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Recall the definition of a symplectic groupoid. Roughly this is a Lie groupoid such that the object manifold is Poisson, and the arrow manifold is symplectic such that the symplectic form is compatible with the groupoid structure. There is naturally a forgetful 2-functor $SymplGpd \to LieGpd$.

Clearly not every Lie groupoid is in the image of this functor. Pick, for instance a manifold which is not symplectic and consider the trivial Lie groupoid with this as objects. But given that a Lie groupoid is in the image of the forgetful 2-functor, what does the fibre over it look like? To ask a question off the top of my head, would it be too much to ask that symplectic groupoids with identical underlying Lie groupoids are Morita equivalent (in the sense of Xu see e.g. this or this, or this for the quasisymplectic version, in case that helps.)

I'm cheating a bit here, because I am not specifying the 1- and 2-arrows of the 2-category of symplectic groupoids. I do this so that answers can clarify what these might be and how this relates to my question.

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# What are the possible symplectic structures on a given Lie groupoid?

Recall the definition of a symplectic groupoid. Roughly this is a Lie groupoid such that the object manifold is Poisson, and the arrow manifold is symplectic such that the symplectic form is compatible with the groupoid structure. There is naturally a forgetful 2-functor $SymplGpd \to LieGpd$.

Clearly not every Lie groupoid is in the image of this functor. Pick, for instance a manifold which is not symplectic and consider the trivial Lie groupoid with this as objects. But given that a Lie groupoid is in the image of the forgetful 2-functor, what does the fibre over it look like? To ask a question off the top of my head, would it be too much to ask that symplectic groupoids with identical underlying Lie groupoids are Morita equivalent (in the sense of Xu see e.g. this or this for the quasisymplectic version, in case that helps.)

I'm cheating a bit here, because I am not specifying the 1- and 2-arrows of the 2-category of symplectic groupoids. I do this so that answers can clarify what these might be and how this relates to my question.