Enriched Koszul duality for dg categories

It is well known that the category of small dg categories \mathsf{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 \mathsf{ptdCoa}^{*} over a field \mathbf{k} and show that the Quillen equivalence relating it to \mathsf{dgCat} is monoidal. We also show that \mathsf{dgCat} is a \mathsf{ptdCoa}^{*} -enriched model category. As a consequence, the homotopy category of \mathsf{dgCat} is closed monoidal and is equivalent as a closed monoidal category to the homotopy category of \mathsf{ptdCoa}^{*} . In particular, this gives a conceptual construction of a derived internal hom in \mathsf{dgCat} which we establish over a general PID. This proves Kontsevich’s characterization of the internal hom in terms of A_{\infty} -functors. As an application we obtain a new description of simplicial mapping spaces in \mathsf{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.