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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.

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    $\begingroup$ Yes, I'd say this is classical by now and goes back to the thesis of Spitzweck. Under some extra assumptions (e.g. monoid axiom) you don't need X to be cofibrant. With only your hypotheses, have a look at Chapter 13 of Fresse, Benoit. Modules over Operads and Functors. Vol. 1967. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009. ams.org/mathscinet-getitem?mr=2494775. $\endgroup$ – Fernando Muro Jan 17 at 23:39
  • $\begingroup$ I couldn't find it as such in Spitzweck's thesis, that's why I asked. But I forgot about Fresse's book, I'll go check there. $\endgroup$ – Simon Henry Jan 18 at 0:16
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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.

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    $\begingroup$ Thanks, I think that's exactly what I nedded ! "This results is true" is maybe a bit exaggerated though : the condition in your paper seems quite different from the guess I made. (So, unless I'm missing something I was probably wrong). $\endgroup$ – Simon Henry Jan 18 at 3:09

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