Double Groupoids and the Symplectic Category

TL;DR

This paper introduces symplectic hopfoids as groupoid-like structures in symplectic geometry, establishing a one-to-one correspondence with symplectic double groupoids and generalizing previous results in the field.

Contribution

It defines symplectic hopfoids and proves their equivalence to symplectic double groupoids, extending the concept to double Lie groupoids and relations in smooth manifolds.

Findings

Symplectic hopfoids correspond bijectively to symplectic double groupoids.

The core of a symplectic double groupoid can be realized as a symplectic quotient.

The cotangent functor relates double Lie groupoids to symplectic double groupoids.

Abstract

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on the fact that one can realize the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.