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Homotopy Colimits of DG Categories and Fukaya Categories
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We construct a new cylinder object for semifree differential graded (dg) categories in the category of dg categories. Using this, we give a practical formula computing homotopy colimits of semifree dg categories. Combining it with the result of Ganatra, Pardon, and Shende, we get a formula computing wrapped Fukaya categories of Weinstein manifolds using their sectorial coverings. This formula has lots of applications including a practical computation of the wrapped Fukaya category of any cotangent bundle or plumbing space. In this paper, we compute wrapped Fukaya categories of cotangent bundles of lens spaces using their Heegaard decomposition. From the computation, we show that the endomorphism algebra of the cotangent fibre is a full invariant of the homotopy type of lens spaces.
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Simple homotopy theory for Fukaya categories
Isomorphisms in the Fukaya category of a Weinstein manifold with vanishing first Chern class automatically have trivial Whitehead torsion, so isomorphic Lagrangians are simple homotopy equivalent.
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