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am starting model structure on dg-algebras over an operad
added the details for the existence of the model structure and its simplicial enrichment by
$Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(\Delta^n))) \,.$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.
I realize that it is not true that Hinich gives the $sSet_{Quillen}$-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 $sSet$ to one of simplicial sets one would need an isomorphism of forms on simplicial sets $\Omega^\bullet_{poly}(S \times T) \simeq \Omega^\bullet_{poly}(S) \otimes \Omega^\bullet_{poly}(T)$, but that’s only a quasi-iso.
So all there is with this is that the simplicial hom-set
$Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A , B \otimes \Omega^\bullet_{poly}(\Delta[n])))$has the right connected components when $A$ is cofibrant, in that
$Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(A,B) \,.$But I am not sure what the remaining homotopy type of $Alg_P(A,B)$ is doing.
But I am not sure what the remaining homotopy type of $Alg_P(A,B)$ 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 $A$ the simplicial dg-algebra
$s A : [n] \mapsto A \otimes_k \Omega^\bullet_{poly}(\Delta^n)$is a simplicial resolution – a right framing in the terminology of Hovey’s book .
The polynomial forms are acyclic, so $const A \to s A$ is a weak equivalence.
It remains to show that $s A$ is Reedy fibrant. If I see correctly, its matching object is
$(match s A)_r = A \otimes \Omega^\bullet_{poly}(\partial \Delta^r) \,.$So we need to show that $(s A)_r \to (match s A)$ hence $\Omega^\bullet_{poly}(\partial \Delta[r] \hookrightarrow \Delta[r])$ is a fibration in $cdgAlg_k$.
But this follows by the the fact that we have the standard Quillen adjunction of rational homotopy theory $\Omega^\bullet_{poly} : sSet \to dgcAlg_k^{op}$ (for $k$ of characteristic 0) which means in particular that $\Omega^\bullet_{poly}$ sends monomorphisms of simplicial sets to surjections of dg-algebras.
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.
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