Simplicial enrichment on unbounded algebras over an operad

Question

In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get too technical, we can essentially assume that we have algebras in complexes over a ring containing the rationals.

There is, however, a difficulty. That is, this enrichment only really works for finite simplicial sets, i.e., the isomorphism (in Lemma 4.8.3) $$ \underline{\textrm{Hom}}(A,\Omega(W)\otimes B)\cong \textrm{sSet}(W,\underline{\textrm{Hom}}(A,B)) $$ holds only provided that the simplicial set $W$ is finite. Here $\Omega$ denotes the usual PL-de Rham algebra.

Is there a conceptual obstruction to promoting this structure to a bona fide simplicial enrichment?

Since I am really only interested in the commutative operad, the answer for this case will already be a great help. Also, if there are some other references for the unbounded case, this would also be really great.

Answer 1

There is no obstruction. If $M$ is a simplicial monoidal model category, and $O$ is an operad in $M$, then the category of $O$-algebras is simplicially enriched, tensored, and cotensored. If it's a model category then it's a simplicial model category. For $O = Com$, a reference is page 5 of my paper on commutative monoids, though this was known long before. For a general operad $O$, this is already in Spitzweck's thesis. A published reference is page 4 of this paper by Gutierrez et al.

So the next question is: if $M$ is the category of (bounded or unbounded) chain complexes of $R$-modules, with the projective model structure, is $M$ a simplicial model category? And the answer is yes. I learned this while writing a paper with Boris Chorny (see Sec 4.2 of this). It's spelled out in this paper.