how to describe perverse schober on $(D, 0)$? it is a machine that, input a disk with stop / singular Lagrangian skeleton / holomorphic function on an open subset, and output a category; input a morphism of object, output a functor; finally, input a 2-morphisms between 1-morphism, and output a natural transformation.

but the problem is, what is a 2-mor in the category of disk with stops? I can view disk with stops as Lagrangian in $T^*X$, or constructible sheaves in X. But these are 1-categories, hom between these objects are just set, at best chain complexes.

How can we move these sectors? closed embedding, open restriction, and then non-char deformation. cobordism as higher morphism doesn't give me much at all.

How to use SOD? Does SOD give me SES of id -> UV -> T? Maybe we just say, 2-perv is data plus condition. this is like saying, endomorphism algebra has some generators and relations, that is to say, endomorphism algebra is the quotient of some free algebra by some 2-sided ideal. So to define action of the quotient algebra on a vector space, we just need to ask the free algebra act, and such that the relation is satisfied. Instead of saying some linear combination of the free algebra is zero, we say some chain complex of object is acyclic. The question is, how do you build that chain complex.

i want to translate, wedrich-dyckerhoff, into our own language.

First, what is Beck-Chevalley stuff? I am reading page 10 of DW. There are two ways to go from $(2,1)$ to $(1,2)$ partition on the line, one is break-then-merge, the other is merge-then-break. The two ways are different, obviously. And even better, there is a relation between the two ways, how? We have adjunctions, in our setting, splitting is the right adjoint (splitting is restriction) of merge. (we can have various versions of adjoints, don't worry, or limit oneself).

We have these two adjunctions

• [ (2) -> (1,1) -> (2) ] ==> [(2) -id-> (2) ]
• [(1, 1) -id-> (1,1)] ==> [ (1, 1) -> (2) -> (1,1) ]

What's the condition of an $A_1$-schober? cofiber of the 2nd guy is an auto-equivalence. Fiber of the first guy is also an equivalence.

If we want to match with the diagram DW consider, we need to identify the left-adjoint with merge, and right-adjoint with split

We are going to use these adjunctions to get the sweeping move

• (2,1) -> (1,1,1) -> (1,2)
• (2,1) -> (1,1,1) -> (1,2) -> (3) -> (1,2)
• (2,1) -> (1,1,1) -> (2,1) -> (3) -> (1,2)
• (2,1) -> (3) -> (1,2)

what are we trying to say now? we can say

• (a+b,c) -> (a,b,c) -> (a,b+c)
• (a+b,c) -> (a,b,c) -> (a,b+c) -> (a+b+c) -> (a, b+c)
• (a+b,c) -> (a,b,c) -> (a+b,c) -> (a+b+c) -> (a, b+c)
• (a+b,c) -> (a+b+c) -> (a,b+c)

There is a natural transformation from the top row to the bottom row, but

Given this commutative square (1,1,1) -> (2,1) or (1,2) -> (3), it is neither a pullback square or a pushforward square.

let me read the construction 3.5. OK, you have something X, living over the $k$-cube. You don't want a section, no. you want a 'fibered map'

you first map a thickened cube, to the cube, where the initial time slice maps to the origin, and the final slice map to identity. Then you want to map the thickend cube to X, so that fiber goes to fiber. what's going on here? You want these arrows, along the time direction, to be 'cocartesian', meaning having the initial-factor-through property, as GPT teaches me.

What is the so called inflation-deflation, on pg 13? it is push-forward from the origin fiber, or pullback to the origin fiber

let me try to understand (3.1.4). We start life with a cube worth of categories, and functors lining the edges. We assume there is a biCartesian fibration, which means there exists (up to contractible choices) a unique lift of an edge if we fix the starting point, or the ending point.

We look at a square in the cube, and project the cube to the square. Say a 3-dim cube, so we are left with 1-dim interval in the fiber. We start with the fiber in the upper right corner. we start with the initial node in that fiber, run co-cartesian extension to the full fiber, then for each fiber position, we have a natural transformation. what's wrong with that? why we need to do deflation? ok, whatever. we still get [1]^{n-1}, in which one direction is in the diagonal base direction, and (n-2) is the fiber direction. And this cube is in the category of functors from this tip to that tip.

the total fiber of (3.1.5) as the BC defect. Question, if we permute the ordering of the index, is it still the same thing? why the first two indices?

If we want to use 'right-adjoint' for the merging, then we are not using $NH_2 \otimes_{1,1} M$ to do extension, rather we are doing $Hom_{1,1}(NH_2, M)$

OK, maybe I want to say, there are two ways of doing restriction, yes. there is a 'left-adjoint restriction', which is, you Reeb flow to the stop, cut, then un-flow a bit. ok, i will do that.

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proving isomorphism

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Cech

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The 3d GMW theory.

In Kapranov-Kontsevich-Soibelmann paper, 10 years ago, they mentioned that it is possible to consider marked polytope in $\R^3$. There is a $E_3$-algebra controlling the deformation of $E_2$-algebra, and there can be coefficient enhancing all these. I want to understand what precisely is the statement.

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• Discussion with Xin Jin about her proof

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