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3 2d-mappings
 3.1 Morphisms of 2d-objects
  3.1-1 Source
 3.2 Morphisms of pre-crossed modules
  3.2-1 IsXModMorphism

  3.2-2 IsInjective

  3.2-3 XModMorphism
 3.3 Morphisms of pre-cat1-groups
  3.3-1 Cat1Morphism
 3.4 Operations on morphisms
  3.4-1 Order

  3.4-2 Kernel

3 2d-mappings

3.1 Morphisms of 2d-objects

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 2dObjects are implemented as 2dMappings. These have a pair of 2d-objects 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 )
> IsCat1Morphism( map )( property )
> IsPreXModMorphism( map )( property )
> IsPreCat1Morphism( 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 \sigma = \rho \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 )
> IsEndomorphism2dObject( map )( property )
> IsAutomorphism2dObject( 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 )
> InclusionMorphism2dObjects( X1, S1 )( operation )
> InnerAutomorphismXMod( X1, r )( operation )
> IdentityMapping( X1 )( attribute )
> IsomorphismPermObject( 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) ]
  [ (1,2,4,3) ]
: Range = Source
: Source Homomorphism maps source generators to:
  [ (5,9,8,7,6) ]
: Range Homomorphism maps range generators to:
  [ (1,2,4,3) ]
gap> IsAutomorphism2dObject( mor1 );
true 
gap> Print( RepresentationsOfObject(mor1), "\n" );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "Is2dMappingRep" ]
gap> Print( KnownPropertiesOfObject(mor1), "\n" );
[ "IsTotal", "IsSingleValued", "IsInjective", "IsSurjective", "Is2dMapping",
  "IsPreXModMorphism", "IsXModMorphism", "IsEndomorphism2dObject",
  "IsAutomorphism2dObject" ]
gap> Print( KnownAttributesOfObject(mor1), "\n" );
[ "Name", "Range", "Source", "SourceHom", "RangeHom" ]

3.3 Morphisms of pre-cat1-groups

A morphism of pre-cat1-groups from mathcalC_1 = (e_1;t_1,h_1 : G_1 -> R_1) to mathcalC_2 = (e_2;t_2,h_2 : G_2 -> R_2) is a pair (gamma, rho) where gamma : G_1 -> G_2 and rho : R_1 -> R_2 are homomorphisms satisfying

h_2 \gamma = \rho h_1, \qquad t_2 \gamma = \rho t_1, \qquad e_2 \rho = \gamma e_1.

3.3-1 Cat1Morphism
> Cat1Morphism( args )( function )
> Cat1MorphismByHoms( C1, C2, gamma, rho )( operation )
> PreCat1Morphism( args )( function )
> PreCat1MorphismByHoms( P1, P2, gamma, rho )( operation )
> InclusionMorphism2dObjects( C1, S1 )( operation )
> InnerAutomorphismCat1( C1, r )( operation )
> IdentityMapping( C1 )( attribute )
> IsmorphismPermObject( obj )( function )
> SmallerDegreePerm2dObject( obj )( function )

The global function IsomorphismPermObject calls IsomorphismPermPreCat1, 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 SmallerDegreePerm2dObject. Names are assigned automatically.


gap> iso2 := IsomorphismPermObject( C2 );
[[s3c4=>s3] => [Ps3c4=>Ps3]]
gap> Display( iso2 );
Morphism of cat1-groups :-
: Source = [s3c4=>s3] with generating sets:
  [ f1, f2, f3, f4 ]
  [ f1, f2 ]
:  Range = [Ps3c4=>Ps3] with generating sets:
  [ ( 5, 9)( 6,10)( 7,11)( 8,12), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12),
  ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)
    (11,12) ]
  [ (2,3), (1,2,3) ]
: Source Homomorphism maps source generators to:
  [ ( 5, 9)( 6,10)( 7,11)( 8,12), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12),
  ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)
    (11,12) ]
: Range Homomorphism maps range generators to:
  [ (2,3), (1,2,3) ]

3.4 Operations on morphisms

3.4-1 Order
> Order( auto )( attribute )
> CompositionMorphism( map2, map1 )( operation )

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


gap> Order( mor1 );
2
gap> GeneratorsOfGroup( d16 );
[ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ]
gap> d8 := Subgroup( d16, [ c^2, d ] );;
gap> c4 := Subgroup( d8, [ c^2 ] );;
gap> SetName( d8, "d8" );  SetName( c4, "c4" );
gap> X16 := XModByNormalSubgroup( d16, d8 );
[d8->d16]
gap> X8 := XModByNormalSubgroup( d8, c4 );
[c4->d8]
gap> IsSubXMod( X16, X8 );
true
gap> incd8 := InclusionMorphism2dObjects( X16, X8 );
[[c4->d8] => [d8->d16]]
gap> rho := GroupHomomorphismByImages( d16, d16, [c,d], [c,d^(c^2)] );;
gap> sigma := GroupHomomorphismByImages( d8, d8, [c^2,d], [c^2,d^(c^2)] );;
gap> mor := XModMorphismByHoms( X16, X16, sigma, rho );
[[d8->d16] => [d8->d16]]
gap> comp := incd8 * mor;
[[c4->d8] => [d8->d16]]
gap> comp = CompositionMorphism( mor, incd8 );
true

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) );;
gap> i2 := Subgroup( c2, [()] );;
gap> X9 := XModByNormalSubgroup( c2, i2 );;
gap> sigma9 := GroupHomomorphismByImages( c4, i2, [c^2], [()] );;
gap> rho9 := GroupHomomorphismByImages( d8, c2, [c^2,d], [(),(19,20)] );;
gap> mor9 := XModMorphism( X8, X9, sigma9, rho9 );
[[c4->d8] => [..]]
gap> K9 := Kernel( mor9 );
[Group( [ (11,13,15,17)(12,14,16,18) ] )->Group( [ (11,13,15,17)(12,14,16,18)
 ] )]
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