  
  [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  groupsand  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;6Ythe actor crossed square of a crossed module;[133X
  
  [30X    [33X[0;6Ycrossed squares 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.[133X
  
  [33X[0;0YMany  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;0YThe  current  versions  are 2.22 for [5XGAP[105X 4.6, released on January 25th 2013;
  and 2.41 for [5XGAP[105X 4.7, released on 5th June 2015.[133X
  
  [33X[0;0YThe package may be obtained as a compressed tar file [11Xxmod-2.41.tar.gz[111X by ftp
  from  one  of the sites with a [5XGAP[105X 4 archive, or from the Bangor Mathematics
  website             pages             using             the             URL:
  [7Xhttp://www.maths.bangor.ac.uk/chda/gap4/xmod/xmod241.html[107X[133X
  
  [33X[0;0YCrossed  modules  and  cat1-groups are special types of [13X2-dimensional groups[113X
  [Bro82],  [BHS11],  and  are  implemented as [10X2dDomains[110X and [10X2dGroups[110X having a
  [10XSource[110X and a [10XRange[110X.[133X
  
  [33X[0;0YThe  package  divides  into four 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 deals 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  fourth 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;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;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). Cat2-groups, and
  conversion betwen these and crossed squares, are also planned.[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 [5XGpd[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, it is possible to check the correct installation
  by running the test suite of the package with the following command.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XReadPackage( "xmod", "tst/testall.g" );[127X[104X
    [4X[28X+ Testing examples in Chapter 2 of the XMod manual[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/[107X.[133X
  
