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.

Notes

Sources

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

• Victor Guillemin and Shlomo Sternberg, Some problems in integral geometry and some related problems in microlocal analysis, American Journal of Mathematics 101 (1979), 915–955.

• Fourier integral operator

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