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Weinstein manifolds are an important class of symplectic manifolds $W$ equipped with some additional data. Starkston showed that Weinstein manifolds in dimension 2 and 4 can be studied via skeleta with "arboreal singularities". For every tree $T$, Nadler defined an arboreal singularity $L_T$ and showed the microlocal sheaf category associated to $L_T$ is the dg-derived category of modules over $T$. From this perspective, if $W$ has arboreal skeleton, then it is natural to associate to $W$ a diagram of dg-categories of modules, and then take the homotopy limit of this diagram. I will refer to the result as $\mu Sh(W)$.

I will describe the diagrams arising in dimension 2 and how they are encoded by ribbon trivalent graphs. Generalizing work of Karabaş, I will give a formula for the homotopy limit of any diagram obtained in this way, via Reedy model structures. Using this formula, I will sketch a proof that, in dimension 2, $\mu Sh(W)$ is a Weinstein homotopy invariant.