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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 be a field of char 0 and . Then (everything unbounded) has the standard projective model structure with fibrations the degreewise surjections.

   I want to transfer along

   Does preserve filtered colimits? We do have fibrant replacement and path object functor on the left. So if the transferred model structure exists, 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)
   • PermaLink
   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 there is a functor

   that sends a complex of -modules to the symmetric tensor dg-algebra over (under) that is

   I was looking for conditions under wich this preserved weak equivalences. But I think I found an answer that works for my purpose.