$\mathcal A_{\infty}$-functors and homotopy theory of dg-categories

Abstract

In this paper we prove that Töen’s derived enrichment of the model category of dg-categories defined by Tabuada, is computed by the dg-category of \mathcal A_{\infty} -functors. This approach was suggested by Kontsevich. We further put this construction into the framework of (\infty, 2) -categories. Namely, we enhance the categories dgCat and \mathcal A_{\infty} Cat, of dg and \mathcal A_{\infty} -categories, to (\infty, 2) -categories using the nerve construction of [4] and the \mathcal A_{\infty} -formalism. We prove that the (\infty, 1) -truncation of to the (\infty, 2) -category of dg-categories is a model for the simplicial localization at the model structure of Tabuada. As an application, we prove that the homotopy groups of the mapping space of endomorphisms at the identity functor in the (\infty, 2) -category of \mathcal A_{\infty} -categories compute the Hochschild cohomology.