Homotopical algebra for Lie algebroids

Abstract

We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and $L_\infty$-algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. that have a null-homotopic anchor map. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids.