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Alp<br /> Ömer Halisdemir University<br /> Art and Science Faculty<br /> Mathematics Department<br /> Nigde<br /> Turkey<br />p ><p ><b >b >br />Email: <span class="URL" ><a href="mailto:aodabas@ogu.edu.tr" >aodabas@ogu.edu.tr </a ></span >br />Address : <br />Dr. A . Odabas <br /> Osmangazi University <br /> Arts and Sciences Faculty <br /> Department of Mathematics and Computer Science <br /> Eskisehir <br /> Turkey<br />p ><p ><b >b >p >p ><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a ></p >p >The <strong class="pkg" >XMod</strong > package provides functions for computation with</p >ul >li ><p >finite crossed modules of groups and cat1-groups, and morphisms of these structures;</p >li >li ><p >finite pre -crossed modules, pre -cat1-groups, and their Peiffer quotients;</p >li >li ><p >isoclinism classes of groups and crossed modules;</p >li >li ><p >derivations of crossed modules and sections of cat1-groups;</p >li >li ><p >crossed squares and their morphisms, including the actor crossed square of a crossed module;</p >li >li ><p >crossed modules of finite groupoids (experimental version).</p >li >ul >p ><strong class="pkg" >XMod</strong > was originally implemented in 1996 using the <strong class="pkg" >GAP</strong >3 language, when the second author was studying for a Ph.D. <a href="chapBib.html#biBA1" >[Alp97]</a > in the School of Mathematics and Computer Science at Bangor University.</p >p >In April 2002 the first and third parts were converted to <strong class="pkg" >GAP</strong >4 , the pre 2 .001 was released. The final two parts, covering derivations, sections and actors, were included in the January 2004 release 2 .002 for <strong class="pkg" >GAP</strong > 4 .4 .</p >p >In October 2015 functions for computing isoclinism classes of crossed modules, written by Alper Odabaş and Enver Uslu, were added. These are contained in Chapter <a href="chap4.html#X802AFE8E7EDB435E" ><span class="RefLink" >4 </span ></a >, and are described in detail in the paper <a href="chapBib.html#biBIOU1" >[IOU16]</a >.</p >p >Bug reports, suggestions and comments are, of course, welcome. Please submit an issue at <span class="URL" ><a href="https://github.com/gap-packages/xmod/issues/ " >https://github.com/gap-packages/xmod/issues/</a ></span > or send an email to the first author at <span class="URL" ><a href="mailto:cdwensley@btinternet.com" >cdwensley@btinternet.com</a ></span >.</p >p ><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a ></p >p >© 1996 -2025 , Chris Wensley et al.</p >p >The <strong class="pkg" >XMod</strong > package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option ) any later version.</p >p ><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a ></p >p >This documentation was prepared using the <strong class="pkg" >GAPDoc</strong > <a href="chapBib.html#biBGAPDoc" >[LN17]</a > and <strong class="pkg" >AutoDoc</strong > <a href="chapBib.html#biBAutoDoc" >[GH17]</a > packages.</p >p >The procedure used to produce new releases uses the package <strong class="pkg" >GitHubPagesForGAP</strong > <a href="chapBib.html#biBGitHubPagesForGAP" >[Hor17]</a > and the package <strong class="pkg" >ReleaseTools</strong >.</p >p ><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a ></p >div class="contents" >a id="contents" name="contents" ></a ></h3>div class="ContChap" ><a href="chap1.html#X7DFB63A97E67C0A1" >1 <span class="Heading" >Introduction</span ></a >div >div class="ContChap" ><a href="chap2.html#X7EB8288E8424F39F" >2 <span class="Heading" >2 d-groups : crossed modules and cat<span class="SimpleMath" >^1 </span >-groups</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7BAD9A7F7AFEEC89" >2 .1 <span class="Heading" >Constructions for crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7C8175AE7F76B586" >2 .1 -1 XMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83050ED686776933" >2 .1 -2 XModByNormalSubgroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X867B2D53832EF05E" >2 .1 -3 XModByTrivialAction</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X78B14FDA817CCEEF" >2 .1 -4 XModByAutomorphismGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7D0F6FAA7AF69844" >2 .1 -5 XModByCentralExtension</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X84FA2B0A795B6997" >2 .1 -6 XModByPullback</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X824631577864961E" >2 .1 -7 XModByAbelianModule</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X81704DFB795C0D29" >2 .1 -8 DirectProduct</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X790248A67CB9C33A" >2 .1 -9 Source </a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7AF6602C87845F1D" >2 .1 -10 ImageElmXModAction</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7846A7D37957B89E" >2 .1 -11 Size2d</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X85516B19803C01C0" >2 .1 -12 Name</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7CF622538749FE73" >2 .2 <span class="Heading" >Properties of crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7E77E6B881B1CE50" >2 .2 -1 IsXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7884284383284A87" >2 .2 -2 SubXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7D8165F77B23BCF6" >2 .2 -3 KernelCokernelXMod</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7D435B6279032D4D" >2 .3 <span class="Heading" >Pre -crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8487BE427858C5C9" >2 .3 -1 PreXModByBoundaryAndAction</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8527F4C07A8F359E" >2 .3 -2 PeifferSubgroup</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7AAABC1D7E110988" >2 .4 <span class="Heading" >Cat<span class="SimpleMath" >^1 </span >-groups and pre -cat<span class="SimpleMath" >^1 </span >-groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7CF4C37F87D27EBA" >2 .4 -1 Cat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7C4FFC4086531157" >2 .4 -2 Source </a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X79944C7B87F767FD" >2 .4 -3 DiagonalCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X79385660821E54A3" >2 .4 -4 TransposeCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X87544FAD873672E1" >2 .4 -5 Cat1GroupByPeifferQuotient</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X85D9C5F881DBA9FC" >2 .4 -6 SubCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7CE4F14585F6D473" >2 .4 -7 DirectProduct</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8317816A8361F88C" >2 .5 <span class="Heading" >span class="SimpleMath" >^1 </span >-groups and pre -cat<span class="SimpleMath" >^1 </span >-groups span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X78E03FAB84A57D03" >2 .5 -1 IsCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B7CF88F83B0129D" >2 .5 -2 IsPreCat1GroupWithIdentityEmbedding</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X82F10A59867C765D" >2 .5 -3 Cat1GroupOfXMod</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X80D6CB4080417BFA" >2 .6 <span class="Heading" >Enumerating cat<span class="SimpleMath" >^1 </span >-groups with a given source </span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7BDEBBF17CE6A6D4" >2 .6 -1 AllCat1GroupsWithImage</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FBFC8C87FC1AC5A" >2 .6 -2 AllCat1GroupsMatrix</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FEB2FCE7D9ADA85" >2 .6 -3 AllCat1GroupsIterator</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7C6346A17FEEDFA1" >2 .6 -4 CatnGroupNumbers</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7A6A70BD86DE458D" >2 .7 <span class="Heading" >Selection of a small cat<span class="SimpleMath" >^1 span >-group</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B8E67D880E380C8" >2 .7 -1 Cat1Select</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8614CDCF8063117F" >2 .8 <span class="Heading" >More functions for crossed modules and cat<span class="SimpleMath" >^1 </span >-groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7831DB527CF9DD57" >2 .8 -1 IdGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83BBC6818168C282" >2 .8 -2 IsSubXMod</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7CFAB044817E5E91" >2 .9 <span class="Heading" >The group groupoid associated to a cat<span class="SimpleMath" >^1 </span >-group</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7AF5AF668331321E" >2 .9 -1 GroupGroupoid</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8578AB6D7C1FC4F3" >2 .9 -2 GroupGroupoidElement</a ></span >div ></div >div >div class="ContChap" ><a href="chap3.html#X815144D67C1D1AE3" >3 <span class="Heading" >2 d-mappings</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7BBEA95E7AE1F317" >3 .1 <span class="Heading" >Morphisms of 2 -dimensional groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7FFD094F7FFB1F17" >3 .1 -1 Source </a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X78CADE4D7EB1EA44" >3 .2 <span class="Heading" >Morphisms of pre -crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X82B912B18127A42A" >3 .2 -1 IsXModMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7E078F497F4EFA9F" >3 .2 -2 IsInjective</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7CEABD6487CF2A38" >3 .2 -3 XModMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X854FC0C781AD62EC" >3 .2 -4 IsomorphismPerm2DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X87BCAAF787A7FF69" >3 .2 -5 MorphismOfPullback</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7B9D3C1F7A395FF2" >3 .3 <span class="Heading" >Morphisms of pre -cat<span class="SimpleMath" >^1 </span span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C47D0EC782D4C40" >3 .3 -1 IsCat1GroupMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7D7459E67B568B44" >3 .3 -2 Cat1GroupMorphismOfXModMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C6AF7C285D546B2" >3 .3 -3 IsomorphismPermObject</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X837B0299846C2391" >3 .3 -4 SmallerDegreePermutationRepresentation2DimensionalGroup</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7B09A28579707CAF" >3 .4 <span class="Heading" >Operations on morphisms</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X811F886081AAB95F" >3 .4 -1 CompositionMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X83C3A2478159DE76" >3 .4 -2 Kernel</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X79C47E3D7855A117" >3 .5 <span class="Heading" >Quasi-isomorphisms</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X86F08B1981618400" >3 .5 -1 QuotientQuasiIsomorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C4C1C587B8932A7" >3 .5 -2 SubQuasiIsomorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X83365AF2812E5C04" >3 .5 -3 QuasiIsomorphism</a ></span >div ></div >div >div class="ContChap" ><a href="chap4.html#X802AFE8E7EDB435E" >4 <span class="Heading" >Isoclinism of groups and crossed modules</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7E373BF3836B3A9C" >4 .1 <span class="Heading" >More operations for crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X873ED97185D9176E" >4 .1 -1 FactorPreXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8591E25680C5C575" >4 .1 -2 IntersectionSubXMods</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7E20208279038BB8" >4 .1 -3 Displacement</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X86ACB83E7D70C625" >4 .1 -4 CommutatorSubXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X86E0804B780A7FD6" >4 .1 -5 DerivedSubXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X85640DD17F5A2949" >4 .1 -6 FixedPointSubgroupXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B57446086BA1BF0" >4 .1 -7 CentreXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X814D9E1E78EEE665" >4 .1 -8 CentralQuotient</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F4222757B0E08B6" >4 .1 -9 IsAbelian2DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X87C524C08588AAC0" >4 .1 -10 LowerCentralSeriesOfXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7C67623F797A0301" >4 .1 -11 IsomorphismXMods</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81EE2188863E6E85" >4 .1 -12 AllXMods</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7B0D511A82FD945E" >4 .2 <span class="Heading" >Isoclinism for groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B0D511A82FD945E" >4 .2 -1 Isoclinism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7C72991985B58DB8" >4 .2 -2 IsStemDomain</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X82DD52587F81C95C" >4 .2 -3 IsoclinicRank</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X81338C977972AD83" >4 .3 <span class="Heading" >Isoclinism for crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81338C977972AD83" >4 .3 -1 Isoclinism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7E86DCB083CA5915" >4 .3 -2 IsStemDomain</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X820C412679910975" >4 .3 -3 IsoclinicRank</a ></span >div ></div >div >div class="ContChap" ><a href="chap5.html#X85CD9A43847AE1B8" >5 <span class="Heading" >Whitehead group of a crossed module</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7C01AE7783898705" >5 .1 <span class="Heading" >Derivations and Sections</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X83EC6F7780F5636E" >5 .1 -1 DerivationByImages</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7F119F9580C150B1" >5 .1 -2 PrincipalDerivation</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7E1C72897CD31D66" >5 .1 -3 SectionByHomomorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X87D9F7257DFF0236" >5 .1 -4 IdentityDerivation</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7AD6E23F8254F400" >5 .1 -5 WhiteheadProduct</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X861A52407D3C627D" >5 .2 <span class="Heading" >Whitehead Monoids and Groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X788884E48534F7CB" >5 .2 -1 AllDerivations</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7CB1614E7EC58A84" >5 .2 -2 WhiteheadMonoidTable</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X84CD856C84BDB019" >5 .2 -3 RegularDerivations</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X8452E45878691CD3" >5 .2 -4 PrincipalDerivations</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7E53AF1884B2D03D" >5 .3 <span class="Heading" >Endomorphisms determined by a derivation</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X86AF32CA84217C46" >5 .3 -1 SourceEndomorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X84463DE2872AA709" >5 .3 -2 RangeEndomorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7E27AE6478566A94" >5 .3 -3 Object2dEndomorphism</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X820501BF83D1D6D7" >5 .4 <span class="Heading" >Whitehead groups for cat<span class="SimpleMath" >^1 </span >-groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X813CAA17855172E4" >5 .4 -1 AllSections</a ></span >div ></div >div >div class="ContChap" ><a href="chap6.html#X84C872BB7F1E5F25" >6 <span class="Heading" >Actors of 2 d-groups</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7B853602873FC7AB" >6 .1 <span class="Heading" >Actor of a crossed module</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X80F121357F06E72D" >6 .1 -1 AutomorphismPermGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X790EBC7C7D320C03" >6 .1 -2 WhiteheadXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X85CF21F57F0F1329" >6 .1 -3 XModCentre</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X81BFAD86831097E3" >6 .2 <span class="Heading" >Actor of a cat<span class="SimpleMath" >^1 </span >-group</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X82097EE1866D0C2B" >6 .2 -1 ActorCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7FE056707BB983B3" >6 .2 -2 Actor</a ></span >div ></div >div >div class="ContChap" ><a href="chap7.html#X8339DF98872D2E1C" >7 <span class="Heading" >Induced constructions</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X80D3C8F97A10D5E5" >7 .1 <span class="Heading" >Coproducts of crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7C01F5D98046E44B" >7 .1 -1 CoproductXMod</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7966FF497C36C465" >7 .2 <span class="Heading" >Induced crossed modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X874CB2A278AADE3A" >7 .2 -1 InducedXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B24D47F8078540F" >7 .2 -2 AllInducedXMods</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X814A695779706E22" >7 .3 <span class="Heading" >Induced cat<span class="SimpleMath" >^1 </span >-groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7BCE57BE7F6E6B08" >7 .3 -1 InducedCat1Group</a ></span >div ></div >div >div class="ContChap" ><a href="chap8.html#X780368C083C76EDC" >8 <span class="Heading" >Crossed squares and Cat<span class="SimpleMath" >^2 </span >-groups</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7C4AFE8D85848C8F" >8 .1 <span class="Heading" >Definition of a crossed square a crossed <span class="SimpleMath" >n</span >-cube of groups</span ></a >span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X820A7D30847BC828" >8 .2 <span class="Heading" >Constructions for crossed squares</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X866A7FAC7FCB62C2" >8 .2 -1 CrossedSquareByXMods</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7FA98BD47FF9B044" >8 .2 -2 Size3d</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7896DAF786F46234" >8 .2 -3 CrossedSquareByNormalSubgroups</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7FA367977B1895A7" >8 .2 -4 CrossedSquareByNormalSubXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X833362FE87ED3C48" >8 .2 -5 ActorCrossedSquare</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X82E4EA93824F7B26" >8 .2 -6 CrossedSquareByAutomorphismGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X839E065783795CB8" >8 .2 -7 CrossedSquareByPullback</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7F5554907AF73190" >8 .2 -8 CrossedSquareByXModSplitting</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X87FBE3CE87DC8CD5" >8 .2 -9 CrossedSquare</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7F94830681EA19BE" >8 .2 -10 Transpose3DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7C2647CB82DDD065" >8 .2 -11 CentralQuotient</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8645AA3686F126D5" >8 .2 -12 IsCrossedSquare</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X828CFC5A83097189" >8 .2 -13 Up2DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7F81DA90820F4405" >8 .2 -14 IsSymmetric3DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7AE671C7798F99FD" >8 .2 -15 Crossed Pairing</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X79A59CED7C69BF18" >8 .3 <span class="Heading" >Substructures of Crossed Squares</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X83F16E94857407F3" >8 .3 -1 SubCrossedSquare</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7D0D7F787D438D9A" >8 .3 -2 TrivialSub3DimensionalGroup</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X78A79A7E85128C7B" >8 .4 <span class="Heading" >Morphisms of crossed squares</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X83E733547A14FD61" >8 .4 -1 CrossedSquareMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DE8173F80E07AB1" >8 .4 -2 Source </a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8284240C7B9BB783" >8 .4 -3 IsCrossedSquareMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8614E38E7A67690E" >8 .4 -4 InclusionMorphismHigherDimensionalDomains</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X86D5AA247B64ED51" >8 .5 <span class="Heading" >Definitions and constructions for cat<span class="SimpleMath" 2 </span >-groups and their morphisms span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X849D845586F92444" >8 .5 -1 Cat2Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X81BD4011837BCC2E" >8 .5 -2 Up2DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X861BA02C7902A4F4" >8 .5 -3 DirectProduct</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X85713B1C7C34324E" >8 .5 -4 DisplayLeadMaps</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X864B346686AAA522" >8 .5 -5 Transpose3DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X83616C237F4745FB" >8 .5 -6 Cat2GroupMorphism</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7D46CB517FCAA83B" >8 .5 -7 Cat2GroupOfCrossedSquare</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X82F896C7851294B7" >8 .5 -8 Subdiagonal2DimensionalGroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DB082D282C52855" >8 .5 -9 SubCat2Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X838423B47FEE09B2" >8 .5 -10 TrivialSubCat2Group</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X80FB2B328578DE42" >8 .6 <span class="Heading" >Enumerating cat<span class="SimpleMath" >^2 </span >-groups with a given source </span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7D421DE57B44F37A" >8 .6 -1 AllCat2GroupsWithImagesIterator</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X783F7FEB7B79B062" >8 .6 -2 AllCat2GroupsWithFixedUp</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X84636E047CDF2DF3" >8 .6 -3 AllCat2GroupsMatrix</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7EFCF9697E845B2C" >8 .6 -4 AllCat2GroupsIterator</a ></span >div ></div >div >div class="ContChap" ><a href="chap9.html#X7DBA3A7E81C71A64" >9 <span class="Heading" >Cat<span class="SimpleMath" >^3 </span >-groups and Crossed cubes</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7CC52AF4840F478E" >9 .1 <span class="Heading" >Functions for (pre -)cat<span class="SimpleMath" >^3 </span >-groups</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7828496F7D72E232" >9 .1 -1 Cat3Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85A5A6967D942463" >9 .1 -2 Front3DimensionalGroup</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X80E074E37D02B2F6" >9 .2 <span class="Heading" >Enumerating cat<span class="SimpleMath" >^3 </span >-groups with a given source </span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X83E4A7367DD13598" >9 .2 -1 AllCat3GroupTriples</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7F8538B580847268" >9 .3 <span class="Heading" >span class="SimpleMath" >^n</span >-groups and their morphisms span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X81C3D39A81B20D76" >9 .3 -1 PreCatnGroup</a ></span >div ></div >div >div class="ContChap" ><a href="chap10.html#X80B3A81B7E5CA3A9" >10 <span class="Heading" >Crossed modules of groupoids</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X847F4ED77F50528C" >10 .1 <span class="Heading" >Constructions for crossed modules of groupoids</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X78F89CAB7A281B8F" >10 1 -1 PreXModWithObjectsByBoundaryAndAction</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X86CD034F82F5F029" >10 1 -2 SinglePiecePreXModWithObjects</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7B76F2BF82E075FF" >10 1 -3 IsXModWithObjects</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X858EB4F97D04D012" >10 1 -4 IsPermPreXModWithObjects</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X797B1CD07C3682EE" >10 1 -5 Root2dGroup</a ></span >div ></div >div >div class="ContChap" ><a href="chap11.html#X83B7E8A287C9284A" >11 <span class="Heading" >Double Groupoids</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X87AC8EF586C35CD4" >11 .1 <span class="Heading" >Constructions for Double Groupoids</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X78936F448231692E" >11 1 -1 SinglePieceDoubleGroupoid</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X823A3A7481B90EB7" >11 1 -2 SquareOfArrows</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7D3737FA7E9E3ECA" >11 1 -3 HorizontalProduct</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X873F01287A2DC41F" >11 1 -4 VerticalProduct</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7D69B5A680FE4C81" >11 .2 <span class="Heading" >Conversion of Basic Double Groupoids</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7CB177EF78B559DB" >11 2 -1 EnhancedBasicDoubleGroupoid</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X853B15F483477D5C" >11 .3 <span class="Heading" >Commutative double groupoids</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7DC35C557E498880" >11 3 -1 DoubleGroupoidWithZeroBoundary</a ></span >div ></div >div >div class="ContChap" ><a href="chap12.html#X7DD7F0847FF2B96C" >12 <span class="Heading" >Applications</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap12.html#X8575260A80F735BD" >12 .1 <span class="Heading" >Free Loop Spaces</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap12.html#X87781C76804E783E" >12 1 -1 LoopClasses</a ></span >div ></div >div >div class="ContChap" ><a href="chap13.html#X81EC8C8A82C15298" >13 <span class="Heading" >Interaction with HAP </span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap13.html#X865CE53A827FBE6F" >13 .1 <span class="Heading" >Calling HAP functions</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13.html#X8699357D7DC6279E" >13 1 -1 SmallCat1Group</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13.html#X7B00E3FB82DC305D" >13 1 -2 CatOneGroupToXMod</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap13.html#X84B7160284FD454A" >13 1 -3 IdCat1Group</a ></span >div ></div >div >div class="ContChap" ><a href="chap14.html#X810FFB1C8035C8BE" >14 <span class="Heading" >Utility functions</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14.html#X7C9734B880042C73" >14 .1 <span class="Heading" >Mappings</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14.html#X7F8E297F7C84DE51" >14 1 -1 InclusionMappingGroups</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14.html#X81D29E737F3D4878" >14 1 -2 InnerAutomorphismsByNormalSubgroup</a ></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14.html#X7FC631B786C1DC8B" >14 1 -3 IsGroupOfAutomorphisms</a ></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap14.html#X852BD9CA84C2AFF0" >14 .2 <span class="Heading" >Abelian Modules</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap14.html#X806DEFCC859BB4F1" >14 2 -1 AbelianModuleObject</a ></span >div ></div >div >div class="ContChap" ><a href="chap15.html#X810C43BC7F63C4B4" >15 <span class="Heading" >Development history</span ></a >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15.html#X7ACE7E8384B73156" >15 .1 <span class="Heading" >Changes from version to version</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X848198AA862249C4" >15 1 -1 <span class="Heading" >Version 1 for <strong class="pkg" >GAP</strong > 3 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X7CF8E72D80AAB54F" >15 1 -2 <span class="Heading" >Version 2 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X7F9CE0487BB6F660" >15 1 -3 <span class="Heading" >Version 2 .001 for <strong class="pkg" >GAP</strong > 4 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X7966FF497C36C465" >15 1 -4 <span class="Heading" >Induced crossed modules</span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X7E0B70FD82DC5BA8" >15 1 -5 <span class="Heading" >Versions 2 .002 -- 2 .006 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X7F6E650E85384C25" >15 1 -6 <span class="Heading" >Versions 2 .007 -- 2 .010 </span ></a >span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15.html#X80A8A3FB82048ADD" >15 .2 <span class="Heading" >Versions for <strong class="pkg" >GAP</strong > [4 .5 4 .12 ]</span ></a >span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X794BBE42839F2E18" >15 2 -1 <span class="Heading" >AllCat1s</span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X78C26CC27D48B1A8" >15 2 -2 <span class="Heading" >Versions 2 .43 - 2 .56 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X8715310378F0F8D2" >15 2 -3 <span class="Heading" >Version 2 .61 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X87EE8E70786CAF46" >15 2 -4 <span class="Heading" >Versions 2 .63 - 2 .74 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X85F63D6979B72CA5" >15 2 -5 <span class="Heading" >Versions 2 .75 - 2 .85 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X81023EBF7CD2352E" >15 2 -6 <span class="Heading" >Versions 2 .86 - 2 .91 </span ></a >span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap15.html#X87062F217BAC0B6E" >15 2 -7 <span class="Heading" >Versions from 2 .92 </span ></a >span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap15.html#X83D1530487593182" >15 .3 <span class="Heading" >What needs doing next?</span ></a >span >div >div >div class="ContChap" ><a href="chapBib.html" ><span class="Heading" >References</span ></a ></div >div class="ContChap" ><a href="chapInd.html" ><span class="Heading" >Index</span ></a ></div >br />div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a > <a href="chap0.html#contents" >[Contents]</a > <a href="chap1.html" >[Next Chapter]</a > </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a > <a href="chap1.html" >1 </a > <a href="chap2.html" >2 </a > <a href="chap3.html" >3 </a > <a href="chap4.html" >4 </a > <a href="chap5.html" >5 </a > <a href="chap6.html" >6 </a 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