Pseudocolimits of Small Filtered Diagrams of Internal Categories

Abstract

Pseudocolimits are formal gluing constructions that combine objects in a category indexed by a pseudofunctor. When the objects are categories and the domain of the pseudofunctor is small and filtered it is known in SGA4 that the pseudocolimit can be computed by taking the Grothendieck construction of the pseudofunctor and inverting the class of cartesian arrows with respect to the canonical fibration. In this thesis we present a set of conditions on an ambient category E for defining the Grothendieck construction as an oplax colimit and another set of conditions on E along with conditions on an internal category, C, and a map picking out arrows in C that allow us to translate the axioms for a category of (right) fractions, and construct an internal category of (right) fractions. We combine these results in a suitable context to compute the pseudocolimit of a small filtered diagram of internal categories.