Enriched Koszul duality for dg categories

Julian Holstein ; Andrey Lazarev

Documenta mathematica, Tome 30 (2025) no. 4, pp. 755-786

Résumé

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 A∞​-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.

Détail

DOI : 10.4171/dm/1002

Classification : 18N40, 18D20, 18M70
Mots-clés : DG categories, coalgebras, monoidal categories, model categories, enriched categories, bar-construction, cobar-construction