• 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, then I can give you a formula, to build up a thing in the original world, a Frankenstein monster (no, need to find a better name). But, one still need to verify that, this monster, when it come back, induces this original comodule. Good, then we don't need to use Barr-Beck-Lurie (or, this is a concrete way of using BBL)

Let's consider the reconstruction functor. First, in the commutative case. I am afraid of the infinite wrapping case, and I am not sure why it is useful in real application. So, why cannot you just add up the pushforward everywhere? Is it that simple?

Well, let's check. say you have 3 patches, you add up the contribution from patches (of course, to form a chain complex), then you can restrict to each patch and see, if they recover the original thing, and sadly, or gladly, you discovered that. On the first patch, you get $M_1 + T_{12} M_2 + T_{13} M_3$ on level 1, and then $M_{12}, M_{23}$ on level 2, and $M_{123}$ on level $3$. Something about contractiblity is important. I guess you can say, you check on the stalk level, and they all match up. But, there is no stalk in this world. So, how do you know that they do match up? Well, you can say, let's check on $M_{123}$, all good? then let's check on $U_{12}$, the part that is not in $U_{123}$, hmm? what does that even mean? don't worry let's see. What do we get? All the big patch, gives $M_{12}, M_{12}, M_{123}$, which sort of renders the intersection guy obsolete, but, that's the point.

Let's try two patch first. We have reconstructed guy $$ (j_{12})_* M_{12} \to (j_1)_* M_1 + (j_2)_* M_2 $$ then, we need to check that there is no 'dead loop', or 'infinite junk generator', so we restrict to $U_1$, we get $$ (j^1)^*(j_{12})_* M_{12} \to (j^1)^*(j_1)_* M_1 + (j^1)^*(j_2)_* M_2$$ Now, two things we need to use $(j^1)^*(j_1)_* M_1 = M_1$, and $(j^1)^*(j_2)_* M_2 = (j^1)^*(j_{12})_* M_{12}$. Hey, look, the second condition is exactly the so called Beck-Chevalley condition, indeed, you don't need to know it before hand, you will discover it naturally. LHS is like, you start from $U_2$, you go up and back to $U_1$, RHS is like, you start from $U_2$, you go down to the intersection, you include up, and then come back. You need to say, this extra pushforward to the big space, does not change you.

Is that a bit sad? You went to a bigger stage, but when you come back down, you don't change? No, let's not make sentimental analogies.

Can I say that, $Q_i$ give you permanent damage? yes, sadly so. In your lifetime, you only need to apply $Q_i$ once, nothing else will restore your vision beyond $Q_i$. A limb cut is a limb cut. (hey, don't be too sentimental!).

OK, with that in place, I believe now you can do whatever you want.

False. You cannot just remember the top layer cover. That's the bad thing of playing too much with the Venn diagram, or matching in set. The gluing data is essential. Consider the line bundle on $\P^1$. How do we reconstruct $O(1)$ from line bundle on $\P^1$? One way of doing so is like, you remember the object (object itself, not some isomorphism class) when restricted to each patch, and ask that they glue honestly. Sure, you can say, here is “restriction of $O(1)$ to $\C^*$, and here is the restriction of $O(2)$ to $\C^*$, no, don't mistake them from one another, they had different origin. ” I would say, that's only conceptually useful, but not really. What we are doing? We are considering 'sheaf of sections'. Indeed, the global section of sheaves of $\Gamma(\C^*, O(2))$ and $\Gamma(\C^*, O(1))$ are apriori non-comparable, withtout a morphism.

When we say, glue, we mean on each local piece, there are some standard building block. But, there are interesting automorphisms. When stuff from more global places restricts to local piece, we need to specify, how do we 'standardize' it.