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Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if and only if $U(f)$ is such in $M$. I am interested in the property:

(*) Every cofibrant $O$-algebra $A$ forgets to a cofibrant object $U(A)$ in $M$.

If $O$ is a $\Sigma$-cofibrant operad, this is automatic. If $O$ is $Com$, then (*) is true if $M$ satisfies the strong commutative monoid axiom. Now suppose $O(n)$ is cofibrant in $M$ for every $n$. This case is harder. It should be known for positive flat model structures, but I don't know a reference.

There are multiple choices for closed symmetric monoidal categories of spectra (most famously: S-modules, symmetric spectra, and orthogonal spectra), and many model structures in the latter two. In the standard stable model structure, the $Com$ operad is not admissible, because of a well-known obstruction due to Gaunce Lewis (because the unit is cofibrant). For this reason, Mandell, May, Schwede, and Shipley invented the "positive stable model structure" to make the unit not be cofibrant. In this model, a cofibration must be an isomorphism in level 0. In this model, there is a model structure on commutative ring spectra, but cofibrant commutative ring spectra need not forget to positive cofibrant spectra (See Prop 4.2 of Shipley's paper).

Shipley (following an idea of Jeff Smith) introduced a new model structure, nowadays commonly called the "positive flat stable model structure" by first enlarging the class of cofibrations of the stable model structure (to $S\otimes M$ where $S$ is the sphere spectrum and $M$ is the class of monomorphisms of symmetric sequences of simplicial sets), and then moving to the positive model structure by insisting cofibrations are isomorphisms in level 0. See her Theorem 3.2. Stolz did the same for orthogonal spectra and $G$-equivariant orthogonal spectra for a compact Lie group $G$. Again, commutative monoids have a transferred model structure, but now the forgetful functor takes a cofibrant commutative monoid to a positive flat cofibrant spectrum (Shipley Prop 4.1, Stolz Theorem 1.3.29). Stolz proved the same for the $G$-equivariant case, again with the positive flat stable model structure (but not for the complete model structure needed in Hill, Hopkins, Ravenel).

Pereira (Theorem 1.5) proved (*) for the positive flat stable model structure on symmetric spectra. Pavlov and Scholbach (4.4) extended this to colored operads. This begs a few questions:

  1. Is (*) true for $G$-equivariant symmetric spectra?

  2. Is (*) true for the positive flat stable model structure on orthogonal spectra?

  3. Is (*) true in some appropriate complete model structure on $G$-equivariant orthogonal spectra? Probably it would need to be positive, flat, and complete. Note, in the new book, Hill, Hopkins, and Ravenel write equifibrant instead of complete.

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  • $\begingroup$ Pereira claims at the bottom of page 5 of his paper that the answer to (1) is "yes". But the follow-up paper never appeared. I did write to Markus Hausmann to ask. $\endgroup$ Commented Jun 28, 2022 at 13:34

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I asked this question over a year ago, when revising my paper (joint with Donald Yau) Smith Ideals of Operadic Algebras in Monoidal Model Categories. In the end, I thought this question was relevant enough to write some evidence in favor of it, and to leave it as Problems 6.2.8, 6.2.9, 6.2.10 in the final, published version of that paper. I certainly hope someone will come along someday and solve those problems (and then for free they'll learn about Smith ideals in those categories). In writing these up as problems in that paper, I was following my own advice from this mathoverflow post, advice that I myself learned from Mark Hovey Clark Barwick.

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