${\mathcal{A}}_{\infty }$-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.