Peng Zhou

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====== 2026-04-12 a new possibility ======

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: 
  * general bar gluing follows from double-stopped bar gluing, using stop removal
  * fuksym with extra stops admits sod description. 
  * colimit of local sod is global sod.