Some useful ``categories" in symplectic geometry, candidates for being the domains of quantization functors, are ones in which the morphisms X --> Y between symplectic manifolds are relations, rather than maps.  These are submanifolds of X x Y having nice geometric properties with respect to the product of the symplectic form on X and the negative of the symplectic form on Y.

An obstruction to getting actual categories is that the set-theoretic composition of relations does not preserve the class of manifolds, due to possible failures of transversality.

In this talk, I will describe several approaches to resolving the transversality problem, concentrating on the linear case.  Although the composition of linear relations is always linear, the composition operation itself fails to be continuous until it is modified to take nontransversality into account.

The talk will be based in part on work with David Li-Bland and Jonathan Lorand, available on the arXiv.