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Let $\Omega^{*}_{\text{poly}}\: : \: sSet\to dg_{\geq 0}Comm_{+}$ be the polynomial De Rham functor on simplicial sets. I have the following questions

1) When we have a quasi-isomorphism between $\Omega^{*}_{\text{poly}}\left(K\times L\right)$ and $\Omega^{*}_{\text{poly}}\left(K\right)\otimes \Omega^{*}_{\text{poly}}\left( L\right)$?

2) When we have an ISOMORPHISM?(Conjecture: if $K$ or $L$ is a finite simplicial set)

Here $dg_{\geq 0}Comm_{+}$ is the category of commutative unitary cochain differential graded algebras over a field of char zero.

Thanks!!!

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  • $\begingroup$ For which $K$ and $L$ have you been able to verify this tensor formula? $\endgroup$
    – Yemon Choi
    Commented Dec 4, 2014 at 18:53
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    $\begingroup$ I think that 1) is a consequence of the Eilenberg-Zilber theorem and the quasi-isomorphism between the polynomial differential forms and the rational cochains. For 2), why would you want an isomorphism, quasi-isomorphism seems the more natural thing to ask. $\endgroup$
    – Matthias Wendt
    Commented Dec 4, 2014 at 20:10
  • $\begingroup$ I was interested to find some special cases such that 2) is true. But using the Eilenberg Zielber theorem I think that 2) is always false, even with the standard simplicial sets $X=\Delta^{n}$ and $Y=\Delta^{m}$. $\endgroup$
    – Cepu
    Commented Dec 6, 2014 at 12:32

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