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. We illuminate the implicit use of recursion in Leinster's definition, generating the higher dimensions by a process of repeated enrichment. The key fact is that if V is a locally presentable and extensive category, then so are the categories of small V-graphs and small V-categories. Iterating, this produces categories of n-dimensional V-graphs and strict n-dimensional V-categories that are also locally presentable and extensive. We show that the free strict n-dimensional V-category monad on the category of n-dimensional V-graphs is cartesian. This, along with results due to Garner, allows us to follow Batanin and Leinster's approach for defining weak n-categories. In the case that V = Set, the resulting definition of weak n-dimensional V-category agrees with Leinster's definition.