Enriched Koszul duality for dg categories

Abstract

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 ${\mathsf{ptdCoa}}^{\ast }$ 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 ${\mathsf{ptdCoa}}^{\ast }$-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 ${\mathsf{ptdCoa}}^{\ast }$. 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.