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

TL;DR

The paper establishes that diagrammatic $( ablafty,n)$-categories, modeled as diagrammatic sets with a Gray product defined by Day convolution, form a monoidal model category under the Gray product. It achieves this by first proving a monoidal structure for the coinductive $( ablafty,n)$-model structure on marked diagrammatic sets and then transferring the monoidality to diagrammatic sets via a Quillen equivalence. A key technical achievement is showing that tensoring a Gray product with an equivalence yields an equivalence, and that left or right tensoring with the walking equivalence provides a functorial cylinder, with the opposite functor acting as a Quillen self-equivalence. These results jointly yield a robust monoidal framework for diagrammatic higher categories, enabling coherent Gray-product operations and duality-preserving constructions in this setting.

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.

Theorems & Definitions (124)

• Theorem
• Theorem
• Lemma 1.2: Retract Lemma
• proof
• Remark 1.4
• Proposition 1.5: Small object argument
• proof
• Remark 1.6
• Remark 1.11
• Lemma 1.13