Localizations of the Category of $A_{\infty }$ Categories and Internal Homs

Abstract

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_{\infty }$ 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 $A_{\infty }$ category of $A_{\infty }$ 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 $A_{\infty }$ functors between them.