Symplectic category

In mathematics, Weinstein's '''symplectic category''' is (roughly) a [[category (mathematics)|category]] whose objects are [[symplectic manifold]]s and whose morphisms are '''canonical relations''', inclusions of [[Lagrangian submanifold]]s ''L'' into <math>M \times N^{-}</math>, 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<ref>He means [[geometric quantization]].</ref> 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 {{reflist}} ;Sources *{{cite arXiv |last=Weinstein |first=Alan |authorlink=Alan Weinstein |eprint=0911.4133 |title=Symplectic Categories |year=2009|class=math.SG }}

== Further reading == *[[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.

== See also == *[[Fourier integral operator]]

[[Category:Category theory]] [[Category:Symplectic geometry]]