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 $X^{Delta_{Diff}^bullet}$ of smooth singular simplices on $X$,




• the infinitesimal singular simplicial complex $X^{(Delta^bullet_{inf})}$ of infinitesimal singular simplices.

There is a canonical injection $X^{(Delta^bullet_{inf})} to X^{Delta^bullet_{Diff}}$. We may take degreewise (internally, i.e. smoothly) functions on these, to get the cosimplicial algebras $[X^{Delta^bullet_{inf}},R]$ and $[X^{Delta^bullet_{Diff}},R]$.

The normalized cochain complex of chains on $[X^{Delta^bullet_{Diff}},R]$ is the complex of smooth singular cochains.

The normalized cochain complex of chains on $[X^{Delta^bullet_{inf}},R]$ 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

$$
N^bullet([X^{Delta^bullet_{Diff}},R] to [X^{Delta^bullet_{inf}},R])
=
C^bullet_{smooth}(X) to Omega_{dR}^bullet(X)
$$

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.