<h4>5.1 <span class="Heading">Derivations and Sections</span></h4>
<p>The Whitehead monoid <span class="SimpleMath">Der(calX)</span> of <span class="SimpleMath">calX</span> was defined in <a href="chapBib.html#biBW2">[Whi48]</a> to be the monoid of all <em>derivations</em> from <span class="SimpleMath">R</span> to <span class="SimpleMath">S</span>, that is the set of all maps <span class="SimpleMath">χ : R -> S</span>, with <em>Whitehead product</em> <span class="SimpleMath">⋆</span> (on the <em>right</em>) satisfying:</p>
<p>It easily follows that <span class="SimpleMath">χ 1 = 1</span> and <span class="SimpleMath">χ(r^-1) = ((χ r)^-1)^r^-1}</span>, and that the zero map is the identity for this composition. Invertible elements in the monoid are called <em>regular</em>. The Whitehead group of <span class="SimpleMath">calX</span> is the group of regular derivations in <span class="SimpleMath">Der(calX )</span>. In the next chapter the <em>actor</em> of <span class="SimpleMath">calX</span> is defined as a crossed module whose source and range are permutation representations of the Whitehead group and the automorphism group of <span class="SimpleMath">calX</span>.</p>
<p>The construction for cat<span class="SimpleMath">^1</span>-groups equivalent to the derivation of a crossed module is the <em>section</em>. The monoid of sections of <span class="SimpleMath">calC = (e;t,h : G -> R)</span> is the set of group homomorphisms <span class="SimpleMath">ξ : R -> G</span>, with Whitehead multiplication <span class="SimpleMath">⋆</span> (on the <em>right</em>) satisfying:</p>
<p class="pcenter">
{\bf Sect\ 1}: t \circ \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).
</p>
<p>The embedding <span class="SimpleMath">e</span> is the identity for this composition, and <spanclass="SimpleMath">h(ξ_1 ⋆ ξ_2) = (h ξ_1)(h ξ_2)</span>. A section is <em>regular</em> when <span class="SimpleMath">h ξ</span> is an automorphism, and the group of regular sections is isomorphic to the Whitehead group.</p>
<p>If <span class="SimpleMath">ϵ</span> denotes the inclusion of <span class="SimpleMath">S = ker t</span> in <span class="SimpleMath">G</span> then <span class="SimpleMath">∂ = h ϵ : S -> R</span> and</p>
<p class="pcenter">
\xi r ~=~ (e r)(\epsilon \chi r),
\quad\mbox{which equals}\quad
(r, \chi r) ~\in~ R \ltimes S,
</p>
<p>determines a section <span class="SimpleMath">ξ</span> of <span class="SimpleMath">calC</span> in terms of the corresponding derivation <span class="SimpleMath">χ</span> of <span class="SimpleMath">calX</span>, and conversely.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationByImages</code>( <var class="Arg">X0</var>, <var class="Arg">ims</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDerivation</code>( <var class="Arg">map</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUp2DimensionalMapping</code>( <var class="Arg">chi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpGeneratorImages</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpImagePositions</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Object2d</code>( <var class="Arg">chi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationImage</code>( <var class="Arg">chi</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">χ</span> is stored like a group homomorphisms by specifying the images of the generating set <code class="code">StrongGeneratorsStabChain( StabChain(R) )</code> of the range <span class="SimpleMath">R</span>. This set of images is stored as the attribute <code class="code">UpGeneratorImages</code> of <span class="SimpleMath">χ</span>. The function <code class="code">IsDerivation</code> is automatically called to check that this procedure is well-defined. The attribute <code class="code">Object2d</code><span class="SimpleMath">(χ)</span> returns the underlying crossed module.</p>
<p>Images of the remaining elements may be obtained using axiom <span class="SimpleMath">Der 1</span>. <code class="code">UpImagePositions(chi)</code> is the list of the images under <span class="SimpleMath">χ</span> of <code class="code">Elements(R)</code> and <code class="code">DerivationImage(chi,r)</code> returns <span class="SimpleMath">χ r</span>.</p>
<p>In the following example a cat<span class="SimpleMath">^1</span>-group <code class="code">C3</code> and the associated crossed module <code class="code">X3</code> are constructed, where <code class="code">X3</code> is isomorphic to the inclusion of the normal cyclic group <code class="code">c3</code> in the symmetric group <code class="code">s3</code>. The derivation <span class="SimpleMath">χ_1</span> maps <code class="code">c3</code> to the identity and the other <span class="SimpleMath">3</span> elements to <span class="SimpleMath">(1,2,3)(4,6,5)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SectionByHomomorphism</code>( <var class="Arg">C</var>, <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSection</code>( <var class="Arg">xi</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UpHomomorphism</code>( <var class="Arg">xi</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SectionByDerivation</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationBySection</code>( <var class="Arg">xi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Sections <em>are</em> group homomorphisms but, although they do not require a special representation, one is provided in the category <code class="code">IsUp2DimensionalMapping</code> having attributes <code class="code">Object2d</code>, <code class="code">UpHomomorphism</code> and <code class="code">UpGeneratorsImages</code>. Operations <code class="code">SectionByDerivation</code> and <code class="code">DerivationBySection</code> convert derivations to sections, and vice-versa, calling <code class="func">Cat1GroupOfXMod</code> (<a href="chap2.html#X82F10A59867C765D"><span class="RefLink">2.5-3</span></a>) and <code class="func">XModOfCat1Group</code> (<a href="chap2.html#X82F10A59867C765D"><span class="RefLink">2.5-3</span></a>) automatically.</p>
<p>Two strategies for calculating derivations and sections are implemented, see <a href="chapBib.html#biBAW1">[AW00]</a>. The default method for <code class="func">AllDerivations</code> (<a href="chap5.html#X788884E48534F7CB"><span class="RefLink">5.2-1</span></a>) 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 early versions of this package, the default method for <code class="code">AllSections( <C> )</code> was to compute all endomorphisms on the range group <code class="code">R</code> of <code class="code">C</code> as possibilities for the composite <span class="SimpleMath">h ξ</span>. 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 <code class="func">SectionByDerivation</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityDerivation</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentitySection</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The identity derivation maps the range group to the identity subgroup of the source, while the identity section is just the range embedding considered as a section.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadProduct</code>( <var class="Arg">upi</var>, <var class="Arg">upj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadOrder</code>( <var class="Arg">up</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The <code class="code">WhiteheadProduct</code>, defined in section <a href="chap5.html#X7C01AE7783898705"><span class="RefLink">5.1</span></a>, may be applied to two derivations to form <span class="SimpleMath">χ_i ⋆ χ_j</span>, or to two sections to form <span class="SimpleMath">ξ_i ⋆ ξ_j</span>. The <code class="code">WhiteheadOrder</code> of a regular derivation <span class="SimpleMath">χ</span> is the smallest power of <span class="SimpleMath">χ</span>, using this product, equal to the <code class="func">IdentityDerivation</code> (<a href="chap5.html#X87D9F7257DFF0236"><span class="RefLink">5.1-4</span></a>).</p>
<h4>5.2 <span class="Heading">Whitehead Monoids and Groups</span></h4>
<p>As mentioned at the beginning of this chapter, the Whitehead monoid <span class="SimpleMath">Der(calX)</span> of <span class="SimpleMath">calX</span> is the monoid of all derivations from <span class="SimpleMath">R</span> to <span class="SimpleMath">S</span>. Monoids of derivations have representation <code class="code">IsMonoidOfUp2DimensionalMappingsObj</code>. Multiplication tables for Whitehead monoids enable the construction of transformation representations.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImagesList</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DerivationClass</code>( <var class="Arg">mon</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ImagesTable</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Using our example <code class="code">X3</code> we find that there are just nine derivations. Here <code class="code">AllDerivations</code> returns a collection of up mappings with attributes:</p>
<ul>
<li><p><code class="code">Object2d</code> - the crossed modules <span class="SimpleMath">calX</span>;</p>
</li>
<li><p><code class="code">ImagesList</code> - a list, for each derivation, of the images of the generators of the range group;</p>
</li>
<li><p><code class="code">DerivationClass</code> - the string "all"; other classes include "regular" and "principal";</p>
</li>
<li><p><code class="code">ImagesTable</code> - this is a table whose <span class="SimpleMath">[i,j]</span>-th entry is the position in the list of elements of <span class="SimpleMath">S</span> of the image under the <span class="SimpleMath">i</span>-th derivation of the <span class="SimpleMath">j</span>-th element of <span class="SimpleMath">R</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadMonoidTable</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadTransformationMonoid</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <code class="code">WhiteheadMonoidTable</code> of <span class="SimpleMath">calX</span> is the multiplication table whose <span class="SimpleMath">[i,j]</span>-th entry is the position <span class="SimpleMath">k</span> in the list of derivations of the Whitehead product <span class="SimpleMath">χ_i*χ_j = χ_k</span>.</p>
<p>Using the rows of the table as transformations, we may construct the <code class="code">WhiteheadTransformationMonoid</code> of <span class="SimpleMath">calX</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadGroupTable</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadPermGroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWhiteheadPermGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadRegularGroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadGroupIsomorphism</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="code">RegularDerivations</code> are those derivations which are invertible in the monoid. The multiplication table for a Whitehead group - a subtable of the Whitehead monoid table - enables the construction of a permutation representation <code class="code">WhiteheadPermGroup</code> <span class="SimpleMath">WcalX</span> of <span class="SimpleMath">calX</span>. This group satisfies the property <code class="code">IsWhiteheadPermGroup</code>. and <span class="SimpleMath">calX</span> is its attribute <code class="code">Object2d</code>.</p>
<p>Of the nine derivations of <code class="code">X3</code> just six are regular. The associated group is isomorphic to the symmetric group <code class="code">s3</code>.</p>
<p>Taking the rows of the <code class="code">WhiteheadGroupTable</code> as permutations, we may construct the <code class="code">WhiteheadRegularGroup</code> of <span class="SimpleMath">calX</span>. Then, seeking a <code class="code">SmallerDegreePermutationRepresentation</code>, we obtain the <code class="code">WhiteheadGroupIsomorphism</code> whose image is the <code class="code">WhiteheadPermGroup</code> of <span class="SimpleMath">calX</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalDerivations</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrincipalDerivationSubgroup</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WhiteheadHomomorphism</code>( <var class="Arg">X0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The principal derivations form a subgroup of the Whitehead group. The <code class="code">PrincipalDerivationSubgroup</code> is the corresponding subgroup of the <code class="code">WhiteheadPermGroup</code>.</p>
<p>The Whitehead homomorphism <span class="SimpleMath">η : S -> WcalX, s ↦ η_s</span> for <span class="SimpleMath">calX</span> maps the source group of <span class="SimpleMath">calX</span> to the Whitehead group of <span class="SimpleMath">calX</span>.</p>
<p><strong class="button">Exercise:</strong><span class="SimpleMath">~</span> Use the two crossed module axioms to show that <span class="SimpleMath">η_s_1 ⋆ η_s_2 = η_s_1s_2</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SourceEndomorphism</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">χ</span> of <span class="SimpleMath">calX = (∂ : S -> R)</span> determines an endomorphism <span class="SimpleMath">σ_χ : S -> S,~ s ↦ s(χ ∂ s)</span>. We may verify that <span class="SimpleMath">σ_χ</span> is a homomorphism by:</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Object2dEndomorphism</code>( <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>A derivation <span class="SimpleMath">χ</span> of <span class="SimpleMath">calX = (∂ : S -> R)</span> determines an endomorphism <span class="SimpleMath">α_χ : calX -> calX</span> whose source and range endomorphisms are given by the previous two operations.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := Object2dEndomorphism( chi2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( alpha2 );</span>
Morphism of crossed modules :-
: Source = [c3->s3] with generating sets:
[ (1,2,3)(4,6,5) ]
[ (4,5,6), (2,3)(5,6) ]
: Range = Source
: Source Homomorphism maps source generators to:
[ (1,3,2)(4,5,6) ]
: Range Homomorphism maps range generators to:
[ (4,6,5), (2,3)(4,6) ]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSections</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RegularSections</code>( <var class="Arg">C0</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>These operations are currently obtained by running the equivalent operation for derivations and then converting the result to sections.</p>
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