Selective Categories and Linear Canonical Relations

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 SLREL of linear canonical relations and the result WW(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 SLREL itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts.