Friday, 20 July 2012

at.algebraic topology - Infinity de Rham quasi-isomorphism

Yes.



Here is one way to see it:



before passing to dg-algebras, let's look at cosimplicial algebras and then later apply the normalized cochain (Moore) complex functor.



Work in a smooth (oo,1)-topos, modeled by simplicial presheaves on a site of smooth loci. In there, we have for every manifold X



  • the singular simplicial complex XDeltabDiffullet of smooth singular simplices on X,


  • the infinitesimal singular simplicial complex X(Deltabulletinf) of infinitesimal singular simplices.


There is a canonical injection X(Deltabulletinf)toXDeltabulletDiff. We may take degreewise (internally, i.e. smoothly) functions on these, to get the cosimplicial algebras [XDeltabulletinf,R] and [XDeltabulletDiff,R].



The normalized cochain complex of chains on [XDeltabulletDiff,R] is the complex of smooth singular cochains.



The normalized cochain complex of chains on [XDeltabulletinf,R] turns out to be, by some propositions by Anders Kock, to be the deRham algebra, as discussed a bit at differential forms in synthetic differential geometry.



Therefore under the ordinary Dold-Kan correspondence we have a canonical morphism



Nbullet([XDeltabulletDiff,R]to[XDeltabulletinf,R])=Cbulletsmooth(X)toOmegabdRullet(X)



which is an equivalence of cochain complexes. But there is a refinement of the Dold-Kan correspondence the monoidal Dold-Kan correspondence. And this says that this functor is also a weak equivalence of oo-monoid objects.

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