objects are symplectic manifolds ;
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 is transverse.
Kitchloo defines the stable symplectic category , which has as objects symplectic manifolds, and morphisms are certain Thom spectra associated to Lagrangian correspondences , where denotes the conjugate with symplectic form . One can view this as a category of symplectic motives.
Considering an oriented version of the category , there is a canonical fiber functor , and one may consider the motivic Galois group? of monoidal automorphisms of . It turns out to have a natural subgroup which is isomorphic to the quotient of the Grothendieck-Teichmüller group.