The folk model category structure on strict $\omega$-categories is monoidal

TL;DR

This paper proves that the folk model category structure on strict ω-categories is monoidal with respect to Gray tensor and join, leading to a monoidal model structure on strict ω-groupoids.

Contribution

It establishes the monoidality of the folk model structure on strict ω-categories for Gray tensor and join, and introduces a compatible tensor product for strict (m,n)-categories.

Findings

The folk model category structure is monoidal for Gray tensor and join.

Gray tensor induces a tensor product on strict (m,n)-categories.

The monoidal model structure satisfies the monoid axiom.

Abstract

We prove that the folk model category structure on the category of strict $\omega$-categories, introduced by Lafont, M\'etayer and Worytkiewicz, is monoidal, first, for the Gray tensor product and, second, for the join of $\omega$-categories, introduced by the first author and Maltsiniotis. We moreover show that the Gray tensor product induces, by adjunction, a tensor product of strict $(m,n)$-categories and that this tensor product is also compatible with the folk model category structure. In particular, we get a monoidal model category structure on the category of strict $\omega$-groupoids. We prove that this monoidal model category structure satisfies the monoid axiom, so that the category of Gray monoids, studied by the second author, bears a natural model category structure.