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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 $$(G \Rightarrow R), (G \Rightarrow Q)$$ and $$(G \Rightarrow H)$$, but it is more convenient to think of $$\calE$$ as generated by two cat$$^2$$-groups

• front$$(\calE)$$, generated by $$(G \Rightarrow R)$$ and $$(G \Rightarrow Q)$$;

• left$$(\calE)$$, generated by $$(G \Rightarrow Q)$$ and $$(G \Rightarrow H)$$.

Because the tail, head and embedding maps all commute, it follows that up$$(\calE)$$, generated by $$(G \Rightarrow H)$$ and $$(G \Rightarrow R)$$, 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 (8.5-4) 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[6], all1[8] );;
gap> C811 := Cat2Group( all1[8], all1[11] );;
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[6], all1[8], all1[11] );
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[11], all1[6] );;
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 )

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[46];
[ 5, 7, 11 ]
gap> all1 := AllCat1Groups( d12 );;
gap> Cat3Group( all1[5], all1[7], all1[11] );
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 $$\{i,j\} \subseteq A \subseteq [n],~ i \neq j$$, 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 $$\calC$$ 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 $$\calC_{[n],i} ~=~ (e_{[n],i};\; t_{[n],i},\; h_{[n],i} \ :\ G_{[n]} \to G_{[n] \setminus \{i\}})~$$ 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 $$\calC$$ consists of two cat$$^{(n-1)}$$-groups:

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

• $$\calB$$ 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 $$t,h : \calA \to \calB$$ and $$e : \calB \to \calA$$ subject to the following conditions:

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

##### 9.3-1 PreCatnGroup
 ‣ PreCatnGroup( L ) ( operation )
 ‣ CatnGroup( L ) ( operation )

The operation (Pre)CatnGroup expects as input a list of cat$$^1$$-groups.


gap> PC4 := PreCatnGroup( [ all1[5], all1[7], all1[11], all1[12] ] );
(pre-)cat4-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) ] )]
4 : [d12 => Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] )]
gap> IsCatnGroup( PC4 );
true
gap> HigherDimension( PC4 );
5


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