Koszul duality for Lie algebroids

TL;DR

This paper establishes a deep connection between dg-Lie algebroids and formal moduli problems in derived deformation theory, revealing new equivalences and geometric interpretations in the context of dg-algebras.

Contribution

It proves an equivalence between homotopy theories of formal moduli problems and dg-Lie algebroids, and describes their representation categories geometrically.

Findings

Equivalence between formal moduli problems and dg-Lie algebroids.

Representation category of dg-Lie algebroids extends quasi-coherent sheaves.

Geometric description of the extension via pro-coherent sheaves.

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.