Gray products of diagrammatic $(\infty, n)$-categories

Abstract

For each $n \in \mathbb{N} \cup \{\infty\}$, diagrammatic sets admit a model structure whose fibrant objects are the diagrammatic $(\infty, n)$- categories. They also support a notion of Gray product given by the Day convolution of a monoidal structure on their base category. The goal of this article is to show that the model structures are monoidal with respect to the Gray product. On the way to the result, we also prove that the Gray product of any cell and an equivalence is again an equivalence. Finally, we show that tensoring on the left or the right with the walking equivalence is a functorial cylinder for the model structures, and that the functor sending a diagrammatic set to its opposite is a Quillen self-equivalence.