Some insights on bicategories of fractions - III

TL;DR

This paper investigates conditions under which bicategories of fractions are equivalent, providing necessary and sufficient criteria for such equivalences induced by pseudofunctors between bicategories.

Contribution

It establishes precise conditions for when bicategories of fractions are equivalent, extending understanding of their structural relationships.

Findings

Characterizes when a pseudofunctor induces an equivalence of bicategories of fractions.

Provides necessary and sufficient conditions for bicategory equivalences.

Clarifies the relationship between different bicategories of fractions.

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 equivalence of bicategories from $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$ to $\mathscr{B}[\mathbf{W}_{\mathscr{B}}^{-1}]$. In particular, this gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$ to any given bicategory $\mathscr{B}$.