GAP (XModAlg) - Chapter 1: Introduction

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1 Introduction

### 1 Introduction

In 1950 S. MacLane and J.H.C. Whitehead, [Whi49] suggested that crossed modules modeled homotopy \(2\)-types. Later crossed modules have been considered as \(2\)-dimensional groups, [Bro82], [Bro87]. The commutative algebra version of this construction has been adapted by T. Porter, [AP96], [Por87]. This algebraic version is called combinatorial algebra theory, which contains potentially important new ideas (see [AP96], [AP98], [AE03]).

A share package XMod, [AOUW17], [AW00], was prepared by M. Alp and C.D. Wensley for the GAP computational group theory language, initially for GAP3 then revised for GAP4. The \(2\)-dimensional part of this programme contains functions for computing crossed modules and cat\(^{1}\)-groups and their morphisms [AOUW17].

This package includes functions for computing crossed modules of algebras, cat\(^{1}\)-algebras and their morphisms by analogy with computational group theory. 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 \(\mathcal{K}(.):Gr\rightarrow Alg\) is left adjoint to the unit group functor \(\mathcal{U}(.):Alg\rightarrow Gr\).

The categories `XModAlg` (crossed modules of algebras) and `Cat1Alg` (cat\(^{1}\)-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, [Oda09] and described in detail in the paper [AO16].

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.

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