# Poisson manifolds and their associated stacks

## TL;DR

This paper constructs a stack from any integrable Poisson manifold and demonstrates that symplectic Morita equivalence corresponds precisely to the isomorphism of these stacks, linking geometric and categorical perspectives.

## Contribution

It introduces a novel stack-theoretic framework for integrable Poisson manifolds and characterizes symplectic Morita equivalence through stack isomorphisms.

## Key findings

- Associated stacks classify integrable Poisson manifolds up to Morita equivalence
- Stack isomorphism corresponds to symplectic Morita equivalence
- Provides a categorical perspective on Poisson geometry

## Abstract

We associate to any integrable Poisson manifold a stack, i.e. a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic.