discussed with Hayata and Nicolo

True statements:

• Given a Liouville domain, the space of Liouville structure is convex, hence contractible.
• Given any two Liouville structures on the same domain, we can find a canonical path connecting them.
• Given any two Liouville structures on a Liouville manifold (completion from a domain), we can find a diffeotopy, such that $h^* \lambda_1 - \lambda_0 = dh$ is exact and $dh$ vanishes outside a compact set. (note, $h$ may not be compactly supported, do we care? this problem only arise at 1 complex dim, so shouldn't be hard)
• All the GPS construction are done with contractible choices.

OK, great, what do I have now? I have a Weinstein manifold, and a collection of open submanifold, with nice intersections (don't worry about corner). also consider their intersections.

Each space is equipped with a convex set of choice of Liouville structures. So, roughly speaking, the Fukaya category is independent of the choice of Liouville structure there. We associate to each patch, the Fukaya category of its Liouville manifold completion. Do we fix a representative? We don't.

If you wish, fix any choice as you wish. But the correct thing and easier thing is: don't choose, keep them all. infinity category allows you to do that!

If one subset sits inside another subset, then we can consider Liouville structure on the big guy that happens to restrict to Liouville structure on the small guy. assume that it is not empty.

A point is not a point! A point is a contractible set! better, a convex set.

Then, there is a functor from a nice representative of the big guy to a nice representative of the small guy. Don't choose, keep all choices.

Then, the crucial thing, that distinguishes our setup with the compact (non-exact) symp manifold setup, is that, our big boss, the original Weinstien manifold, maps to each and everyone of them! all the arrow are nice.

So, here is the question: how do we check that they are equivalent?