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<Chapter Label="Intro">

<Heading>Introduction</Heading>

In 1950 S. MacLane and J.H.C. Whitehead, <Cite Key="whitehead"/>
suggested that crossed modules modeled homotopy <M>2</M>-types. 
Later crossed modules have been considered as 
<M>2</M><E>-dimensional groups</E>, <Cite Key="brown1"/>, <Cite Key="brown2"/>. 
The commutative algebra version of this construction has been adapted by 
T. Porter, <Cite Key="arvasi2"/>, <Cite Key="porter1"/>. 
This algebraic version is called <E>combinatorial algebra theory</E>, 
which contains potentially important new ideas 
(see <Cite Key="arvasi2"/>, <Cite Key="arvasi3"/>, <Cite Key="arvasi4"/>).
<P/>
A share package <Package>XMod</Package>, <Cite Key="alp3"/>, 
<Cite Key="alp2"/>, was prepared by 
M. Alp and C.D. Wensley for the &GAP; computational group theory language, 
initially for &GAP;3 then revised for &GAP;4.  
The <M>2</M>-dimensional part of this programme contains functions for computing
crossed modules and cat<M>^{1}</M>-groups and their morphisms  
<Cite Key="alp3"/>.
<P/>
This package includes functions for computing crossed modules of algebras, 
cat<M>^{1}</M>-algebras and their
morphisms by analogy with <E>computational group theory</E>. 
We will concentrate on group rings over of abelian groups over finite fields because these algebras are conveniently implemented in &GAP;.
The tools needed are the group algebras in which the group algebra functor 
<M>\mathcal{K}(.):Gr\rightarrow Alg</M> 
is left adjoint to the unit group functor 
<M>\mathcal{U}(.):Alg\rightarrow Gr</M>.
<P/>
The categories <C>XModAlg</C> (crossed modules of algebras) 
and <C>Cat1Alg</C> (cat<M>^{1}</M>-algebras) are equivalent, 
and we include functions to convert objects and morphisms between them. 
The algorithms implemented in this package are analyzed in
A. Odabas's Ph.D. thesis, <Cite Key="aodabas1"/> 
and described in detail in the paper <Cite Key="arvasi_odabas" />.
<P/>
There are aspects of commutative algebras for which no &GAP; functions yet exist, for example semidirect products. 
We have included here functions for all homomorphisms of algebras. 
</Chapter>
