Monoidal Dold–Kan correspondence for non-connected CDGA

Question

Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?

I understand that there is a technical problem with the original proof due to Quillen given in "Rational Homotopy Theory" (Remark on p.223). However, I don't understand what is the conceptual reason for this.

Edit: In the initial post I mistakenly used the term connective to denote CDGAs that have the ground field in dim 0 and 0 in negative degrees. It was pointed out by Ben Wieland in the comments that these algebras should be called connected instead.

Answer 1

The answer to the question:

Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?

is that in fact the Dold-Kan correspondence can be extended to this setting.

The classical Dold-Kan correspondence is an equivalence of categories between the category $ch^+$ of non-negatively graded chain complexes of abelian groups, and the category $sAb$ of simplicial abelian groups. The normalization functor $N: sAb \to ch^+$ is monoidal. So is the inverse functor $\Gamma: ch^+\to sAb$, thanks to the Alexander-Whitney map. The natural isomorphism $N\Gamma \cong Id_{Ch^+}$ is monoidal, but the natural isomorphism $\Gamma N \cong Id_{sAb}$ is not, as explained in the paper Equivalences of monoidal model categories by Schwede and Shipley.

The Dold-Kan equivalence can be defined for any commutative ring $R$, analogously to the classical case $R = \mathbb{Z}$, leading to a Quillen equivalence between non-negatively graded $R$-modules and simplicial $R$-modules (where $R$ is regarded as a monoid in the category of simplicial abelian groups). This Quillen equivalence is weak monoidal rather than strong monoidal. The monoidality is enough to induce an adjoint pair on categories of $O$-algebras, for any operad $O$ (e.g., the commutative monoid operad).

Let $k$ be a field of characteristic zero. Then the Quillen equivalence $(N,\Gamma)$ is nice in the sense of Definition 3.5.5 of the paper Homotopical Adjoint Lifting Theorem by me and Donald Yau. This is the content of Example 3.5.6. Consequently, Corollary 5.4.1 proves that the induced adjunction between CDGAs over $k$ and simplicial commutative $k$-algebras is a Quillen equivalence. Note that we place no restriction at all on the values of the CDGAs in degree zero, so we're in the "non-connected" setting of the question.

For completeness, let me unpack why this setting is "nice":

1. In $\mathcal{N} = ch^+_k$ (with the projective model structure, which here coincides with the injective model structure), for any symmetric group $\Sigma_n$, every object of $\mathcal{N}^{\Sigma_n}$ is projectively cofibrant (essentially, $\Sigma_n$ acts freely due to the characteristic being zero). So all the conditions of niceness are automatic for $\mathcal{N}$.
2. Let $\mathcal{M}$ denote the category of simplicial $k$-modules. Whenever $X\in \mathcal{M}^{\Sigma_n}$ is cofibrant, so is the object of $\Sigma_n$-coinvariants $X_{\Sigma_n}$. Furthermore, the domains of the generating cofibrations of $\mathcal{M}$ are cofibrant.
3. For cofibrant objects $W$ and $X$, the map $N^2: N(W\otimes X)\to NW \otimes NX$ is a weak equivalence and remains a weak equivalence after taking $\Sigma_n$-coinvariants (if $W$ and $X$ have $\Sigma_n$-actions), again, because $\mathcal{N}$ is so nicely behaved.
4. For any $X\in \mathcal{M}^{\Sigma_n}$ cofibrant in $\mathcal{M}$, if $f$ is a (trivial) cofibration, then so is $X\otimes_{\Sigma_n} f^{\Box n}$, where $f^{\Box n}$ is the $n$-fold iterated pushout product. This condition guarantees that the homotopy theory of $\mathcal{M}$ plays nicely with symmetric powers, so that commutative monoids (and algebras over more general operads) inherit transferred model structures. It holds because we're in the characteristic zero setting.

As for the proof of Corollary 5.4.1, it comes down to Theorem 4.2.1, which works by a transfinite induction over the $Sym$ functor, where $Sym^n(X) = X^{\otimes n}/\Sigma_n$. Because each of these functors preserves weak equivalences between cofibrant objects (again using the characteristic zero assumption), the transfinite induction works and proves the relevant morphism is a weak equivalence, to know that the induced adjoint pair is a Quillen equivalence.