Cubes are dense in $(\mathrm{\infty },\mathrm{\infty })$-categories

We show that the strict 1-category $\mathrm{\square }$ 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 $(\mathrm{\infty },1)$-category ${\mathsf{C}\mathsf{a}\mathsf{t}}_{\omega }$ of weak $(\mathrm{\infty },\mathrm{\infty })$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal’s category $\mathrm{\Theta }$ is contained in the idempotent completion of $\mathrm{\square }$, and in fact that the idempotent completion of $\mathrm{\square }$ 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 $(\mathrm{\infty },n)$-categories.