<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%--> <!-- --> <!-- intro.xml XMOD documentation Chris Wensley --> <!-- & Murat Alp --> <!-- --> <!-- $Id: intro.xml,v 2.12 2008/11/24 gap Exp $ --> <!-- --> <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <?xml version="1.0" encoding="ISO-8859-15"?> <!-- $Id: intro.xml,v 2.12 Exp $ --> <Chapter Label="Intro"> <Heading>Introduction</Heading> The &XMod; package provides functions for computation with <List> <Item> finite crossed modules and cat1-groups, and morphisms of these structures; </Item> <Item> finite pre-crossed modules, pre-cat1-groups, and their Peiffer quotients; </Item> <Item> derivations of crossed modules and sections of cat1-groups; </Item> <Item> the actor crossed square of a crossed module; and </Item> <Item> crossed squares and their morphisms (experimental version). </Item> </List> It is loaded with the command <Example> <![CDATA[ gap> LoadPackage( "xmod" ); ]]> </Example> <P/> The term crossed module was introduced by J. H. C. Whitehead in <Cite Key="W2" />, <Cite Key="W1" />. Loday, in <Cite Key="L1" />, reformulated the notion of a crossed module as a cat1-group. Norrie <Cite Key="N1" />, <Cite Key="N2" /> and Gilbert <Cite Key="G1" /> have studied derivations, automorphisms of crossed modules and the actor of a crossed module, while Ellis <Cite Key="E1" /> has investigated higher dimensional analogues. Properties of induced crossed modules have been determined by Brown, Higgins and Wensley in <Cite Key="BH1" />, <Cite Key="BW1" /> and <Cite Key="BW2" />. For further references see <Cite Key="AW1" />, where we discuss some of the data structures and algorithms used in this package, and also tabulate isomorphism classes of cat1-groups up to size <M>30</M>. <P/> &XMod; was originally implemented in 1997 using the &GAP; 3 language. In April 2002 the first and third parts were converted to &GAP; 4, the pre-structures were added, and version 2.001 was released. The final two parts, covering derivations, sections and actors, were included in the January 2004 release 2.002 for &GAP; 4.4. The current version is 2.12, released on 24th November 2008. <P/> Many of the function names have been changed during the conversion, for example <C>ConjugationXMod</C> has become <C>XModByNormalSubgroup</C>. For a list of name changes see the file <F>names.pdf</F> in the <F>doc</F> directory. <P/> Crossed modules and cat1-groups are special types of <E>2-dimensional groups</E> <Cite Key="B82"/> and are implemented as <C>2dObjects</C> having a <C>Source</C> and a <C>Range</C>. See the file <F>notes.pdf</F> in the <F>doc</F> directory for an introductory account of these algebraic gadgets. <P/> The package divides into four parts, all converted from &GAP; 3 to the &GAP; 4.4 release. <P/> The first part is concerned with the standard constructions for pre-crossed modules and crossed modules; together with direct products; normal sub-crossed modules; and quotients. Operations for constructing pre-cat1-groups and cat1-groups, and for converting between cat1-groups and crossed modules, are also included. <P/> The second part is concerned with <E>morphisms</E> of (pre-)crossed modules and (pre-)cat1-groups, together with standard operations for morphisms, such as composition, image and kernel. <P/> The third part deals with the equivalent notions of <E>derivation</E> for a crossed module and <E>section</E> for a cat1-group, and the monoids which they form under the Whitehead multiplication. <P/> The fourth part deals with actor crossed modules and actor cat1-groups. For the actor crossed module <M>{\rm Act}(\mathcal{X})</M> of a crossed module <M>\mathcal{X}</M> we require representations for the Whitehead group of regular derivations of <M>\mathcal{X}</M> and for the group of automorphisms of <M>\mathcal{X}</M>. The construction also provides an inner morphism from <M>\mathcal{X}</M> to <M>{\rm Act}(\mathcal{X})</M> whose kernel is the centre of <M>\mathcal{X}</M>. <P/> From version 2.007 there are experimental functions for <E>crossed squares</E> and their morphisms, structures which arise as <M>3</M>-dimensional groups. Examples of these are inclusions of normal sub-crossed modules, and the inner morphism from a crossed module to its actor. <P/> The package may be obtained as a compressed tar file <File>xmod.2.11.tar.gz</File> by ftp from one of the sites with a &GAP; 4 archive, or from the Bangor Mathematics web site whose URL is: <URL>http://www.informatics.bangor.ac.uk/public/mathematics/chda/</URL> <P/> The following constructions were not in the &GAP; 3 version of the package: sub-2d-object functions, functions for pre-crossed modules and the Peiffer subgroup of a pre-crossed module, and the associated crossed modules. The source and range groups in these constructions are no longer required to be permutation groups. <P/> Future plans include the implementation of <E>group-graphs</E> which will provide examples of pre-crossed modules (their implementation will require interaction with graph-theoretic functions in &GAP; 4). Cat2-groups, and conversion betwen these and crossed squares, are also planned. <P/> The equivalent categories <C>XMod</C> (crossed modules) and <C>Cat1</C> (cat1-groups) are also equivalent to <C>GpGpd</C>, the subcategory of group objects in the category <C>Gpd</C> of groupoids. Finite groupoids have been implemented in Emma Moore's package <Package>Gpd</Package> <Cite Key="M1"/> for groupoids and crossed resolutions. <P/> <Index Key="InfoXMod"><C>InfoXMod</C></Index> In order that the user has some control of the verbosity of the &XMod; package's functions, an <C>InfoClass</C> <C>InfoXMod</C> is provided (see Chapter <C>ref:Info Functions</C> in the &GAP; Reference Manual for a description of the <C>Info</C> mechanism). By default, the <C>InfoLevel</C> of <C>InfoXMod</C> is <C>0</C>; progressively more information is supplied by raising the <C>InfoLevel</C> to <C>1</C>, <C>2</C> and <C>3</C>. <Example> <![CDATA[ gap> SetInfoLevel( InfoXMod, 1); #sets the InfoXMod level to 1 ]]> </Example> Once the package is loaded, it is possible to check the correct installation by running the test suite of the package with the following command. (The test file itself is <F>tst/xmod_manual.tst</F>.) <Example> <![CDATA[ gap> ReadPackage( "xmod", "tst/testall.g" ); + Testing all example commands in the XMod manual + GAP4stones: 0 true ]]> </Example> Additional information can be found on the <E>Computational Higher-dimensional Discrete Algebra</E> web site at <URL>http://www.informatics.bangor.ac.uk/public/mathematics/chda/</URL> </Chapter>