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<!--  up2.xml             XMOD documentation              Chris Wensley  -->
<!--                                                        & Murat Alp  -->
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<!--  $Id: up2.xml,v 2.007 2006/02/24 gap Exp $                          -->
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<?xml version="1.0" encoding="ISO-8859-15"?>
  <!-- $Id: up2.xml,v 2.07  Exp $ -->

<Chapter Label="chap-up2">
<Heading>Derivations and Sections</Heading>
<Index>up 2d-mapping of 2d-object</Index>

<Section><Heading>Whitehead Multiplication</Heading>
<Index>Whitehead multiplication</Index>

<ManSection>
   <Prop Name="IsDerivation"
         Arg="map" />
   <Prop Name="IsSection"
         Arg="map" />
   <Prop Name="IsUp2dMapping"
         Arg="map" />
<Description>
<Index>derivation, of crossed module</Index>
<Index>Whitehead monoid</Index>
The Whitehead monoid  <M>{\rm Der}(\mathcal{X})</M> of <M>\mathcal{X}</M> 
was defined in <Cite Key="W2" /> to be the monoid of all <E>derivations</E> 
from <M>R</M> to <M>S</M>, that is the set of all maps  
<M>\chi : R \to S</M>,  with <E>Whitehead multiplication</E>  
<M>\star</M> (on the <E>right</E>) satisfying:
<Display>
{\bf Der\ 1}:  \chi(qr) ~=~ (\chi q)^{r} \; (\chi r),
\qquad
{\bf Der\ 2}: (\chi_1 \star \chi_2)(r) 
    ~=~ (\chi_2 r)(\chi_1 r)(\chi_2 \partial \chi_1 r).
\end{eqnarray*}
</Display>
The zero map is the identity for this composition.
<Index>regular derivation</Index>
Invertible elements in the monoid are called <E>regular</E>.
<Index>Whitehead group</Index>
The Whitehead group of <M>\mathcal{X}</M> 
is the group of regular derivations in <M>{\rm Der}(\mathcal{X} )</M>. 
In the next chapter the <E>actor</E> of <M>\mathcal{X}</M> 
is defined as a crossed module whose source and range are permutation 
representations of the Whitehead group and the automorphism group of 
<M>\mathcal{X}</M>.
<P/>
<Index>section, of cat1-group</Index>
The construction for cat1-groups equivalent to the derivation of a
crossed module is the <E>section</E>. 
<Index>Whitehead multiplication</Index>
The monoid of sections of  <M>\mathcal{C} = (e;t,h : G \to R)</M>  
is the set of group homomorphisms
<M>\xi : R \to G</M>, with Whitehead multiplication  <M>\star</M>,  
(on the <E>right</E>)  satisfying:
<Display>
{\bf Sect\ 1}: t \xi ~=~ {\rm id}_R,
\quad
{\bf Sect\ 2}: (\xi_1 \star \xi_2)(r) 
                ~=~ (\xi_1 r)(e h \xi_1 r)^{-1}(\xi_2 h \xi_1 r) 
                ~=~ (\xi_2 h \xi_1 r)(e h \xi_1 r)^{-1}(\xi_1 r).
</Display>
The embedding  <M>e</M>  is the identity for this composition,
and  <M>h(\xi_1 \star \xi_2) = (h \xi_1)(h \xi_2)</M>.
A section is  <E>regular</E>  when  <M>h \xi</M>  is an automorphism, and 
the group of regular sections is isomorphic to the Whitehead group.
<P/>
If  <M>\epsilon</M> denotes the inclusion of  
<M>S = {\rm ker} t</M>  in  <M>G</M>
then  <M>\partial = h \epsilon : S \to R</M> and
<Display>
\xi r ~=~ (e r)(e \chi r) ~=~ (r, \chi r)
</Display>
determines a section  <M>\xi</M>  of  <M>\mathcal{C}</M>  
in terms of the corresponding derivation  <M>\chi</M>  of
<M>\mathcal{X}</M>, and conversely.
</Description>
</ManSection>


<ManSection>
   <Oper Name="DerivationByImages"
         Arg="X0,ims" />
   <Attr Name="Object2d"
         Arg="chi" />
   <Attr Name="GeneratorImages"
         Arg="chi" />
<Description>
Derivations are stored like group homomorphisms by specifying the images
of a generating set.  
Images of the remaining elements may then be obtained using axiom  
{\bf Der\ 1}.  
The function <C>IsDerivation</C> is automatically called to 
check that this procedure is well-defined.
<P/>
In the following example a cat1-group <C>C3</C> and the associated 
crossed module  <C>X3</C>  are constructed, 
where <C>X3</C> is isomorphic to the inclusion of the normal
cyclic group <C>c3</C> in the symmetric group <C>s3</C>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> g18 := Group( (1,2,3), (4,5,6), (2,3)(5,6) );;
gap> SetName( g18, "g18" );
gap> gen18 := GeneratorsOfGroup( g18 );;
gap> g1 := gen18[1];;  g2 := gen18[2];;  g3 := gen18[3];;
gap> s3 := Subgroup( g18, gen18{[2..3]} );;
gap> SetName( s3, "s3" );;
gap> t := GroupHomomorphismByImages( g18, s3, gen18, [g2,g2,g3] );;
gap> h := GroupHomomorphismByImages( g18, s3, gen18, [(),g2,g3] );;
gap> e := GroupHomomorphismByImages( s3, g18, [g2,g3], [g2,g3] );;
gap> C3 := Cat1( t, h, e );
[g18=>s3]
gap> SetName( Kernel(t), "c3" );;
gap> X3 := XModOfCat1( C3 );;
gap> Display( X3 );
Crossed module [c3->s3] :-
: Source group has generators:
  [ ( 1, 2, 3)( 4, 6, 5) ]
: Range group has generators:
  [ (4,5,6), (2,3)(5,6) ]
: Boundary homomorphism maps source generators to:
  [ (4,6,5) ]
: Action homomorphism maps range generators to automorphisms:
  (4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
  (2,3)(5,6) --> { source gens --> [ (1,3,2)(4,5,6) ] }
  These 2 automorphisms generate the group of automorphisms.
: associated cat1-group is [g18=>s3]

gap> imchi := [ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ];;
gap> chi := DerivationByImages( X3, imchi );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ],
[ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ] )
]]>
</Example>

<ManSection>
   <Oper Name="SectionByImages"
         Arg="C, ims" />
   <Oper Name="SectionByDerivation"
         Arg="chi" />
   <Oper Name="DerivationBySection"
         Arg="xi" />
<Description>
Sections <E>are</E> group homomorphisms, 
so do not need a special representation.
Operations <C>SectionByDerivation</C> and <C>DerivationBySection</C> 
convert derivations to sections, and vice-versa, 
calling <C>Cat1OfXMod</C> and <C>XModOfCat1</C> automatically.
<P/>
Two strategies for calculating derivations and sections are implemented,
see <Cite Key="AW1" />.
The default method for <C>AllDerivations</C> is to search for all possible
sets of images using a backtracking procedure, and when all the derivations
are found it is not known which are regular.
In the &GAP;3 version of this package, the default method for 
<C>AllSections( &lt;C&gt; )</C> was to compute all endomorphisms
on the range group <C>R</C> of <C>C</C> as possibilities 
for the composite <M>h \xi</M>.
A backtrack method then found possible images for such a section.
In the current version the derivations of the associated crossed module
are calculated, and these are all converted to sections
using <C>SectionByDerivation</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> xi := SectionByDerivation( chi );
[ (4,5,6), (2,3)(5,6) ] -> [ (1,2,3), (1,2)(4,6) ]
]]>
</Example>
</Section>



<Section><Heading>Whitehead Groups and Monoids</Heading>

<ManSection>
   <Attr Name="RegularDerivations"
         Arg="X0" />
   <Attr Name="AllDerivations"
         Arg="X0" />
   <Attr Name="RegularSections"
         Arg="C0" />
   <Attr Name="AllSections"
         Arg="C0" />
   <Attr Name="ImagesList"
         Arg="obj" />
   <Attr Name="ImagesTable"
         Arg="obj" />
<Description>
There are two functions to determine the elements of the Whitehead group 
and the Whitehead monoid of <M>\mathcal{X}0</M>,
namely <C>RegularDerivations</C> and <C>AllDerivations</C>.  
(The functions <C>RegularSections</C> and <C>AllSections</C> 
perform corresponding tasks for a cat1-group.)
<P/>
Using our example  <C>X3</C> 
we find that there are just nine derivations,
six of them regular, and the associated group is isomorphic to <C>s3</C>.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> all3 := AllDerivations( X3 );;
gap> imall3 := ImagesList( all3 );; Display( imall3 );
[ [ (), () ],
  [ (), ( 1, 2, 3)( 4, 6, 5) ],
  [ (), ( 1, 3, 2)( 4, 5, 6) ],
  [ ( 1, 2, 3)( 4, 6, 5), () ],
  [ ( 1, 2, 3)( 4, 6, 5), ( 1, 2, 3)( 4, 6, 5) ],
  [ ( 1, 2, 3)( 4, 6, 5), ( 1, 3, 2)( 4, 5, 6) ],
  [ ( 1, 3, 2)( 4, 5, 6), () ],
  [ ( 1, 3, 2)( 4, 5, 6), ( 1, 2, 3)( 4, 6, 5) ],
  [ ( 1, 3, 2)( 4, 5, 6), ( 1, 3, 2)( 4, 5, 6) ]
  ]
gap> KnownAttributesOfObject( all3 );
[ "Object2d", "ImagesList", "AllOrRegular", "ImagesTable" ]
gap> Display( ImagesTable( all3 ) );
[ [  1,  1,  1,  1,  1,  1 ],
  [  1,  1,  1,  2,  2,  2 ],
  [  1,  1,  1,  3,  3,  3 ],
  [  1,  2,  3,  1,  2,  3 ],
  [  1,  2,  3,  2,  3,  1 ],
  [  1,  2,  3,  3,  1,  2 ],
  [  1,  3,  2,  1,  3,  2 ],
  [  1,  3,  2,  2,  1,  3 ],
  [  1,  3,  2,  3,  2,  1 ] ]
]]>
</Example>


<ManSection>
   <Oper Name="CompositeDerivation"
         Arg="chi1,chi2" />
   <Attr Name="ImagePositions"
         Arg="chi" />
   <Oper Name="CompositeSection"
         Arg="xi1,xi2" />
<Description>
The Whitehead product <M>\chi_1 \star \chi_2</M> is implemented as 
<C>CompositeDerivation( &lt;chi1&gt;, &lt;chi2&gt; )</C>.
The composite of two sections is just the composite of homomorphisms.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> reg3 := RegularDerivations( X3 );;
gap> imder3 := ImagesList( reg3 );;
gap> chi4 := DerivationByImages( X3, imder3[4] );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ ( 1, 3, 2)( 4, 5, 6), () ] )
gap> chi5 := DerivationByImages( X3, imder3[5] );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ ( 1, 3, 2)( 4, 5, 6), ( 1, 2, 3)( 4, 6, 5) ] )
gap> im4 := ImagePositions( chi4 );
[ 1, 3, 2, 1, 3, 2 ] 
gap> im5 := ImagePositions( chi5 );
[ 1, 3, 2, 2, 1, 3 ] 
gap> chi45 := chi4 * chi5;
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ (), ( 1, 2, 3)( 4, 6, 5) ] )
gap> im45 := ImagePositions( chi45 );
[ 1, 1, 1, 2, 2, 2 ]  
gap> pos := Position( imder3, GeneratorImages( chi45 ) );
2
]]>
</Example>

<ManSection>
   <Attr Name="WhiteheadGroupTable"
         Arg="X0" />
   <Attr Name="WhiteheadMonoidTable"
         Arg="X0" />
   <Attr Name="WhiteheadPermGroup"
         Arg="X0" />
   <Attr Name="WhiteheadTransMonoid"
         Arg="X0" />
<Description>
Multiplication tables for the Whitehead group or monoid
enable the construction of permutation or transformation representations.
</Description>
</ManSection>

<Example>
<![CDATA[
gap> wgt3 := WhiteheadGroupTable( X3 );; Display( wgt3 );
[ [  1,  2,  3,  4,  5,  6 ],
  [  2,  3,  1,  5,  6,  4 ],
  [  3,  1,  2,  6,  4,  5 ],
  [  4,  6,  5,  1,  3,  2 ],
  [  5,  4,  6,  2,  1,  3 ],
  [  6,  5,  4,  3,  2,  1 ] ]
gap> wpg3 := WhiteheadPermGroup( X3 );
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
gap> wmt3 := WhiteheadMonoidTable( X3 );; Display( wmt3 );
[ [  1,  2,  3,  4,  5,  6,  7,  8,  9 ],
  [  2,  3,  1,  5,  6,  4,  8,  9,  7 ],
  [  3,  1,  2,  6,  4,  5,  9,  7,  8 ],
  [  4,  4,  4,  4,  4,  4,  4,  4,  4 ],
  [  5,  5,  5,  5,  5,  5,  5,  5,  5 ],
  [  6,  6,  6,  6,  6,  6,  6,  6,  6 ],
  [  7,  9,  8,  4,  6,  5,  1,  3,  2 ],
  [  8,  7,  9,  5,  4,  6,  2,  1,  3 ],
  [  9,  8,  7,  6,  5,  4,  3,  2,  1 ] ]
gap> wtm3 := WhiteheadTransMonoid( X3 );
Monoid( [ Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ),
  Transformation( [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ] ),
  Transformation( [ 3, 1, 2, 6, 4, 5, 9, 7, 8 ] ),
  Transformation( [ 4, 4, 4, 4, 4, 4, 4, 4, 4 ] ),
  Transformation( [ 5, 5, 5, 5, 5, 5, 5, 5, 5 ] ),
  Transformation( [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ] ),
  Transformation( [ 7, 9, 8, 4, 6, 5, 1, 3, 2 ] ),
  Transformation( [ 8, 7, 9, 5, 4, 6, 2, 1, 3 ] ),
  Transformation( [ 9, 8, 7, 6, 5, 4, 3, 2, 1 ] ) ], ... )
]]>
</Example>

</Section>

</Chapter>
