Localizations of the Category of A_∞Categories and Internal Homs

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital A_∞categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the A_∞category of A_∞functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital A_∞functors between them.