How strict is strictification?

TL;DR

This paper investigates the strictification process in 3-dimensional category theory, revealing that the adjunction between 2-categories and bicategories can be strictly enriched, simplifying the understanding of higher categorical structures.

Contribution

It demonstrates that the strictification adjunction can be strictly enriched over the multicategory of bicategories, providing new insights into the structure of tricategories and their relationships.

Findings

The adjunction can be strictly enriched over the symmetric multicategory of bicategories.

The adjunction underlies an adjunction of bicategory-enriched symmetric multicategories.

Introduction of the symmetric closed multicategory of pseudo double categories.

Abstract

The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the $\mathbf{Gray}$-category of $2$-categories and the tricategory of bicategories. We show that -- far from requiring the full weakness provided by the definitions of tricategory theory -- this adjunction can be $\textit{strictly}$ enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appendix introduces the symmetric closed multicategory of pseudo double categories, into which Verity's symmetric multicategory of bicategories embeds fully.