Derived moduli spaces and homotopy types of algebraic varieties

The cohomology of an algebraic variety comes naturally equiped with additional structures (e.g. Hodge structures, Galois actions ...), enconding interesting geometrical and arithmetical informations (e.g. Torelli theorems, periods maps, number of points over finite fields ...). The purpose of this talk is to present non-abelian versions of this general picture. In particular, I will explain how algebraic varieties have homotopy types as well as moduli spaces of non-abelian cohomology, and how both come equipped with additional structures. Finally, if time allows, I will present some relations between homotopy types of algebraic varieties, moduli spaces of non-abelian cohomology and derived moduli spaces of local systems.