Symmetry in the cubical Joyal model structure

TL;DR

This paper investigates the properties of the cubical Joyal model structure on cubical sets, demonstrating its cartesian monoidal nature and the symmetry of the geometric product up to weak equivalence.

Contribution

It introduces a combinatorial approach to compare cubical sets with and without symmetries and proves the symmetric monoidal property of the geometric product in this context.

Findings

Cubical Joyal model structures are cartesian monoidal.

The geometric product is symmetric up to natural weak equivalence.

Induced model structures for (,1)-categories are obtained on cubical sets with symmetries.

Abstract

We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we prove that the cubical Joyal model structures on categories of cubical sets with connections are cartesian monoidal. Our techniques also allow us to prove that the geometric product of cubical sets (with or without connections) is symmetric up to natural weak equivalence in the cubical Joyal model structure, and to obtain induced model structures for $(\infty,1)$-categories on cubical sets with symmetries.