Poisson $C^{\mathrm{\infty }}$-schemes

We introduce Poisson $C^{\mathrm{\infty }}$-rings and Poisson local $C^{\mathrm{\infty }}$-ringed spaces. We show that the spectrum of a Poisson $C^{\mathrm{\infty }}$-ring is an affine Poisson $C^{\mathrm{\infty }}$-scheme. We then discuss applications that include singular symplectic and Poisson reductions, singular quasi-Poisson reduction, coisotropic reduction and Poisson-Dirac subschemes.