A cubical model for $(\mathrm{\infty },n)$-categories

We propose a new model for the theory of $(\mathrm{\infty },n)$-categories (including the case $n=\mathrm{\infty }$) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our model is shown to be monoidal with respect to suitably defined (lax and pseudo) Gray tensor products; in particular, these tensor products are both associative and biclosed. Furthermore, we show that the triangulation functor to pre-complicial sets is a left Quillen functor and is strong monoidal with respect to both Gray tensor products.