  
  
                                      [1X[5XXMod[105X[101X
  
  
                     [1XCrossed modules and cat1-groups in [5XGAP[105X[101X
  
  
                                  Version 2.41
  
  
                                   05/06/2015
  
  
                                   Murat Alp
  
                                 Chris Wensley
  
  
  
  Murat Alp
      Email:    [7Xmailto:muratalp@nigde.edu.tr[107X
      Address:  [33X[0;14YNigde Üniversitesi,[133X
                [33X[0;14YFen Edebiyat Fakültesi, Matematik Bölümü[133X
                [33X[0;14YNigde, Turkey.[133X
  
  
  Chris Wensley
      Email:    [7Xmailto:c.d.wensley@bangor.ac.uk[107X
      Homepage: [7Xhttp://pages.bangor.ac.uk/~mas023/[107X
      Address:  [33X[0;14YSchool of Computer Science, Bangor University,[133X
                [33X[0;14YDean Street, Bangor, Gwynedd, LL57 1UT, U.K.[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[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;6Ythe actor crossed square of a crossed module.[133X
  
  [33X[0;0YInitial,  experimental  versions  of the following are also included in this
  version:[133X
  
  [30X    [33X[0;6Ycrossed squares and their morphisms;[133X
  
  [30X    [33X[0;6Ycrossed modules of groupoids.[133X
  
  [33X[0;0Y[5XXMod[105X  was  originally  implemented in 1997 using the [5XGAP[105X3 language. when the
  first author was studying for a Ph.D. [Alp97] in Bangor.[133X
  
  [33X[0;0YIn  April  2002  the  first  and  third  parts  were  converted to [5XGAP[105X4, 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;0YVersion 2.19 was released on 9th June 2012 for the first release of [5XGAP[105X 4.5.[133X
  
  [33X[0;0YThe current version is 2.41, released 5th June 2015 for [5XGAP[105X 4.7.[133X
  
  [33X[0;0YBug  reports,  suggestions  and  comments  are,  of  course, welcome. Please
  contact the second author at [7Xmailto:c.d.wensley@bangor.ac.uk[107X.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  1997-2015  Murat  Alp  and  Chris Wensley. [5XXMod[105X is free software; you can
  redistribute  it  and/or modify it under the terms of the GNU General Public
  License  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  as  published  by  the Free
  Software  Foundation;  either  version 2 of the License, or (at your option)
  any later version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThis  documentation was prepared with the [5XGAPDoc[105X package of Frank Lübeck and
  Max Neunhöffer.[133X
  
  [33X[0;0YThe  first  author  wishes to acknowledge support from Dumlupinar University
  and the Turkish government.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (XMod)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0Y2d-groups : crossed modules and cat1-groups[133X
    2.1 [33X[0;0YConstructions for crossed modules[133X
      2.1-1 XMod
      2.1-2 Source
      2.1-3 Size
      2.1-4 SubXMod
    2.2 [33X[0;0YPre-crossed modules[133X
      2.2-1 PreXModByBoundaryAndAction
      2.2-2 PeifferSubgroup
      2.2-3 IsPermXMod
    2.3 [33X[0;0YCat1-groups and pre-cat1-groups[133X
      2.3-1 Source
      2.3-2 Cat1
      2.3-3 Cat1OfXMod
    2.4 [33X[0;0YSelection of a small cat1-group[133X
      2.4-1 Cat1Select
      2.4-2 AllCat1sBasic
  3 [33X[0;0Y2d-mappings[133X
    3.1 [33X[0;0YMorphisms of 2d-groups[133X
      3.1-1 Source
    3.2 [33X[0;0YMorphisms of pre-crossed modules[133X
      3.2-1 IsXModMorphism
      3.2-2 IsInjective
      3.2-3 XModMorphism
    3.3 [33X[0;0YMorphisms of pre-cat1-groups[133X
      3.3-1 IsCat1Morphism
    3.4 [33X[0;0YOperations on morphisms[133X
      3.4-1 CompositionMorphism
      3.4-2 Kernel
  4 [33X[0;0YDerivations and Sections[133X
    4.1 [33X[0;0YWhitehead Multiplication[133X
      4.1-1 IsDerivation
      4.1-2 DerivationByImages
      4.1-3 SectionByImages
    4.2 [33X[0;0YWhitehead Groups and Monoids[133X
      4.2-1 RegularDerivations
      4.2-2 CompositeDerivation
      4.2-3 WhiteheadGroupTable
  5 [33X[0;0YActors of 2d-groups[133X
    5.1 [33X[0;0YActor of a crossed module[133X
      5.1-1 WhiteheadXMod
      5.1-2 XModCentre
  6 [33X[0;0YInduced Constructions[133X
    6.1 [33X[0;0YInduced crossed modules[133X
      6.1-1 InducedXMod
      6.1-2 AllInducedXMods
  7 [33X[0;0YCrossed squares and their morphisms[133X
    7.1 [33X[0;0YConstructions for crossed squares[133X
      7.1-1 XSq
      7.1-2 IsXSq
      7.1-3 Up2dGroup
    7.2 [33X[0;0YMorphisms of crossed squares[133X
      7.2-1 Source
      7.2-2 IsXSqMorphism
  8 [33X[0;0YCrossed modules of groupoids[133X
    8.1 [33X[0;0YConstructions for crossed modules of groupoids[133X
      8.1-1 PreXModWithObjectsObj
  9 [33X[0;0YUtility functions[133X
    9.1 [33X[0;0YPrinting Lists[133X
      9.1-1 PrintListOneItemPerLine
    9.2 [33X[0;0YInclusion and Restriction Mappings[133X
      9.2-1 InclusionMappingGroups
      9.2-2 InnerAutomorphismByNormalSubgroup
    9.3 [33X[0;0YAbelian Modules[133X
      9.3-1 AbelianModuleObject
    9.4 [33X[0;0YDistinct and Common Representatives[133X
      9.4-1 DistinctRepresentatives
  10 [33X[0;0YDevelopment history[133X
    10.1 [33X[0;0YChanges from version to version[133X
      10.1-1 [33X[0;0YVersion 1 for [5XGAP[105X 3[133X
      10.1-2 [33X[0;0YVersion 2[133X
      10.1-3 [33X[0;0YVersion 2.001 for [5XGAP[105X 4[133X
      10.1-4 [33X[0;0YInduced crossed modules[133X
      10.1-5 [33X[0;0YVersions 2.002 -- 2.006[133X
      10.1-6 [33X[0;0YVersions 2.007 -- 2.010[133X
    10.2 [33X[0;0YVersions for [5XGAP[105X 4.5, 4.6 and 4.7[133X
      10.2-1 [33X[0;0YAllCat1s[133X
      10.2-2 [33X[0;0YVersion 2.41[133X
    10.3 [33X[0;0YWhat needs doing next?[133X
  
  
  [32X
