# koszul duality and algebras over operads

Given a pair of Koszul dual algebras, say $S^*(V)$ and $\bigwedge^*(V^*)$ for some vector space $V$, one obtains a triangulated equivalence between their bounded derived categories of finitely-generated graded modules.

Given a pair of Koszul dual operads, say the Lie and commutative operads, what is the precise analogue of a derived equivalence between their categories of algebras?

The situation for graded modules over a pair of Koszul dual algebras is more complicated, actually. What the question says is true for Koszul algebras $A$ and $A^!$ provided that $A$ is Noetherian and $A^!$ is finite-dimensional (including the case of the symmetric and exterior algebras) but not otherwise. In general one can say that the unbounded derived categories of positively graded modules with finite-dimensional components over $A$ and $A^!$ are anti-equivalent. The subcategories of complexes of positively graded modules bounded separately in every grading in these unbounded derived categories are also anti-equivalent.
With algebras over operads, the analogue of the equivalence for graded modules involves DG-algebras with an additional positive grading (there being only the ground field $k$ in the additional grading $0$ and nothing in the negative additional grading), with the additional grading preserved by the differential. The Koszul duality is an anti-equivalence between the localizations of the categories of DG-algebras of this kind, with every component of fixed additional grading being a bounded complex of finite-dimensional vector spaces, by quasi-isomorphisms. For some operads (e.g., for Lie and Com) one has to assume the field $k$ to have characteristic $0$, while for some others (e.g., Ass) one doesn't.