Homotopy theory of $dg$ categories via localizing pairs and Drinfeld’s $dg$ quotient

Using localizing pairs and Drinfeld’s $dg$ quotient we construct a new Quillen model for the homotopy theory of $dg$ categories. We prove that, in contrast with the original model, this new Quillen model carries a natural closed symmetric monoidal structure. As an application, we obtain a simple construction of the internal $Hom$-objects and a conceptual characterization of Toën’s previous work. Making use of this new Quillen model, Lowen has recently developed a derived deformation theory.

Keywords

$dg$ category, localization pair, Drinfeld $dg$ quotient, Quillen model category

2010 Mathematics Subject Classification

18D20, 18G55

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Published 1 January 2010