 CrossedSquareByNormalSubXMod[101X
  
  [33X[1;0Y[29X[2XCrossedSquareByNormalSubXMod[102X( [3XX0[103X, [3XX1[103X ) [32X operation[133X
  
  [33X[0;0YIf  [22XmathcalX_1  =  (∂_1  :  S_1  ->  R_1)[122X  is a normal sub-crossed module of
  [22XmathcalX_0  = (∂_0 : S_0 -> R_0)[122X then the inclusion morphism gives a crossed
  square with crossed pairing[133X
  
  
  [24X[33X[0;6Y\boxtimes   \;:\;   R_1   \times   S_0  \to  S_1,  \quad  (r_1,s_0)  \mapsto
  (s_0^{-1})^{r_1} s_0.[133X
  
  [124X
  
  [33X[0;0YThe   example  constructs  the  same  crossed  square  as  in  the  previous
  subsection.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XX20 := XModByNormalSubgroup( d20, d10a );; [127X[104X
    [4X[25Xgap>[125X [27XX10 := XModByNormalSubgroup( d10b, c5d );; [127X[104X
    [4X[25Xgap>[125X [27Xok := IsNormalSub2DimensionalDomain( X20, X10 ); [127X[104X
    [4X[28Xtrue [128X[104X
    [4X[25Xgap>[125X [27XXS20 := CrossedSquareByNormalSubXMod( X20, X10 ); [127X[104X
    [4X[28X[  c5d -> d10a ][128X[104X
    [4X[28X[   |      |   ][128X[104X
    [4X[28X[ d10b -> d20  ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-3 ActorCrossedSquare[101X
  
  [33X[1;0Y[29X[2XActorCrossedSquare[102X( [3XX0[103X ) [32X operation[133X
  
  [33X[0;0YThe  actor  [22XmathcalA(mathcalX_0)[122X  of  a  crossed  module [22XmathcalX_0[122X has been
  described in Chapter 5 (see [2XActorXMod[102X ([14X6.1-2[114X)). The crossed pairing is given
  by[133X
  
  
  [24X[33X[0;6Y\boxtimes \;:\; R \times W \,\to\, S, \quad (\chi,r) \,\mapsto\, \chi r~.[133X
  
  [124X
  
  [33X[0;0YThis is implemented as [10XActorCrossedSquare(X0);[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[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   right = Actor[d10a->d20][128X[104X
    [4X[28X    down = Norrie[d10a->d20][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-4 CrossedSquareByAutomorphismGroup[101X
  
  [33X[1;0Y[29X[2XCrossedSquareByAutomorphismGroup[102X( [3XG[103X ) [32X operation[133X
  
  [33X[0;0YFor [22XG[122X a group let [22XInn(G)[122X be its inner automorphism group and [22XAut(G)[122X its full
  automorphism   group.   Then   there   is   a  crossed  square  with  groups
  [22X[G,Inn(G),Inn(G),Aut(G)][122X  where the upper and left boundaries are the maps [22Xg
  ↦ ι_g[122X, where [22Xι_g[122X is conjugation of [22XG[122X by [22Xg[122X, and the right and down boundaries
  are inclusions. The crossed pairing is gived by [22Xι_g ⊠ ι_h = [g,h][122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XAXS20 := CrossedSquareByAutomorphismGroup( d20 );[127X[104X
    [4X[28X[      d20 -> Inn(d20) ][128X[104X
    [4X[28X[     |          |     ][128X[104X
    [4X[28X[ Inn(d20) -> Aut(d20) ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XStructureDescription( AXS20 );[127X[104X
    [4X[28X[ "D20", "D10", "D10", "C2 x (C5 : C4)" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-5 CrossedSquareByPullback[101X
  
  [33X[1;0Y[29X[2XCrossedSquareByPullback[102X( [3XX1[103X, [3XX2[103X ) [32X operation[133X
  
  [33X[0;0YIf  crossed  modules [22XmathcalX_1 = (ν : N -> P)[122X and [22XmathcalX_2 = (μ : M -> P)[122X
  have  a  common range [22XP[122X, let [22XL[122X be the pullback of [22X{ν,μ}[122X. Then [22XN[122X acts on [22XL[122X by
  [22X(n,m)^n'  =  (n^n',m^ν n')[122X, and [22XM[122X acts on [22XL[122X by [22X(n,m)^m' = (n^μ m', m^m')[122X. So
  [22X(π_1 : L -> N)[122X and [22X(π_2 : L -> M)[122X are crossed modules, where [22Xπ_1,π_2[122X are the
  two projections. The crossed pairing is given by:[133X
  
  
  [24X[33X[0;6Y\boxtimes  \;:\;  N  \times  M  \to L, \quad (n,m) \mapsto (n^{-1}n^{\mu m},
  (m^{-1})^{\nu n}m) .[133X
  
  [124X
  
  [33X[0;0YThe  second  example uses the central extension crossed module [10XX12=(D12->S3)[110X
  constructed  in  subsection  ([2XXModByCentralExtension[102X ([14X2.1-5[114X)), with pullback
  group [10XD12xC2[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdn := Down2DimensionalGroup( XSconj );;[127X[104X
    [4X[25Xgap>[125X [27Xrt := Right2DimensionalGroup( XSconj );;[127X[104X
    [4X[25Xgap>[125X [27XXSP := CrossedSquareByPullback( dn, rt ); [127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( XSP );                  [127X[104X
    [4X[28X[ "C5", "D10", "D10", "D20" ][128X[104X
    [4X[25Xgap>[125X [27XXS12 := CrossedSquareByPullback( X12, X12 );; [127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( XS12 );                  [127X[104X
    [4X[28X[ "C2 x C2 x S3", "D12", "D12", "S3" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-6 CrossedSquareByXModSplitting[101X
  
  [33X[1;0Y[29X[2XCrossedSquareByXModSplitting[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YFor  [22XmathcalX  = (∂ : S -> R)[122X let [22XQ[122X be the image of [22X∂[122X. Then [22X∂ = ∂' ∘ ι[122X where
  [22X∂'  : S -> Q[122X and [22Xι[122X is the inclusion of [22XQ[122X in [22XR[122X. The diagonal of the square is
  then the initial [22XmathcalX[122X, and the crossed pairing is given by commutators.[133X
  
  [33X[0;0YA particular case is when [22XS[122X is an [22XR[122X-module [22XA[122X and [22X∂[122X is the zero map.[133X
  
  
  [24X[33X[0;6Y\vcenter{\xymatrix{  &  &  S  \ar[rr]^{\partial'}  \ar[dd]_{\partial'}  && Q
  \ar[dd]^{\iota}  && A \ar[rr]^0 \ar[dd]_0 && 1 \ar[dd]^{\iota} & \\ & & && &
  && \\ & & Q \ar[rr]_{\iota} && R && 1 \ar[rr]_{\iota} && R }}[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xk4 := Group( (1,2), (3,4) );;[127X[104X
    [4X[25Xgap>[125X [27XAX4 := XModByAutomorphismGroup( k4 );;[127X[104X
    [4X[25Xgap>[125X [27XX4 := Image( IsomorphismPermObject( AX4 ) );;[127X[104X
    [4X[25Xgap>[125X [27XXSS4 := CrossedSquareByXModSplitting( X4 );;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( XSS4 );[127X[104X
    [4X[28X[ "C2 x C2", "1", "1", "S3" ][128X[104X
    [4X[25Xgap>[125X [27XXSS20 := CrossedSquareByXModSplitting( X20 );;[127X[104X
    [4X[25Xgap>[125X [27Xup20 := Up2DimensionalGroup( XSS20 );; [127X[104X
    [4X[25Xgap>[125X [27XRange( up20 ) = d10a; [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetName( Range( up20 ), "d10a" ); [127X[104X
    [4X[25Xgap>[125X [27XName( XSS20 ); [127X[104X
    [4X[28X"[d10a->d10a,d10a->d20]"[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-7 CrossedSquare[101X
  
  [33X[1;0Y[29X[2XCrossedSquare[102X( [3Xargs[103X ) [32X function[133X
  
  [33X[0;0YThe  function  [10XCrossedSquare[110X  may  be used to call some of the constructions
  described in the previous subsections.[133X
  
  [30X    [33X[0;6Y[10XCrossedSquare(X0)[110X calls [10XCrossedSquareByXModSplitting[110X.[133X
  
  [30X    [33X[0;6Y[10XCrossedSquare(C0)[110X calls [10XCrossedSquareOfCat2Group[110X.[133X
  
  [30X    [33X[0;6Y[10XCrossedSquare(X0,X1)[110X  calls  [10XCrossedSquareByPullback[110X  when  there is a
        common range.[133X
  
  [30X    [33X[0;6Y[10XCrossedSquare(X0,X1)[110X calls [10XCrossedSquareByNormalXMod[110X when [10XX1[110X is normal
        in [10XX0[110X .[133X
  
  [30X    [33X[0;6Y[10XCrossedSquare(L,M,N,P)[110X calls [10XCrossedSquareByNormalSubgroups[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xdiag := Diagonal2DimensionalGroup( AXS20 );[127X[104X
    [4X[28X[d20->Aut(d20)][128X[104X
    [4X[25Xgap>[125X [27XXSdiag := CrossedSquare( diag );;      [127X[104X
    [4X[25Xgap>[125X [27XStructureDescription( XSdiag );  [127X[104X
    [4X[28X[ "D20", "D10", "D10", "C2 x (C5 : C4)" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-8 Transpose3DimensionalGroup[101X
  
  [33X[1;0Y[29X[2XTranspose3DimensionalGroup[102X( [3XS0[103X ) [32X attribute[133X
  
  [33X[0;0YThe   [13Xtranspose[113X   of  a  crossed  square  [22XmathcalS[122X  is  the  crossed  square
  [22XtildemathcalS}[122X  obtained  by interchanging [22XM[122X with [22XN[122X, [22Xκ[122X with [22Xλ[122X, and [22Xν[122X with [22Xμ[122X.
  The crossed pairing is given by[133X
  
  
  [24X[33X[0;6Y\tilde{\boxtimes}   \;:\;   M  \times  N  \to  L,  \quad  (m,n)  \;\mapsto\;
  m\,\tilde{\boxtimes}\,n := (n \boxtimes m)^{-1}~.[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XXStrans := Transpose3DimensionalGroup( XSconj ); [127X[104X
    [4X[28X[  c5d -> d10b ][128X[104X
    [4X[28X[   |      |   ][128X[104X
    [4X[28X[ d10a -> d20  ][128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-9 CentralQuotient[101X
  
  [33X[1;0Y[29X[2XCentralQuotient[102X( [3XX0[103X ) [32X attribute[133X
  
  [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  up  and down 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[25Xgap>[125X [27XIdGroup( Xn7 );[127X[104X
    [4X[28X[ [ 12, 2 ], [ 24, 5 ] ][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[25Xgap>[125X [27XStructureDescription( CQXn7 );[127X[104X
    [4X[28X[ "C12", "C3", "C4 x S3", "S3" ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-10 IsCrossedSquare[101X
  
  [33X[1;0Y[29X[2XIsCrossedSquare[102X( [3Xobj[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIs3dObject[102X( [3Xobj[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPerm3dObject[102X( [3Xobj[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPc3dObject[102X( [3Xobj[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsFp3dObject[102X( [3Xobj[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPreCrossedSquare[102X( [3Xobj[103X ) [32X property[133X
  
  [33X[0;0YThese  are  the  basic  properties  for  3d-groups,  and  crossed squares in
  particular.[133X
  
  [1X8.2-11 Up2DimensionalGroup[101X
  
  [33X[1;0Y[29X[2XUp2DimensionalGroup[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XLeft2DimensionalGroup[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDown2DimensionalGroup[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRight2DimensionalGroup[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDiagonalAction[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XCrossedPairing[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDiagonal2DimensionalGroup[102X( [3XXS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XName[102X( [3XS0[103X ) [32X method[133X
  
  [33X[0;0YThese are the basic attributes of a crossed square [22XmathcalS[122X. The six objects
  used   in  the  construction  of  [22XmathcalS[122X  are  the  four  crossed  modules
  (2d-groups)  on  the  sides  of  the  square (up; left; right and down); the
  diagonal  action of [22XP[122X on [22XL[122X; and the crossed pairing [22X{M,N} -> L[122X. The diagonal
  crossed module [22X(L -> P)[122X is an additional attribute.[133X
  
  [33X[0;0YCrossed    pairings    have    been    implemented    using   an   operation
  [10XMapping2ArgumentsByFunction[110X. This encodes a map [22X{M,N} -> L[122X as a map [22XM × N ->
  L[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XUp2DimensionalGroup( XSconj );[127X[104X
    [4X[28X[c5d->d10a][128X[104X
    [4X[25Xgap>[125X [27XRight2DimensionalGroup( XSact );[127X[104X
    [4X[28XActor[d10a->d20][128X[104X
    [4X[25Xgap>[125X [27Xdiact := DiagonalAction( XSact );;[127X[104X
    [4X[25Xgap>[125X [27XImageElm( diact, (1,4)(2,3)(6,9)(7,8) );  [127X[104X
    [4X[28X^(1,5,7,3)(2,8,6,10)[128X[104X
    [4X[25Xgap>[125X [27XDiagonal2DimensionalGroup( XSconj );[127X[104X
    [4X[28X[c5d->d20][128X[104X
    [4X[25Xgap>[125X [27XName( XSconj ); [127X[104X
    [4X[28X"[c5d->d10a,d10b->d20]"[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.2-12 ImageElmCrossedPairing[101X
  
  [33X[1;0Y[29X[2XImageElmCrossedPairing[102X( [3XXS[103X, [3Xpair[103X ) [32X operation[133X
  
  [33X[0;0YThis  operation  returns  the  image  when  a  crossed pairing [22X{M,N} -> L[122X is
  applied to the pair [22X[m,n][122X with [22Xm ∈ M,~ n ∈ N[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xxp := CrossedPairing( XSconj );;                            [127X[104X
    [4X[25Xgap>[125X [27XImageElmCrossedPairing( xp,                                 [127X[104X
    [4X[25X>[125X [27X     [ (1,6)(2,5)(3,4)(7,10)(8,9), (1,5)(2,4)(6,9)(7,8) ] );[127X[104X
    [4X[28X(1,7,8,5,3)(2,9,10,6,4)[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 CrossedSquareMorphism[101X
  
  [33X[1;0Y[29X[2XCrossedSquareMorphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCrossedSquareMorphismByXModMorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCrossedSquareMorphismByGroupHomomorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xhoms[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreCrossedSquareMorphismByPreXModMorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreCrossedSquareMorphismByGroupHomomorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xhoms[103X ) [32X operation[133X
  
  [1X8.3-2 Source[101X
  
  [33X[1;0Y[29X[2XSource[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRange[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XUp2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XLeft2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDown2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRight2DimensionalMorphism[102X( [3Xmap[103X ) [32X attribute[133X
  
  [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
  
  [1X8.3-3 IsCrossedSquareMorphism[101X
  
  [33X[1;0Y[29X[2XIsCrossedSquareMorphism[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsPreCrossedSquareMorphism[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsBijective[102X( [3Xmor[103X ) [32X method[133X
  [33X[1;0Y[29X[2XIsEndomorphism3dObject[102X( [3Xmor[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsAutomorphism3dObject[102X( [3Xmor[103X ) [32X property[133X
  
  [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,       [10XRight2DimensionalMorphism(mor)[110X      and
  [10XDown2DimensionalMorphism(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 [27Xup := Up2DimensionalGroup( XSconj );;[127X[104X
    [4X[25Xgap>[125X [27Xlt := Left2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xrt := Right2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xdn := Down2DimensionalGroup( XSconj );; [127X[104X
    [4X[25Xgap>[125X [27Xmup := XModMorphismByGroupHomomorphisms( up, up, idc5d, ad10a );[127X[104X
    [4X[28X[[c5d->d10a] => [c5d->d10a]][128X[104X
    [4X[25Xgap>[125X [27Xmlt := XModMorphismByGroupHomomorphisms( lt, lt, idc5d, ad10b );[127X[104X
    [4X[28X[[c5d->d10b] => [c5d->d10b]][128X[104X
    [4X[25Xgap>[125X [27Xmrt := XModMorphismByGroupHomomorphisms( rt, rt, ad10a, ad20 );[127X[104X
    [4X[28X[[d10a->d20] => [d10a->d20]][128X[104X
    [4X[25Xgap>[125X [27Xmdn := XModMorphismByGroupHomomorphisms( dn, dn, ad10b, ad20 );[127X[104X
    [4X[28X[[d10b->d20] => [d10b->d20]][128X[104X
    [4X[25Xgap>[125X [27Xautoconj := CrossedSquareMorphism( XSconj, XSconj, [mup,mlt,mrt,mdn] );; [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->d10a,d10b->d20][128X[104X
    [4X[28X: Range = [c5d->d10a,d10b->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), ( 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-left: [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-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
  
  [1X8.3-4 InclusionMorphismHigherDimensionalDomains[101X
  
  [33X[1;0Y[29X[2XInclusionMorphismHigherDimensionalDomains[102X( [3Xobj[103X, [3Xsub[103X ) [32X operation[133X
  
  
  [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 pre-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
  
  [33X[1;0Y[29X[2XCat2Group[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPreCat2Group[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XIsCat2Group[102X( [3XC[103X ) [32X property[133X
  [33X[1;0Y[29X[2XPreCat2GroupByPreCat1Groups[102X( [3XL[103X ) [32X operation[133X
  
  [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 [27Xa := (1,2,3,4)(5,6,7,8);;[127X[104X
    [4X[25Xgap>[125X [27Xb := (1,5)(2,6)(3,7)(4,8);; [127X[104X
    [4X[25Xgap>[125X [27Xc := (2,6)(4,8);;[127X[104X
    [4X[25Xgap>[125X [27XG16 := Group( a, b, c );; [127X[104X
    [4X[25Xgap>[125X [27XSetName( G16, "c4c2:c2" );[127X[104X
    [4X[25Xgap>[125X [27Xt1a := GroupHomomorphismByImages( G16, G16, [a,b,c], [(),(),c] );; [127X[104X
    [4X[25Xgap>[125X [27XC1a := PreCat1GroupByEndomorphisms( t1a, t1a );;[127X[104X
    [4X[25Xgap>[125X [27Xt1b := GroupHomomorphismByImages( G16, G16, [a,b,c], [a,(),()] );; [127X[104X
    [4X[25Xgap>[125X [27XC1b := PreCat1GroupByEndomorphisms( t1b, t1b );;[127X[104X
    [4X[25Xgap>[125X [27XC16 := Cat2Group( C1a, C1b );[127X[104X
    [4X[28Xcat2-group with generating (pre-)cat1-groups:[128X[104X
    [4X[28X1 : [c4c2:c2 => Group( [ (), (), (2,6)(4,8) ] )][128X[104X
    [4X[28X2 : [c4c2:c2 => Group( [ (1,2,3,4)(5,6,7,8), (), () ] )][128X[104X
    [4X[25Xgap>[125X [27XIsCat2Group( C16 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCat1Group( Diagonal2DimensionalGroup( C16 ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X8.4-2 Cat2GroupMorphism[101X
  
  [33X[1;0Y[29X[2XCat2GroupMorphism[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCat2GroupMorphismByCat1GroupMorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCat2GroupMorphismByGroupHomomorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreCat2GroupMorphismByPreCat1GroupMorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPreCat2GroupMorphismByGroupHomomorphisms[102X( [3Xsrc[103X, [3Xrng[103X, [3Xmors[103X ) [32X operation[133X
  
  [1X8.4-3 Cat2GroupOfCrossedSquare[101X
  
  [33X[1;0Y[29X[2XCat2GroupOfCrossedSquare[102X( [3Xxsq[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XCrossedSquareOfCat2Group[102X( [3XCC[103X ) [32X attribute[133X
  
  [33X[0;0Y[13XThese functions are very experimental![113X[133X
  
  [33X[0;0YThese  functions  provide  the conversion from crossed square to cat2-group,
  and  conversely.  (They are the 3-dimensional equivalents of [2XCat1GroupOfXMod[102X
  ([14X2.5-2[114X) and [2XXModOfCat1Group[102X ([14X2.5-2[114X).)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XxsC16 := CrossedSquareOfCat2Group( C16 );[127X[104X
    [4X[28Xcrossed square with crossed modules:[128X[104X
    [4X[28X      up = [Group( [ (1,5)(2,6)(3,7)(4,8) ] ) -> Group( [ ( 2, 6)( 4, 8) ] )][128X[104X
    [4X[28X    left = [Group( [ (1,5)(2,6)(3,7)(4,8) ] ) -> Group( [128X[104X
    [4X[28X[ (1,2,3,4)(5,6,7,8), (), () ] )][128X[104X
    [4X[28X   right = [Group( [ ( 2, 6)( 4, 8) ] ) -> Group( () )][128X[104X
    [4X[28X    down = [Group( [ (1,2,3,4)(5,6,7,8), (), () ] ) -> Group( () )][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIdGroup( xsC16 );[127X[104X
    [4X[28X[ [ 2, 1 ], [ 2, 1 ], [ 4, 1 ], [ 1, 1 ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSetName( Source( Right2DimensionalGroup( XSact ) ), "c5:c4" );[127X[104X
    [4X[25Xgap>[125X [27XSetName( Range( Right2DimensionalGroup( XSact ) ), "c5:c4" );[127X[104X
    [4X[25Xgap>[125X [27XName( XSact );[127X[104X
    [4X[28X"[d10a->c5:c4,d20->c5:c4]"[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XC2act := Cat2GroupOfCrossedSquare( XSact );             [127X[104X
    [4X[28Xcat2-group with generating (pre-)cat1-groups:[128X[104X
    [4X[28X1 : [((c5:c4 |X c5:c4) |X (d20 |X d10a))=>(c5:c4 |X c5:c4)][128X[104X
    [4X[28X2 : [((c5:c4 |X c5:c4) |X (d20 |X d10a))=>(c5:c4 |X d20)][128X[104X
    [4X[25Xgap>[125X [27XSize( C2act );[127X[104X
    [4X[28X[ 80000, 400, 400, 20 ][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
  
