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
@article{TAC_2020_35_a20,
author = {Dimitri Ara and Maxime Lucas},
title = {The folk model category structure on strict
\ensuremath{\omega}-categories is monoidal},
journal = {Theory and applications of categories},
pages = {745--808},
year = {2020},
volume = {35},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2020_35_a20/}
}
Dimitri Ara; Maxime Lucas. The folk model category structure on strict ω-categories is monoidal. Theory and applications of categories, Tome 35 (2020), pp. 745-808. http://geodesic.mathdoc.fr/item/TAC_2020_35_a20/