Pseudocolimits of Small Filtered Diagrams of Internal Categories

TL;DR

This paper explores conditions under which pseudocolimits of small filtered diagrams of internal categories can be computed internally, extending classical results with new internal constructions and axioms.

Contribution

It provides new internal conditions and constructions for computing pseudocolimits of filtered diagrams of internal categories, expanding the theoretical framework.

Findings

Established conditions for internal categories of elements as oplax colimits

Presented axioms for internal categories of fractions using weak-equivalences

Combined conditions to compute pseudocolimits internally in a suitable context

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 has been known since Exppose 6 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. This paper is a reformatted version of a MSc thesis submitted and defended at Dalhousie University in August 2022. The first part presents a set of conditions for defining an internal category of elements of a diagram of internal categories and proves it is the oplax colimit. The second part presents a set of conditions on an ambient category and an internal category with an object of weak-equivalences that allows an internal description of the axioms for a category of (right) fractions and a definition of the internal category of (right) fractions when all the conditions and axioms are satisfied. These are combined to present a suitable context for computing the pseudocolimit of a small filtered diagram of internal categories.