Lax functorialities of the comma construction for ω-categories

Abstract

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