  We give a definition of coisotropic morphisms of shifted Poisson (i.e. $P_n$)
algebras which is a derived version of the classical notion of coisotropic
submanifolds. Using this we prove that an intersection of coisotropic morphisms
of shifted Poisson algebras carries a Poisson structure of shift one less.
Using an interpretation of Hamiltonian spaces as coisotropic morphisms we show
that the classical BRST complex computing derived Poisson reduction coincides
with the complex computing coisotropic intersection. Moreover, this picture
admits a quantum version using brace algebras and their modules: the quantum
BRST complex is quasi-isomorphic to the complex computing tensor product of
brace modules.
