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1. • CommentRowNumber1.
   • CommentAuthorMatanP
   • CommentTimeJul 17th 2012
   • PermaLink
   Author: MatanP
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorMatanP
   • CommentTimeJul 18th 2012
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   Author: MatanP
   Format: MarkdownItexSorry 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorMatanP
   • CommentTimeJul 19th 2012
   • (edited Jul 19th 2012)
   • PermaLink
   Author: MatanP
   Format: MarkdownItexI now see that the introduction of [BZFN](http://arxiv.org/pdf/0805.0157v5.pdf) 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 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 ?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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?

   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?

5. • CommentRowNumber5.
   • CommentAuthorMatanP
   • CommentTimeJul 22nd 2012
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   Author: MatanP
   Format: MarkdownItexSorry for the bad English -- I meant stub. As for the math., Toen and Vezzosi (and [BZFN](http://arxiv.org/pdf/0805.0157v5.pdf) ) 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.

   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.