nLab Weinstein symplectic category




When symplectic geometry is used to model mechanics in physics, then a symplectic manifold (X,ω)(X,\omega) encodes the phase space of a mechanical system and a symplectomorphism

ϕ:(X 1,ω 1)(X 2,ω 2) \phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)

encodes a process undergone by this system, for instance the time evolution induced by a Hamiltonian vector field.

However, this is too restrictive for a notion of a morphism. Indeed, even at the level of symplectic vector spaces, a symplectic morphism ϕ:(V,ω)(W,ω)\phi: (V,\omega)\to (W,\omega') is required to satisfy

ϕ *ω=ω. \phi^*\omega'=\omega.

This implies that for any element vker(ϕ)v\in\text{ker}(\phi), it holds that ω(v,w)=ω(f(v),f(w))=0\omega(v,w) = \omega'(f(v),f(w)) = 0, and since ω\omega is non-degenerate ϕ\phi must be injective.

Now the graph of a symplectomorphism ϕ\phi is a Lagrangian submanifold of the Cartesian product space X 1×X 2X_1 \times X_2 regarded as a symplectic manifold with symplectic form p 1 *ω 1p 2 *ω 2p_1^\ast \omega_1 - p_2^\ast \omega_2. In other words, a symplectomorphism ϕ\phi as above constitutes a Lagrangian correspondence between (X 1,ω 1)(X_1,\omega_1) and (X 2,ω 2)(X_2, \omega_2). See for instance (Cattaneo-Mnev-Reshetikhin 12) for a review.

This suggests that instead of the category whose objects are symplectic manifolds and whose morphisms are symplectomorphisms, one might consider a kind of category of correspondences whose objects are symplectic manifolds, and whose morphisms include Lagrangian correspondences, so that composition is given by forming the fiber product along adjacent legs of correspondences.

Alan Weinstein called this would-be category the symplectic category and suggested that it is the natural domain for geometric quantization.

However, take at face value, symplectic manifolds with Lagrangian correspondences between them do not quite form a category, since the usual composition is only well-defined when the intersection of L 1×L 2X 1×Δ(X 2)×X 3L_1 \times L_2 \cap X_1 \times \Delta(X_2) \times 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).


Prequantum correspondences

A refinement of the symplectic category to prequantum geometry is the following (see S 13).

Write BU(1) conn\mathbf{B}U(1)_{conn} for the moduli stack of smooth circle group-principal connections. Write Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids, and SmoothGrpd /BU(1) connSmooth\infty Grpd_{/\mathbf{B}U(1)_{conn}} for the corresponding slice (∞,1)-topos. Finally write

Corr 1(SmoothGrpd,BU(1) conn) Corr_1(Smooth\infty Grpd,\mathbf{B}U(1)_{conn})

for the (∞,1)-category of correspondences in SmoothGrpd /BU(1) connSmooth\infty Grpd_{/\mathbf{B}U(1)_{conn}}.

An object in here is a prequantum geometry (X,)(X,\nabla) given by a map

X BU(1) conn. \array{ X \\ \downarrow^{\mathrlap{\nabla}} \\ \mathbf{B}U(1)_{conn} } \,.

Under the curvature map F ():BU(1) connΩ cl 2F_{(-)} \colon \mathbf{B}U(1)_{conn} \to \Omega^2_{cl} this maps to a presymplectic structure

(X,ω)=(X,F ). (X,\omega) = (X, F_{\nabla}) \,.

If here ω\omega is non-degenerate, this is a symplectic structure as in Weinstein’s symplectic category.

Moreover, a morphism (X 1, 1)(X 2, 2)(X_1,\nabla_1) \to (X_2,\nabla_2) is a diagram of the form

Z i 1 i 2 X 1 η X 2 1 2 BU(1) conn, \array{ && Z \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ X_1 && \swArrow_{\eta} && X_2 \\ & {}_{\mathllap{\nabla_1}}\searrow && \swarrow_{\mathrlap{\nabla_2}} \\ && \mathbf{B}U(1)_{conn} } \,,

hence a correspondence space (smooth \infty-groupoid) ZZ over XX and YY together with an equivalence in an (∞,1)-category

η:i 2 * 2i 1 * 1. \eta \colon i_2^\ast \nabla_2 \stackrel{\simeq}{\to} i_1^\ast \nabla_1 \,.

On the underlying curvatures this implies that

i 2 *ω 2=i 1 *ω 1. i_2^\ast \omega_2 = i_1^\ast \omega_1 \,.

Hence if ZX×YZ \to X \times Y is a maximal inclusion with this property, the above diagram is a prequantization of a morphism in the Weinstein symplectic category.

Motivic stabilization

Nitu Kitchloo defines the stable symplectic category 𝕊\mathbb{S}, which has as objects symplectic manifolds, and morphisms are certain Thom spectra

associated to Lagrangian correspondences M¯×N\overline{M} \times N, where M¯\overline{M} denotes the conjugate with symplectic form ω-\omega. One can view this as a category of symplectic motives.

Considering an oriented version of the category 𝕊\mathbb{S}, there is a canonical fiber functor F:M𝕊(pt,M)F : M \mapsto \mathbb{S}(pt, M), and one may consider the motivic Galois group GG of monoidal automorphisms of FF. (Kitchloo 12, question 8.6, p. 19).

It turns out to have a natural subgroup which is isomorphic to the quotient of the Grothendieck-Teichmüller group.


The way that Lagrangian correspondences encode symplectomorphisms in symplectic geometry and hence evolution in mechanics is reviewed (and put in the broader context of BV-BRST formalism) in

In his work on Fourier integral operators,

  • Lars Hörmander, Fourier Integral Operators I., Acta Math. 127 (1971) 79–183.



  • Victor Maslov, Theory of Perturbations and Asymptotic Methods, (in Russian) Moskov. Gos. Univ., Moscow (1965).

observed that, under a transversality assumption, the set-theoretic composition of two Lagrangian submanifolds is again a Lagrangian submanifold, and that this composition is a “classical limit” of the composition of certain linear operators.

Shortly thereafter,

  • Jędrzej Śniatycki, W.M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J. 22 (1972) 267-275

defined symplectic relations as isotropic submanifolds of products and showed that this class of relations was closed under “clean” composition. Following in part some (unpublished) ideas of Alan Weinstein,

observed that the linear canonical relations (i.e., lagrangian subspaces of products of symplectic vector spaces) could be considered as the morphisms of a category, and they constructed a partial quantization of this category (in which lagrangian subspaces are enhanced by halfdensities.) The quantization of the linear symplectic category was part of a larger project of quantizing canonical relations (enhanced with extra structure, such as half-densities) in a functorial way, and this program was set out more formally

See also:

  • W.M.Tulczyjew, S.Zakrzewski, The category of Fresnel kernels, J. Geom. Phys. 1:3, 1984, 79–120 doi

Lecture notes reviewing these developments include

from the introduction of which parts of the commented list of references above is taken. Further review includes

  • Santiago Canez, Double Groupoids, Orbifolds, and the Symplectic Category (arXiv:1105.2592)

Further refinements in higher category theory:

A closed symmetric monoidal category version of the symplectic category and the observation that this hence is a categorical semantics for quantum logic qua linear logic is in

  • Sergey Slavnov, From proof-nets to bordisms: the geometric meaning of multiplicative connectives, Mathematical Structures in Computer Science 15:06 (2005) 1151–1178

  • Sergey Slavnov, Geometrical semantics for linear logic (multiplicative fragment), Theoretical Computer Science 357, no. 1–3 (2006) 215–229 doi

Remarks about refinements to correspondences of smooth \infty-groupoids in the slice over prequantum moduli is in

String diagrams for the linear and affine Weinstein category using graphical linear algebra

Last revised on June 2, 2024 at 22:33:21. See the history of this page for a list of all contributions to it.