• Comonad
• What is the ind-completion of Fukaya category?

Suppose we have a B-side category, and we have a bunch of open subsets, then we can glue quasi-coherent sheaves. I would just take direct sum of each pieces, and cancel the over-count by over-count. That's that.

Similar tricks better work on the A-side as well. So, how does it work? Suppose I have a cylinder, then I cut it into many pieces, and I want to glue it back. What do I do?

If I think of decomposition, then for a given Lagrangian, I will just restrict it to the cover (what if the boundary of the cover does not interact well with the Lagrangian?) does that matter? Do I really have Viterbo restriction? Let's not worry about those technical details. Then, we do the restriction, and have a collection of compatible objects in each piece. Descent data.

Local module over local algebra.

Can we invent something like, “gentle stop removal”? In the sense that, we let an object do the stop cross-over just once. like we only open the gate for one second, when it returns we need to open the gate again? This would avoid using 'ind-completion'.

Anyway, we do have a functor from the original guy, to the world of comodule. Better, it is a functor for free. We now need to check that, it is fully faithful, and it is essentially surjective. Well, I have never worried about the second thing, essentially surjectivity. Can we really obtain more than what we started from?

This is where the 'reconstruction' functor should play a role. In the sense that, if you give me a comodule