Symplectic Categories

TL;DR

This paper explores the structure of symplectic categories by examining canonical relations, addressing composition issues due to nontransversality, and discussing potential solutions involving restrictions or generalizations of lagrangian objects.

Contribution

It analyzes the challenges in forming a symplectic category with canonical relations and proposes methods to overcome nontransversality problems.

Findings

Canonical relations compose well under transversality.

Nontransversality prevents forming a strict category.

Possible remedies include restricting to certain lagrangian submanifolds or generalizing lagrangian objects.

Abstract

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.