I've been trying to understand better the relation between the basic blocks of derived algebraic geometry. More precisely, I'm trying to understand the relation between the DG approach, the spectral approach and the simplicial approach. There's these excellent answers in What is a simplicial commutative ring from the point of view of homotopy theory? . However I still have some doubts.

Let $R$ be a commutative ring. In page 46 of Lurie's thesis (http://dspace.mit.edu/handle/1721.1/30144), he says that there are functors $$R\text{-Alg}^{{\Delta}^{op}} \xrightarrow{\phi} \text{comm. DG} \ R\text{-Alg} \xrightarrow{\psi} E_{\infty} \ R\text{-Alg}$$ and that if $R$ is a $\mathbb{Q}$-algebra, then both arrows are an equivalence of categories when the codomain of $\phi$ is restricted to the connectives objects.

Furthermore, in the above cited question, Lurie says that, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic.

Where can I find a proof for these facts? Furthermore, if possible, could someone, please, describe explicitly these functors briefly? I believe the first functor $\phi$ is just the ordinary Dold-Kan correspondence however I'm not sure since I've never seen this correspondence for $R$-algebras when $R$ is not a field...

So, summarizing the question:

1)What's $\phi$ and $\psi$?

2)Why, when $R$ is a $\mathbb{Q}$-algebra and $\psi$ is restricted in the codomain to the connective algebras, $\phi$ and $\psi$ are equivalences?

3)Why, when $R$ is a field, the composition $\psi\phi$ is monadic and comonadic?

Thanks in advance.

  • 1
    $\begingroup$ I believe 2 and 3 are answered in DAG5, Prop 4.1.11, modulo the Barr-Beck theorem from Higher Algebra (there is no need for $R$ to be a field, it seems). $\endgroup$ – Marc Hoyois Sep 4 '15 at 1:13
  • $\begingroup$ @MarcHoyois Thanks again. How can you conclude the monadicity and comonadicity from this proposition. Or, equivalently (by Beck's theorem), how can you conclude that $\psi \phi$ creates (co)equalizers of $\psi \phi$-split pairs? Furthermore is $\phi$ just the ordinary "monoidal" Dold-Kan correspondence? $\endgroup$ – user40276 Sep 4 '15 at 3:00
  • $\begingroup$ Barr-Beck doesn't require creation, existence+preservation suffices. Your guess as to what $\phi$ is is as good as mine. $\endgroup$ – Marc Hoyois Sep 5 '15 at 0:53
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    $\begingroup$ I think you misread: he says that, for $R$ a $\mathbb{Q}$-algebra, $\psi$ is an equivalence (without connectivity restrictions), and $\phi$ is an equivalence onto its essential image which consists of the bounded below ("connective") chain complexes. $$$$ In the DAG reference given by Marc above, he goes directly from the simplicial into the spectra world. To see a comparison of $E_\infty$ $R$-algebras with commutative dgas, see Higher Algebra, Proposition $\endgroup$ – Bruno Stonek Feb 22 '17 at 9:00
  • $\begingroup$ @BrunoStonek Thanks for the reference. Sorry, I mistyped $\psi$ instead of $\phi$. I will correct it. $\endgroup$ – user40276 May 6 '17 at 23:23

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