Higher Dimensional Categories: Induction on Extensivity

In this paper, we explore, enrich, and otherwise mildly generalise a prominent definition of weak n-category by Batanin, as refined by Leinster, to give a definition of weak n-dimensional V-category, with a view to applications in programming semantics. We require V to be locally presentable and to be (infinitarily) extensive, a condition which ensures that coproducts are suitably well-behaved. Our leading example of such a V is the category ω-Cpo, ω-Cpo-enriched bicategories already having been used in denotational semantics.