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    • CommentRowNumber1.
    • CommentAuthorBruno Stonek
    • CommentTimeDec 1st 2021

    There is a sentence that is confusing me. I suspect it’s a mistake, or at least I don’t know how to make it work as stated. Right after Remark 2.9 here Introduction+to+Stable+homotopy+theory+–+1-1

    It is stated that Σpq1𝕊XˆXΣpq𝕊 is a homotopy cofiber sequence. However, the left square above that is not a homotopy pushout. You cannot replace that disk by a point, just as in the definition of a classical CW-complex you cannot replace the disks by points.

    I’m sorry if this is not the right place to write about this!

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2021

    I think the answer is:

    The 1-categorical pushout diagram with the disk is a model for the -categorical homotopy pushout with that disk replaced by a point.

    Does that answer the question? Otherwise, you may please have to say again what issue you see.

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeDec 1st 2021

    The surrounding context looks like it’s talking about the model category rather than the -category it presents, so I think that whole section should be interpreted as talking about the model category.

    The thing I think #1 overlooks is that if one leg of the span is a cofibration between cofibrant objects, and the third vertex is cofibrant, you can replace it with anything else that is weakly equivalent and also cofibrant.

    In particular, the homotopy cofiber of a cofibration between cofibrant objects can be given by taking the pushout with the point.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2021

    Yes, that’s what I mean (saying “-” instead of “homotopy-“). This is what the Rem. 2.9 in question is referring to in parenthesis: where it says “hence its homotopy cofiber (def.)”. Here “def.” is def. 4.16 in Part I of the notes. More on this is in Prop. 3.1 at homotopy pullback.

    If that is what #1 is asking about, let me know which kind of clarifying sentence you’d like to see added to the notes.

    • CommentRowNumber5.
    • CommentAuthorBruno Stonek
    • CommentTimeDec 2nd 2021
    • (edited Dec 2nd 2021)

    Thanks for the replies. Hurkyl, I agree that would work. A closely related, alternative justification: the stable model category of spectra is proper, so a pushout is a homotopy pushout as soon as one of the legs is a cofibration, which is obviously the case here.

    I don’t immediately see which of the two justifications would be more appropriate, or less circular.

    I would say that a justification along these lines would clarify the exposition.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2021

    The justification as such is already given, as pointed out in #4. What you seem to want is more prose to go with it.

    • CommentRowNumber7.
    • CommentAuthorBruno Stonek
    • CommentTimeDec 2nd 2021

    That’s true. I personally think a more verbose explanation would not be pointless: see my confusion in #1!

    Now I’m confused by something else there :) Shouldn’t the first term of the homotopy cofiber sequence actually be 𝕊pq1𝕊q? The second summand appears because of the added disjoint basepoint: Σ+Sp1Σ(Sp1S0).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2021
    • (edited Dec 2nd 2021)

    I personally think a more verbose explanation would not be pointless: see my confusion in #1!

    Yes, I understand, but I am trying to make you help me see how it should be expanded to be more helpful, since to my eyes all comments are already there. :-) That happens, of course, the author of a text sometimes can’t step back from it far enough. But anyway, I can try to expand. Later, when I have a quiet moment.

    Now I’m confused by something else there

    True, that’s more of a gap in the commentary: Namely the bottom left object is not actually a replacement of the point, either, but is itself a wedge sum of such with 𝕊q. So that pushout square on the left is really the one we have been discussing, wedged with the cofiber square of the identity on 𝕊q.

    Okay, I’ll look into clarifying that remark a little more. When I find a minute. Thanks for bringing this up!

    • CommentRowNumber9.
    • CommentAuthorBruno Stonek
    • CommentTimeDec 2nd 2021
    • (edited Dec 2nd 2021)

    Ah yes, of course, the bottom left object also has the +, so the left square doesn’t obviously present a homotopy cofiber sequence.

    Thanks for looking at it, looking forward to your edit.

    Oh, and as to how to make the previous part clearer: I think it’s in Hurkyl’s comment or mine above, i.e. explicitly remark that those pushouts are actually homotopy pushouts because such-and-such.