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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2011

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2011

    let’s see, I guess we can go via the transferred model struccture.

    Let kk be a field of char 0 and AcdgAlg kA \in cdgAlg_k. Then AModA Mod (everything unbounded) has the standard projective model structure with fibrations the degreewise surjections.

    I want to transfer along

    cgdAlg AUFAMod. cgdAlg_{A} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} A Mod \,.

    Does UU preserve filtered colimits? We do have fibrant replacement and path object functor () kΩ ([0,1])(-)\otimes_k \Omega^\bullet([0,1]) on the left. So if the transferred model structure exists, FF is left Quillen, which would be good enough for me, probably.

    • CommentRowNumber3.
    • CommentAuthorjim_stasheff
    • CommentTimeMar 17th 2011
    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?
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2011
    • (edited Mar 17th 2011)

    For AcdgAlg kA \in cdgAlg_k there is a functor

    Sym A:AModcdgAlg A Sym_A : A Mod \to cdgAlg_A

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

    Sym AV=AVV A symV. 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.