Crossed squares were introduced by Guin-Wal\'ery 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,...,n}. We use the n=2 version of the definition of crossed n-cube as given by Ellis and Steiner [ES87].
A crossed square mathcalS consists of the following:
Groups S_J for each of the four subsets J ⊆ [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_∅ on S_{1}, S_{2} and S_[2] which determine actions of S_{1} on S_{2} and S_[2] via dot∂_1 and actions of S_{2} on S_{1} and S_[2] via dot∂_2~;
a function ⊠ : S_{1} × S_{2} -> S_[2].
The following axioms must be satisfied for all l ∈ S_[2], m,m_1,m_2 ∈ S_{1}, n,n_1,n_2 ∈ S_{2}, p ∈ S_∅:
the homomorphisms ddot∂_1, ddot∂_2 preserve the action of S_∅~;
each of
\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_∅);
⊠ is a crossed pairing:
(m_1m_2 ⊠ n) = (m_1 ⊠ n)^m_2 (m_2 ⊠ n),
(m ⊠ n_1n_2) = (m ⊠ n_2) (m ⊠ n_1)^n_2,
(m ⊠ n)^p = (m^p ⊠ n^p);
ddot∂_1 (m ⊠ n) = (n^-1)^m n \quad \mbox{and} \quad ddot∂_2 (m ⊠ n) = m^-1 m^n,
(m ⊠ ddot∂_1 l) = (l^-1)^m l \quad \mbox{and} \quad (ddot∂_2 l ⊠ n) = l^-1 l^n.
Note that the actions of S_{1} on S_{2} and S_{2} on S_{1} via S_∅ 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.
Analogously to the data structure used for crossed modules, crossed squares are implemented as 3d-groups. When times allows, cat2-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 do.
‣ XSq( args ) | ( function ) |
‣ XSqByNormalSubgroups( P, N, M, L ) | ( operation ) |
‣ ActorXSq( X0 ) | ( operation ) |
‣ Transpose3dGroup( S0 ) | ( attribute ) |
‣ Name( S0 ) | ( attribute ) |
Here are some standard examples of crossed squares.
If M, N are normal subgroups of a group P, and L = M ∩ N, then the four inclusions, L -> N,~ L -> M,~ M -> P,~ N -> P, together with the actions of P on M, N and L given by conjugation, and the 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
is a crossed square. This construction is implemented as XSqByNormalSubgroups(P,N,M,L);.
The actor mathcalA(mathcalX_0) of a crossed module mathcalX_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 ActorXSq( X0 );.
The transpose of mathcalS is the crossed square tildemathcalS} obtained by interchanging S_{1} with S_{2}, ddot∂_1 with ddot∂_2, and dot∂_1 with dot∂_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> c := (11,12,13,14,15,16);; gap> d := (12,16)(13,15);; gap> cd := c*d;; gap> d12 := Group( [ c, d ] );; gap> s3a := Subgroup( d12, [ c^2, d ] );; gap> s3b := Subgroup( d12, [ c^2, cd ] );; gap> c3 := Subgroup( d12, [ c^2 ] );; gap> SetName( d12, "d12"); SetName( s3a, "s3a" ); gap> SetName( s3b, "s3b" ); SetName( c3, "c3" ); gap> XSconj := XSqByNormalSubgroups( d12, s3b, s3a, c3 ); [ c3 -> s3b ] [ | | ] [ s3a -> d12 ] gap> Name( XSconj ); "[c3->s3b,s3a->d12]" gap> XStrans := Transpose3dGroup( XSconj ); [ c3 -> s3a ] [ | | ] [ s3b -> d12 ] gap> X12 := XModByNormalSubgroup( d12, s3a ); [s3a->d12] gap> XSact := ActorXSq( X12 ); crossed square with: up = Whitehead[s3a->d12] left = [s3a->d12] down = Norrie[s3a->d12] right = Actor[s3a->d12]
‣ IsXSq( obj ) | ( property ) |
‣ Is3dObject( obj ) | ( property ) |
‣ IsPerm3dObject( obj ) | ( property ) |
‣ IsPc3dObject( obj ) | ( property ) |
‣ IsFp3dObject( obj ) | ( property ) |
‣ IsPreXSq( obj ) | ( property ) |
These are the basic properties for 3d-groups, and crossed squares in particular.
‣ Up2dGroup( XS ) | ( attribute ) |
‣ Left2dGroup( XS ) | ( attribute ) |
‣ Down2dGroup( XS ) | ( attribute ) |
‣ Right2dGroup( XS ) | ( attribute ) |
‣ DiagonalAction( XS ) | ( attribute ) |
‣ XPair( XS ) | ( attribute ) |
‣ ImageElmXPair( XS, pair ) | ( operation ) |
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; 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> Up2dGroup( XSconj ); [c3->s3b] gap> Right2dGroup( XSact ); Actor[s3a->d12] gap> xpconj := XPair( XSconj );; gap> ImageElmXPair( xpconj, [ (12,16)(13,15), (11,16)(12,15)(13,14) ] ); (11,15,13)(12,16,14) gap> diag := DiagonalAction( XSact ); [ (2,3)(5,6), (1,2)(4,6) ] -> [ [ (11,13,15)(12,14,16), (12,16)(13,15) ] -> [ (11,15,13)(12,16,14), (12,16)(13,15) ], [ (11,13,15)(12,14,16), (12,16)(13,15) ] -> [ (11,15,13)(12,16,14), (11,13)(14,16) ] ]
This section describes an initial implementation of morphisms of (pre-)crossed squares.
‣ Source( map ) | ( attribute ) |
‣ Range( map ) | ( attribute ) |
‣ Up2dMorphism( map ) | ( attribute ) |
‣ Left2dMorphism( map ) | ( attribute ) |
‣ Down2dMorphism( map ) | ( attribute ) |
‣ Right2dMorphism( map ) | ( attribute ) |
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.
‣ IsXSqMorphism( map ) | ( property ) |
‣ IsPreXSqMorphism( map ) | ( property ) |
‣ IsBijective( mor ) | ( property ) |
‣ IsEndomorphism3dObject( mor ) | ( property ) |
‣ IsAutomorphism3dObject( mor ) | ( property ) |
A morphism mor between two pre-crossed squares mathcalS_1 and mathcalS_2 consists of four crossed module morphisms Up2dMorphism( mor ), mapping the Up2dGroup of mathcalS_1 to that of mathcalS_2, Left2dMorphism( mor ), Down2dMorphism( mor ) and Right2dMorphism( 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 IsXSqMorphism does not perform all the required checks.
gap> ad12 := GroupHomomorphismByImages( d12, d12, [c,d], [c,d^c] );; gap> as3a := GroupHomomorphismByImages( s3a, s3a, [c^2,d], [c^2,d^c] );; gap> as3b := GroupHomomorphismByImages( s3b, s3b, [c^2,cd], [c^2,cd^c] );; gap> idc3 := IdentityMapping( c3 );; gap> upconj := Up2dGroup( XSconj );; gap> leftconj := Left2dGroup( XSconj );; gap> downconj := Down2dGroup( XSconj );; gap> rightconj := Right2dGroup( XSconj );; gap> up := XModMorphismByHoms( upconj, upconj, idc3, as3b ); [[c3->s3b] => [c3->s3b]] gap> left := XModMorphismByHoms( leftconj, leftconj, idc3, as3a ); [[c3->s3a] => [c3->s3a]] gap> down := XModMorphismByHoms( downconj, downconj, as3a, ad12 ); [[s3a->d12] => [s3a->d12]] gap> right := XModMorphismByHoms( rightconj, rightconj, as3b, ad12 ); [[s3b->d12] => [s3b->d12]] gap> autoconj := XSqMorphism( XSconj, XSconj, up, left, down, right );; gap> ord := Order( autoconj );; gap> Display( autoconj ); Morphism of crossed squares :- : Source = [c3->s3b,s3a->d12] : Range = [c3->s3b,s3a->d12] : order = 3 : up-left: [ [ (11,13,15)(12,14,16) ], [ (11,13,15)(12,14,16) ] ] : up-right: [ [ (11,13,15)(12,14,16), (11,16)(12,15)(13,14) ], [ (11,13,15)(12,14,16), (11,12)(13,16)(14,15) ] ] : down-left: [ [ (11,13,15)(12,14,16), (12,16)(13,15) ], [ (11,13,15)(12,14,16), (11,13)(14,16) ] ] : down-right: [ [ (11,12,13,14,15,16), (12,16)(13,15) ], [ (11,12,13,14,15,16), (11,13)(14,16) ] ] gap> KnownPropertiesOfObject( autoconj ); [ "IsTotal", "IsSingleValued", "IsInjective", "IsSurjective", "Is3dMapping", "IsPreXSqMorphism", "IsXSqMorphism", "IsEndomorphism3dObject" ] gap> IsAutomorphism3dObject( autoconj ); true
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