Enriched Koszul duality for dg categories

TL;DR

This paper establishes a monoidal model structure for pointed curved coalgebras and demonstrates a monoidal Quillen equivalence with dg categories, providing new insights into their homotopy theory and internal homs.

Contribution

It constructs a monoidal model structure on pointed curved coalgebras and proves a monoidal Quillen equivalence with dg categories, enabling new conceptual tools.

Findings

Homotopy category of dgCat is closed monoidal.

Derived internal hom in dgCat is constructed.

Homotopy groups of mapping spaces relate to Hochschild cohomology.

Abstract

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* 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. As an application we obtain a new description of simplicial mapping spaces in dgCat and a calculation of their homotopy groups in terms of Hochschild cohomology groups, reproducing and slightly generalizing well-known results of To\"en. Comparing our approach to To\"en's, we also obtain a description of the core of Lurie's dg nerve in terms of the ordinary nerve of a discrete category.