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3 2d-mappings
 3.1 Morphisms of 2d-groups
 3.2 Morphisms of pre-crossed modules
 3.3 Morphisms of pre-cat1-groups
 3.4 Operations on morphisms

3 2d-mappings

3.1 Morphisms of 2d-groups

This chapter describes morphisms of (pre-)crossed modules and (pre-)cat1-groups.

3.1-1 Source
‣ Source( map )( attribute )
‣ Range( map )( attribute )
‣ SourceHom( map )( attribute )
‣ RangeHom( map )( attribute )

Morphisms of 2d-groups are implemented as 2d-mappings. These have a pair of 2d-groups as source and range, together with two group homomorphisms mapping between corresponding source and range groups. These functions return fail when invalid data is supplied.

3.2 Morphisms of pre-crossed modules

3.2-1 IsXModMorphism
‣ IsXModMorphism( map )( property )
‣ IsPreXModMorphism( map )( property )

A morphism between two pre-crossed modules \(\mathcal{X}_{1} = (\partial_1 : S_1 \to R_1)\) and \(\mathcal{X}_{2} = (\partial_2 : S_2 \to R_2)\) is a pair \((\sigma, \rho)\), where \(\sigma : S_1 \to S_2\) and \(\rho : R_1 \to R_2\) commute with the two boundary maps and are morphisms for the two actions:

\[ \partial_2 \circ \sigma ~=~ \rho \circ \partial_1, \qquad \sigma(s^r) ~=~ (\sigma s)^{\rho r}. \]

Thus \(\sigma\) is the SourceHom and \(\rho\) is the RangeHom. When \(\mathcal{X}_{1} = \mathcal{X}_{2}\) and \(\sigma, \rho\) are automorphisms then \((\sigma, \rho)\) is an automorphism of \(\mathcal{X}_1\). The group of automorphisms is denoted by \({\rm Aut}(\mathcal{X}_1 )\).

3.2-2 IsInjective
‣ IsInjective( map )( property )
‣ IsSurjective( map )( property )
‣ IsSingleValued( map )( property )
‣ IsTotal( map )( property )
‣ IsBijective( map )( property )
‣ IsEndo2dMapping( map )( property )

The usual properties of mappings are easily checked. It is usually sufficient to verify that both the SourceHom and the RangeHom have the required property.

3.2-3 XModMorphism
‣ XModMorphism( args )( function )
‣ XModMorphismByHoms( X1, X2, sigma, rho )( operation )
‣ PreXModMorphism( args )( function )
‣ PreXModMorphismByHoms( P1, P2, sigma, rho )( operation )
‣ InclusionMorphism2dDomains( X1, S1 )( operation )
‣ InnerAutomorphismXMod( X1, r )( operation )
‣ IdentityMapping( X1 )( attribute )
‣ IsomorphismPerm2dGroup( obj )( function )
‣ IsomorphismPc2dGroup( obj )( function )

These are the constructors for morphisms of pre-crossed and crossed modules.

In the following example we construct a simple automorphism of the crossed module X1 constructed in the previous chapter.


gap> sigma1 := GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ]
        [ (5,9,8,7,6) ] );;
gap> rho1 := IdentityMapping( Range( X1 ) );
IdentityMapping( PAut(c5) )
gap> mor1 := XModMorphism( X1, X1, sigma1, rho1 );
[[c5->PAut(c5))] => [c5->PAut(c5))]] 
gap> Display( mor1 );
Morphism of crossed modules :-
: Source = [c5->PAut(c5))] with generating sets:
  [ (5,6,7,8,9) ]
  [ (1,2,3,4) ]
: Range = Source
: Source Homomorphism maps source generators to:
  [ (5,9,8,7,6) ]
: Range Homomorphism maps range generators to:
  [ (1,2,3,4) ]
gap> IsAutomorphism2dDomain( mor1 );
true 
gap> Order( mor1 );
2
gap> RepresentationsOfObject( mor1 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "Is2dMappingRep" ]
gap> KnownPropertiesOfObject( mor1 );
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsTotal", 
  "IsSingleValued", "IsInjective", "IsSurjective", "RespectsMultiplication", 
  "IsPreXModMorphism", "IsXModMorphism", "IsEndomorphism2dDomain", 
  "IsAutomorphism2dDomain" ]
gap> KnownAttributesOfObject( mor1 );
[ "Name", "Order", "Range", "Source", "SourceHom", "RangeHom" ]

3.3 Morphisms of pre-cat1-groups

A morphism of pre-cat1-groups from \(\mathcal{C}_1 = (e_1;t_1,h_1 : G_1 \to R_1)\) to \(\mathcal{C}_2 = (e_2;t_2,h_2 : G_2 \to R_2)\) is a pair \((\gamma, \rho)\) where \(\gamma : G_1 \to G_2\) and \(\rho : R_1 \to R_2\) are homomorphisms satisfying

\[ h_2 \circ \gamma ~=~ \rho \circ h_1, \qquad t_2 \circ \gamma ~=~ \rho \circ t_1, \qquad e_2 \circ \rho ~=~ \gamma \circ e_1. \]

3.3-1 IsCat1Morphism
‣ IsCat1Morphism( map )( property )
‣ IsPreCat1Morphism( map )( property )
‣ Cat1Morphism( args )( function )
‣ Cat1MorphismByHoms( C1, C2, gamma, rho )( operation )
‣ PreCat1Morphism( args )( function )
‣ PreCat1MorphismByHoms( P1, P2, gamma, rho )( operation )
‣ InclusionMorphism2dDomains( C1, S1 )( operation )
‣ InnerAutomorphismCat1( C1, r )( operation )
‣ IdentityMapping( C1 )( attribute )
‣ SmallerDegreePerm2dDomain( obj )( function )

The global function IsomorphismPermObject calls IsomorphismPerm2dGroup, which constructs a morphism whose SourceHom and RangeHom are calculated using IsomorphismPermGroup on the source and range. Similarly SmallerDegreePermutationRepresentation is used on the two groups to obtain SmallerDegreePerm2dDomain. Names are assigned automatically.


gap> iso2 := IsomorphismPerm2dGroup( C2 );
[[G2=>d12] => [..]]
gap> Display( iso2 );
Morphism of cat1-groups :- 
: Source = [G2=>d12] with generating sets:
  [ f1, f2, f3, f4, f5, f6, f7 ]
  [ f1, f2, f3 ]
:  Range = P[G2=>d12] with generating sets:
  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), 
  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), 
  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)
    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)
    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)
    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), 
  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)
    (16,23,27) ]
  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ]
: Source Homomorphism maps source generators to:
  [ ( 6,12)( 8,15)( 9,16)(11,19)(13,26)(14,22)(17,27)(18,25)(20,21)(23,24), 
  ( 2, 3)( 5,10)( 9,16)(11,18)(17,23)(19,25)(24,27), 
  ( 4, 5, 7,10)( 6, 9,12,16)( 8,11,14,18)(13,17,20,23)(15,19,22,25)
    (21,24,26,27), ( 4, 6, 7,12)( 5, 9,10,16)( 8,13,14,20)(11,17,18,23)
    (15,21,22,26)(19,24,25,27), ( 4, 7)( 5,10)( 6,12)( 8,14)( 9,16)(11,18)
    (13,20)(15,22)(17,23)(19,25)(21,26)(24,27), ( 1, 2, 3), 
  ( 4, 8,15)( 5,11,19)( 6,13,21)( 7,14,22)( 9,17,24)(10,18,25)(12,20,26)
    (16,23,27) ]
: Range Homomorphism maps range generators to:
  [ (2,6)(3,5), (1,2,3,4,5,6), (1,3,5)(2,4,6) ]

3.4 Operations on morphisms

3.4-1 CompositionMorphism
‣ CompositionMorphism( map2, map1 )( operation )

Composition of morphisms (written (<map1> * <map2>) when maps act on the right) calls the CompositionMorphism function for maps (acting on the left), applied to the appropriate type of 2d-mapping.


gap> H2 := Subgroup(G2,[G2.3,G2.4,G2.6,G2.7]);  SetName( H2, "H2" );
Group([ f3, f4, f6, f7 ])
gap> c6 := Subgroup( d12, [b,c] );  SetName( c6, "c6" );
Group([ f2, f3 ])
gap> SC2 := Sub2dGroup( C2, H2, c6 );
[H2=>c6]
gap> IsCat1( SC2 );
true
gap> inc2 := InclusionMorphism2dDomains( C2, SC2 );
[[H2=>c6] => [G2=>d12]]
gap> CompositionMorphism( iso2, inc );                  
[[H2=>c6] => P[G2=>d12]]

3.4-2 Kernel
‣ Kernel( map )( operation )
‣ Kernel2dMapping( map )( attribute )

The kernel of a morphism of crossed modules is a normal subcrossed module whose groups are the kernels of the source and target homomorphisms. The inclusion of the kernel is a standard example of a crossed square, but these have not yet been implemented.


gap> c2 := Group( (19,20) );                                    
Group([ (19,20) ])
gap> X0 := XModByNormalSubgroup( c2, c2 );  SetName( X0, "X0" );
[Group( [ (19,20) ] )->Group( [ (19,20) ] )]
gap>  SX2 := Source( X2 );;
gap> genSX2 := GeneratorsOfGroup( SX2 ); 
[ f1, f4, f5, f7 ]
gap> sigma0 := GroupHomomorphismByImages(SX2,c2,genSX2,[(19,20),(),(),()]);
[ f1, f4, f5, f7 ] -> [ (19,20), (), (), () ]
gap> rho0 := GroupHomomorphismByImages(d12,c2,[a1,a2,a3],[(19,20),(),()]);
[ f1, f2, f3 ] -> [ (19,20), (), () ]
gap> mor0 := XModMorphism( X2, X0, sigma0, rho0 );;           
gap> K0 := Kernel( mor0 );
[Group( [ <identity> of ..., f4, f5, f7 ] )->Group( 
[ <identity> of ..., f2, f3 ] )]
 ] )]
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