  We show that the strict 1-category $\square$ of cubes -- defined to be the
full subcategory of strict $\omega$-categories whose objects are the Gray
tensor powers of the arrow category -- are dense in the $(\infty,1)$-category
$\mathsf{Cat}_\omega$ of weak $(\infty,\infty)$-categories, in both
Rezk-complete and incomplete variants. More precisely, we show that Joyal's
category $\Theta$ is contained in the idempotent completion of $\square$, and
in fact that the idempotent completion of $\square$ is closed under suspensions
and wedge sums. This result extends a theorem of Campbell and Maehara in
dimension 2. Following Campbell and Maehara's program, we will in future work
apply this result to give a new construction of the Gray tensor product of weak
$(\infty,n)$-categories.

Category Theory