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<!-- --> <!-- intro.xml XMod documentation Chris Wensley --> <!-- --> <!-- Copyright (C) 1996-2021, Chris Wensley et al, --> <!-- --> <!-- ------------------------------------------------------------------- --> <?xml version="1.0" encoding="UTF-8"?> <Chapter Label="Intro"> <Heading>Introduction</Heading> The &XMod; package provides functions for computation with <List> <Item> finite crossed modules of groups 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> isoclinism of groups and crossed modules; </Item> <Item> the actor crossed square of a crossed module; </Item> <Item> crossed squares, cat2-groups, and their morphisms (experimental version); </Item> <Item> crossed modules of groupoids (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. Many of the function names have been changed during the conversion, for example <C>ConjugationXMod</C> has become <Ref Oper="XModByNormalSubgroup"/>. For a list of name changes see the file <F>names.pdf</F> in the <F>doc</F> directory. <P/> In October 2015 Alper Odabaş and Enver Uslu were added to the list of package authors. Their functions for computing isoclinism classes of groups and crossed modules are contained in Chapter <Ref Chap="chap-isclnc" />, and are described in detail in their paper <Cite Key="IOU1" />. <P/> The package may be obtained as a compressed tar file <File>XMod-version.number.tar.gz</File> by ftp from one of the following sites: <List> <Item> the &XMod; GitHub release site: <URL>https://github.com/gap-packages.github.io/xmod/</URL>. </Item> <Item> any &GAP; archive, e.g. <URL>https://www.gap-system.org/Packages/packages.html</URL>; </Item> </List> The package also has a GitHub repository at: <URL>https://github.com/gap-packages/xmod/</URL>. <P/> Crossed modules and cat1-groups are special types of <E>2-dimensional groups</E> <Cite Key="B82"/>, <Cite Key="brow:hig:siv"/>, and are implemented as <C>2DimensionalDomains</C> and <C>2DimensionalGroups</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 eight parts. 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 is the most recent part of the package, introduced in October 2015. Additional operations and properties for crossed modules are included in Section <Ref Sect="sect-more-xmod-ops" />. Then, in <Ref Sect="sect-isoclinic-groups" /> and <Ref Sect="sect-isoclinic-xmods" /> there are functions for isoclinism of groups and crossed modules. <P/> The fourth part is concerned 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 fifth part deals with actor crossed modules and actor cat1-groups. For the actor crossed module <M>{\rm Act}(\calX)</M> of a crossed module <M>\calX</M> we require representations for the Whitehead group of regular derivations of <M>\calX</M> and for the group of automorphisms of <M>\calX</M>. The construction also provides an inner morphism from <M>\calX</M> to <M>{\rm Act}(\calX)</M> whose kernel is the centre of <M>\calX</M>. <P/> The sixth part, which remains under development, contains functions to compute induced crossed modules. <P/> Since 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 eighth part has some experimental functions for crossed modules of groupoids, interacting with the package <Package>groupoids</Package>. Much more work on this is needed. <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). There are also plans to implement cat2-groups, and conversion betwen these and crossed squares. <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>groupoids</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>. <P/> <Example> <![CDATA[ gap> SetInfoLevel( InfoXMod, 1); #sets the InfoXMod level to 1 ]]> </Example> <P/> Once the package is loaded, the manual <Code>doc/manual.pdf</Code> can be found in the documentation folder. The <Code>html</Code> versions, with or without MathJax, should be rebuilt as follows: <P/> <Example> <![CDATA[ gap> ReadPackage( "xmod, "makedoc.g" ); ]]> </Example> <P/> It is possible to check that the package has been installed correctly by running the test files: <P/> <Example> <![CDATA[ gap> ReadPackage( "xmod", "tst/testall.g" ); #I Testing .../pkg/xmod/tst/gp2obj.tst ... ]]> </Example> <P/> Additional information can be found on the <E>Computational Higher-dimensional Discrete Algebra</E> website at: <URL>https://github.com/cdwensley</URL>. </Chapter> ¤ Dauer der Verarbeitung: 0.10 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28