Derived mapping spaces of $\mathrm{\infty }$-categories

We prove the Derived Mapping Space Lemma, which generalizes the central theorem of Cisinski’s work on calculus of fractions for $\mathrm{\infty }$-categories, and allows us to provide a unified framework for analyzing mapping spaces in localizations of ($\mathrm{\infty }$-)categories. As an application, we give a sufficient condition for when a cubical or simplicial category is the localization of its underlying category at homotopy equivalences.