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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 12th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItexThe nLab page [[de Rham theorem]] mentions the A_∞ de Rham theorem, which can be formulated by saying that the HR-algebras Hom(Σ^∞ X, HR) and the HR-algebra coming from the real dga Ω^*(X) are naturally weakly equivalent. In fact, the Eilenberg-MacLane spectrum HR is an E_∞-spectrum, hence so is the mapping spectrum Hom(Σ^∞ X, HR) for any smooth manifold M, in fact it is an E_∞ HR-algebra. On the other hand, differential forms form a real commutative dga, which therefore gives another E_∞ HR-algebra. Surely somebody has already proved that these two HR-algebras are naturally equivalent, so what is the reference for this?

   The nLab page de Rham theorem mentions the A_∞ de Rham theorem, which can be formulated by saying that the HR-algebras Hom(Σ^∞ X, HR) and the HR-algebra coming from the real dga Ω^*(X) are naturally weakly equivalent.

   In fact, the Eilenberg-MacLane spectrum HR is an E_∞-spectrum, hence so is the mapping spectrum Hom(Σ^∞ X, HR) for any smooth manifold M, in fact it is an E_∞ HR-algebra.

   On the other hand, differential forms form a real commutative dga, which therefore gives another E_∞ HR-algebra.

   Surely somebody has already proved that these two HR-algebras are naturally equivalent, so what is the reference for this?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 13th 2015
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   Author: Urs
   Format: MarkdownItexThat's at least implicit in [arXiv:1311.3188](http://arxiv.org/abs/1311.3188) in the discussion of Deligne cohomology: in the extreme case where the truncation degree is such that the differential spectrum comes out equivalent to the plain spectrum it arises as the pullback of the equivalence from the de Rham complex to $H\mathbb{R}$ (which in the opposite extreme is the map from just closed forms whose pullback gives the genuine Deligne complex). Unfortunately the arXiv version of that section has essentially no details. I think in the published version this was expanded on. In any case, in this same context the statement must also be at least implicit in Hopkins-Singer.

   That’s at least implicit in arXiv:1311.3188 in the discussion of Deligne cohomology: in the extreme case where the truncation degree is such that the differential spectrum comes out equivalent to the plain spectrum it arises as the pullback of the equivalence from the de Rham complex to $H\mathbb{R}$ (which in the opposite extreme is the map from just closed forms whose pullback gives the genuine Deligne complex). Unfortunately the arXiv version of that section has essentially no details. I think in the published version this was expanded on. In any case, in this same context the statement must also be at least implicit in Hopkins-Singer.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 3rd 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItexI guess the easiest way to see this result is to apply the theorem of Morel-Voevodsky and Dugger: given an **R**-local ∞-sheaf on the site of smooth manifolds with values in any presentable ∞-category C, the canonical natural transformation F→Map(−,F(pt)) is an equivalence. In our case we can substitute F=Ω, the de Rham functor, and C=**R**-CDGA, the ∞-category of real cdgas (aka H**R**-algebras) We obtain that Ω(X) is equivalent to Map(X,Ω(pt)) and Ω(pt)=H**R**. To apply the MVD-theorem we need to know that Ω is **R**-invariant (trivial) and that it is an ∞-sheaf, which is the content of Weil's Čech—de Rham theory. By the way, I cannot find anything about Weil's de Rham descent theorem on nLab; I could create a stub article, but I have to know that I didn't miss anything in my search.

   I guess the easiest way to see this result is to apply the theorem of Morel-Voevodsky and Dugger: given an R-local ∞-sheaf on the site of smooth manifolds with values in any presentable ∞-category C, the canonical natural transformation F→Map(−,F(pt)) is an equivalence.

   In our case we can substitute F=Ω, the de Rham functor, and C=R-CDGA, the ∞-category of real cdgas (aka HR-algebras) We obtain that Ω(X) is equivalent to Map(X,Ω(pt)) and Ω(pt)=HR.

   To apply the MVD-theorem we need to know that Ω is R-invariant (trivial) and that it is an ∞-sheaf, which is the content of Weil’s Čech—de Rham theory.

   By the way, I cannot find anything about Weil’s de Rham descent theorem on nLab; I could create a stub article, but I have to know that I didn’t miss anything in my search.