Coisotropic calculus and Poisson groupoids

Lagrangian submanifolds play a special role in the geometry of symplectic manifolds. From the point of view of quantization theory, or simply a cate- gorical approach to symplectic geometry [Gu-S2], [W3], lagrangian submanifolds are the "elements" of symplectic manifolds. Since the canonical transformations between symplectic manifolds P₁ and P₂ are those whose graphs are lagrangian in P₂ P₁⁻ (the $"-,$ , indicating that the symplectic structure on P₁ has been multiplied by $-1$ ), one calls arbitrary lagrangian submanifolds of a product P₂ P₁⁻ canonical relations. It turns out that, under a transversality or clean intersection assumption, the composition of canonical relations is again canonical.