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I know there exists a groupoid-valued prestack parameterizing (connected) triangulated dg-categories (ie the points of this moduli are not objects of a fixed dg-category, but rather dg-categories themselves).

Has anything been done for stable $\infty$-categories?

[Connectedness is a condition on Hochschild cohomology: $HH^{<0}=0$ and $HH^0=k$.]

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    $\begingroup$ Rathen than answering, I'm interested in what you know :) What is a connected triangulated dg-category and where can I find references about their moduli stack? $\endgroup$ – Fernando Muro May 21 '15 at 23:11
  • $\begingroup$ Exactly what you mean "parametrizing stable $\infty$-categories"? Usually you get prestacks when your type of objects depend on a parameter. Do you mean $R$-enriched stable categories (where $R$ varies in rings)? $\endgroup$ – Denis Nardin May 22 '15 at 16:49
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    $\begingroup$ If you just want a prestack, rather than say a geometric stack, I don't see the relevance of connectedness or of the dg vs stable distinction... $\endgroup$ – David Ben-Zvi May 22 '15 at 16:50
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    $\begingroup$ @AdeelKhan I must've confused myself regarding connectedness. The reference I had in mind is a D-stack $Cat^{CW}_{[n,0]}$ buried somewhere in Toen-Vezzosi's HAGII. Though I remember they comment on the fact that this is not very useful in the algebro-geometric context, as in general D(X) for a smooth and projective variety X will not be a point of it (something to do with the boundedness condition [n,0], I don't remember the details, sorry!). $\endgroup$ – pro May 25 '15 at 2:54
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    $\begingroup$ @pro, thanks. I know that moduli stacks of dg-categories are discussed in To\"en's beautiful paper on derived Azumaya algebras, which has been generalized to a spectral stack parametrizing R-linear categories for R a commutative ring spectrum by Antieau-Gepner. I don't think they discuss the type of issue you are interested in there, though. $\endgroup$ – AAK May 25 '15 at 7:14

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