Internalising the theory of weak complicial sets III

Dominic Verity – 16 November 2005

Our intuition that the theory of weak complicial sets is really some kind of weak omega-category theory leads us to hope that these structures behave as if they were quasi-categories enriched in weak complicial sets. We make this intuition precise and discuss some of the theory of these latter structures. Just as the theory of 2-categories is illuminated by regarding them as certain kinds of categories internal to Cat (vertically discrete double categories), it is convenient here to study the theory of quasi-categories internal to a combinatorial and complicially enriched Quillen model category. We generalise the theory of categorical discrete fibrations to this context and use it to provide motivation for a Yoneda embedding style coherence theorem which demonstrates that every weak complicial set enriched quasi-category may be replaced (up to weak equivalence) by a (strictly) enriched category. This latter result may be regarded as a natural generalisation of the Gordon-Power-Street coherence result relating tricategories and Gray-enriched categories.