Category of symplectic manifolds and symplectic maps

Definition

The category of symplectic manifolds used here has finite-dimensional Hausdorff second-countable symplectic manifolds without boundary $(M,\omega_M)$ as objects and symplectic maps as morphisms. Disconnected objects are allowed when their dimensions are globally bounded. Thus a morphism

f:(M,\omega_M)\longrightarrow(N,\omega_N)

is a smooth map satisfying

f^*\omega_N=\omega_M.

Identity maps are symplectic, and pullback functoriality gives

(g\circ f)^*\omega_P=f^*(g^*\omega_P)=\omega_M,

so these data do form a category.

Nondegeneracy forces $df$ to be injective at every point. Morphisms in this category are therefore smooth immersions and may increase dimension; they are not assumed to be diffeomorphisms. An isomorphism in the category is exactly a symplectomorphism.

Maximal subgroupoid

Keeping every object but only the isomorphisms gives the maximal subgroupoid of symplectic manifolds and symplectomorphisms. The automorphism group of $(M,\omega)$ in this subgroupoid is the symplectomorphism group

\operatorname{Symp}(M,\omega).

This subgroupoid should not be called a “symplectic groupoid.” In standard symplectic geometry, a symplectic groupoid is a Lie groupoid whose arrow manifold carries a compatible multiplicative symplectic form, a substantially different structure.

Terminology

Some sources reserve “symplectic map” for a symplectomorphism. Under that convention, their category is already a groupoid. The present corpus uses the broader pullback-preserving convention, under which a symplectic embedding of positive codimension is a morphism.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §§1.1–1.2, symplectic maps and submanifolds.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 1, symplectic manifolds and maps.