  The purpose of this paper is to investigate the definition of symplectic
structure on a smooth stratified pseudomanifold in the framework of local
$\C^{\infty}$-ringed space theory. We introduce a sheaf-theoretic definition of
symplectic form and cohomologically symplectic structure on smooth stratified
pseudomanifolds. In particular, we give an indirect definition of symplectic
form on the quotient space of a smooth $G$-stratified pseudomanifold. Based on
the structure theorem of singular symplectic quotients by Sjamaar--Lerman, we
show that the singular reduced space $M_{0}=\mu^{-1}(0)/G$ of a symplectic
Hamiltonian $G$-manifold $(M,\omega,G,\mu)$ admits a natural (indirect)
symplectic form and a unique cohomologically symplectic structure.
