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In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of S-modules as a model for the stable homotopy category. The category of S-modules is a closed symmetric monoidal model category whose monoidal product descends to the usual product on the stable homotopy category. Its unit is the sphere spectrum. All objects are fibrant but the unit is not cofibrant. Every operad O is admissible, meaning the category of O-algebras has a transferred model structure where a morphism $f$ of O-algebras is a weak equivalence (resp. fibration) if and only if $U(f)$ is in S-modules. A reference is Proposition 1.5 in "Moduli Spaces of Commutative Ring Spectra" by Goerss and Hopkins. The proof uses that S-modules have a structured interval object.

For the operad $O = Ass$, Theorem VII.6.2 in EKMM proves that if $A$ is a cofibrant S-algebra then the unit $S \to A$ is a cofibration of $S$-modules. Just after, they write "In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups."

The property that I want, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. A related notion, studied by Pavlov and Scholbach, is to say that an operad $O$ is strongly admissible, meaning that, if $a: A\to A'$ is a cofibration of $O$-algebras with $A$ cofibrant, then $U(a)$ is a cofibration and $U(A)$ is cofibrant. So this implies what I want, and even more.

Unfortunately, as I'll point out below, in the category of S-modules with the EKMM model structure, the commutative monoid operad does not have the property I want. S is a cofibrant commutative monoid that is not cofibrant as an S-module.

There might still be a hope that $\Sigma$-cofibrant operads are strongly admissible, but normally proving this requires the unit to be cofibrant. In particular, I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then we know that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

(1) Can we tweak the EKMM model structure on S-modules in some way so that cofibrant commutative monoids forget to cofibrant S-modules?

(2) Can we do the same for some class of operads? Offhand, I don't even know that it's true for the class of $\Sigma$-cofibrant operads, which should be the simplest situation to check.

In 1997, Elmendorf, Kriz, Mandell, and May wrote a book Rings, Modules, and Algebras in Stable Homotopy Theory in which they introduced the category of $S$-modules as a model for the stable homotopy category. The category of $S$-modules is a closed symmetric monoidal model category whose monoidal product descends to the usual product on the stable homotopy category. Its unit is the sphere spectrum. All objects are fibrant but the unit is not cofibrant. Every operad $O$ is admissible, meaning the category of $O$-algebras has a transferred model structure where a morphism $f$ of $O$-algebras is a weak equivalence (resp. fibration) if and only if $U(f)$ is in $S$-modules. A reference is Proposition 1.5 in "Moduli Spaces of Commutative Ring Spectra" by Goerss and Hopkins. The proof uses that $S$-modules have a structured interval object.

For the operad $O = Ass$, Theorem VII.6.2 in EKMM proves that if $A$ is a cofibrant $S$-algebra then the unit $S \to A$ is a cofibration of $S$-modules. Just after, they write "In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups."

The property that I want, that a cofibrant $O$-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. A related notion, studied by Pavlov and Scholbach, is to say that an operad $O$ is strongly admissible, meaning that, if $a: A\to A'$ is a cofibration of $O$-algebras with $A$ cofibrant, then $U(a)$ is a cofibration and $U(A)$ is cofibrant. So this implies what I want, and even more.

Unfortunately, in the category of $S$-modules with the EKMM model structure, the commutative monoid operad does not have the property I want. $S$ is a cofibrant commutative monoid that is not cofibrant as an $S$-module. We have the same problem with $O = Ass$. Interestingly, because $S$ is not cofibrant in $S$-mod, the associative operad is not $\Sigma$-cofibrant, as $Ass(n)$ is a coproduct of copies of $S$, one for each $\sigma$ in the symmetric group $\Sigma_n$.

There might still be a hope that $\Sigma$-cofibrant operads are strongly admissible, but normally proving this requires the unit to be cofibrant. In particular, I don't know whether $S$-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then we know that cofibrant $O$-algebras forget to cofibrant $S$-modules for operads $O$ whose spaces $O(n)$ are cofibrant.

(1) Can we tweak the EKMM model structure on $S$-modules in some way so that cofibrant commutative monoids forget to cofibrant $S$-modules?

(2) Can we do the same for some class of operads? Maybe in the case of $\Sigma$-cofibrant operads, we don't have to tweak the EKMM model structure at all. If so, I'd love to know.

In modern terminology, this result proves that the operad $O = Ass$ is strongly admissible, a notion studied by Pavlov and Scholbach.

The property that I want, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. Unfortunately, to get from "O is strongly admissible" to "cofibrant O-algebras forget to cofibrant underlying objects" Pavlov and Scholbach must assume the unit of the base model category $M$ is cofibrant, so this doesn't help with S-modules. I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then the latter reference proves that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

It is worth remarking that one really does need to tweak the EKMM model structure to get a positive answer to (1), because S is a cofibrant commutative monoid that is not cofibrant as an S-module.

(2) Can we do the same for some class of operads? Offhand, I don't even know that it's true for the class of $\Sigma$-cofibrant operads.

The property that I want, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. A related notion, studied by Pavlov and Scholbach, is to say that an operad $O$ is strongly admissible, meaning that, if $a: A\to A'$ is a cofibration of $O$-algebras with $A$ cofibrant, then $U(a)$ is a cofibration and $U(A)$ is cofibrant. So this implies what I want, and even more.

Unfortunately, as I'll point out below, in the category of S-modules with the EKMM model structure, the commutative monoid operad does not have the property I want. S is a cofibrant commutative monoid that is not cofibrant as an S-module.

There might still be a hope that $\Sigma$-cofibrant operads are strongly admissible, but normally proving this requires the unit to be cofibrant. In particular, I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then we know that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

(2) Can we do the same for some class of operads? Offhand, I don't even know that it's true for the class of $\Sigma$-cofibrant operads, which should be the simplest situation to check.

For the operad $O = Ass$, a cofibrant algebra is cofibrant as an S-module, by Theorem VII.6.2 in EKMM. Just after, they write "In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups."

This property, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. A related concept is that a cofibration of O-algebras with cofibrant source forgets to a cofibration of spectra. This property is studied by Pavlov and Scholbach, and they say O is strongly admissible when it's satisfied. But, to get from "strongly admissible" to "cofibrant O-algebras forget to cofibrant underlying objects" they assume the unit of the base model category $M$ is cofibrant, so this doesn't help with S-modules. I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then the latter reference proves that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

For the operad $O = Ass$, Theorem VII.6.2 in EKMM proves that if $A$ is a cofibrant S-algebra then the unit $S \to A$ is a cofibration of $S$-modules. Just after, they write "In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups."

In modern terminology, this result proves that the operad $O = Ass$ is strongly admissible, a notion studied by Pavlov and Scholbach.

The property that I want, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. Unfortunately, to get from "O is strongly admissible" to "cofibrant O-algebras forget to cofibrant underlying objects" Pavlov and Scholbach must assume the unit of the base model category $M$ is cofibrant, so this doesn't help with S-modules. I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of Bousfield Localization and Algebras over Colored Operads. If they do, then the latter reference proves that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

It is worth remarking that one really does need to tweak the EKMM model structure to get a positive answer to (1), because S is a cofibrant commutative monoid that is not cofibrant as an S-module.