  
  [1X3. 2d-mappings[0X
  
  
  [1X3.1 Morphisms of 2d-objects[0X
  
  This    chapter   describes   morphisms   of   (pre-)crossed   modules   and
  (pre-)cat1-groups.
  
  [1X3.1-1 Source[0X
  
  [2X> Source( [0X[3Xmap[0X[2X ) ___________________________________________________[0Xattribute
  [2X> Range( [0X[3Xmap[0X[2X ) ____________________________________________________[0Xattribute
  [2X> SourceHom( [0X[3Xmap[0X[2X ) ________________________________________________[0Xattribute
  [2X> RangeHom( [0X[3Xmap[0X[2X ) _________________________________________________[0Xattribute
  
  Morphisms  of  [10X2dObjects[0X are implemented as [10X2dMappings[0X. 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 [10Xfail[0X when invalid data is supplied.
  
  
  [1X3.2 Morphisms of pre-crossed modules[0X
  
  [1X3.2-1 IsXModMorphism[0X
  
  [2X> IsXModMorphism( [0X[3Xmap[0X[2X ) ____________________________________________[0Xproperty
  [2X> IsCat1Morphism( [0X[3Xmap[0X[2X ) ____________________________________________[0Xproperty
  [2X> IsPreXModMorphism( [0X[3Xmap[0X[2X ) _________________________________________[0Xproperty
  [2X> IsPreCat1Morphism( [0X[3Xmap[0X[2X ) _________________________________________[0Xproperty
  
  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  [10XSourceHom[0X  and  $\rho$  is  the  [10XRangeHom[0X.  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 ${\rm Aut}(\mathcal{X}_1 ).$
  
  [1X3.2-2 IsInjective[0X
  
  [2X> IsInjective( [0X[3Xmap[0X[2X ) _______________________________________________[0Xproperty
  [2X> IsSurjective( [0X[3Xmap[0X[2X ) ______________________________________________[0Xproperty
  [2X> IsSingleValued( [0X[3Xmap[0X[2X ) ____________________________________________[0Xproperty
  [2X> IsTotal( [0X[3Xmap[0X[2X ) ___________________________________________________[0Xproperty
  [2X> IsBijective( [0X[3Xmap[0X[2X ) _______________________________________________[0Xproperty
  [2X> IsEndomorphism2dObject( [0X[3Xmap[0X[2X ) ____________________________________[0Xproperty
  [2X> IsAutomorphism2dObject( [0X[3Xmap[0X[2X ) ____________________________________[0Xproperty
  
  The  usual  properties  of  mappings  are  easily  checked.  It  is  usually
  sufficient  to  verify  that  both  the  [10XSourceHom[0X and the [10XRangeHom[0X have the
  required property.
  
  [1X3.2-3 XModMorphism[0X
  
  [2X> XModMorphism( [0X[3Xargs[0X[2X ) _____________________________________________[0Xfunction
  [2X> XModMorphismByHoms( [0X[3XX1, X2, sigma, rho[0X[2X ) ________________________[0Xoperation
  [2X> PreXModMorphism( [0X[3Xargs[0X[2X ) __________________________________________[0Xfunction
  [2X> PreXModMorphismByHoms( [0X[3XP1, P2, sigma, rho[0X[2X ) _____________________[0Xoperation
  [2X> InclusionMorphism2dObjects( [0X[3XX1, S1[0X[2X ) ____________________________[0Xoperation
  [2X> InnerAutomorphismXMod( [0X[3XX1, r[0X[2X ) __________________________________[0Xoperation
  [2X> IdentityMapping( [0X[3XX1[0X[2X ) ___________________________________________[0Xattribute
  [2X> IsomorphismPermObject( [0X[3Xobj[0X[2X ) _____________________________________[0Xfunction
  
  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 [10XX1[0X constructed in the previous chapter.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> sigma1 := GroupHomomorphismByImages( c5, c5, [ (5,6,7,8,9) ][0X
    [4X        [ (5,9,8,7,6) ] );;[0X
    [4Xgap> rho1 := IdentityMapping( Range( X1 ) );[0X
    [4XIdentityMapping( PAut(c5) )[0X
    [4Xgap> mor1 := XModMorphism( X1, X1, sigma1, rho1 );[0X
    [4X[[c5->PAut(c5))] => [c5->PAut(c5))]] [0X
    [4Xgap> Display( mor1 );[0X
    [4XMorphism of crossed modules :-[0X
    [4X: Source = [c5->PAut(c5))] with generating sets:[0X
    [4X  [ (5,6,7,8,9) ][0X
    [4X  [ (1,2,4,3) ][0X
    [4X: Range = Source[0X
    [4X: Source Homomorphism maps source generators to:[0X
    [4X  [ (5,9,8,7,6) ][0X
    [4X: Range Homomorphism maps range generators to:[0X
    [4X  [ (1,2,4,3) ][0X
    [4Xgap> IsAutomorphism2dObject( mor1 );[0X
    [4Xtrue [0X
    [4Xgap> Print( RepresentationsOfObject(mor1), "\n" );[0X
    [4X[ "IsComponentObjectRep", "IsAttributeStoringRep", "Is2dMappingRep" ][0X
    [4Xgap> Print( KnownPropertiesOfObject(mor1), "\n" );[0X
    [4X[ "IsTotal", "IsSingleValued", "IsInjective", "IsSurjective", "Is2dMapping",[0X
    [4X  "IsPreXModMorphism", "IsXModMorphism", "IsEndomorphism2dObject",[0X
    [4X  "IsAutomorphism2dObject" ][0X
    [4Xgap> Print( KnownAttributesOfObject(mor1), "\n" );[0X
    [4X[ "Name", "Range", "Source", "SourceHom", "RangeHom" ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X3.3 Morphisms of pre-cat1-groups[0X
  
  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.
  
  
  [1X3.3-1 Cat1Morphism[0X
  
  [2X> Cat1Morphism( [0X[3Xargs[0X[2X ) _____________________________________________[0Xfunction
  [2X> Cat1MorphismByHoms( [0X[3XC1, C2, gamma, rho[0X[2X ) ________________________[0Xoperation
  [2X> PreCat1Morphism( [0X[3Xargs[0X[2X ) __________________________________________[0Xfunction
  [2X> PreCat1MorphismByHoms( [0X[3XP1, P2, gamma, rho[0X[2X ) _____________________[0Xoperation
  [2X> InclusionMorphism2dObjects( [0X[3XC1, S1[0X[2X ) ____________________________[0Xoperation
  [2X> InnerAutomorphismCat1( [0X[3XC1, r[0X[2X ) __________________________________[0Xoperation
  [2X> IdentityMapping( [0X[3XC1[0X[2X ) ___________________________________________[0Xattribute
  [2X> IsmorphismPermObject( [0X[3Xobj[0X[2X ) ______________________________________[0Xfunction
  [2X> SmallerDegreePerm2dObject( [0X[3Xobj[0X[2X ) _________________________________[0Xfunction
  
  The  global  function  [10XIsomorphismPermObject[0X  calls  [10XIsomorphismPermPreCat1[0X,
  which  constructs  a  morphism  whose  [10XSourceHom[0X and [10XRangeHom[0X are calculated
  using    [10XIsomorphismPermGroup[0X   on   the   source   and   range.   Similarly
  [10XSmallerDegreePermutationRepresentation[0X  is  used on the two groups to obtain
  [10XSmallerDegreePerm2dObject[0X. Names are assigned automatically.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> iso2 := IsomorphismPermObject( C2 );[0X
    [4X[[s3c4=>s3] => [Ps3c4=>Ps3]][0X
    [4Xgap> Display( iso2 );[0X
    [4XMorphism of cat1-groups :-[0X
    [4X: Source = [s3c4=>s3] with generating sets:[0X
    [4X  [ f1, f2, f3, f4 ][0X
    [4X  [ f1, f2 ][0X
    [4X:  Range = [Ps3c4=>Ps3] with generating sets:[0X
    [4X  [ ( 5, 9)( 6,10)( 7,11)( 8,12), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12),[0X
    [4X  ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)[0X
    [4X    (11,12) ][0X
    [4X  [ (2,3), (1,2,3) ][0X
    [4X: Source Homomorphism maps source generators to:[0X
    [4X  [ ( 5, 9)( 6,10)( 7,11)( 8,12), ( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12),[0X
    [4X  ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)[0X
    [4X    (11,12) ][0X
    [4X: Range Homomorphism maps range generators to:[0X
    [4X  [ (2,3), (1,2,3) ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X3.4 Operations on morphisms[0X
  
  [1X3.4-1 Order[0X
  
  [2X> Order( [0X[3Xauto[0X[2X ) ___________________________________________________[0Xattribute
  [2X> CompositionMorphism( [0X[3Xmap2, map1[0X[2X ) _______________________________[0Xoperation
  
  Composition  of  morphisms, written [10X(<map1> * <map2>)[0X for maps acting of the
  right,  calls  the [10XCompositionMorphism[0X function for maps acting on the left,
  applied to the appropriate type of 2d-mapping.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> Order( mor1 );[0X
    [4X2[0X
    [4Xgap> GeneratorsOfGroup( d16 );[0X
    [4X[ (11,12,13,14,15,16,17,18), (12,18)(13,17)(14,16) ][0X
    [4Xgap> d8 := Subgroup( d16, [ c^2, d ] );;[0X
    [4Xgap> c4 := Subgroup( d8, [ c^2 ] );;[0X
    [4Xgap> SetName( d8, "d8" );  SetName( c4, "c4" );[0X
    [4Xgap> X16 := XModByNormalSubgroup( d16, d8 );[0X
    [4X[d8->d16][0X
    [4Xgap> X8 := XModByNormalSubgroup( d8, c4 );[0X
    [4X[c4->d8][0X
    [4Xgap> IsSubXMod( X16, X8 );[0X
    [4Xtrue[0X
    [4Xgap> incd8 := InclusionMorphism2dObjects( X16, X8 );[0X
    [4X[[c4->d8] => [d8->d16]][0X
    [4Xgap> rho := GroupHomomorphismByImages( d16, d16, [c,d], [c,d^(c^2)] );;[0X
    [4Xgap> sigma := GroupHomomorphismByImages( d8, d8, [c^2,d], [c^2,d^(c^2)] );;[0X
    [4Xgap> mor := XModMorphismByHoms( X16, X16, sigma, rho );[0X
    [4X[[d8->d16] => [d8->d16]][0X
    [4Xgap> comp := incd8 * mor;[0X
    [4X[[c4->d8] => [d8->d16]][0X
    [4Xgap> comp = CompositionMorphism( mor, incd8 );[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X3.4-2 Kernel[0X
  
  [2X> Kernel( [0X[3Xmap[0X[2X ) ___________________________________________________[0Xoperation
  [2X> Kernel2dMapping( [0X[3Xmap[0X[2X ) __________________________________________[0Xattribute
  
  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.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> c2 := Group( (19,20) );;[0X
    [4Xgap> i2 := Subgroup( c2, [()] );;[0X
    [4Xgap> X9 := XModByNormalSubgroup( c2, i2 );;[0X
    [4Xgap> sigma9 := GroupHomomorphismByImages( c4, i2, [c^2], [()] );;[0X
    [4Xgap> rho9 := GroupHomomorphismByImages( d8, c2, [c^2,d], [(),(19,20)] );;[0X
    [4Xgap> mor9 := XModMorphism( X8, X9, sigma9, rho9 );[0X
    [4X[[c4->d8] => [..]][0X
    [4Xgap> K9 := Kernel( mor9 );[0X
    [4X[Group( [ (11,13,15,17)(12,14,16,18) ] )->Group( [ (11,13,15,17)(12,14,16,18)[0X
    [4X ] )][0X
  [4X------------------------------------------------------------------[0X
  
