Crossed squares and their morphisms
crossed module
cat2-group
3d-object

Crossed squares were introduced by Guin-Wal\'ery and Loday (see, for example, ) 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 .

A crossed square \mathcal{R} consists of the following: Groups R_J for each of the four subsets J \subseteq [2]; a commutative diagram of group homomorphisms: \ddot{\partial}_1 : R_{[2]} \to R_{\{2\}}, \quad \ddot{\partial}_2 : R_{[2]} \to R_{\{1\}}, \quad \dot{\partial}_1 : R_{\{1\}} \to R_{\emptyset}, \quad \dot{\partial}_2 : R_{\{2\}} \to R_{\emptyset}; actions of R_{\emptyset} on R_{\{1\}}, R_{\{2\}} and R_{[2]} which determine actions of R_{\{1\}} on R_{\{2\}} and R_{[2]} via \dot{\partial}_1 and actions of R_{\{2\}} on R_{\{1\}} and R_{[2]} via \dot{\partial}_2~; a function ~ \boxtimes : R_{\{1\}} \times R_{\{2\}} \to R_{[2]}~. The following axioms must be satisfied for all l \in R_{[2]},\; m,m_1,m_2 \in R_{\{1\}},\; n,n_1,n_2 \in R_{\{2\}},\; p \in R_{\emptyset}~: the homomorphisms \ddot{\partial}_1, \ddot{\partial}_2 preserve the action of R_{\emptyset}~; each of \ddot{\mathcal{R}}_1 = (\ddot{\partial}_1 : R_{[2]} \to R_{\{2\}}),~ \ddot{\mathcal{R}}_2 = (\ddot{\partial}_2 : R_{[2]} \to R_{\{1\}}), ~ \dot{\mathcal{R}}_1 = (\dot{\partial}_1 : R_{\{1\}} \to R_{\emptyset}),~ \dot{\mathcal{R}}_2 = (\dot{\partial}_2 : R_{\{2\}} \to R_{\emptyset}), and the diagonal \mathcal{R}_{12} = (\partial_{12} := \dot{\partial}_1\ddot{\partial}_2 = \dot{\partial}_2\ddot{\partial}_1 : R_{[2]} \to R_{\emptyset}) are crossed modules (with actions via R_{\emptyset}); crossed pairing \boxtimes is a \emph{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 R_{\{1\}} on R_{\{2\}} and R_{\{2\}} on R_{\{1\}} via R_{\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~.

Constructions for crossed squares

Analogously to the data structure used for crossed modules, crossed squares are implemented as 3d-objects. 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.

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, 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 \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 ActorXSq( X0 );. The transpose of \mathcal{R} is the crossed square \tilde{\mathcal{R}} obtained by interchanging R_{\{1\}} with R_{\{2\}}, \ddot{\partial}_1 with \ddot{\partial}_2, and \dot{\partial}_1 with \dot{\partial}_2. The crossed pairing is given by \tilde{\boxtimes} \;:\; R_{\{2\}} \times R_{\{1\}} \to R_{[2]}, \quad (n,m) \;\mapsto\; n\,\tilde{\boxtimes}\,m := (m \boxtimes n)^{-1}~. The following constructions will be implemented in the next release. If M, N are ordinary P-modules and A is an arbitrary abelian group on which P acts trivially, then there is a crossed square with sides 0 : A \to N,\quad 0 : A \to M,\quad 0 : M \to P,\quad 0 : N \to P. For a group L, the automorphism crossed module {\rm Act}\ L = (\iota : L \to {\rm Aut}\ L) splits to form the square with (\iota_1 : L \to {\rm Inn}\ L) on two sides, and (\iota_2 : {\rm Inn}\ L \to {\rm Aut}\ L) on the other two sides, where \iota_1 maps l \in L to the inner automorphism \beta_l : L \to L,\; l^{\prime} \mapsto l^{-1}l^{\prime}l, and $\iota_2$ is the inclusion of {\rm Inn}\ L in {\rm Aut}\ L. The actions are standard, and the crossed pairing is \boxtimes \;:\; {\rm Inn}\ L \times {\rm Inn}\ L \to L, \quad (\beta_l, \beta_{l^{\prime}}) \;\mapsto\; [l, l^{\prime}]~.
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 := Transpose3dObject( 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] right = Actor[s3a->d12] down = Norrie[s3a->d12]
These are the basic properties for 3dobjects, and crossed squares in particular.
In this implementation the attributes used in the construction of a crossed square XS are the four crossed modules (2d-objects) 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> Up2dObject( XSconj ); [c3->s3b] gap> Right2dObject( XSact ); Actor[s3a->d12] gap> xpconj := XPair( XSconj );; gap> ImageElmXPair( xpconj, [ (1,6)(2,5)(3,4), (2,6)(3,5) ] ); (1,3,5)(2,4,6) gap> diag := DiagonalAction( XSact ); [ (2,3)(6,8)(7,9), (1,2)(4,6)(5,7) ] -> [ [ (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) ] ]
Morphisms of crossed squares
morphism of 3d-object
crossed square morphism
3d-mapping
This section describes an initial implementation of morphisms of (pre-)crossed squares.

Morphisms of 3dObjects are implemented as 3dMappings. These have a pair of 3d-objects 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.
A morphism mor between two pre-crossed squares \mathcal{R}_{1} and \mathcal{R}_{2} consists of four crossed module morphisms Up2dMorphism( mor ), mapping the Up2dObject of \mathcal{R}_1 to that of \mathcal{R}_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 := Up2dObject( XSconj );; gap> leftconj := Left2dObject( XSconj );; gap> downconj := Down2dObject( XSconj );; gap> rightconj := Right2dObject( 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