Models for classifying spaces and derived deformation theory

Using the theory of extensions of L ∞ algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley–Eilenberg and Harrison cohomology. We also investigate the algebraic structure of the Chevalley–Eilenberg complexes of L ∞ algebras and show that they possess, along with the Gerstenhaber bracket, an L ∞ structure that is homotopy abelian.