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I looked at the stab on monoidal monoidal Dold-Kan correspondence and was wondering if one could get a Quillen equivalence via a DK-correspondence (or alike) from Cdga/k = the category of (cochain) commutative dg algebras concentrated in non-positive degrees over a ring k of characteristic zero to the category simplicial commutative k-algebras.
The stab only talks about char 0 field and doesn’t mention an equivalence in this case (but rather a correspondence) and on the other hand it seems that Toen’s DAG uses this implicitly.
Sorry for the confusion. I now see that the only gap between the nLab entry and the equivalence used in Toen and Vezzosi’s DAG is that they only assume k is a ring of char 0. It seems that the Quillen equivalence in the case of char 0 field (in “Rational homotopy theory”) extends for the more general case but I want to be sure. We can then add it to the nLab entry and make the connection with DAG.
I now see that the introduction of BZFN says that the Dold-Kan becomes an equivalence between simplicial commutative k-algebras and connected cdga’s for k a $\mathbb{Q}$-algebra. Is it ok to add it to monoidal Dold-Kan correspondence ?
I looked at the stab on monoidal monoidal Dold-Kan correspondence
The stab only talks about char 0
Do you mean “stab” or “stub”? Could you say what you mean by the word?
Sorry for the bad English – I meant stub. As for the math., Toen and Vezzosi (and BZFN ) say that the Quillen equivalence works for k a $\mathbb{Q}$ algebra which I guess is what most people call a ring of charachteristic 0. I’ll try to see if Quillen’s original argument is valid as is for that case but I wanted to see if someone knows it already.
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