Weak Complicial Sets A Simplicial Weak ω-Category Theory Part II: Nerves of Complicial Gray-Categories

This paper continues the development of a simplicial theory of weak ω-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of homotopy theory and the 3-categorical Gray-categories of weak 3-category theory. We derive a simplicial nerve construction, which is closely related to Cordier and Porter’s homotopy coherent nerve, and show that this faithfully represents complicial Gray-categories as weak complicial sets. The category of weak complicial sets may itself be canonically enriched to a complicial Gray-category whose homsets are higher generalisations of the bicategory of homomorphisms, strong transformations and modifications. By applying our nerve construction to this structure, we demonstrate that the totality of all (small) weak complicial sets and their structural morphisms at higher dimensions form a richly structured (large) weak complicial set.