The Gray tensor product of $(\mathrm{\infty },n)$-categories

In this note, we leverage the author’s pasting theorem for $(\mathrm{\infty },n)$-categories to construct new models of $(\mathrm{\infty },n)$-categories for all $n\le \mathrm{\infty }$, as presheaves on certain categories of computads. Among these new models are some which facilitate a construction of the Gray tensor product of $(\mathrm{\infty },n)$-categories Day convolution and reflection, which we carry out here. After constructing this Gray tensor product of $(\mathrm{\infty },n)$-categories, we characterize it via several model-independent universal properties.