Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{ X}$ on a manifold with corners $X$, we can interpret $\OO_{ X}(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this way, a poisson structure on $(X, \OO_{ X})$ is a sheaf morphism $$ \{ -, -\}: \OO_{ X} \times \OO_{ X} \longrightarrow \OO_{ X} $$ which satisfies the Leibniz rule an also the Jacobi Identity.
Categories: math.AG