Symplectic Microgeometry III: Monoids

TL;DR

This paper establishes an equivalence between three categories related to symplectic microgeometry, connecting Poisson manifolds, microgroupoids, and monoids within the microsymplectic framework.

Contribution

It demonstrates the categorical equivalence of Poisson manifolds, symplectic microgroupoids, and monoids in the microsymplectic category, unifying these structures.

Findings

Categories are equivalent as symmetric monoidal categories

Poisson manifolds correspond to microgroupoids and monoids

Provides a unified categorical framework for symplectic microgeometry

Abstract

We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.