<!-- ------------------------------------------------------------------- --> <!-- --> <!-- convert.xml XModAlg documentation Z. Arvasi --> <!-- & A. Odabas --> <!-- Copyright (C) 2014-2025, Z. Arvasi & A. Odabas, --> <!-- Osmangazi University, Eskisehir, Turkey --> <!-- --> <!-- ------------------------------------------------------------------- -->
<?xmlversion="1.0"encoding="UTF-8"?>
<Chapter Label="chap-convert">
<Heading>Conversion between cat1-algebras and crossed modules</Heading>
<Section>
<Heading>Equivalent Categories</Heading>
The categories <M>\mathbf{Cat1Alg}</M> (cat<M>^{1}</M>-algebras)
and <M>\mathbf{XModAlg}</M> (crossed modules)
are naturally equivalent <Cite Key="ellis1"/>.
This equivalence is outlined in what follows.
For a given crossed module <M>(\partial : S \rightarrow R)</M>
we can construct the semidirect product <M>R \ltimes S</M>
thanks to the action of <M>R</M> on <M>S</M>.
If we define <M>t,h : R \ltimes S \rightarrow R</M>
and <M>e : R \rightarrow R \ltimes S</M> by
<Display>
t(r,s) = r, \qquad
h(r,s) = r + \partial(s), \qquad
e(r) = (r,0),
</Display>
respectively, then
<M>\mathcal{C} = (e;t,h : R \ltimes S \rightarrow R)</M>
is a cat<M>^{1}-</M>algebra.
<P/>
Notice that <M>h</M> <E>is</E> an algebra homomorphism, since:
<Display>
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).
</Display>
<P/>
Conversely, for a given cat<M>^{1}</M>-algebra
<M>\mathcal{C}=(e;t,h : A \rightarrow R)</M>,
the map <M>\partial : \ker t \rightarrow R</M> is a crossed module,
where the action is multiplication action by <M>eR</M>, and
<M>\partial</M> is the restriction of <M>h</M> to <M>\ker t</M>.
<P/>
Since all of these operations are linked to the functions
<Ref Oper="Cat1Algebra"/> and <Ref Oper="XModAlgebra"/>,
they can be performed by calling these two functions.
We may also use the function <Ref Oper="Cat1Algebra"/>
instead of the operation <Ref Oper="Cat1AlgebraSelect"/>.
<ManSection>
<Oper Name="Cat1AlgebraOfXModAlgebra"
Arg="X0" />
<Oper Name="PreCat1AlgebraOfPreXModAlgebra"
Arg="X0" />
<Description>
These operations are used for constructing a cat<M>^{1}</M>-algebra
from a given crossed module of algebras.
As an example we use the crossed module <C>XAB</C> constructed in section
<Ref Sect="XModAlgebraByIdeal"/>.
</Description>
</ManSection>
<ManSection>
<Oper Name="XModAlgebraOfCat1Algebra"
Arg="C" />
<Oper Name="PreXModAlgebraOfPreCat1Algebra"
Arg="C" />
<Description>
These operations are used for constructing a crossed module of algebras
from a given cat<M>^{1}</M>-algebra.
The example uses the cat<M>^1</M>-algebra <C>C3</C>
constructed in section <Ref Sect="SubCat1Algebra"/>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> X6 := XModAlgebraOfCat1Algebra( C6 );
[ <algebra of dimension 3 over GF(2)> -> <algebra of dimension 3 over GF(2)> ]
gap> Display( X6 );
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 ... ]
]]>
</Example>
</Section>
</Chapter>
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