  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.
