  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YThe [5XXMod[105X package provides functions for computation with[133X
  
  [30X    [33X[0;6Yfinite  crossed  modules  of  groups and cat1-groups, and morphisms of
        these structures;[133X
  
  [30X    [33X[0;6Yfinite   pre-crossed   modules,  pre-cat1-groups,  and  their  Peiffer
        quotients;[133X
  
  [30X    [33X[0;6Yderivations of crossed modules and sections of cat1-groups;[133X
  
  [30X    [33X[0;6Yisoclinism of groups and crossed modules;[133X
  
  [30X    [33X[0;6Ythe actor crossed square of a crossed module;[133X
  
  [30X    [33X[0;6Ycrossed   squares,  cat2-groups,  and  their  morphisms  (experimental
        version);[133X
  
  [30X    [33X[0;6Ycrossed modules of groupoids (experimental version).[133X
  
  [33X[0;0YIt is loaded with the command[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "xmod" ); [127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  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 [22X30[122X.[133X
  
  [33X[0;0Y[5XXMod[105X  was  originally implemented in 1997 using the [5XGAP[105X 3 language. In April
  2002  the  first and third parts were converted to [5XGAP[105X 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  [5XGAP[105X 4.4. Many of the function names have been changed during the
  conversion, for example [10XConjugationXMod[110X has become [10XXModByNormalSubgroup[110X. For
  a list of name changes see the file [11Xnames.pdf[111X in the [11Xdoc[111X directory.[133X
  
  [33X[0;0YIn  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 [14X4[114X, and are described in detail
  in their paper [IOU16].[133X
  
  [33X[0;0YThe current version is 2.64 for [5XGAP[105X 4.8, released on 30th November 2017.[133X
  
  [33X[0;0YThe package may be obtained as a compressed tar file [11Xxmod-2.64.tar.gz[111X by ftp
  from one of the following sites:[133X
  
  [30X    [33X[0;6Yany                  [5XGAP[105X                 archive,                 e.g.
        [7Xhttp://www.gap-system.org/Packages/packages.html[107X;[133X
  
  [30X    [33X[0;6Ythe                            Bangor                            site:
        [7Xhttp://www.maths.bangor.ac.uk/chda/gap4/xmod/xmod.html[107X;[133X
  
  [30X    [33X[0;6Ythe package GitHub repository: [7Xhttps://github.com/gap-packages/xmod[107X.[133X
  
  [33X[0;0YCrossed  modules  and  cat1-groups are special types of [13X2-dimensional groups[113X
  [Bro82],   [BHS11],   and   are   implemented   as  [10X2DimensionalDomains[110X  and
  [10X2DimensionalGroups[110X having a [10XSource[110X and a [10XRange[110X.[133X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0YThe  second  part  is  concerned with [13Xmorphisms[113X of (pre-)crossed modules and
  (pre-)cat1-groups,  together with standard operations for morphisms, such as
  composition, image and kernel.[133X
  
  [33X[0;0YThe 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  [14X4.1[114X. Then, in [14X4.2[114X and [14X4.3[114X there are functions for isoclinism of
  groups and crossed modules.[133X
  
  [33X[0;0YThe fourth part is concerned with the equivalent notions of [13Xderivation[113X for a
  crossed module and [13Xsection[113X for a cat1-group, and the monoids which they form
  under the Whitehead multiplication.[133X
  
  [33X[0;0YThe  fifth  part deals with actor crossed modules and actor cat1-groups. For
  the  actor  crossed  module  [22XAct(mathcalX)[122X  of  a crossed module [22XmathcalX[122X we
  require  representations  for  the Whitehead group of regular derivations of
  [22XmathcalX[122X  and  for  the group of automorphisms of [22XmathcalX[122X. The construction
  also  provides an inner morphism from [22XmathcalX[122X to [22XAct(mathcalX)[122X whose kernel
  is the centre of [22XmathcalX[122X.[133X
  
  [33X[0;0YThe  sixth  part,  which  remains  under  development, contains functions to
  compute induced crossed modules.[133X
  
  [33X[0;0YSince version 2.007 there are experimental functions for [13Xcrossed squares[113X and
  their morphisms, structures which arise as [22X3[122X-dimensional groups. Examples of
  these  are  inclusions of normal sub-crossed modules, and the inner morphism
  from a crossed module to its actor.[133X
  
  [33X[0;0YThe  eighth  part  has  some  experimental  functions for crossed modules of
  groupoids, interacting with the package [5Xgroupoids[105X. Much more work on this is
  needed.[133X
  
  [33X[0;0YFuture  plans  include the implementation of [13Xgroup-graphs[113X which will provide
  examples   of   pre-crossed   modules  (their  implementation  will  require
  interaction  with  graph-theoretic functions in [5XGAP[105X 4). There are also plans
  to implement cat2-groups, and conversion betwen these and crossed squares.[133X
  
  [33X[0;0YThe  equivalent categories [10XXMod[110X (crossed modules) and [10XCat1[110X (cat1-groups) are
  also  equivalent  to [10XGpGpd[110X, the subcategory of group objects in the category
  [10XGpd[110X  of  groupoids.  Finite  groupoids have been implemented in Emma Moore's
  package [5Xgroupoids[105X [Moo01] for groupoids and crossed resolutions.[133X
  
  [33X[0;0YIn  order  that  the  user  has  some  control  of the verbosity of the [5XXMod[105X
  package's functions, an [10XInfoClass[110X [10XInfoXMod[110X is provided (see Chapter [10Xref:Info
  Functions[110X  in  the  [5XGAP[105X  Reference  Manual  for  a  description  of the [10XInfo[110X
  mechanism).  By  default, the [10XInfoLevel[110X of [10XInfoXMod[110X is [10X0[110X; progressively more
  information is supplied by raising the [10XInfoLevel[110X to [10X1[110X, [10X2[110X and [10X3[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel( InfoXMod, 1); #sets the InfoXMod level to 1[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOnce  the  package  is loaded, the manual [10Xdoc/manual.pdf[110X can be found in the
  documentation  folder. The [10Xhtml[110X versions, with or without MathJax, should be
  rebuilt as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XReadPackage( "xmod, "makedocrel.g" ); [127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  possible  to  check that the package has been installed correctly by
  running the test files:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XReadPackage( "xmod", "tst/testall.g" );[127X[104X
    [4X[28X#I  Testing .../pkg/xmod/tst/gp2obj.tst [128X[104X
    [4X[28X... [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAdditional  information can be found on the [13XComputational Higher-dimensional
  Discrete                 Algebra[113X                 website                 at:
  [7Xhttp://pages.bangor.ac.uk/~mas023/chda/intro.html[107X.[133X
  
  [33X[0;0YThis  version  2.61  had  to  be  released  in a hurry when required package
  [5Xgroupoids[105X  was  renamed as [5Xgroupoids[105X. 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.[133X
  
