Cotangent Microbundle Category, I

TL;DR

This paper introduces the cotangent microbundle category, a monoidal category linking monoids to Poisson manifolds and symplectic groupoids, and connects it to Kontsevich's star-product for real-analytic Poisson structures.

Contribution

It defines the cotangent microbundle category, establishes its monoidal structure, and shows how monoids correspond to Poisson manifolds and symplectic groupoids, linking to deformation quantization.

Findings

Monoids in MiC induce Poisson manifolds with symplectic groupoids.

Monoid morphisms correspond to Poisson maps.

Semi-classical Kontsevich star-product produces monoids in MiC.

Abstract

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.