Enriched Koszul duality for dg categories

It is well known that the category of small dg categories dgCat, though it is monoidal, does not form a monoidal model category. In this paper we construct a monoidal model structure on the category of pointed curved coalgebras ptdCoa over a field k and show that the Quillen equivalence relating it to dgCat is monoidal. We also show that dgCat is a ptdCoa-enriched model category. As a consequence, the homotopy category of dgCat is closed monoidal and is equivalent as a closed monoidal category to the homotopy category of ptdCoa. In particular, this gives a conceptual construction of a derived internal hom in dgCat which we establish over a general PID. This proves Kontsevich’s characterization of the internal hom in terms of A1-functors. As an application we obtain a new description of simplicial mapping spaces in dgCat (over a field) and a calculation of their homotopy groups in terms of Hochschild cohomology groups, reproducing a well-known results of Toën. Comparing our approach to Toën’s, we also obtain a description of the core of Lurie’s dg nerve in terms of the ordinary nerve of a discrete category.