A Quillen adjunction between globular and complicial approaches to $(\mathrm{\infty },n)$-categories

We prove the compatibility between the suspension construction and the complicial nerve of $\omega$-categories. As a motivating application, we produce a Quillen pair between the models of $(\mathrm{\infty },n)$-categories given by Rezk’s complete Segal ${\mathrm{\Theta }}_n$-spaces and Verity’s $n$-complicial sets.