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<p id="mathjaxlink" class="pcenter"><a href="chap5_mj.html">[MathJax on]</a></p>
<p><a id="X7D65751085F46462" name="X7D65751085F46462"></a></p>
<div class="ChapSects"><a href="chap5.html#X7D65751085F46462">5 <span class="Heading">Conversion between cat1-algebras and crossed modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8617844F86989A78">5.1 <span class="Heading">Equivalent Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A7328237B8ABDD1">5.1-1 Cat1AlgebraOfXModAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EDFD11181CE143B">5.1-2 XModAlgebraOfCat1Algebra</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Conversion between cat1-algebras and crossed modules</span></h3>

<p><a id="X8617844F86989A78" name="X8617844F86989A78"></a></p>

<h4>5.1 <span class="Heading">Equivalent Categories</span></h4>

<p>The categories <span class="SimpleMath">mathbfCat1Alg</span> (cat<span class="SimpleMath">^1</span>-algebras) and <span class="SimpleMath">mathbfXModAlg</span> (crossed modules) are naturally equivalent <a href="chapBib.html#biBellis1">[Ell88]</a>. This equivalence is outlined in what follows. For a given crossed module <span class="SimpleMath">(∂ : S → R)</span> we can construct the semidirect product <span class="SimpleMath">R ⋉ S</span> thanks to the action of <span class="SimpleMath">R</span> on <span class="SimpleMath">S</span>. If we define <span class="SimpleMath">t,h : R ⋉ S → R</span> and <span class="SimpleMath">e : R → R ⋉ S</span> by</p>

<p class="pcenter">
t(r,s) = r, \qquad
h(r,s) = r + \partial(s), \qquad
e(r) = (r,0),
</p>

<p>respectively, then <span class="SimpleMath">mathcalC = (e;t,h : R ⋉ S → R)</span> is a cat<span class="SimpleMath">^1-</span>algebra.</p>

<p>Notice that <span class="SimpleMath">h</span> <em>is</em> an algebra homomorphism, since:</p>

<p class="pcenter">
h(r_1r_2,~ r_1 \cdot s_2 + r_2 \cdot s_1 + s_1s_2)
~=~
r_1r_2 + r_1(\partial s_2) + r_2(\partial s_1) + (\partial s_1)(\partial s_2)
~=~
(r_1 + \partial s_1)(r_2 + \partial s_2).
</p>

<p>Conversely, for a given cat<span class="SimpleMath">^1</span>-algebra <span class="SimpleMath">mathcalC=(e;t,h : A → R)</span>, the map <span class="SimpleMath">∂ : ker t → R</span> is a crossed module, where the action is multiplication action by <span class="SimpleMath">eR</span>, and <span class="SimpleMath">∂</span> is the restriction of <span class="SimpleMath">h</span> to <span class="SimpleMath">ker t</span>.</p>

<p>Since all of these operations are linked to the functions <code class="func">Cat1Algebra</code> (<a href="chap3.html#X7B761CD9812972F6"><span class="RefLink">3.1-1</span></a>) and <code class="func">XModAlgebra</code> (<a href="chap4.html#X813D94F97D8E71A8"><span class="RefLink">4.1-1</span></a>), they can be performed by calling these two functions. We may also use the function <code class="func">Cat1Algebra</code> (<a href="chap3.html#X7B761CD9812972F6"><span class="RefLink">3.1-1</span></a>) instead of the operation <code class="func">Cat1AlgebraSelect</code> (<a href="chap3.html#X82EC94BA7E7F8DEA"><span class="RefLink">3.1-3</span></a>).</p>

<p><a id="X7A7328237B8ABDD1" name="X7A7328237B8ABDD1"></a></p>

<h5>5.1-1 Cat1AlgebraOfXModAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cat1AlgebraOfXModAlgebra</code>( <var class="Arg">X0</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">‣ PreCat1AlgebraOfPreXModAlgebra</code>( <var class="Arg">X0</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These operations are used for constructing a cat<span class="SimpleMath">^1</span>-algebra from a given crossed module of algebras. As an example we use the crossed module <code class="code">XAB</code> constructed in section <a href="chap4.html#X7B31475D7C030075"><span class="RefLink">4.1-2</span></a>.</p>

<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">Cn := Cat1AlgebraOfXModAlgebra( Xn );</span>
[An |X Bn -> An]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( Cn );</span>
Cat1-algebra [An |X Bn => An] :-
:  range algebra has generators:
[
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), Z(5)^0 ] ],
  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
: tail homomorphism maps source generators to:
[
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), Z(5)^0 ] ],
  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
: head homomorphism maps source generators to:
[
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), Z(5)^0 ] ],
  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), Z(5)^0, Z(5)^3 ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ],
  [ [ 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5) ],
      [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ]
: range embedding maps range generators to:    [ v.1, v.2 ]
: kernel has generators:  [ v.4, v.5 ]

</pre></div>

<p>As a second example, we convert the crossed module <span class="SimpleMath">X4</span> constructed in section <a href="chap4.html#X78400B837A2C8FB9"><span class="RefLink">4.1-8</span></a></p>

<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">C3 := Cat1AlgebraOfXModAlgebra( X3 );</span>
[A3 |X GR(c3) -> A3]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C3 );                 </span>
Cat1-algebra [A3 |X GR(c3) => A3] :-
: range algebra has generators:[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
: tail homomorphism maps source generators to:
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
  [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ],
  [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]
: head homomorphism maps source generators to:
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
  [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
  [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ] ]
: range embedding maps range generators to:    [ v.1 ]
: kernel has generators:  [ v.4, v.5, v.6 ]

</pre></div>

<p><a id="X7EDFD11181CE143B" name="X7EDFD11181CE143B"></a></p>

<h5>5.1-2 XModAlgebraOfCat1Algebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XModAlgebraOfCat1Algebra</code>( <var class="Arg">C</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">‣ PreXModAlgebraOfPreCat1Algebra</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These operations are used for constructing a crossed module of algebras from a given cat<span class="SimpleMath">^1</span>-algebra. The example uses the cat<span class="SimpleMath">^1</span>-algebra <code class="code">C3</code> constructed in section <a href="chap3.html#X86E99B197E920C21"><span class="RefLink">3.1-4</span></a>.</p>

<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">X6 := XModAlgebraOfCat1Algebra( C6 );</span>
[ <algebra of dimension 3 over GF(2)> -> <algebra of dimension 3 over GF(2)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( X6 ); </span>
Crossed module [..->..] :-
: Source algebra has generators:
  [ (Z(2)^0)*()+(Z(2)^0)*(4,5), (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,2,3)(4,5),
  (Z(2)^0)*(1,3,2)+(Z(2)^0)*(1,3,2)(4,5) ]
: Range algebra has generators:
  [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3) ]
: Boundary homomorphism maps source generators to:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]

</pre></div>

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