Localizations of the categories of $A_{\mathrm{\infty }}$ categories and internal Homs over a ring

We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of cohomologically unital, of unital and of strictly unital $A_{\mathrm{\infty }}$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. The result is proven at the $\mathrm{\infty }$-categorical level by considering the natural $\mathrm{\infty }$-categorical models of the categories above. As an application of the techniques we develop to compare the localizations mentioned above, we provide a new proof of the existence of internal Homs for the homotopy category of dg categories in terms of the category of (strictly) unital $A_{\mathrm{\infty }}$ functors. This yields a complete proof of a claim by Kontsevich and Keller.