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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2010
    • (edited Mar 14th 2013)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2010
    • (edited Dec 9th 2010)

    added the details for the existence of the model structure and its simplicial enrichment by

    Alg P(A,B):=([n]Hom Alg P(A,BΩ poly (Δ n))). Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(\Delta^n))) \,.

    Hinich’s proof of the enrichment is just a pointer to the old Bousfield-Gugenheim article. As far as i can see, they consider the case of non-positively/negatively garded chain complexes, though, whereas Hinich uses unbounded chain complexes. But I guess it’s obvious that the proof of the pushout-product axiom goes through to the unbounded case immediately.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2010
    • (edited Dec 9th 2010)

    I realize that it is not true that Hinich gives the sSet QuillensSet_{Quillen}-enrichment of unbounded dg-algebras. It comes pretty close, but some parts are missing:

    he shows the copowering only over finite simplicial sets (since the tensor product of the algebras only commutes with finite products of algebras) and for these only externally (the defining natural isomorphism only at the level of the underlying sets).

    And I don’t see a way to fix this. To lift the natural isomorphism defining the copowering over sSetsSet to one of simplicial sets one would need an isomorphism of forms on simplicial sets Ω poly (S×T)Ω poly (S)Ω poly (T)\Omega^\bullet_{poly}(S \times T) \simeq \Omega^\bullet_{poly}(S) \otimes \Omega^\bullet_{poly}(T), but that’s only a quasi-iso.

    So all there is with this is that the simplicial hom-set

    Alg P(A,B):=([n]Hom Alg P(A,BΩ poly (Δ[n]))) Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A , B \otimes \Omega^\bullet_{poly}(\Delta[n])))

    has the right connected components when AA is cofibrant, in that

    Ho(Alg P)(A,B)π 0Alg P(A,B). Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(A,B) \,.

    But I am not sure what the remaining homotopy type of Alg P(A,B)Alg_P(A,B) is doing.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2010

    But I am not sure what the remaining homotopy type of Alg P(A,B)Alg_P(A,B) is doing.

    So I am trying to fill in the proof that this is indeed the correct derived hom-space.

    As referenced there, we need to show that for any cdg-algebra AA the simplicial dg-algebra

    sA:[n]A kΩ poly (Δ n) s A : [n] \mapsto A \otimes_k \Omega^\bullet_{poly}(\Delta^n)

    is a simplicial resolution – a right framing in the terminology of Hovey’s book .

    The polynomial forms are acyclic, so constAsAconst A \to s A is a weak equivalence.

    It remains to show that sAs A is Reedy fibrant. If I see correctly, its matching object is

    (matchsA) r=AΩ poly (Δ r). (match s A)_r = A \otimes \Omega^\bullet_{poly}(\partial \Delta^r) \,.

    So we need to show that (sA) r(matchsA)(s A)_r \to (match s A) hence Ω poly (Δ[r]Δ[r])\Omega^\bullet_{poly}(\partial \Delta[r] \hookrightarrow \Delta[r]) is a fibration in cdgAlg kcdgAlg_k.

    But this follows by the the fact that we have the standard Quillen adjunction of rational homotopy theory Ω poly :sSetdgcAlg k op\Omega^\bullet_{poly} : sSet \to dgcAlg_k^{op} (for kk of characteristic 0) which means in particular that Ω poly \Omega^\bullet_{poly} sends monomorphisms of simplicial sets to surjections of dg-algebras.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2013

    have added to model structure on dg-algebras over an operad Hinich’s theorem on the Quillen equivalences between the algebras between quasi-isomorphic operads, hence in particular the rectification of homotopy algebras.