# Symplectic category

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In mathematics, Weinstein's **symplectic category** is (roughly) a [category](/source/Category_(mathematics)) whose objects are [symplectic manifolds](/source/Symplectic_manifold) and whose morphisms are **canonical relations**, inclusions of [Lagrangian submanifolds](/source/Lagrangian_submanifold) *L* into M × N − {\displaystyle M\times N^{-}} , where the superscript minus means minus the given symplectic form (for example, the graph of a [symplectomorphism](/source/Symplectomorphism); hence, minus). The notion was introduced by [Alan Weinstein](/source/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](/source/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**

1. **[^](#cite_ref-1)** He means [geometric quantization](/source/Geometric_quantization).

**Sources**

- [Weinstein, Alan](/source/Alan_Weinstein) (2009). "Symplectic Categories". [arXiv](/source/ArXiv_(identifier)):[0911.4133](https://arxiv.org/abs/0911.4133) [[math.SG](https://arxiv.org/archive/math.SG)].

## Further reading

- [Victor Guillemin](/source/Victor_Guillemin) and [Shlomo Sternberg](/source/Shlomo_Sternberg), *Some problems in integral geometry and some related problems in microlocal analysis*, [American Journal of Mathematics](/source/American_Journal_of_Mathematics) **101** (1979), 915–955.

## See also

- [Fourier integral operator](/source/Fourier_integral_operator)

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