My question is the following: is there an analog of Morita theorem in the simplicial setting?

I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories of simplicial modules $Mod(A)$ and $Mod(B)$ are equivalent (or maybe equivalent as simplicial categories?). Is there a result similar to the classical Morita theorem which gives the criterion for two rings to be Morita equivalent?

I tried to google, but I couldn't find anything useful. So I think maybe such an extension is just some trivial formality? Or maybe no one considered that?

Thank you very much!

  • $\begingroup$ I think the version with simplicially enriched equivalences can be proved in exactly the same way. But perhaps the real question is whether there is a homotopical version of the Morita theorem. $\endgroup$ – Zhen Lin Nov 15 '13 at 0:19
  • $\begingroup$ @ZhenLin Can you, please, give a reference? Have you seen it somewhere? To be honest, I do not want to repeat the proof of classical Morita theorem, even if it is possible. At least, I don't want to do that if there is a chance that it is already written somewhere =) $\endgroup$ – Sasha Patotski Nov 15 '13 at 0:24
  • $\begingroup$ I have not seen a proof, but it seems reasonable. $\endgroup$ – Zhen Lin Nov 15 '13 at 8:23
  • $\begingroup$ @ZhenLin, by a homotopical version do you mean what To\"en calls "derived Morita theory" for DG categories? $\endgroup$ – user314 Nov 15 '13 at 11:17
  • 1
    $\begingroup$ That sounds like what I imagined might exist! $\endgroup$ – Zhen Lin Nov 15 '13 at 11:34

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