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<!--  map2.xml            XMOD documentation              Chris Wensley  -->
<!--                                                        & Murat Alp  -->
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<!--  $Id: map2.xml,v 2.007 2006/02/15 gap Exp $                         -->
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<?xml version="1.0" encoding="ISO-8859-15"?>
  <!-- $Id: map2.xml,v 2.07  Exp $ -->

<Chapter Label="chap-map2">
<Heading>2d-mappings</Heading>
<Index>2d-mapping</Index>

<Section><Heading>Morphisms of 2d-objects</Heading>

<Index>morphism of 2d-object</Index>
<Index>crossed module morphism</Index>
This chapter describes morphisms of (pre-)crossed modules
and (pre-)cat1-groups.

<ManSection>
   <Attr Name="Source"
         Arg="map" />
   <Attr Name="Range"
         Arg="map" />
   <Attr Name="SourceHom"
         Arg="map" />
   <Attr Name="RangeHom"
         Arg="map" />
<Description>
Morphisms of <C>2dObjects</C> are implemented as <C>2dMappings</C>.
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 <C>fail</C> when invalid data is supplied.
</Description>
</ManSection>
</Section>


<Section><Heading>Morphisms of pre-crossed modules</Heading>

<Index>morphism</Index>
<ManSection>
   <Prop Name="IsXModMorphism"
         Arg="map" />
   <Prop Name="IsCat1Morphism"
         Arg="map" />
   <Prop Name="IsPreXModMorphism"
         Arg="map" />
   <Prop Name="IsPreCat1Morphism"
         Arg="map" />
<Description>
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:
<Display>
\partial_2 \sigma = \rho \partial_1, \qquad
\sigma(s^r) = (\sigma s)^{\rho r}.
</Display>
Thus  $\sigma$ is the <C>SourceHom</C>  and  $\rho$  is the <C>RangeHom</C>.
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 ${\rm Aut}(\mathcal{X}_1 ).$ 
</Description>
</ManSection>


<ManSection>
   <Prop Name="IsInjective"
         Arg="map" />
   <Prop Name="IsSurjective"
         Arg="map" />
   <Prop Name="IsSingleValued"
         Arg="map" />
   <Prop Name="IsTotal"
         Arg="map" />
   <Prop Name="IsBijective"
         Arg="map" />
   <Prop Name="IsEndomorphism2dObject"
         Arg="map" />
   <Prop Name="IsAutomorphism2dObject"
         Arg="map" />
<Description>
The usual properties of mappings are easily checked.
It is usually sufficient to verify that both the <C>SourceHom</C> and the
<C>RangeHom</C> have the required property.
</Description>
</ManSection>

<ManSection>
   <Func Name="XModMorphism"
         Arg="args" />
   <Oper Name="XModMorphismByHoms"
         Arg="X1,X2,sigma,rho" />
   <Func Name="PreXModMorphism"
         Arg="args" />
   <Oper Name="PreXModMorphismByHoms"
         Arg="P1,P2,sigma,rho" />
   <Oper Name="InclusionMorphism2dObjects"
         Arg="X1,S1" />
   <Oper Name="InnerAutomorphismXMod"
         Arg="X1,r" />
   <Attr Name="IdentityMapping"
         Arg="X1" />
   <Func Name="IsomorphismPermObject"
         Arg="obj" />
<Description>
These are the constructors for morphisms of pre-crossed and crossed modules.
<P/>
In the following example we construct a simple automorphism of 
the crossed module  <C>X1</C>  constructed in the previous chapter.
</Description>
</ManSection>

<Index>display a 2d-mapping</Index>
<Example>
<![CDATA[
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" ]
]]>
</Example>
</Section>


<Section Label="sect-mor-pre-cat1">
<Heading>Morphisms of pre-cat1-groups</Heading>

A morphism of pre-cat1-groups from 
<M>\mathcal{C}_1 = (e_1;t_1,h_1 : G_1 \to R_1)</M>
to  <M>\mathcal{C}_2 = (e_2;t_2,h_2 : G_2 \to R_2)</M>  
is a pair  <M>(\gamma, \rho)</M>  where
<M>\gamma : G_1 \to G_2</M>  and  <M>\rho : R_1 \to R_2</M>  
are homomorphisms satisfying
<Display>
h_2 \gamma = \rho h_1~, \qquad %
t_2 \gamma = \rho t_1~, \qquad %
e_2 \rho = \gamma e_1~\.
</Display>

<ManSection>
   <Func Name="Cat1Morphism"
         Arg="args" />
   <Oper Name="Cat1MorphismByHoms"
         Arg="C1,C2,gamma,rho" />
   <Func Name="PreCat1Morphism"
         Arg="args" />
   <Oper Name="PreCat1MorphismByHoms"
         Arg="P1,P2,gamma,rho" />
   <Oper Name="InclusionMorphism2dObjects"
         Arg="C1,S1" />
   <Oper Name="InnerAutomorphismCat1"
         Arg="C1,r" />
   <Attr Name="IdentityMapping"
         Arg="C1" />
   <Func Name="IsmorphismPermObject"
         Arg="obj" />
   <Func Name="SmallerDegreePerm2dObject"
         Arg="obj" />
<Description>
The global function <C>IsomorphismPermObject</C> 
calls <C>IsomorphismPermPreCat1</C>, 
which constructs a morphism whose <C>SourceHom</C> and <C>RangeHom</C> 
are calculated using <C>IsomorphismPermGroup</C> on the source and range.  
Similarly <C>SmallerDegreePermutationRepresentation</C> is used on the
two groups to obtain <C>SmallerDegreePerm2dObject</C>.
Names are assigned automatically.
</Description>
</ManSection>

<Example>
<![CDATA[
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) ]
]]>
</Example>
</Section>



<Section Label="sect-oper-mor">
<Heading>Operations on morphisms</Heading>
<Index>operations on morphisms</Index>

<ManSection>
   <Attr Name="Order"
         Arg="auto" />
   <Oper Name="CompositionMorphism"
         Arg="map2,map1" />
<Description>
Composition of morphisms, written <C>(&lt;map1&gt; * &lt;map2&gt;)</C> 
for maps acting of the right,
calls the <C>CompositionMorphism</C> function for maps acting on the left,
applied to the appropriate type of 2d-mapping.
</Description>
</ManSection>

<Example>
<![CDATA[
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
]]>
</Example>

<ManSection>
   <Oper Name="Kernel"
         Arg="map" />
   <Attr Name="Kernel2dMapping"
         Arg="map" />
<Description>
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.
</Description>
</ManSection>

<Example>
<![CDATA[
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)
 ] )]
]]></Example>

</Section>

</Chapter>
