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As a result of a Cafe discussion with Kathryn Hess, we learned and recorded at monoidal Dold-Kan correspondence that the normalized Moore complex functor from simplicial abelian groups to chain complexes is both lax and colax monoidal (and in fact Frobenius). Since this functor is an equivalence of categories, it is in particular both a left and a right adjoint, so it seems that it should follow by doctrinal adjunction that its inverse functor, from chain complexes to simplicial abelian groups, is also both lax and colax monoidal. But I don’t see that written down anywhere; does it sound right?
It would be good to eventually improve the nLab entry on these points. It is easy to forget the intricate details of all this stuff after a while, and the entry presently does not live up to fully refreshing memory.
Okay, so let me look at Schwede-Shipley again:
That $N$ (Moore chains) is monoidal and lax monoidal is on page 295 (9 of 48).
That the co-monoidal structure on $N$ may be turned into a monoidal structure on $\Gamma$ (the functor going the other way) is on the bottom of that page.
The explicit construction is the first displayed math on the following page 296.
I don’t know about doctrinal adjunctions. Is that construction on 296 what you would want to see?
P.S. Maybe not relevant for your question, but just for completeness: the issue with the monoidal DK-correspondence is not the failure of both functors to be monoidal, but to form a monoidal adjunction (Remark 2.14, page 13). And what really matters is that we have a Quillen adjunction . It is a lucky and somewhat misleading coincidence that the ordinary DK-correspondence is on top of being a Quillen equivalence also a plain equivalence of categories.
In Schwede Shipley, to get the Quillen equivalence between simplicial rings and chain dg-algebras, the DK-functor $\Gamma$ is replaced by something else. In Cortinas et al, to get the Quillen equivalence between co-simplicial rings and co-chain dg-algebras, the Moore functor $N$ is replaced by something else.
Is that construction on 296 what you would want to see?
Yes – they’re taking the mate of $A W$ with respect to the adjunctions $M N \dashv M \Gamma$ and $N \dashv \Gamma$, where $M$ is the monoidal category monad, by pasting $A W$ with $\eta$ and $M \epsilon$.
Great, thank you; I should have known to look at Schwede-Shipley.
I found the remarks about symmetry at monoidal Dold-Kan correspondence to be somewhat opaque, so I clarified them a bit. (I hope I got it right: N is lax symmetric monoidal, but colax non-symmetric monoidal, but colax $E_\infty$ monoidal.) Along the way I felt the need to write monoidal adjunction.
I found the remarks about symmetry at monoidal Dold-Kan correspondence to be somewhat opaque,
My fault.
so I clarified them a bit.
Thanks!
I am wondering what you are working on related to monoidal Dold-Kan…
Nothing; someone just asked me a question about it and I wanted to point them to the nLab page. (-:
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