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1. • CommentRowNumber1.
   • CommentAuthorBruno Stonek
   • CommentTimeDec 1st 2021
   • PermaLink
   Author: Bruno Stonek
   Format: MarkdownItexThere 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 $\Sigma^{p-q-1}\mathbb{S} \to X \to \hat{X} \to \Sigma^{p-q}\mathbb{S}$ 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!

   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 $\Sigma^{p-q-1}\mathbb{S} \to X \to \hat{X} \to \Sigma^{p-q}\mathbb{S}$ 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!

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI think the answer is: The 1-categorical pushout diagram with the disk is a model for the $\infty$-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.

   I think the answer is:

   The 1-categorical pushout diagram with the disk is a model for the $\infty$-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.

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeDec 1st 2021
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   Author: Hurkyl
   Format: MarkdownItexThe surrounding context looks like it's talking about the model category rather than the $\infty$-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.

   The surrounding context looks like it’s talking about the model category rather than the $\infty$-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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's what I mean (saying "$\infty$-" instead of "homotopy-"). This is what the [Rem. 2.9](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra) in question is referring to in parenthesis: where it says "hence its homotopy cofiber ([def.](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1#StrictModelStructureCellAttachmentToSpectra))". Here "def." is [def. 4.16](https://ncatlab.org/nlab/show/Introduction+to+Homotopy+Theory#HomotopyFiber) in Part I of the notes. More on this is in [Prop. 3.1](https://ncatlab.org/nlab/show/homotopy+pullback#HomotopyPullbackByOrdinaryPullback) 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.

   Yes, that’s what I mean (saying “$\infty$-” 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.

5. • CommentRowNumber5.
   • CommentAuthorBruno Stonek
   • CommentTimeDec 2nd 2021
   • (edited Dec 2nd 2021)
   • PermaLink
   Author: Bruno Stonek
   Format: MarkdownItexThanks 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe justification as such is already given, as pointed out in #4. What you seem to want is more prose to go with it.

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

7. • CommentRowNumber7.
   • CommentAuthorBruno Stonek
   • CommentTimeDec 2nd 2021
   • PermaLink
   Author: Bruno Stonek
   Format: MarkdownItexThat'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 $\mathbb{S}^{p-q-1}\vee \mathbb{S}^{-q}$? The second summand appears because of the added disjoint basepoint: $\Sigma^{\infty}_+S^{p-1}\simeq \Sigma^\infty(S^{p-1}\vee S^0)$.

   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 $\mathbb{S}^{p-q-1}\vee \mathbb{S}^{-q}$? The second summand appears because of the added disjoint basepoint: $\Sigma^{\infty}_+S^{p-1}\simeq \Sigma^\infty(S^{p-1}\vee S^0)$.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2021
   • (edited Dec 2nd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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 $\mathbb{S}^{-q}$. So that pushout square on the left is really the one we have been discussing, wedged with the cofiber square of the identity on $\mathbb{S}^{-q}$. Okay, I'll look into clarifying that remark a little more. When I find a minute. Thanks for bringing this up!

   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 $\mathbb{S}^{-q}$. So that pushout square on the left is really the one we have been discussing, wedged with the cofiber square of the identity on $\mathbb{S}^{-q}$.

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

9. • CommentRowNumber9.
   • CommentAuthorBruno Stonek
   • CommentTimeDec 2nd 2021
   • (edited Dec 2nd 2021)
   • PermaLink
   Author: Bruno Stonek
   Format: MarkdownItexAh 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.

   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.