GAP (XMod) - Chapter 3: 2d-mappings

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### 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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