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Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic
invariant of high interest in mirror symmetry and geometric representation
theory. This paper informally sketches how, in analogy with Morse homology, the
Fukaya category might result from gluing together Fukaya categories of
Weinstein cells. This can be formalized by a recollement pattern for Lagrangian
branes parallel to that for constructible sheaves. Assuming this structure, we
exhibit the Fukaya category as the global sections of a sheaf on the conic
topology of the Weinstein manifold. This can be viewed as a symplectic analogue
of the well-known algebraic and topological theories of (micro)localization
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