Cubical n-Categories and Finite Limits Theories

TL;DR

This paper explores constructing cubical n-categories through models of finite limits theories, extending double bicategories to n-tuple bicategories and examining simpler definitions of weak cubical n-categories.

Contribution

It introduces a method to build cubical n-categories by iterating models of finite limits theories, extending existing bicategory constructions to higher dimensions.

Findings

Extended double bicategories to n-tuple bicategories.

Provided a framework for constructing weak cubical n-categories.

Connected models of finite limits theories with higher-dimensional category structures.

Abstract

This note informally describes a way to build certain cubical n-categories by iterating a process of taking models of certain finite limits theories. We base this discussion on a construction of "double bicategories" as bicategories internal to Bicat, and see how to extend this to n-tuple bicategories (and similarly for tricategories etc.) We briefly consider how to reproduce "simpler" definitions of weak cubical n-category from these.