Some insights on bicategories of fractions - III

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 $\mathscr{B}$.