Re: reference: normal categorical subgroup?

[Ronnie Brown <mas010@bangor.ac.uk>]

A definition of normal subcrossed module was given by Kathy Norrie in
her thesis, (see the references to her work in two papers in the April,
2003, issue of Applied Categorical Structures). Of course crossed modules
are equivalent to cat^1-groups, (G,s,t), so it is a nice exercise to do
the translation.  The expected answer is (H,s',t') such that H is normal
in G and invariant under s,t.

However categorical groups are different to cat^1-groups, so it is a nice
exercise to solve this problem. It would also be good to characterise
groupoid objects internal to categorical groups, since these
should generalise the congruences for those objects, and hopefully lead to
a definition of free groupoid in categorical groups, and so to notions
of free resolutions.

For cat^1-groups (and crossed modules) there is a nice notion of induced
object (see papers by Brown and Wensley in TAC, for example) which is
required to compute the 2-type of a mapping cone induced on classifying
spaces by a morphism of groups.

Best regards

Ronnie


On Thu, 5 Jun 2003, Marco Mackaay wrote:

> To all category theorists,
>
> I'm looking for a reference to the definition of a normal categorical
> subgroup, i.e. the right kind of subgroupoid of a categorical group for
> taking the quotient. I know the definition, but I have no reference. Does
> anyone know a published origin of the definition?
>
> Best wishes,
>
> Marco Mackaay
>
>
>
>
>
>
>
>

Prof R. Brown,
School of Informatics, Mathematics Division,
University of Wales, Bangor
Dean St., Bangor,
Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474|office:     382681
fax: +44 1248 361429
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
(Links to survey articles:
Higher dimensional group theory
Groupoids and crossed objects in algebraic topology)

 Centre for the Popularisation of Mathematics
 Raising Public Awareness of Mathematics CDRom
 Symbolic Sculpture and Mathematics:
 http://www.cpm.informatics.bangor.ac.uk/centre/index.html