Symplectic category

In mathematics, Weinstein's symplectic category is (roughly) a category whose objects are symplectic manifolds and whose morphisms are canonical relations, inclusions of Lagrangian submanifolds L into \( {\displaystyle M\times N^{-}} \), where the superscript minus means minus the given symplectic form (for example, the graph of a symplectomorphism; hence, minus). The notion was introduced by Alan Weinstein, according to whom "Quantization problems[1] suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms." The composition of canonical relations is given by a fiber product.

Strictly speaking, the symplectic category is not a well-defined category (since the composition may not be well-defined) without some transversality conditions.
References

Notes

He means geometric quantization.

Sources

Weinstein, Alan (2009). "Symplectic Categories". arXiv:0911.4133.