Some insights on bicategories of fractions: representations and compositions of 2-morphisms

TL;DR

This paper simplifies the construction of bicategories of fractions by removing the need for the axiom of choice under certain conditions, making the process more accessible and paving the way for further research.

Contribution

It provides a simplified description of the equivalence relation and compositions in bicategories of fractions, reducing reliance on the axiom of choice.

Findings

Simplified the construction of bicategories of fractions.

Removed the need for the axiom of choice in certain cases.

Facilitated future work on pseudofunctors and bicategory equivalences.

Abstract

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathcal{C}$ together with a suitable class of morphisms $\mathbf{W}$, one can construct a bicategory $\mathcal{C}[\mathbf{W}^{-1}]$, where all the morphisms of $\mathbf{W}$ are turned into internal equivalences, and that is universal with respect to this property. Most of the descriptions leading to this construction were long and heavily based on the axiom of choice. In this paper we considerably simplify the description of the equivalence relation on $2$-morphisms and the constructions of associators, vertical and horizontal compositions in $\mathcal{C}[\mathbf{W}^{-1}]$, thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and compositions will also play a crucial role in two forthcoming papers about pseudofunctors and equivalences between bicategories of fractions.