The Three F's for Bicategories I: Localization by Fractions is Exact

TL;DR

This paper explores how filteredness, fractions, and fibrations interact in bicategories, providing explicit formulas and proving the exactness of the canonical pseudo-functor into a bicategory of fractions.

Contribution

It generalizes classical category results to bicategories, offering explicit formulas for filtered pseudo-colimits and demonstrating the exactness of the canonical pseudo-functor.

Findings

Explicit formula for filtered pseudo-colimits in bicategories

Computation of hom-categories in bicategories of fractions

Proof that the canonical pseudo-functor is exact

Abstract

We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories indexed by a bicategory, and we use it to compute the hom-categories of a bicategory of fractions. As a consequence, we show that the canonical pseudo-functor into a bicategory of fractions is exact.