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The bounded derived category $D^b\mathrm{Coh}(X)$ is the homotopy category of a stable $\infty$-category $\mathbb{D}^b\mathrm{Coh}(X)$. Apparently there are reasons, such as "nonfunctoriality of the cone", why it is sometimes better to consider $\mathbb{D}^b\mathrm{Coh}(X)$; in particular, it is the correct category to use from the derived point of view (in the sense of derived algebraic geometry).

The thing is, in homological mirror symmetry, the B-side is (often) given by $D^b\mathrm{Coh}(X)$. Is there expected to be an upgrade of HMS where the B-side is instead the full $\infty$-category $\mathbb{D}^b\mathrm{Coh}(X)$? Presumably this would involve exhibiting the (derived, split-closed, ...) Fukaya category as the homotopy category of a stable $\infty$-category equivalent to $\mathbb{D}^b\mathrm{Coh}$ of the mirror.

Has there been any work done in this direction? If not, is there a good reason why it wouldn't be very interesting?

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  • $\begingroup$ The main reason is the lack of definition of this "A-side" $\endgroup$
    – Denis T
    Commented Jul 27 at 13:01
  • $\begingroup$ @DenisT do you know of any candidate definitions experts may be thinking about? $\endgroup$
    – andres
    Commented Aug 21 at 22:44

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Yes. The full stable infinity category is the right answer. The reason is that both sides are topological quantum field theories. In two dimensions, which is the important case here, Costello showed that these are $A_\infty$-categories, and Lurie showed(ish) that TQFTs are $(\infty,n)$-categories more generally.

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