<!-- ------------------------------------------------------------------- --> <!-- --> <!-- intro.xml XMod documentation Chris Wensley --> <!-- --> <!-- Copyright (C) 1996-2021, Chris Wensley et al, --> <!-- --> <!-- ------------------------------------------------------------------- -->
<?xmlversion="1.0"encoding="UTF-8"?>
<Chapter Label="Intro">
<Heading>Introduction</Heading>
The &XMod; package provides functions for computation with
<List>
<Item>
finite crossed modules of groups and cat1-groups,
and morphisms of these structures;
</Item>
<Item>
finite pre-crossed modules, pre-cat1-groups, and their Peiffer quotients;
</Item>
<Item>
derivations of crossed modules and sections of cat1-groups;
</Item>
<Item>
isoclinism of groups and crossed modules;
</Item>
<Item>
the actor crossed square of a crossed module;
</Item>
<Item>
crossed squares, cat2-groups, and their morphisms (experimental version);
</Item>
<Item>
crossed modules of groupoids (experimental version).
</Item>
</List>
It is loaded with the command
<Example>
<![CDATA[
gap> LoadPackage( "xmod" );
]]>
</Example>
<P/>
The term crossed module was introduced by J. H. C. Whitehead in
<Cite Key="W2" />, <Cite Key="W1" />.
Loday, in <Cite Key="L1" />,
reformulated the notion of a crossed module as a cat1-group.
Norrie <Cite Key="N1" />, <Cite Key="N2" />
and Gilbert <Cite Key="G1" /> have studied derivations,
automorphisms of crossed modules and the actor of a crossed module,
while Ellis <Cite Key="E1" />
has investigated higher dimensional analogues.
Properties of induced crossed modules have been determined by
Brown, Higgins and Wensley in <Cite Key="BH1" />,
<Cite Key="BW1" /> and <Cite Key="BW2" />.
For further references see <Cite Key="AW1" />,
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 <M>30</M>.
<P/>
&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 <C>ConjugationXMod</C>
has become <Ref Oper="XModByNormalSubgroup"/>.
For a list of name changes see the file <F>names.pdf</F>
in the <F>doc</F> directory.
<P/>
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 <Ref Chap="chap-isclnc" />,
and are described in detail in their paper <Cite Key="IOU1" />.
<P/>
The package may be obtained as a compressed tar file
<File>XMod-version.number.tar.gz</File>
by ftp from one of the following sites:
<List>
<Item>
the &XMod; GitHub release site:
<URL>https://github.com/gap-packages.github.io/xmod/</URL>.
</Item>
<Item>
any &GAP; archive, e.g.
<URL>https://www.gap-system.org/Packages/packages.html</URL>;
</Item>
</List>
The package also has a GitHub repository at:
<URL>https://github.com/gap-packages/xmod/</URL>.
<P/>
Crossed modules and cat1-groups are special types of
<E>2-dimensional groups</E> <Cite Key="B82"/>, <Cite Key="brow:hig:siv"/>,
and are implemented as <C>2DimensionalDomains</C> and <C>2DimensionalGroups</C>
having a <C>Source</C> and a <C>Range</C>. <!-- See the file <F>notes.pdf</F> in the <F>doc</F> directory --> <!-- for an introductory account of these algebraic gadgets. -->
<P/>
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.
<P/>
The second part is concerned with <E>morphisms</E> of (pre-)crossed modules
and (pre-)cat1-groups, together with standard operations for morphisms,
such as composition, image and kernel.
<P/>
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 <Ref Sect="sect-more-xmod-ops" />.
Then, in <Ref Sect="sect-isoclinic-groups" />
and <Ref Sect="sect-isoclinic-xmods" />
there are functions for isoclinism of groups and crossed modules.
<P/>
The fourth part is concerned with the equivalent notions of
<E>derivation</E> for a crossed module and <E>section</E> for a cat1-group,
and the monoids which they form under the Whitehead multiplication.
<P/>
The fifth part deals with actor crossed modules and actor cat1-groups.
For the actor crossed module <M>{\rm Act}(\calX)</M> of a
crossed module <M>\calX</M> we require representations
for the Whitehead group of regular derivations of <M>\calX</M>
and for the group of automorphisms of <M>\calX</M>.
The construction also provides an inner morphism from <M>\calX</M>
to <M>{\rm Act}(\calX)</M>
whose kernel is the centre of <M>\calX</M>.
<P/>
The sixth part, which remains under development, contains
functions to compute induced crossed modules.
<P/>
Since version 2.007 there are experimental functions for
<E>crossed squares</E> and their morphisms,
structures which arise as <M>3</M>-dimensional groups.
Examples of these are inclusions of normal sub-crossed modules,
and the inner morphism from a crossed module to its actor.
<P/>
The eighth part has some experimental functions for crossed modules
of groupoids, interacting with the package <Package>groupoids</Package>.
Much more work on this is needed.
<P/>
Future plans include the implementation of <E>group-graphs</E>
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.
<P/>
The equivalent categories <C>XMod</C> (crossed modules) and
<C>Cat1</C> (cat1-groups) are also equivalent to <C>GpGpd</C>,
the subcategory of group objects in the category <C>Gpd</C> of groupoids.
Finite groupoids have been implemented in Emma Moore's package
<Package>groupoids</Package> <Cite Key="M1"/>
for groupoids and crossed resolutions.
<P/>
<Index Key="InfoXMod"><C>InfoXMod</C></Index>
In order that the user has some control of the verbosity of the
&XMod; package's functions,
an <C>InfoClass</C> <C>InfoXMod</C> is provided
(see Chapter <C>ref:Info Functions</C> in the &GAP; Reference Manual for
a description of the <C>Info</C> mechanism).
By default, the <C>InfoLevel</C> of <C>InfoXMod</C> is <C>0</C>;
progressively more information is supplied by raising
the <C>InfoLevel</C> to <C>1</C>, <C>2</C> and <C>3</C>.
<P/>
<Example>
<![CDATA[
gap> SetInfoLevel( InfoXMod, 1); #sets the InfoXMod level to 1
]]>
</Example>
<P/>
Once the package is loaded, the manual <Code>doc/manual.pdf</Code>
can be found in the documentation folder.
The <Code>html</Code> versions, with or without MathJax,
should be rebuilt as follows:
<P/>
<Example>
<![CDATA[
gap> ReadPackage( "xmod, "makedoc.g" );
]]>
</Example>
<P/>
It is possible to check that the package has been installed correctly
by running the test files:
<P/>
<Example>
<![CDATA[
gap> ReadPackage( "xmod", "tst/testall.g" );
#I Testing .../pkg/xmod/tst/gp2obj.tst
...
]]>
</Example>
<P/>
Additional information can be found on the
<E>Computational Higher-dimensional Discrete Algebra</E> website at:
<URL>https://github.com/cdwensley</URL>.
</Chapter>
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