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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 17th 2011
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   Author: Urs
   Format: MarkdownItexunder 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 17th 2011
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   Author: Urs
   Format: MarkdownItexlet'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.

   let’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.

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeMar 17th 2011
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   Author: jim_stasheff
   Format: Textsends a chain complex to the free dg-algebra over it meaning a resolution of the chain complex? do you ask for both the free dg-algebra and the free graded commutative version?

   sends a chain complex to the free dg-algebra over it
   
   meaning a resolution of the chain complex?
   
   do you ask for both the free dg-algebra and the free graded commutative version?
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 17th 2011
   • (edited Mar 17th 2011)
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   Author: Urs
   Format: MarkdownItexFor $A \in cdgAlg_k$ 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.

   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.