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under which general conditions does the map that sends a chain complex to the free dg-algebra over it preserve quasi-isomorphisms?
I need this for chain complexes of modules over a fixed cdg-algebra (over the ground field) and cdg-algebras over that cdg-algebra. But any related info would be welcome.
letâ€™s see, I guess we can go via the transferred model struccture.
Let $k$ be a field of char 0 and $A \in cdgAlg_k$. Then $A Mod$ (everything unbounded) has the standard projective model structure with fibrations the degreewise surjections.
I want to transfer along
$cgdAlg_{A} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} A Mod \,.$Does $U$ preserve filtered colimits? We do have fibrant replacement and path object functor $(-)\otimes_k \Omega^\bullet([0,1])$ on the left. So if the transferred model structure exists, $F$ is left Quillen, which would be good enough for me, probably.
For $A \in cdgAlg_k$ there is a functor
$Sym_A : A Mod \to cdgAlg_A$that sends a complex $V$ of $A$-modules to the symmetric tensor dg-algebra over (under) $A$ that is
$Sym_A V = A \oplus V \oplus V \otimes_A^{sym} V \oplus \cdots \,.$I was looking for conditions under wich this preserved weak equivalences. But I think I found an answer that works for my purpose.
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