Cubes are dense in $(\infty,\infty)$-categories

TL;DR

This paper proves that the category of cubes is dense in the category of weak $( ext{infinity}, ext{infinity})$-categories, extending previous results and enabling new constructions of tensor products in higher category theory.

Contribution

It demonstrates that the strict cube category is dense in the $( ext{infinity}, ext{infinity})$-category of weak categories, extending dimension 2 results and facilitating future tensor product constructions.

Findings

Joyal's category $ heta$ is in the idempotent completion of $ox$

Idempotent completion of $ox$ is closed under suspensions and wedge sums

Extension of Campbell and Maehara's theorem to higher dimensions

Abstract

We show that the strict 1-category $\square$ of cubes -- defined to be the full subcategory of strict $\omega$-categories whose objects are the Gray tensor powers of the arrow category -- are dense in the $(\infty,1)$-category $\mathsf{Cat}_\omega$ of weak $(\infty,\infty)$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal's category $\Theta$ is contained in the idempotent completion of $\square$, and in fact that the idempotent completion of $\square$ is closed under suspensions and wedge sums. This result extends a theorem of Campbell and Maehara in dimension 2. Following Campbell and Maehara's program, we will in future work apply this result to give a new construction of the Gray tensor product of weak $(\infty,n)$-categories.