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I am trying to read Seidel's book, and I am confused about the "derived" Fukaya category. If I understand properly, one starts from the $A_{\infty}$-category $\mathfrak{F}(M)$, enlarges it to the $A_{\infty}$-category $Tw(\mathfrak{F}(M)$) of twisted complexes, and then, through $H^0$, recovers from this an honest category, $D(\mathfrak{F}(M))$, and calls this the derived Fukaya category.

It is unclear to me how this construction relates to the usual notion of derived categories (if it does at all!). In the classical sense, one starts from an abelian category, and one keeps the same complexes at all stages to arrive at the derived category; only the hom-sets are altered.

Could someone expand on the connections between the two notions? Since the homological mirror symmetry conjecture is based on a "derived equivalence" of categories, one of them being a honest derived category, there are certainly some things to be said.

Thanks in advance.

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    $\begingroup$ For any $A_\infty$ category $C$, $H^0(Tw(C))$ is the full subcategory of the derived category $D(C)$ of the $A_\infty$-category $C$ spanned by (some or all, depending on whether $C$ is idempotent complete) compact objects. This is a well established notion. BTW, when you construct the derived category of an abelian category you do not keep the same objects all the time, you consider complexes in the abelian category. $\endgroup$ – Fernando Muro Nov 4 '13 at 22:49
  • $\begingroup$ @FernandoMuro I am not familiar with the notion of the derived category of an $A_{\infty}$-category. Do you have some standard references on the subject at hand? $\endgroup$ – A.P. Nov 4 '13 at 23:02
  • $\begingroup$ See for instance Lefevre-Hasegawa's thesis $\endgroup$ – Fernando Muro Nov 4 '13 at 23:03
  • $\begingroup$ (WHich you can get from arXiv, for example) $\endgroup$ – Mariano Suárez-Álvarez Nov 4 '13 at 23:15
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You should consider what happens if you have a generator E, e.g. an object which generates the whole category under cones, shifts and summands of your category of coherent sheaves. All this says is that the derived Fukaya category doesn't have a natural t-structure. There are many such structures which are transparent on one side of mirror symmetry but not on the other. For example, DCoh(X) is a monoidal cat...

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