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