Cotangent Microbundle Category, I

We define a local version of the extended symplectic category, the cotangent microbundle category, MiC, which turns out to be a true monoidal category. We show that a monoid in this category induces a Poisson manifold together with the local symplectic groupoid integrating it. Moreover, we prove that monoid morphisms produce Poisson maps between the induced Poisson manifolds in a functorial way. This gives a functor between the category of monoids in MiC and the category of Poisson manifolds and Poisson maps. Conversely, the semi-classical part of the Kontsevich star-product associated to a real-analytic Poisson structure on an open subset of R^n produces a monoid in MiC.