Lax functorialities of the comma construction for $ω$-categories

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.

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