# Free extension of algebra for an operad

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.

• 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. – Fernando Muro Jan 17 at 23:39
• 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. – Simon Henry Jan 18 at 0:16