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John Wiltshire-Gordon and I were reading the nlab article on the monoidal Dold-Kan correspondence and we are puzzled by the following statement:
“The two functors in the Dold-Kan correspondence individually respect these monoidal structures, in the sense that they are lax monoidal functors.”
This seems to us to be false. Let $D_1$ be the chain complex (indexed from $0$) which is $\mathbb{Q}$ in degree $1$ and $0$ in every other degree. Then $D_1 \otimes D_1$ (using the monoidal structure on chain complexes, by taking the total complex of the double complex) is the chain complex which is $\mathbb{Q}$ in degree $2$ and $0$ in every other degree. Applying $\Gamma$, we get a certain simplicial vector space which has dimension $\binom{n-1}{2}$ in degree $n$.
On the other hand $\Gamma(D_1)$ is a simplicial vector space which has dimension $n$ in degree $n$ and $\Gamma(D_1) \otimes \Gamma(D_1)$ (using the monomial structure on simplicial vector spaces, by pulling back along the diagonal map) has dimension $n^2$ in degree $n$. So $\Gamma(D_1 \otimes D_1)$ is not isomorphic to $\Gamma(D_1) \otimes \Gamma(D_1)$. Of course, the two are homotopy equivalent.
Are we missing something dumb, or is this an error in the article?
It is true that the Dold–Kan functors are not strong monoidal functors. But the article only claims that they are lax monoidal functors. (I don’t know the details of the latter claim.)
Yes, that distinction between lax and strong monoidalness drives the whole topic of the monoidal Dold-Kan correspondence. There is a remark in the entry amplifying this. Presently it says this:
The upshot is that $N$ and $G$ are both pretty close to being strong monoidal functors, but fail to be so. If they were, the monoidal Dold-Kan correspondence would be a simple corollary of the Dold-Kan correspondence and would hold at the level of 1-categories.
Explicitly, the failure of $N$ to be strong monoidal is in that the Eilenberg-Zilber map is (on normalized chain complexes) a right inverse to the Alexander-Whitney map, but not a left inverse. But it is a homotopy-inverse: because the components of the Alexander-Whitney map are (as discussed there) quasi-isomorphisms. By 2-out-of-3 it follows that also the EZ-maps are quasi-isomorphisms and that these are indeed inverse to the AW map in the homotopy category of chain complexes (the derived category).
Therefore we expect that the monoidal Dold-Kan correspondence holds, while not necessarily at the level of ordinary categories, at least at the level of homotopical categories. This is indeed the case, as discussed below.
I have now added a warning and a pointer to this remark from the earlier sentence that #1 pointed out.
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