I am thinking about proving multiplicatve - multiplicative HMS, with Spencer, and with quantization.
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.
new space and new function.
I want to study $[1]-(1)$ quiver. In the sense of how to see it as framed zastava space.
our goal is to prove bar gluing, namely colimit of a bar diagram is the desired Fukaya category. VS provides a new method, let's see how it works.
we added two stops to the picture. and split the picture into left middle and right. middle can map to left and right.
If we look directly at the FukSym of the glued surface, we found it admits a triangular poset, labelled by $(l,m,r)$, with $n=l+m+r$, with relation generated by $(l,m,r) \to (l+1,m-1,r)$ and $(l,m,r) \to (l,m-1,r+r)$. It probably is not hard to identify these subcategories, and show the semi-orthogonality according to poset, but generation might be not so easy. We need bend and break argument.
Next, if we look at the bar diagram's term. we still get a bunch of term, except we have further decomposition of the $(a; m_1, \cdots, m_k; b)$ term. We rewrite $a$ and $b$ factor using SOD. We could. Now these arrows in the diagram are kinda easy, all fully faithful.
Indeed, in the end, we want to say, the colimit of that diagram of sod, equal to the final sod.
so there are three ingredients:
In the nicest setting, max of smooth psh function is still psh, but with kink when the dominant term switch over. To solve this problem, people developed softmax, which is a smearing of max function. When we softmax a bunch of psh function, the outcome is smooth and psh.
Another application is the following: suppose we have a bunch of locally defined psh function $u_\alpha$, living on some locally finite open cover $\Omega_\alpha$ (say extending continuously to the closure of $\Omega_\alpha$). If we take max of these whole collection of functions, that certainly does not make sense. If we take max at each point $z$, then the problem is that if $z$ moves out of the boundary of certain $\Omega_\beta$, $u_\beta$ will suddenly not avaiable for doing max, it would be a disaster when the 'weight bearer' of the group suddenly leave, we would have a cliff fall over. The only case where everything is safe, is when $u_\beta$ is already relatively 'retired' near the boundary $\Omega_\beta$, as the real work is taken up by some other $u_\alpha$, then there is no problem. Then, you can take pointwise max, the thing will still be continuous psh.
Now, we don't want to do convolution to regularize continuous psh. We want to do softmax. The problem with softmax is that, each term needs a room of epsilon to smooth over. This is usually no problem, the fuzzy uncertainty for $u_\beta$ by $\eta_\beta$ is tolerable, if the bump-up of $u_\beta$ still won't catch the low-day of the best $u_\alpha$, then it is safe to retire, byebye safe trip.
Next, we consider Richberg's theorem. Input a strictly psh function on a manifold $X$
I want a wiki / blog tool, that is online and easy to use and share.
overleaf is good, but not good at sharing or updating.
notability is good and smooth, but not good at sharing.
In order to prove some Liouville pair is a Weinstein pair, we need to know that the stop is good.
If our space is like $\R \times \R_-$, one factor of space is cutting off some factor, but leaving some other factors intact. How does the fiber look like? It would be just the stop-fiber from the relevant factor, times the entire space from the non-partipating factor. So to show the Weinstein-ness, one just need to show that the two factors are. Now, where is the participating factor? It is about some smoothable function, that only involves center of mass variables. So, it is as if in the cotangent bundle case.
Now, how about the other factors? What do we need to show? What do we have already? Those other factor is locally a product, which we assume is Weinstein already.
The first time I heard about this phrase, it is used by Prof Z about some work by Prof B. It is half joking, half disapproval. One either needs to painstakingly verify lots of details, or one can wave ones' way out. Through my academic career, I have seen many handwaver, but also more solid prover.
What should I say? The handwaver usually are better at convince an idea, a desire, a plan, which is most of the time what the reader want. The next step is for the reader to believe or not. If the reader know it must be true, no doubt about it, so fine, we can skip the verification step. The prover, taking much more pain, to show all things works, but people usually just glance over without even bother to read or follow.
What should I be? I cannot be 100% prover. I will knowingly leave a gap, a jumpable gap, not requiring to use your wing-of-belief to fly over, just for the sake of speed. Since I don't have much time. Then, should the proof be considered as finished? For the sake of writing the paper and submit it, yes. For the sake of understanding how it works, maybe not.
So, the answer is no. You should write a clear statement with a clear proof. Put doubt and confusion in the world is negative contribution.
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?
Here is some vague thought about 3d mirror symmetry (following Ben G and Justin H).
Let $G$ and $G^L$ be Langlands dual varieties. Assume $(G, M)$ is S-dual to $(G^L, M^L)$, where $M$ is some nice G-Ham space, same for $M^L$. Example $$ M = pt, \quad M^L= Whit_{G^L}(T^*G^L) $$ where $Whit_{G^L}$ is one-side symp reduction by $U$ with generic character.
We can consider endomorphism type spaces $$ M/G \times_{T^*[1]BG} M/G \leftrightarrow M^L/G^L \times_{T^*[1]BG^L} M^L/G^L$$ In this example, we should get what? The space $T^*(BG)$ 3d mirror to $Whit_{G^L}^2(T^*G^L)$.
OK, if $M$ is a usual 0-shifted symp stack, then $[M/G]$ should be viewed as a 1-shifted Lagrangian in the 1-shifted cotangent bundle $T^*[1]BG = [g^*/G]$, that's what G-Ham space gives you. So that the intersection is like 0-shifted symp space.
And the terminology about Coulomb branch, and bi-Whittaker reduction is that $$ M_C(G, M) = Whit_{G^L}(M^L) = Whit_{G^L}(T^*G^L) \times_{T^*[1]BG^L} M^L/G^L. $$ $$ M_H(G, M) = M / / G = 0/G \times_{T^*[1]BG} M/G $$ $$ M_H(G, M) \leftrightarrow M_C(G, M) $$
So Coulomb branch 3d mirror to Higgs branch is a special case of relative Langlands, where one slot is about the basic dual setup.