2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72252The 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.15 pages, xypicCategory TheoryAlgebraic Topology18D05,18D35,55U99text