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Let $R,S$ be dg-algebras and $f:R \rightarrow S$ be a quasi-isomorphism. Then $f$ induces an equivalence between their derived categories of dg-modules.

If in addition $R,S$ are graded commutative dg-algebras, then their derived categories are equipped with tensor products and the equivalence respects them.

Now suppose one has only a Zig-Zag of quasiisomorphisms, with $S,R$ still graded commutative, but $X$ is not:

$$R \leftarrow X \rightarrow S$$

This induces still an equivalence between the derived cateogies of $R$ and $S$. However will this equivalence be compatible with the monoidal structure?

Put in another way: Let $X$ be a noncommutative dg-algebra, which is known to be quasiisomorphic to a commutative one. Is there a natural monoidal structure on the derived category of $X$?

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In the context of your last paragraph (and working over a field, for safely reasons): $X$ will be commutative up to coherent homotopies, that is, an $E_\infty$-algebra. If you construct the derived category of $X$ taking into account the higher structure, I am pretty sure tha answer is yes. – Mariano Suárez-Alvarez Feb 7 '12 at 18:52

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