The Grothendieck Construction in Enriched, Internal and ∞-Category Theory

Abstract

The Grothendieck construction takes a prestack (or pseudofunctor) B^op → Cat and returns a cartesian fibration over B. Classically, this construction works for categories with sets of morphisms. Enriched categories have morphisms belonging to another monoidal category V, while internal categories require the objects to also belong to V. Many concepts from ordinary (i.e. Set-based) category theory generalize well to enriched and internal category theory, but fibrations and the Grothendieck construction are not one of them. This is especially true if the monoidal product on V is not given by the cartesian product, such as when V = Vect. In this thesis, we generalize prestacks to V-enriched and V-internal categories, where V is non-cartesian, and develop a Grothendieck construction for them. As an application, when V = sSet, we obtain a version of the ∞-categorical Grothendieck construction and show that it is equivalent to existing ∞-categorical constructions.