The Gray Product of $(\mathrm{\infty },n)$-Categories via Lax Grids

We introduce a new model for $(\mathrm{\infty },n)$-categories as Segal sheaves on lax grids, which are pasting diagrams of lax cubes. This model allows for a direct construction of the Gray tensor product via Day convolution. We show that this agrees with Campion’s construction of the Gray tensor product. These results will be applied in future work to equip the higher categories of cobordisms with a Gray-algebra structure given by the cartesian product of manifolds.