  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.
