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Portugaliae Mathematica

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Volume 67, Issue 2, 2010, pp. 261–278
DOI: 10.4171/PM/1866

Published online: 2010-04-27

Symplectic categories

Alan Weinstein[1]

(1) University of California, Berkeley, United States

Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms. These relations compose well when a transversality condition is satisfied, but the failure of the most general compositions to be smooth manifolds means that the canonical relations do not comprise the morphisms of a category.

We discuss several existing and potential remedies to the nontransversality problem. Some of these involve restriction to classes of lagrangian submanifolds for which the transversality property automatically holds. Others involve allowing lagrangian “objects” more general than submanifolds.

Keywords: Symplectic manifold, lagrangian submanifold, canonical relation, category, quantization

Weinstein Alan: Symplectic categories. Port. Math. 67 (2010), 261-278. doi: 10.4171/PM/1866