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Is the Strøm model category left proper? I know that pushout along cofibrations of homotopy equivalences of the form are again homotopy equivalences. (e.g. Hatcher 0.17) Maybe the proof directly generalizes, haven’t checked.
Hi Urs, all objects are cofibrant, so yes :-).
Ah, right. Thanks.
Just a quick note that May and Sigurdsson’s proof is basically completely invalid (as has been acknowledged by May), so I’d suggest either to add a cautionary note, or remove the link completely. Riehl and Barthel’s paper is not really a review, it does the hard work of actually giving a proof! Looking at the history, Mike I think added that link to May and Sigurdsson’s work way back in 2010, before the erroneous nature of May and Sigurdsson’s proof was pointed out.
Hi Richard, it sounds like you’re the one best in position to make corresponding changes to the entry. Or do you want me to do it?
Thanks!
I have touched the text a little, adding mentioning of your name, since I see now from p. 2 in Barthel & Riehl that you are the one who spotted the issue.
Which issue, actually? It remains a little mysterious. Are details of the problems in the published articles discussed anywhere? Are there errata?
What’s the scope of the issue within “Parametrized Homotopy Theory”? Is it just a proof that is wrong but fixable, or does there remain a wrong statement?
Barthel and Riehl discuss Cole’s mistake in §6.1 of their paper.
Thanks. So I have expanded the pointer re the mistake to:
see Barthel & Riehl, p. 2 and Rem 5.12 and Sec. 6.1
So this gives an idea of the technical origin of the mistake, but what’s it’s global impact? Does Barthel & Riehl’s fix salvage all the articles (and books) that are affected?
If we can, we should add a comment on this. Currently the entry ends with claiming a major problem in a series of published articles and books, leaving the reader to wonder if these can safely be cited at all. If the claim is that there is not erroneous propositions left in this published liteature, just an erroneous argument that has meanwhile been fixed, then we should make that explicit in the entry.
Does Barthel & Riehl’s fix salvage all the articles (and books) that are affected?
They provide a proof for Cole’s conjecture (replacing his incorrect argument), so I would answer this question in the affirmative.
Thanks. If that’s the case, the last paragraph (here) should be adjusted, since it gives a different impression.
I would say that one should simply refer to and cite the work of Barthel and Riehl for what is true. Cole does not make a conjecture as such, he tries to carry out a certain construction, which does not quite work and cannot work. Barthel and Riehl adjust the construction slightly, but it is not some trivial modification (they are quite modest in how they present things in the paper); Tobias worked hard to get things to work, enlisting Emily’s help for the algebraic factorisation framework (some other people were involved in discussions along the way, including Bill Richter). For the general results, the hypotheses of Barthel and Riehl are not exactly the same as those in the papers of May et al and Cole. For the classical case (Strøm model structure on topological spaces), there is nothing wrong with Strøm’s original proof, so of course the statement is fine there, it is only a question of whether the new proof works, which is not the case for Cole/May et al’s argument, and is the case for Barthel and Riehl’s argument.
Okay, I have reworded somewhat to tone it down, not to make it sound like a claim that a list of articles in the published literature remain wrong even with the Barthel-Riehl patch applied. On the other hand, I don’t know if it’s true, maybe there are issues left, but I can’t tell for sure from this conversation: What you, Richard said can be interpreted in both ways (you seem to make strong such claims, but now you point me to Barthel & Riehl, who don’t make such strong claims, even if one might sense them between the lines, but who knows).
I just think if it’s true that there are problems left in the literature, then, while it’s most worthwhile to point them out, it needs to be claimed unambiguously, as its not a small claim. For instance, May & Ponto’s “More concise…” does not cite Cole’s “Many homotopy categories…” and so if the entry wants to alert the reader that there is a problem in May & Ponto even in view of Barthel & Riehl (and yes, if so that would be a service to the community to say it) then a tad of further explanation needs to be added.
What you have after #15 looks good to me. The problems in May and Ponto are in Chapter 17, specifically the proof of Lemma 17.1.7 I think.
Apologies if what I wrote is unclear, I am not sure how to make it clearer: one should trust Barthel and Riehl, and if one wishes to use some result from one of the materials co-authored by May (other than the existence of the Strøm model structure, which is certainly fine from Strøm’s original work), then one should check that the hypotheses of Barthel and Riehl apply. I don’t see this as a strong claim, just a statement of fact (which May himself has acknowledged, though I don’t know why no erratum has been written; he referred to the mistake once on Math Overflow, years ago, as an error ’almost no reader will spot’, though I can’t find that comment now).
Okay, thanks.
In #14 you said that “the hypotheses are different”, which I take as saying that statements in May’s articles on this may actually be wrong (at least remain unproven) unless some fine-print is adjusted?
Concretely: the sliced/parameterized version of the model structure in May-Sigurdsson: Does it have a proof? Does Barthel-Riehl’s patch apply? If so, do definitions/hypotheses need to be adjusted first?
(I am not saying you need to know or say all this, but just to indicate which questions I am left with after this conversation.)
In #14 you said that “the hypotheses are different”, which I take as saying that statements in May’s articles on this may actually be wrong (at least remain unproven) unless some fine-print is adjusted?
The relevant part of the article of Barthel and Riehl would be Definition 5.16, which discusses the difference. I actually don’t know to what extent the model structures in May-Sigurdsson are valid; this was not the kind of thing I was interested in at the time. But certainly one needs to be careful and actually establish that Barthel and Riehl’s ’monomorphism hypothesis’ holds; Barthel and Riehl discuss it in some examples in Examples 5.18 and 5.19, but it is reasonably non-trivial, and the case of diagram spectra is only sketched. I’d assume most of the model structures are ultimately fine, though.
The problematic proof in Cole’s paper is that of Proposition 5.3.
Thanks! That’s useful. I have added pointer to that 5.3 to the entry.
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