  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.
