Reedy cofibrancy of the bar construction for algebras

Let $O$ be a $C$-colored operad taking values in a model category $M$ (that may very well have more nice properties if needed; think for the moment of (bounded) cochain complexes over a field). We have the forgetful functor $U : \text{Alg}(O) \to M^C$ from algebras over $O$ to $C$-colored objects in $M$, i.e. functors from the set $C$, seen as discrete category, to $M$. This functor has a left adjoint, the free algebra functor $F: M^C \to \text{Alg}(O)$.

I want the operad $O$ to be admissible, i.e. the adjunction $F\dashv U$ can be used to transfer the model structure from $M^C = \prod_C M$ to $\text{Alg}(O)$. Then weak equivalences and fibrations are defined color-wise, cofibrations are determined by that.

Now the comonad $FU$ in $\text{Alg}(O)$ gives rise to the bar construction, see ncatlab.org/nlab/show/bar+construction

To any $A \in \text{Alg}(O)$ we can associate the corresponding bar construction $\text{Bar}^O A$. This is actually an augmented simplicial object in $O$-algebras, but for the moment I just want to look at the simplicial object.

My question is: Is $\text{Bar}^O A$ Reedy cofibrant (cofibrant in the model category of simplicial $O$-algebras equipped with the Reedy model structure, see Goerss/Jardine's book for example)? Or more precisely, what are requirements on $O$ and $M$ ensuring that this is the case?

Some thoughts on this: To answer this question we have to prove that the latching maps $L_n \text{Bar}^O A \to \text{Bar}^O_n A$ are cofibrations in $O$-algebras.

For $n=0$ this map is $L_0 \text{Bar}^O A = \emptyset \to FA$, which is a cofibration because $F$ is left Quillen. For $n=1$ we get the map $L_1 \text{Bar}^O A = FA \xrightarrow{F(\eta_A)} F^2 A$, where $\eta_A : A \to FA$ is the unit. Again, this is a cofibration. How about the general case? If I use the description of the latching objects in terms of coequalizers as in Goerss/Jardine, I run into problems (already for $n=2$).

Thank you for any comments or pointers to the literature.

• Proposition IX.2.7 of EKMM (Elmendorf-Kriz-Mandell-May) is related to a particular instance of what you're asking for, I think. – Bruno Stonek Jan 4 '18 at 9:24
• Ok, I will try to understand this special case to maybe extract the idea of how to approach the general case. Thank you. – Lukas Woike Jan 4 '18 at 9:52

Check out Pereira's paper Cofibrancy of operadic constructions in positive symmetric spectra. Theorem 1.4 answers your question when $M$ is the positive S-stable model structure on symmetric spectra (generalizing an earlier result of Harper-Hess). The proof proceeds via an analysis of latching objects, and can be generalized to other model categories.

I remember thinking of similar things in my paper with Batanin Left Bousfield Localization and Eilenberg-Moore Categories, in the setting of a general model category $M$. We didn't need the bar construction to be Reedy cofibrant, but we needed something similar in Definition 5.11. I remember analyzing latching spaces in much the same way that Pereira does, when proving that $\Sigma$-cofibrant operads are pointwise Reedy cofibrant. I'll bet you could follow this model to prove that $\Sigma$-cofibrant operads have the property you want.

• If $\Sigma$-cofibrant operads have this desired property, I would already be happy. Thanks you for your answer. It will take me some time to go through these papers, but it sounds promising. – Lukas Woike Jan 4 '18 at 14:16