The low-dimensional structures formed by tricategories

TL;DR

This paper constructs a hierarchy of higher-dimensional categorical structures, demonstrating how tricategories relate to locally cubical bicategories and establishing a tricategory of tricategories.

Contribution

It introduces a new framework connecting tricategories with locally cubical bicategories and proves the existence of a tricategory of tricategories.

Findings

Established a bicategory of tricategories and homomorphisms.

Enriched this bicategory into a locally cubical bicategory.

Proved the existence of a tricategory of tricategories.

Abstract

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.