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.

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

| cite | improve this answer | |
  • $\begingroup$ 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. $\endgroup$ – Lukas Woike Jan 4 '18 at 14:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.