2009/11/20 by Alan Weinstein, Weinstein, Alan · 2 citations
Mathematics · #53D12 #81S10 #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.CT #math.SG #msc:53D12 #msc:81S10
paper · pdf · doi:10.48550/arxiv.0911.4133
16 pages, for proceedings of Geometry Summer School, Lisbon, July 2009
arxiv created 2009/11/20 · openalex publication_date 2009/11/20 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.