chap1.txt

  
  1 Introduction
  
  The XMod package provides functions for computation with
  
      finite  crossed  modules  of  groups and cat1-groups, and morphisms of
        these structures;
  
      finite   pre-crossed   modules,  pre-cat1-groups,  and  their  Peiffer
        quotients;
  
      derivations of crossed modules and sections of cat1-groups;
  
      isoclinism of groups and crossed modules;
  
      the actor crossed square of a crossed module;
  
      crossed   squares,  cat2-groups,  and  their  morphisms  (experimental
        version);
  
      crossed modules of groupoids (experimental version).
  
  It is loaded with the command
  
    Example  
    
    gap> LoadPackage( "xmod" ); 
    
  
  
  The  term  crossed  module  was introduced by J. H. C. Whitehead in [Whi48],
  [Whi49]. Loday, in [Lod82], reformulated the notion of a crossed module as a
  cat1-group.  Norrie  [Nor90],  [Nor87]  and  Gilbert  [Gil90]  have  studied
  derivations,  automorphisms  of  crossed  modules and the actor of a crossed
  module,  while  Ellis [Ell84] has investigated higher dimensional analogues.
  Properties of induced crossed modules have been determined by Brown, Higgins
  and Wensley in [BH78], [BW95] and [BW96]. For further references see [AW00],
  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 30.
  
  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 ConjugationXMod has become XModByNormalSubgroup. For
  a list of name changes see the file names.pdf in the doc directory.
  
  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 4, and are described in detail
  in their paper [IOU16].
  
  The current version is 2.64 for GAP 4.8, released on 30th November 2017.
  
  The package may be obtained as a compressed tar file xmod-2.64.tar.gz by ftp
  from one of the following sites:
  
      any                  GAP                 archive,                 e.g.
        http://www.gap-system.org/Packages/packages.html;
  
      the                            Bangor                            site:
        http://www.maths.bangor.ac.uk/chda/gap4/xmod/xmod.html;
  
      the package GitHub repository: https://github.com/gap-packages/xmod.
  
  Crossed  modules  and  cat1-groups are special types of 2-dimensional groups
  [Bro82],   [BHS11],   and   are   implemented   as  2DimensionalDomains  and
  2DimensionalGroups having a Source and a Range.
  
  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.
  
  The  second  part  is  concerned with morphisms of (pre-)crossed modules and
  (pre-)cat1-groups,  together with standard operations for morphisms, such as
  composition, image and kernel.
  
  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  4.1. Then, in 4.2 and 4.3 there are functions for isoclinism of
  groups and crossed modules.
  
  The fourth part is concerned with the equivalent notions of derivation for a
  crossed module and section for a cat1-group, and the monoids which they form
  under the Whitehead multiplication.
  
  The  fifth  part deals with actor crossed modules and actor cat1-groups. For
  the  actor  crossed  module  Act(mathcalX)  of  a crossed module mathcalX we
  require  representations  for  the Whitehead group of regular derivations of
  mathcalX  and  for  the group of automorphisms of mathcalX. The construction
  also  provides an inner morphism from mathcalX to Act(mathcalX) whose kernel
  is the centre of mathcalX.
  
  The  sixth  part,  which  remains  under  development, contains functions to
  compute induced crossed modules.
  
  Since version 2.007 there are experimental functions for crossed squares and
  their morphisms, structures which arise as 3-dimensional groups. Examples of
  these  are  inclusions of normal sub-crossed modules, and the inner morphism
  from a crossed module to its actor.
  
  The  eighth  part  has  some  experimental  functions for crossed modules of
  groupoids, interacting with the package groupoids. Much more work on this is
  needed.
  
  Future  plans  include the implementation of group-graphs 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.
  
  The  equivalent categories XMod (crossed modules) and Cat1 (cat1-groups) are
  also  equivalent  to GpGpd, the subcategory of group objects in the category
  Gpd  of  groupoids.  Finite  groupoids have been implemented in Emma Moore's
  package groupoids [Moo01] for groupoids and crossed resolutions.
  
  In  order  that  the  user  has  some  control  of the verbosity of the XMod
  package's functions, an InfoClass InfoXMod is provided (see Chapter ref:Info
  Functions  in  the  GAP  Reference  Manual  for  a  description  of the Info
  mechanism).  By  default, the InfoLevel of InfoXMod is 0; progressively more
  information is supplied by raising the InfoLevel to 1, 2 and 3.
  
    Example  
    
    gap> SetInfoLevel( InfoXMod, 1); #sets the InfoXMod level to 1
    
  
  
  Once  the  package  is loaded, the manual doc/manual.pdf can be found in the
  documentation  folder. The html versions, with or without MathJax, should be
  rebuilt as follows:
  
    Example  
    
    gap> ReadPackage( "xmod, "makedocrel.g" ); 
    
  
  
  It  is  possible  to  check that the package has been installed correctly by
  running the test files:
  
    Example  
    
    gap> ReadPackage( "xmod", "tst/testall.g" );
    #I  Testing .../pkg/xmod/tst/gp2obj.tst 
    ... 
    
  
  
  Additional  information can be found on the Computational Higher-dimensional
  Discrete                 Algebra                 website                 at:
  http://pages.bangor.ac.uk/~mas023/chda/intro.html.
  
  This  version  2.61  had  to  be  released  in a hurry when required package
  groupoids  was  renamed as groupoids. As a result some functions for crossed
  squares  and  cat2-groups,  still  under  development, had to be temporarily
  removed,  and  two test files had to be removed from the list of files to be
  tested.