9
$\begingroup$

The paper I'm referring to can be found here, and is absolutely seminal work in area of homotopy theory for operads. It came out in 2003. I remember someone once told me there was a mistake somewhere in this paper, but I can't remember where.

Is there some error in this paper? Is it a big deal?

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?

Improve this question
$\endgroup$
4
  • $\begingroup$ 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. $(S^1)^{\times 2}/\Sigma_2 \simeq S^1 \vee S^1$. $\endgroup$
    – Ricardo Andrade
    Commented May 4, 2013 at 0:55
  • $\begingroup$ @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. $\endgroup$
    – David White
    Commented May 4, 2013 at 1:21
  • $\begingroup$ @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. $\endgroup$
    – Ricardo Andrade
    Commented May 4, 2013 at 4:06
  • 2
    $\begingroup$ 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. $\endgroup$
    – Justin Young
    Commented May 6, 2013 at 11:51

1 Answer 1

Reset to default
10
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Also, welcome to MathOverflow! I guess I'll be seeing you quite soon in Barcelona. Can't wait! $\endgroup$
    – David White
    Commented May 6, 2013 at 14:22
  • $\begingroup$ 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! $\endgroup$
    – Javier J. Gutiérrez
    Commented May 7, 2013 at 14:30

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .