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

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... riction to classes of lagrangian submanifolds for which the transversality property automatically holds. Others involve allowing lagrangian "objects" more general than submanifolds. In his work on Fourier integral operators, Hörmander [12], following Maslov [15], observed that, under a transversality assumption, the set-theoretic composition of two canonical relations is again a canonical relation, and that this composition is a "classical limit" of the composition of certain operators. Shortly thereafter, Sniatycki and Tulczyjew [21] defined symplectic relations as isotropic 1 submanifolds of products and showed that this class of relations was closed under "clean" composition (see Section 2 below). They also observed that the natural relation between a symplectic manifold and the quotient of a submanifold by the kernel of the pulled-back symplectic form is a symplectic relation. Following in part some (unpublished) ideas of the author, Guillemin and Sternberg [9] observed that the linear canonical relations (i.e., lagrangian subspaces of products of symplectic vector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 automorphism groups in this category are the linear symplectic groups, and the restriction of the Guillemin-Sternberg quantization to each such group is a metaplectic representation. On the other hand, the quantization of certain compositions of canonical relations leads to ill-defined operations at the quantum level, such as the evaluation of a delta "function" at its singular point, or the multiplication of delta functions. 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 by the present author in [24] and [25]. It was advocated there that canonical relations should be considered as the morphisms of a "category", and that quantization should be a functor from there to a category of linear spaces and linear maps, consistent with some additional structures. The word "category" appears in quotation marks above because the composition of canonical relations can fail to be a canonical relation, as will be explained in detail below, so we do not have a category. Briefly, there are two problems.