Title: Koszul duality for Lie algebroids,
Abstract: This paper studies the role of dg-Lie algebroids in derived deformation
theory. More precisely, we provide an equivalence between the homotopy theories
of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra
of characteristic zero. At the level of linear objects, we show that the
category of representations of a dg-Lie algebroid is an extension of the
category of quasi-coherent sheaves on the corresponding formal moduli problem.
We describe this extension geometrically in terms of pro-coherent sheaves.