I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable.


If $X$ is a monoid in $C$, and $A \to B$ is a map in $C$, I'm calling free extension of $X$ by $A \to B$ (along some map $A \to X$) the monoid object $X \coprod\_{F(A)} F(B)$ where $F(A)$ and $F(B)$ are the free monoids on $A$ and $B$ and the pushout is a pushout of monoid object (using some arbitrary map $A \to X$).


It can be shown that if $A \to B$ is a (trivial) cofibration and $X$ is a cofibrant as an object of $C$, then the map $X \to X \coprod\_{F(A)} F(B)$ is again a (trivial) cofibration in $C$.


The reason for this is that the map $X \to X \coprod\_{F(A)} F(B)$ can be written as an $\omega$-composition of maps that are obtained by taking iterated "pushout-product" of $A \to B$ with itself and tensor product with $X$ (so in particular, are all cofibrations/trivial cofibrations as soon as $A \to B$ is).


What I wonder about is whether this can be generalized when "monoid" is replaced by $\mathcal{O}$-algebras for a general operad $\mathcal{O}$.


My guess after some computation is that something like the above can be done when $\mathcal{O}$ is $\Sigma$-cofibrant (i.e. cofibrant for the projective model structure on collections) and when $X$ is cofibrant for the projective model structure on $\mathcal{O}(1)$-object (assuming $X$ cofibrant in $C$ do not seem enough in general)


But I feel this might have been treated somewhere or that there might be other set of conditions under which something like this can be done.



Free extension of algebra for an operad
This result is true and is due to Spitzweck, Berger–Moerdijk, Fresse, and Elmendorf–Mandell.
A complete set of references can be found around Proposition 5.7
in the paper <https://arxiv.org/abs/1410.5675>.


