Tricategorical Universal Properties Via Enriched Homotopy Theory

Abstract

We develop the theory of tricategorical limits and colimits, and show that they can be modelled up to biequivalence via certain homotopically well-behaved limits and colimits enriched over the monoidal model category $\mathbf{Gray}$ of $2$-categories and $2$-functors. This categorifies the relationship that bicategorical limits and colimits have with the so called `flexible' enriched limits in $2$-category theory. As examples, we establish the tricategorical universal properties of Kleisli constructions for pseudomonads, Eilenberg-Moore and Kleisli constructions for (op)monoidal pseudomonads, centre constructions for $\mathbf{Gray}$-monoids, and strictifications of bicategories and pseudo-double categories.