The following paper has been put on the arXiv on behalf of Philip Higgins Ronnie Brown Title: Thin Elements and Commutative Shells in Cubical omega-categories Author: Philip J. Higgins Report-no: University of Wales, Bangor, Maths Preprint 04.16 Subj-class: Category Theory; Algebraic Topology MSC-class: 18D05,18D35,55U99 ABSTRACT: The relationships between thin elements, commutative shells and connections in cubical omega-categories are explored by a method which does not involve the use of pasting theory or nerves of omega-categories (both of which were previously needed for this purpose; see previous work by Al-Agl/Brown/Steiner). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 by Brown/Mosa, and Brown/Kamps/Porter. Available from the arXiv math.CT/0408306 Preprint 04.16 on http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/04... -- Professor Emeritus R. Brown, Department of Mathematics, 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: http://www.popmath.org.uk/