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3. Categorical calculations

For applications to mirror symmetry and representation theory, we are interested in better understanding the category of microlocal sheaves along an arboreal Lagrangian, both in specific cases of interest and in the abstract.


    1. Add remark

      Problem 3.1.

      Describe the category of microlocal sheaves along the skeleton of the Weinstein manifold obtained from $T^*\Sigma_g$ by Weinstein handle attachment, as well as the effects of surgery on this category.

        •     There are some interactions between the geometry of certain $n$- and $(n+2)$-dimensional Weinstein manifolds, whose study is motivated for instance by mirror symmetry or the theory of cluster algebras. This is especially interesting in the case $n=4.$

          Add remark

          Problem 3.2.

          [org.aimpl.user:bgammage@math.berkeley.edu] Provide general constructions and explanations describing these sort of relations between skeleta of 4- and 6-dimensional Weinstein manifolds.

            •     It is a well-known "fact" that "every category is a Fukaya category." Can the theory of arboreal spaces be used to give a proof of this assertion?

              Add remark

              Problem 3.3.

              Can any smooth stable $\infty$-category be realized as a finite colimit of categories associated to arboreal singularities? (This turns out to be true for tautological reasons, so we should demand further that such a colimit corresponds to a geometric gluing of the singularities.)

                  Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.