# nLab Weinstein symplectic category

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

Weinstein suggested that geometric quantization should yield a representation of a “category” whose

• objects are symplectic manifolds $\left(X,\omega \right)$;

• morphisms $\left({X}_{1},{\omega }_{1}\right)\to \left({X}_{2},{\omega }_{2}\right)$ are Lagrangian correspondences: Lagrangian submanifolds of $\left({X}_{1},{\omega }_{1}\right)×\left({X}_{2},-{\omega }_{2}\right)$;

and composition is given by taking fiber products of these Lagrangian submanifolds.

However, this is not actually quite a category, since composition is only well-defined when the intersection of ${L}_{1}×{L}_{2}\cap {X}_{1}×\Delta \left({X}_{2}\right)×{X}_{3}$ is transverse.

Proposals for how to rectify this are in (Wehrheim-Woodward) and in (Kitchloo) (by turning this into an (infinity,1)-category).

## Kitchloo approach

Kitchloo defines the stable symplectic category $𝕊$, which has as objects symplectic manifolds, and morphisms are certain Thom spectra associated to Lagrangian correspondences $\overline{M}×N$, where $\overline{M}$ denotes the conjugate with symplectic form $-\omega$. One can view this as a category of symplectic motives.

Considering an oriented version of the category $𝕊$, there is a canonical fiber functor $F:M↦𝕊\left(\mathrm{pt},M\right)$, and one may consider the motivic Galois group? $G$ of monoidal automorphisms of $F$. It turns out to have a natural subgroup which is isomorphic to the quotient of the Grothendieck-Teichmüller group.

## References

Revised on April 17, 2013 18:37:15 by David Corfield (129.12.18.29)