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On this topic I've only seen Ganatra's notes from Paul's Talbot talk (see Scott's answer). An aspect of this, that a Fukaya category can behave in a sheafy way, is part of Nadler's proof that "microlo …

Some blah on symplectic homology vs. cohomology. There's an invariant $SH(M)$ of Liouville domains $M$ which some people call symplectic homology and some symplectic cohomology. This is the direct lim …

As Kevin comments, Hochschild homology and cohomology are defined for any $A_\infty$-category $\mathcal{A}$. That includes Fukaya categories of symplectic manifolds and dg enhancements of the bounded …

Michael's answer describes the sort of thing I had in mind when I brusquely dismissed this as "straightforward" in the earlier, big-picture-style answer that Daniel quoted. (Michael, thanks for explai …

Even without the condition that $HF(L,L)$ is $A_\infty$-isomorphic to $H^\ast(L)$, your conditions are set up in such a way as to disable the standard tricks. This doesn't rule out the existence of ex …

The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomo …

I can't speak for these authors, but what I understand by a "Fourier-Mukai" transform between Fukaya categories is the functor between extended Fukaya categories associated with a Lagrangian correspon …

At first sight, the Fukaya category has obvious cyclic symmetry, because the $A_\infty$ structure maps count points in spaces of rigid pseudo-holomorphic polygons subject to Lagrangian boundary condit …

This sounds like a lovely topic for one or more thesis projects. The relevant definitions are in Seidel's book, as are powerful tools for describing Fukaya categories. The Wehrheim-Woodward functorial …