Previously, we have said that $M_H$ and $M_C$ are $n=0$-shifted symplectic stack. And there is no natural way to shift the $n$ around, so naturally the home for 3d MS is about 0-shifted symp stack.

Now, suppose we are given two hol'c symp Lagrangians, then their intersection is $-1$-shifted symp mfd. $$ X \times_{T^*X} X = T^*[-1] X = Spec_X(Sym(T[1]X)) $$

We somehow need to assign a category to the derived intersection. I don't think that is the way to go, at least not on the 2A-side. One need seriously consider wrapping.

If we consider path space between two Lagrangians, then it carries a natural closed action one-form $\alpha$. Crit cohomology of the path space, $H^*_{crit}(Path(L_1, L_2), \alpha)$, suppose to be Floer cohomology. How to categorify this?

What is the input to a 3d category?