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### 9 Crossed cubes and Cat^3-groups

The term 4d-group refers to a set of equivalent categories of which the most common are the categories of crossed cubes and cat^3-groups. A 4d-mapping is a function between two 4d-groups which preserves all the structure.

The material in this chapter should be considered very experimental. As yet there are no functions for crossed cubes.

#### 9.1 Functions for (pre-)cat^3-groups

We shall use the following standard orientation of a cat^3-group calE on a group G. calE contains 8 groups; 12 cat^1-groups and 6 cat^2-groups forming the vertices; edges and faces of a cube, as shown in the following diagram.

\vcenter{\xymatrix{ & H \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[dr] <+0.6ex> & & & & N \ar[llll] <+0.6ex> \ar[dddd] <+0.6ex> \ar[dddd] <+0.0ex> \ar[dr] <+0.6ex> & \\ & & G \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> & & & & R \ar[llll] <+0.6ex> \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\ & & & & & & \\ & & & & & & \\ & M \ar[uuuu] <+0.6ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[dr] <+0.6ex> & & & & L \ar[uuuu] <+0.6ex> \ar[llll] <+0.6ex> \ar[dr] <+0.6ex> & \\ & & Q \ar[uuuu] <+0.6ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> & & & & P \ar[uuuu] <+0.6ex> \ar[llll] <+0.6ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\ }}

By definition, calE is generated by three commuting cat^1-groups (G ⇒ R), (G ⇒ Q) and (G ⇒ H), but it is more convenient to think of calE as generated by two cat^2-groups

• front(calE), generated by (G ⇒ R) and (G ⇒ Q);

• left(calE), generated by (G ⇒ Q) and (G ⇒ H).

Because the tail, head and embedding maps all commute, it follows that up(calE), generated by (G ⇒ H) and (G ⇒ R), is a third cat^2-group. The three remaining faces (cat^2-groups) right(calE), down(calE) and back(calE) are then easily constructed. We shall always use the order [front,up,left,right,down,back] for the six faces.

##### 9.1-1 Cat3Group
 ‣ Cat3Group( args ) ( function )
 ‣ PreCat3Group( args ) ( function )
 ‣ IsCat3Group( C ) ( property )
 ‣ PreCat3GroupByPreCat2Groups( L ) ( operation )

The global functions Cat3Group and PreCat3Group are normally take as arguments a pair of cat^2-groups or a trio of cat^1-groups. In subsection AllCat2GroupsIterator (8.5-2) the list of pairs CatnGroupLists(d12).pairs contains the three entries [6,8],[8,11] and [6,11]. It follows that the sixth, eighth and eleventh cat^1-groups for d12 generate a cat^3-group.


gap> all1 := AllCat1Groups( d12 );;
gap> C68 := Cat2Group( all1, all1 );;
gap> C811 := Cat2Group( all1, all1 );;
gap> C3Ga := Cat3Group( C68, C811 );
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
gap> C3Gb := Cat3Group( all1, all1, all1 );
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
gap> C3Ga = C3Gb;
true



##### 9.1-2 Front3DimensionalGroup
 ‣ Front3DimensionalGroup( C3 ) ( attribute )
 ‣ Up3DimensionalGroup( C3 ) ( attribute )
 ‣ Left3DimensionalGroup( C3 ) ( attribute )
 ‣ Right3DimensionalGroup( C3 ) ( attribute )
 ‣ Down3DimensionalGroup( C3 ) ( attribute )
 ‣ Back3DimensionalGroup( C3 ) ( attribute )

The six faces of a cat^3-group are stored as these attributes.


gap> C116 := Cat2Group( all1, all1 );;
gap> Up3DimensionalGroup( C3Ga ) = C116;
true



#### 9.2 Enumerating cat^3-groups with a given source

Once the list CatnGroupLists(G).pairs has been obtained we may seek all triples [i,j],[j,k] and [k,i] or [i,k] of pairs in this list and then, for each such triple, construct a cat^3-group generated by the i-th, j-th and k-th cat^1-group on G.

##### 9.2-1 AllCat3GroupTriples
 ‣ AllCat3GroupTriples( G ) ( operation )
 ‣ AllCat3GroupsNumber( G ) ( attribute )
 ‣ AllCat3Groups( G ) ( operation )
 ‣ AllCat3GroupsUpToIsomorphism( G ) ( operation )

The list of triples returned by the operation AllCat3GroupTriples is saved as CatnGroupLists(G).cat3triples. The length of this list is the number of cat^3-groups on G, and is saved as CatnGroupNumbers(G).cat3.

As yet there is no operation AllCat3GroupsUpToIsomorphism(G).


gap> triples := AllCat3GroupTriples( d12 );;
gap> CatnGroupNumbers( d12 ).cat3;
94
gap> triples;
[ 5, 7, 11 ]
gap> all1 := AllCat1Groups( d12 );;
gap> Cat3Group( all1, all1, all1 );
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]



#### 9.3 Definition and constructions for cat^n-groups and their morphisms

In this chapter and the previous one we are interested in cat^2-groups and cat^3-groups, and it is convenient in this section to give the more general definition. There are three equivalent descriptions 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,

\calC_{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 calC_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 calC 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 calC_[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 calC consists of two cat^(n-1)-groups:

• calA with groups G_A, A ⊆ [n-1], and homomorphisms ddott_A,i, ddoth_A,i, ddote_A,i,

• calB 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 : calA -> calB and e : calB -> calA subject to the following conditions:

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

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