The shifted symplectic geometry of derived higher groupoids

The main goal of this work is to introduce derived Lie n-groupoids and their shifted symplectic structures.

We further define shifted lagrangian structures and prove that their composition is well defined under suitable conditions.

As an application, we show that our framework incorporates several reduction procedures at critical values, including: classical Hamiltonian reduction, group valued moment maps, Poisson Lie group valued moment maps and Mikami-Weinstein for proper symplectic groupoids.