## Abstract

We introduce the notion of a symplectic hopfoid, a "groupoid-like" object in the category of symplectic manifolds whose 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 a new realization of 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 groupoidlike objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions. Abstract: We introduce the notion of a symplectic hopfoid, a \groupoid-like" object in the category of symplectic manifolds whose 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 a new realization of 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 groupoidlike objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.

Original language | English (US) |
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Pages (from-to) | 217-250 |

Number of pages | 34 |

Journal | Journal of Geometric Mechanics |

Volume | 10 |

Issue number | 2 |

DOIs | |

State | Published - Jun 18 2018 |

## Keywords

- Canonical relations
- Double groupoids
- Symplectic category.

## ASJC Scopus subject areas

- Mechanics of Materials
- Geometry and Topology
- Control and Optimization
- Applied Mathematics