This chapter describes morphisms of (pre-)crossed modules and (pre-)cat1-groups.
> 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.
> 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 ${\rm Aut}(\mathcal{X}_1 ).$
> 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.
> 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" ] |
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.
> 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) ] |
> 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 |
> 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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