3 2d-mappings

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.

3.1 Morphisms of pre-crossed modules

A morphism between two pre-crossed modules X₁ = (¶₁ : S₁ ® R₁) and X₂ = (¶₂ : S₂ ® R₂) is a pair (s, r), where s: S₁ ® S₂ and r: R₁ ® R₂ commute with the two boundary maps and are morphisms for the two actions:

¶2 s = r¶1,        s(sr) = (ss)rr·

Thus s is the SourceHom and r is the RangeHom.

When X₁ = X₂ and s, r are automorphisms then (s, r) is an automorphism of X₁. The group of automorphisms is denoted by Aut(X₁ )·

The usual properties of mappings may be checked:

Constructors for morphisms of pre-crossed modules include:

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

gap> sigma1 := GroupHomomorphismByImages( c5, c5, [ (1,2,3,4,5) ]
        [ (1,5,4,3,2) ] );;
gap> rho1 := IdentityMapping( Range( X1 ) );
IdentityMapping( PermAut(c5) )
gap> mor1 := XModMorphism( X1, X1, sigma1, rho1 );
[[c5->PermAut(c5)] => [c5->PermAut(c5)]]
gap> Display( mor1 );
Morphism of crossed modules :-
: Source = [c5->PermAut(c5)] with generating sets:
  [ (1,2,3,4,5) ]
  [ (1,2,4,3) ]
: Range = Source
: Source Homomorphism maps source generators to:
  [ (1,5,4,3,2) ]
: Range Homomorphism maps range generators to:
  [ (1,2,4,3) ]
gap> IsAutomorphism( mor1 );
true 

3.2 Morphisms of pre-cat1-groups

A morphism of pre-cat1-groups from C₁ = (e₁;t₁,h₁ : G₁ ® R₁) to C₂ = (e₂;t₂,h₂ : G₂ ® R₂) is a pair (g, r) where g: G₁ ® G₂ and r: R₁ ® R₂ are homomorphisms satisfying

h2 g = rh1 ,        t2 g = rt1 ,        e2 r = ge1 .

The function IsomorphismPermGroup constructs a morphism whose SourceHom and RangeHom are the IsomorphismPermGroups of the source and range.

gap> iso := IsomorphismPermGroup( C2 );
[[..] => [..]]
gap> Display( iso );
Morphism of cat1-groups :-
: Source = [s3c4=>s3] with generating sets:
  [ f1, f2, f3, f4 ]
  [ f1, f2 ]
:  Range = [Group(
[ ( 1,13)( 2,14)( 3,15)( 4,16)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)
    (11,19)(12,20), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,17,21)
    (14,18,22)(15,19,23)(16,20,24), ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)
    (13,15,14,16)(17,19,18,20)(21,23,22,24),
  ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
    (21,22)(23,24) ] )=>Group( [ (1,4)(2,6)(3,5), (1,2,3)(4,5,6)
 ] )] with generating sets:
  [ ( 1,13)( 2,14)( 3,15)( 4,16)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)
    (11,19)(12,20), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,17,21)
    (14,18,22)(15,19,23)(16,20,24), ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)
    (13,15,14,16)(17,19,18,20)(21,23,22,24),
  ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
    (21,22)(23,24) ]
  [ (1,4)(2,6)(3,5), (1,2,3)(4,5,6) ]
: Source Homomorphism maps source generators to:
  [ ( 1,13)( 2,14)( 3,15)( 4,16)( 5,21)( 6,22)( 7,23)( 8,24)( 9,17)(10,18)
    (11,19)(12,20), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)(13,17,21)
    (14,18,22)(15,19,23)(16,20,24), ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)
    (13,15,14,16)(17,19,18,20)(21,23,22,24),
  ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
    (21,22)(23,24) ]
: Range Homomorphism maps range generators to:
  [ (1,4)(2,6)(3,5), (1,2,3)(4,5,6) ]

3.3 Operations on morphisms

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

gap> mor1;
[[c5->PermAut(c5)] => [c5->PermAut(c5)]]
gap> Order(mor1);
2 
gap> mor23;
[[nq8->sl23] => [k4->a4]]
gap> k23 := Kernel( mor23 );
[Group( [ ( 1, 6)( 2, 3)( 4, 5)( 7, 8) ] )->Group(
[ ( 1, 6)( 2, 3)( 4, 5)( 7, 8) ] )]
gap> Display(k23);
Crossed module [..->..] :-
: Source group has generators:
  [ ( 1, 6)( 2, 3)( 4, 5)( 7, 8) ]
: Range group has generators:
  [ ( 1, 6)( 2, 3)( 4, 5)( 7, 8) ]
: Boundary homomorphism maps source generators to:
  [ (1,6)(2,3)(4,5)(7,8) ]
  The automorphism group is trivial
gap> IsNormal( Source(mor23), k23 );
true

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