# American Institute of Mathematical Sciences

June  2018, 10(2): 217-250. doi: 10.3934/jgm.2018009

## Double groupoids and the symplectic category

 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

Received  October 2017 Revised  December 2017 Published  May 2018

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 groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions.

Citation: Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
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##### References:
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