Combinatorial approach to the category $\Theta_0$ of cubical pasting diagrams

TL;DR

This paper develops a combinatorial framework for cubical pasting diagrams and models of cubical weak infinity-categories, proving the monad for cubical strict infinity-categories with connections is cartesian, thus advancing higher category theory.

Contribution

It introduces a detailed combinatorial description of cubical pasting diagrams and proves the monad for cubical strict infinity-categories with connections is cartesian, confirming a prior conjecture.

Findings

Description of the category _ of cubical pasting diagrams

Proof that the monad for cubical strict -categories with connections is cartesian

Framework for cubical weak -categories and -groupoids with connections

Abstract

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are globular weak $(\infty,m)$-categories. Then we describe the combinatorics of the small category $\Theta_0$ whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict $\infty$-categories with connections. We prove that it is a cartesian monad, solving a conjecture in \cite{camark-cub-1}. This puts us in a position to describe the cubical coherator $\Theta^{\infty}_W$ whose set-models are cubical weak $\infty$-categories with connections and the cubical coherator $\Theta^{\infty}_{W^{0}}$ whose set-models are cubical weak $\infty$-groupoids with connections.