  We detail the construction of a weak Poisson bracket over a submanifold of a
smooth manifold M with respect to a local foliation of this submanifold. Such a
bracket satisfies a weak type Jacobi identity but may be viewed as a usual
Poisson bracket on the space of leaves of the foliation. We then lift this weak
Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle,
interpreted as a weak Koszul bracket on differential forms on M. This lift is
achieved by encoding the weak Poisson structure into a homotopy Poisson
structure on an extended manifold, and lifting the Hamiltonian function that
generates this structure. Such a construction has direct physical
interpretation. For a generic gauge system, the submanifold may be viewed as a
stationary surface or a constraint surface, with the foliation given by the
foliation of the gauge orbits. Through this interpretation, the lift of the
weak Poisson structure is simply a lift of the action generating the
corresponding BRST operator of the system.
