Thin Elements and Commutative Shells in Cubical ω-categories

The relationships between thin elements, commutative shells and connections in cubical ω-categories are explored by a method which does not involve the use of pasting theory or nerves of ω-categories (both of which were previously needed for this purpose; see [2], Section 9). 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 in [7, 6].