We prove that the folk model category structure on the category of strict ω-categories, introduced by Lafont, Métayer and Worytkiewicz, is monoidal, first, for the Gray tensor product and, second, for the join of ω-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 ω-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.
Keywords: augmented directed complexes, folk model category structure, Gray tensor product, join, locally biclosed monoidal categories, monoidal model categories, oplax transformations, strict ω-categories, strict ω-groupoids, strict (m, n)-categories
2020 MSC: 18M05, 18N30, 18N40, 55U35
Theory and Applications of Categories, Vol. 35, 2020, No. 21, pp 745-808.