[MathJax off]

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 \(\{1,2,\ldots,n\}\). We use the \(n=2\) version of the definition of crossed \(n\)-cube as given by Ellis and Steiner [ES87].

A crossed square \(\mathcal{S}\) consists of the following:

• groups \(S_J\) for each of the four subsets \(J \subseteq [2]\);

• 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 \(S_{\emptyset}\) on \(S_{\{1\}}, S_{\{2\}}\) and \(S_{[2]}\) which determine actions of \(S_{\{1\}}\) on \(S_{\{2\}}\) and \(S_{[2]}\) via \(\dot{\partial}_1\) and actions of \(S_{\{2\}}\) on \(S_{\{1\}}\) and \(S_{[2]}\) via \(\dot{\partial}_2\;\);

• a function \(\boxtimes : S_{\{1\}} \times S_{\{2\}} \to S_{[2]}\).

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 \(l \in S_{[2]},\; m,m_1,m_2 \in S_{\{1\}},\; n,n_1,n_2 \in S_{\{2\}},\; p \in S_{\emptyset}\):

• the homomorphisms \(\ddot{\partial}_1, \ddot{\partial}_2\) preserve the action of \(S_{\emptyset}\;\);

• 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 \(S_{\emptyset}\));

• \(\boxtimes\) is a crossed pairing:

  • \((m_1m_2 \boxtimes n)\;=\; {(m_1 \boxtimes n)}^{m_2}\;(m_2 \boxtimes n)\),

  • \((m \boxtimes n_1n_2) \;=\; (m \boxtimes n_2)\;{(m \boxtimes n_1)}^{n_2}\),

  • \((m \boxtimes n)^{p} \;=\; (m^p \boxtimes n^p)\);

• \(\ddot{\partial}_1 (m \boxtimes n) \;=\; (n^{-1})^{m}\;n \quad \mbox{and} \quad \ddot{\partial}_2 (m \boxtimes n) \;=\; m^{-1}\;m^{n}\),

• \((m \boxtimes \ddot{\partial}_1 l) \;=\; (l^{-1})^{m}\;l \quad \mbox{and} \quad (\ddot{\partial}_2 l \boxtimes n) \;=\; l^{-1}\;l^n\).

Note that the actions of \(S_{\{1\}}\) on \(S_{\{2\}}\) and \(S_{\{2\}}\) on \(S_{\{1\}}\) via \(S_{\emptyset}\) 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 \(P = S_{\emptyset}\).

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 \(A \subseteq [n]\);

• a commutative diagram of group homomorphisms \(\partial_i : S_A \to S_{A \setminus \{i\}},\; i \in [n]\); with composites \(\partial_B : S_A \to S_{A \setminus B},\; B \subseteq [n]\);

• actions of \(S_{\emptyset}\) on each \(S_A\); and hence actions of \(S_B\) on \(S_A\) via \(\partial_B\) for each \(B \subseteq [n]\);

• functions \(\boxtimes_{A,B} : S_A \times S_B \to S_{A \cup B}, (A,B \subseteq [n])\).

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

Here are some standard examples of crossed squares.

• If \(M, N\) are normal subgroups of a group \(P\), and \(L = M \cap N\), then the four inclusions, \(L \to N,~ L \to M,~ M \to P,~ N \to P\), 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 \(\mathcal{A}(\mathcal{X}_0)\) of a crossed module \(\mathcal{X}_0\) 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 \(\mathcal{S}\) is the crossed square \(\tilde{\mathcal{S}}\) obtained by interchanging \(S_{\{1\}}\) with \(S_{\{2\}}\), \(\ddot{\partial}_1\) with \(\ddot{\partial}_2\), and \(\dot{\partial}_1\) with \(\dot{\partial}_2\). The crossed pairing is given by

  \[ \tilde{\boxtimes} \;:\; S_{\{2\}} \times S_{\{1\}} \to S_{[2]}, \quad (n,m) \;\mapsto\; n\,\tilde{\boxtimes}\,m := (m \boxtimes n)^{-1}~. \]

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

The central quotient of a crossed module \(\mathcal{X} = (\partial : S \to R)\) is the crossed square where:

• the left crossed module is \(\mathcal{X}\);

• the right crossed module is the quotient \(\mathcal{X}/Z(\mathcal{X})\) (see CentreXMod (4.1-7));

• the top and bottom homomorphisms are the natural homomorphisms onto the quotient groups;

• the crossed pairing \(\boxtimes : (R \times F) \to S\), where \(F = \mathop{\textrm{Fix}\rm}(\mathcal{X},S,R)\), is the displacement element \(\boxtimes(r,Fs) = \langle r,s \rangle = (s^{-1})^rs\quad\) (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 \(\unlhd\) X24, constructed in section 4.1.

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

These are the basic properties for 3d-groups, and crossed squares in particular.

8.2-4 Up2DimensionalGroup

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.

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

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

A morphism mor between two pre-crossed squares \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) consists of four crossed module morphisms Up2DimensionalMorphism(mor), mapping the Up2DimensionalGroup of \(\mathcal{S}_1\) to that of \(\mathcal{S}_2\), 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.

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 \(\mathcal{C} = (C_{[2]},C_{\{2\}},C_{\{1\}},C_{\emptyset})\) 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 \(\ddot{\mathcal{C}}_1, \ddot{\mathcal{C}}_2, \dot{\mathcal{C}}_1, \dot{\mathcal{C}}_2\),

• \( \dot{t}_1\circ\ddot{h}_2 = \dot{h}_2\circ\ddot{t}_1, ~ \dot{t}_2\circ\ddot{h}_1 = \dot{h}_1\circ\ddot{t}_2, ~ \dot{e}_1\circ\dot{t}_2 = \ddot{t}_2\circ\ddot{e}_1, ~ \dot{e}_2\circ\dot{t}_1 = \ddot{t}_1\circ\ddot{e}_2, ~ \dot{e}_1\circ\dot{h}_2 = \ddot{h}_2\circ\ddot{e}_1, ~ \dot{e}_2\circ\dot{h}_1 = \ddot{h}_1\circ\ddot{e}_2, \)

• \( \dot{t}_1\circ\ddot{t}_2 = \dot{t}_2\circ\ddot{t}_1 = t_{[2]}, ~ \dot{h}_1\circ\ddot{h}_2 = \dot{h}_2\circ\ddot{h}_1 = h_{[2]}, ~ \dot{e}_1\circ\ddot{e}_2 = \dot{e}_2\circ\ddot{e}_1 = e_{[2]}, \) making the diagonal a cat\(^1\)-group \((e_{[2]}; t_{[2]}, h_{[2]} : C_{[2]} \to C_{\emptyset})\).

It follows from these identities that \((\ddot{t}_1,\dot{t}_1),\,(\ddot{h}_1,\dot{h}_1)\) and \((\ddot{e}_1,\dot{e}_1)\) are morphisms of cat\(^1\)-groups.

Secondly, we give the simplest of the three definitions, adapted from Ellis-Steiner [ES87]. A cat\(^2\)-group \(\mathcal{C}\) consists of groups \(G, R_1,R_2\) and six homomorphisms \(t_1,h_1 : G \to R_2,~ e_1 : R_2 \to G,~ t_2,h_2 : G \to R_1,~ e_2 : R_1 \to G\), satisfying the following axioms for all \(1 \leqslant i \leqslant 2\),

• \( (t_i \circ e_i)r = r,~ (h_i \circ e_i)r = r,~ \forall r \in R_{[2] \setminus \{i\}}, \quad [\ker t_i, \ker h_i] = 1, \)

• \( (e_1 \circ t_1) \circ (e_2 \circ t_2) = (e_2 \circ t_2) \circ (e_1 \circ t_1), \quad (e_1 \circ h_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ h_1), \)

• \( (e_1 \circ t_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ t_1), \quad (e_2 \circ t_2) \circ (e_1 \circ h_1) = (e_1 \circ h_1) \circ (e_2 \circ t_2). \)

Our third definition defines a cat\(^2\)-group as a "cat\(^1\)-group of cat\(^1\)-groups". A cat\(^2\)-group \(\mathcal{C}\) consists of two cat\(^1\)-groups \(\mathcal{C}_1 = (e_1;t_1,h_1 : G_1 \to R_1)\) and \(\mathcal{C}_2 = (e_2;t_2,h_2 : G_2 \to R_2)\) and cat\(^1\)-morphisms \(t = (\ddot{t},\dot{t}),\; h = (\ddot{h},\dot{h}) : \mathcal{C}_1 \to \mathcal{C}_2,\; e = (\ddot{e},\dot{e}) : \mathcal{C}_2 \to \mathcal{C}_1\), 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 \(\ker t = (\ker \ddot{t},\ \ker \dot{t})\), and similarly for \(\ker h\).

8.4-1 Cat2Group

The global functions Cat2Group and PreCat2Group are normally called with a single argument, a list of cat1-groups.

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

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.)

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 \(\mathcal{C}_{A,\{i,j\}}\) for all \(\{i,j\} \subseteq A \subseteq [n],~ i \neq j\), the faces of the cube.

Note that, since the \(t_{A,i}, h_{A,i}\) and \(e_{A,i}\) commute, composite homomorphisms \(t_{A,B}, h_{A,B} : G_A \to G_{A \setminus B}\) and \(e_{A,B} : G_{A \setminus B} \to G_A\) are well defined for all \(B \subseteq A \subseteq [n]\).

Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner [ES87].

A cat\(^n\)-group \(\mathcal{C}\) 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 \(\mathcal{C}_{[n],i} ~=~ (e_{[n],i};\; t_{[n],i},\; h_{[n],i} \ :\ G_{[n]} \to G_{[n] \setminus \{i\}})~\) are commuting cat\(^1\)-groups, so that:

• \( (e_1 \circ t_1) \circ (e_2 \circ t_2) = (e_2 \circ t_2) \circ (e_1 \circ t_1), \quad (e_1 \circ h_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ h_1), \)

• \( (e_1 \circ t_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ t_1), \quad (e_2 \circ t_2) \circ (e_1 \circ h_1) = (e_1 \circ h_1) \circ (e_2 \circ t_2). \)

Our third description defines a cat\(^n\)-group as a "cat\(^1\)-group of cat\(^{(n-1)}\)-groups".

A cat\(^n\)-group \(\mathcal{C}\) consists of two cat\(^{(n-1)}\)-groups:

• \(\mathcal{A}\) with groups \(G_A,\; A \subseteq [n-1]\), and homomorphisms \(\ddot{t}_{A,i}, \ddot{h}_{A,i}, \ddot{e}_{A,i}\),

• \(\mathcal{B}\) with groups \(H_B,\; B \subseteq [n-1]\), and homomorphisms \(\dot{t}_{B,i}, \dot{h}_{B,i}, \dot{e}_{B,i}\), and

• cat\(^{(n-1)}\)-morphisms \(t,h : \mathcal{A} \to \mathcal{B}\) and \(e : \mathcal{B} \to \mathcal{A}\) subject to the following conditions:

  \[ (t \circ e) ~\mbox{and}~ (h \circ e) ~\mbox{are the identity mapping on}~ \mathcal{B}, \qquad [\ker t, \ker h] = \{ 1_{\mathcal{A}} \}. \]

generated by GAPDoc2HTML