Derived coisotropic structures

We define and study coisotropic structures on derived stacks in the framework of shifted Poisson geometry. In particular, we give a presentation of coisotropic structures in terms of relative polyvector fields which shows that the identity morphism carries a unique coisotropic structure. In turn, this gives rise to a nontrivial forgetful map from nn-shifted Poisson structures to (n−1)(n−1)-shifted Poisson structures. We also prove that an intersection of two coisotropic morphisms carries a canonical Poisson structure of shift one less and provide an equivalence between a class of non-degenerate coisotropic morphisms and Lagrangian morphisms