### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 100, 31 pages      arXiv:1401.7302      https://doi.org/10.3842/SIGMA.2014.100
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

### Selective Categories and Linear Canonical Relations

David Li-Bland and Alan Weinstein
Department of Mathematics, University of California, Berkeley, CA 94720 USA

Received July 22, 2014, in final form October 20, 2014; Published online October 26, 2014

Abstract
A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are ''good''. We then apply this notion to the category $\mathbf{SLREL}$ of linear canonical relations and the result ${\rm WW}(\mathbf{SLREL})$ of our version of the WW construction, identifying the morphisms in the latter with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in $\mathbf{SLREL}$ itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.

Key words: symplectic vector space; canonical relation; rigid monoidal category; highly selective category; quantization.

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