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

TL;DR

This paper establishes that the model structures for diagrammatic $( , n)$-categories are monoidal with respect to the Gray product, and explores related properties like equivalences and functorial cylinders.

Contribution

It proves the monoidality of the model structures under the Gray product and analyzes key properties of the Gray product in diagrammatic $( , n)$-categories.

Findings

Gray product of a cell and an equivalence is an equivalence

Tensoring with the walking equivalence acts as a functorial cylinder

Opposite diagrammatic sets form a Quillen self-equivalence

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.