Yuji proposed an interesting construction of category on a disk with stops. The bulk is decorated with some category $C$, and stops are decorated with something else, like $D_1,\cdots, D_n$. Then we have functors $D_i \to C$. We want to take some sort of global section on this.

Where does this come from? Consider a family of LG model over a base $\C$. With total space $X$, function $W$ on it, and in addition, a function $\pi: X \to \C$. OK, you can say that we can combine $W$ and $\pi$ together to have a 2d base, $\C^2_{x,y}$, with some singularity curve $S \In \C^2$. We decree that $Re(y) > R$ is the stop. For example, say $F$ is given by $y^2 = x^3$. And the stop is given by $Re(y) > 10$, and when a singularity falls into the stop. the thing is, instead of integrating out $x$ first, then do $y$, Yuji integrated out $y$ first. That's new and brave! (well maybe we did this as well without realizing it, when we have the $\pi, W$ stuff).

consider a simpler case, $y = x^2$ as singularity, and $Re(y) > 1$ as stop. If we integrate $y$ first, then on the $x$ space, we are left with a cool coefficient system, it would be zero cat when $Re(x^2) > 1$. If we do 'infinitesimal Fukaya category', then we do opposite thimble ending on some singularity. Why the wrapping stops? because upstairs, the seed of the Lagrangian in the singularity $S$ get stopped when wrapping.

Now suppose we have something that is like $\{y=x^2\} \cup \{y=0\}$, so we have something that never escapes. what do we say about the 'nonescaping' one? I want to say, first this one is degenerate.

Let's try another one $\{y=x^2\} \cup \{y=-x^2\}$, right the one that Yuji was considering. The two branches was escaping at different places.

an object is a Lagrangian (or just totally real submanifold), so we have a (constructible) sheaf of category on it, and we want to a global section of object over it.