The Gray tensor product for 2-quasi-categories (Q2217525)

An important construction in the theory of 2-categories is the Gray tensor product, in large part because it allows for a closed monoidal structure on the 2-category of 2-categories. Furthermore, its construction is closely related to a general theory of monads in 2-categories. The purpose of this paper is to develop an analogous notion of the Gray tensor product in the homotopical context of \((\infty,2)\)-categories, specifically in the model of \(\Theta_2\)-sets, or 2-quasi-categories, as developed by Ara. Building on similar constructions of Verity for complicial sets, the author is able to describe a theory of monads and a definition of the Gray tensor product in this setting. Doing so requires overcoming substantial combinatorial obstacles in order to verify, for example, that this tensor product is associative up homotopy. It seems likely that the arguments used in this paper will be useful more generally for anyone working with the combinatorics of \(\Theta_2\)-sets.