  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.
