Basic structures on derived critical loci

We review the derived algebraic geometry of derived zero loci of sections of vector bundles, with particular emphasis on derived critical loci. In particular we some of the structures carried by derived critical loci: the homotopy Batalin-Vilkovisky structure, the action of the 2-monoid of the self-intersection of the zero section, and the derived symplectic structure of degree -1. We also show how this structure exists, more generally, on derived lagrangian intersections inside a symplectic algebraic manifold.