Equivalence of different versions of Gray tensor products and lax transformations for (∞,n)‑categories

Ascertain whether the various definitions of Gray tensor products and lax transformations for (∞,n)‑categories are equivalent, including for n=2 and especially for n>2.

Background

Lax transformations, lax functors, and Gray tensor products are fundamental structures for higher categorical coherence and functoriality. For (∞,2)‑categories, these have been developed via both 2‑fold Segal spaces and scaled simplicial sets, with recent comparison in special cases.

A general equivalence among the different constructions of Gray tensor products and associated lax notions across models and dimensions remains unsettled, posing a significant technical and conceptual challenge particularly for n>2.

References

Many open questions remain, however, in particular regarding the equivalence of different versions of Gray tensor products and lax transformations for (∞,n)-categories, (including for n = 2, but especially for n > 2); see for instance .

— Higher categories (2401.14311 - Haugseng, 25 Jan 2024) in Remark on lax transformations and Gray tensor products, Section “(∞,n)‑categories”