Lax functorialities of the comma construction for $\omega$-categories

Abstract

Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict $\omega$-categories. To state the most general functorialities, we use the language of Gray $\omega$-categories, that is, categories enriched in the category of strict $\omega$-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict $\omega$-categories defines a Gray $\omega$-functor, that is, a morphism of Gray $\omega$-categories. To makes sense of this statement, we prove that slices of Gray $\omega$-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict $\omega$-categories defines a Gray $\omega$-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray $\omega$-functors, which we plan to investigate in future work.