Lax functorialities of the comma construction for $ω$-categories
Dimitri Ara, Léonard Guetta
TL;DR
This work develops a higher-categorical generalization of the Grothendieck construction by framing comma constructions inside Gray $\omega$-categories. It shows that the oplax comma $A \downarrow_C B$ extends to a Gray $\omega$-functor, and that Gray slice categories $\mathbb{C}/c$ exist, enabling coherent handling of higher cones and pasting data. The Grothendieck construction for $\omega$-categories is then defined as a total dual of an oplax comma, yielding a Gray $\omega$-functor that is functorial in both the base category and the diagram; a by-product is a tentative notion of Grothendieck construction for Gray $\omega$-functors. Together, these results provide a unified framework for functoriality of comma and Grothendieck constructions in the setting of strict and Gray $\omega$-categories, with potential applications to higher homotopical properties and type-theoretic models.
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.