We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital 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 category of 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 functors between them.
Cite this article
Alberto Canonaco, Mattia Ornaghi, Paolo Stellari, Localizations of the Category of Categories and Internal Homs. Doc. Math. 24 (2019), pp. 2463–2492DOI 10.4171/DM/731