I am interested in *Lagrangian correspondences* in the context of symplectic manifolds, namely Lagrangian submanifolds $L\_{12}$ of $M\_1\times \bar M\_2$ where $M\_1$ and $M\_2$ are symplectic manifolds with symplectic forms $\omega\_1$ and $\omega\_2$, and $\bar M\_2$ has its symplectic form reversed (so the symplectic form on the product is written $\omega\_1-\omega\_2$, if we omit tedious pullbacks).


As experts know better than I, these correspondences are to be thought of as morphisms in a putative symplectic category. Unfortunately, producing a correspondence $L\_{13}$ given correspondences $L\_{12}$ and $L\_{23}$ is not obvious, unless the latter two intersect transversally.


As of ~10 years ago, there were a few candidate approaches to resolve the issue, which were summarised by Weinstein here:


*Weinstein, Alan*, [**Symplectic categories**](https://doi.org/10.4171/PM/1866), Port. Math. (N.S.) 67, No. 2, 261-278 (2010). [ZBL1193.53173](https://zbmath.org/?q=an:1193.53173).


**What is the state of the art today?** Has there been any progress in finding a useful and versatile definition that does not suffer from (as Weinstein calls it) this nontransversality problem?



Progress on composition of Lagrangian correspondences/definition of symplectic categories?
Though not about solving the nontransversality problem, Fukaya's paper [Unobstructed immersed Lagrangian correspondence and filtered A infinity functor](https://arxiv.org/abs/1706.02131) is the state of the art in why nontransversality isn't a problem for most interesting purposes. It's been known for a long time that generic compositions are immersed, so Fukaya just works with immersed Lagrangians equipped with bounding cochains and shows that this data can be carried through a correspondence to give an $A\_\infty$-functor on Fukaya categories.


You should also check out the work of Nate Bottman, much of which is motivated by trying to construct higher categorical versions of Weinstein's category. The survey article by Abouzaid and Bottman should give you a very up to date picture of the field: [Functoriality in categorical symplectic geometry](https://arxiv.org/abs/2210.11159).


