Some insights on bicategories of fractions - II

TL;DR

This paper investigates conditions for inducing pseudofunctors between bicategories of fractions and provides a simplified description when certain saturation conditions are met.

Contribution

It establishes necessary and sufficient conditions for the existence of induced pseudofunctors between bicategories of fractions and describes these functors explicitly under saturation assumptions.

Findings

Necessary and sufficient conditions for induced pseudofunctors.

A simple description of the induced pseudofunctor under saturation.

Framework applicable to bicategories with specified classes of morphisms.

Abstract

We fix any bicategory $\mathscr{A}$ together with a class of morphisms $\mathbf{W}_{\mathscr{A}}$, such that there is a bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$. Given another such pair $(\mathscr{B},\mathbf{W}_{\mathscr{B}})$ and any pseudofunctor $\mathcal{F}:\mathscr{A}\rightarrow\mathscr{B}$, we find necessary and sufficient conditions in order to have an induced pseudofunctor $\mathcal{G}:\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]\rightarrow \mathscr{B}[\mathbf{W}_{\mathscr{B}}^{-1}]$. Moreover, we give a simple description of $\mathcal{G}$ in the case when the class $\mathbf{W}_{\mathscr{B}}$ is "right saturated".