A-infinity functors and homotopy theory of dg-categories

TL;DR

This paper establishes a deep connection between dg-categories and A-infinity functors using (infinity,2)-categories, providing new insights into their homotopy theory and Hochschild cohomology computations.

Contribution

It proves that Toen's derived enrichment is given by A-infinity functors and embeds this into the (infinity,2)-category framework, advancing the homotopy theory of dg-categories.

Findings

Derived enrichment of dg-categories is computed by A-infinity functors.

The (infinity,1)-truncation models the simplicial localization.

Homotopy groups of endomorphism spaces relate to Hochschild cohomology.

Abstract

In this paper we prove that Toen's derived enrichment of the model category of dg-categories defined by Tabuada, is computed by the dg-category of A-infinity functors. This approach was suggested by Kontsevich. We further put this construction into the framework of (infinity,2)-categories. Namely, we enhance the categories of dg and A-infinity categories, to (infinity,2)-categories. We prove that the (infinity,1)-truncation of to the (infinity,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 (infinity,2)-category of A-infinity categories compute the Hochschild cohomology.