# Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads

The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors.

Is there some place the errors in this paper are enumerated?

An error in a different paper of the authors (but related to this paper) was pointed out and corrected here. Is this the only known error in the paper?

My interest is in building model structures on categories of algebras over an operad. For a long time I thought there weren't any errors in that part of the paper. However, today I realized that in Remark 4.2 they claim that Prop. 4.1 implies every topological operad $P$ is admissible, i.e. $P$-alg inherits a model structure. This makes me suspicious. It would say for one thing that commutative monoids in Top form a model category, i.e. it would give us a model structure on a subcategory of Top where every object is a product of Eilenberg-Maclane spaces. Does that work?

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I sincerely hope you get a good answer to your question. It happens all too often that errors are known, yet they are described nowhere. I just have a minor nitpick. Only a path connected commutative monoid in topological spaces need be weakly equivalent to a product of Eilenberg-MacLane spaces. For example, the free commutative monoid on a space $X$ is the disjoint union of the spaces $X^{\times n}/\Sigma_n$ for $n\in\mathbb{N}$. These quotients will rarely be equivalent to products of Eilenberg-MacLane spaces: e.g. $(S^1)^{\times 2}/\Sigma_2 \simeq S^1 \vee S^1$. –  Ricardo Andrade May 4 '13 at 0:55
@Ricardo: Thanks for your comment. I don't mind the nitpicking at all. Do you happen to know if this model structure exists? For some reason I had it in my brain that it was impossible, but now the evidence is pointing the other way. –  David White May 4 '13 at 1:21
@David: I do not know whether it exists. I am not very familiar with model structures on categories of algebras over operads. I have only read about model structures on categories of algebras over cofibrant operads. Unfortunately, the commutative monoid operad is not at all cofibrant. My only other idea was to try transferring the model structure on simplicial commutative monoids to the category of topological commutative monoids. However, I could not easily see whether the conditions for the existence of the transferred model structure were verified. –  Ricardo Andrade May 4 '13 at 4:06
By studying free algebras, one can prove using direct point set arguments, and knowledge of how to build pushouts in $\mathcal P$ algebras from pushouts in pointed spaces, to prove that the pushout in $\mathcal P$ algebras of the inclusion $\mathbb P(S^n_+) \to \mathbb P(D^{n+1}_+)$ along a map $\mathbb P(S^n_+) \to X$ gives a closed inclusion $X \to Y$. This is enough to get the small object argument working in pointed spaces, so one can do a transfer. –  Justin Young May 6 '13 at 11:51
Karol Szumilo and I have been corresponding about the issue of whether or not topological operads are admissible. I'll add an answer soon. For now, I point out that Theorem 12.2.C in Benoit Fresse's book requires an extra smallness condition in the transfer principle which Berger and Moerdijk never mention. Fresse mentions that this fails for Top, so topological operads don't form a model category (though you do get a semi model category). Because Berger-Moerdijk use the same transfer principle in their argument for admissibility, I'll bet the error is there. –  David White May 8 '13 at 0:27

The only errors that I am aware of in this paper are the following:

• A $G$ missing in the statement of Lemma 5.10 (pointed out by the authors after Lemma 2.5.3 in [1])

• A small mistake in the proof of Proposition 5.1 (also pointed out by the authors and fixed in the Appendix of [2])

[1] I. Moerdijk and C. Berger, The Boardman-Vogt resolution of operads in monoidal model categories, Topology 45 (2006), 807-849.

[2] I. Moerdijk and C. Berger, On the derived category of an algebra over an operad, Georgian Math. J. 16 (2009), 13-28.

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Thanks. I think the second mistake was the one pointed out to me. I knew it was fixed in some appendix but couldn't find it when I searched. Do you happen to know the situation for Commutative Monoids in Top? I feel like this should not form a model category, but Berger and Moerdijk clearly claim it does. If it doesn't, then that would throw doubt on Proposition 4.1 as well. Prop 4.1 is currently the only general theorem I'm aware of whose conclusion is that an operad is admissible. If there's doubt on it, that would be very sad. –  David White May 6 '13 at 14:22
Also, welcome to MathOverflow! I guess I'll be seeing you quite soon in Barcelona. Can't wait! –  David White May 6 '13 at 14:22
I have read the proof of Proposition 4.1 and it looks ok to me, but maybe I am missing something... I am also looking forward to seeing you in Barcelona! –  Javier J. Gutiérrez May 7 '13 at 14:30