I have included a graphics showing the 3d commuting cube pasting diagram that proves the octahedral axiom from the pasting law of homotopy pushouts. here.

]]>I have added statement and proof that homotopy cofiber sequences of spectra coincide with homotopy fiber sequences (here).

(Previously that section only had the statement that the “wrong way” long sequences exist, but not yet that they actually coincide with the respective dual sequences.)

]]>I have now expanded also the section *triangulated structure*. Added a paragraph on the single non-trivial step in the proof of the octahedral axiom in the stable model category.

I discovered a nice note by Andrew Hubery (here) in which a whole bunch of different formulations of the octahedral axiom are proven to be equivalent. One of them (“axiom B” in the note) manifestly axiomatizes just the existence of homotopy pushouts. That is really what one uses, explicitly or implicitly, when proving the octahedral axiom in a stable model category: homotopy pushouts and their pasting law.

]]>As anticipated in #15, after switching to the proof via Bousfield-Friedlander, I have re-installed a much simplified version of the previous proof, now as just a proof of cofibrant generation (here).

I have also expanded various of the little proofs in the section that proves the additivity of the stable homotopy category, here.

]]>I have expanded the section proving the stability of the stable model structure (here). (The previous version didn’t actually prove the isomorphism between standard and “fake” suspension in the stable homotopy category, it just gave the idea.)

]]>I have typed in the remaining lemma and its proof for the statement in #15: Omega-spectrification of topological sequential spectra presrves homotopy pullback squares (here).

]]>Due to the issue mentioned in another thread here, for the moment I have removed the writeup of the proof of the stable model structure on topological sequential spectra that followed MMSS 00.

Instead I have replaced it now by writeup of the proof that uses the Bousfield-Friedlander theorem applied to the (correct) Omega-spectrification functor. This is now in the section

Of course, with that model structure in hand, we immediately prove that otherwise problematic step in the other proof (the “issue” from above). So maybe I’ll re-instantiate the previous proof after all, not as a proof of the model structure as such, but as a proof that it is cofibrantly generated.

]]>I have added statement and proof of the triangulated structure on the stable homotopy category (here).

Then I added statement and proof of the long-in-both-directions homotopy (co-)fiber sequences of spectra, as a general statement from the triangulated category axioms here.

Much of the latter should be moved over to *triangulated category* and only the specialization corollary kept at *model structure on topological sequential spectra*. I’ll do that reorganization later, after adding a bit more accompanying text to the lemmas here and there. But not today.

I have now spelled out the proof, carefully, that the stable homotopy category is additive, here.

]]>In the section proving the stability of the stable model structure (here) I have spelled out the details of the adjunction between ordinary sequential spectra and “$S^2$-spectra” that enter the main proof. (The statement which was found surprising in #6.)

In fact one doesn’t need the full Quillen equivalence here. I gave a more lightweight proof. I am trying to see how far one gets with just the implication

$stable \; weak \; homotopy \; equivalence \;\;\; \Rightarrow \;\;\; stable \; equivalence$without using its converse (which fails for symmetric spectra). And to prove that the “alternative suspension” (which lends itself to proof of stability) is isomorphic in the stable homotopy category to the suspension induced by the standard cylinder, all one needs is the implication this way.

]]>I have added statement and proof here that if a morphism of topological sequential spectra is a stable weak homotopy equivalences then it also is a stable equivalence (in that it induces isos by homming it into Omega-spectra in the homotopy category of the level model structure).

I adapted the general proof from *Model categories of diagram spectra*, but breaking it down to the simple special case of sequential spectra and avoiding discussion of mapping spectra out of free spectra in favor of the familiar $\underset{\longrightarrow}{\lim}_n \Omega^n X[n]$-construction that it comes down to in the sequential case.

Is there a quick argument, not using higher category theoretic technology, that this is indeed the universal stabilization?

]]>I have expanded the discussion of the stability of the model structure by spelling out full proof that not only is it stable, but it is indeed a stabilization of the model structure on topological spaces, culiminating in this corollary, which states that there is a diagram of Quillen adjunctions

$\array{ (Top_{cg}^{\ast/})_{Quillen} & \underoverset{\underoverset{\Omega}{\bot}{\longrightarrow}}{\overset{\Sigma}{\longleftarrow}}{} & (Top^{\ast/}_{cg})_{Quillen} \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ SeqSpec(Top_{cg})_{strict} & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & SeqSpec(Top_{cg})_{strict} \\ {}^{\mathllap{id}}\downarrow \dashv \uparrow^{\mathrlap{id}} && {}^{\mathllap{id}}\downarrow \dashv \uparrow^{\mathrlap{id}} \\ SeqSpec(Top_{cg})_{stable} & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq_{\mathrlap{Q}}} & SeqSpec(Top_{cg})_{stable} } \,,$whose induced diagram of derived functors is of the form

$\array{ Ho(Top^{\ast/}) & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\bot} & Ho(Top^{\ast/}) \\ {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} && {}^{\mathllap{\Sigma^\infty}}\downarrow \dashv \uparrow^{\mathrlap{\Omega^\infty}} \\ Ho(Spectra) & \underoverset {\underset{\Omega}{\longrightarrow}} {\overset{\Sigma}{\longleftarrow}} {\simeq} & Ho(Spectra) }$with top and bottom the correct canonically defined adjunctions.

]]>Oh, OK. I thought it was something to do with even spectra, like $ku$.

]]>Ah, no, that shouldn’t be surprising. In different guise this is an ancient fact. For instance complex Thom spectra all appear as $S^2$-spectra and are then turned into $S^1$-spectra. The Quillen equivalence comes down to the fact that if you take an $S^1$-spectrum first to its underlying $S^2$-spectrum and then go back to an $S^1$-spectrum this way, then the canonical comparison map is clearly an iso on stable homotopy groups, roughly because if we “eventually” reach a stable element, it does not matter whether we do so in steps of one or steps of two. Anyway, a proof of that equivalence (in more generality) is given in that section 10.4 of *Local homotopy theory*.

The Quillen equivalence in your second para in #3

]]>That’s quite surprising!

You may need to help me here: what is quite surprising?

]]>That’s quite surprising! I figured with this whole course you were after some particular calculation, but haven’t yet got a good guess as to what it might be (aside from what we kinda discussed at IHES, the M5 charges).

]]>I have now spelled out the proof (here) that $SeqSpec(Top)_{stable}$ is indeed a stable model category.

The only step that I didn’t make fully explicit yet is the claim at one point that there is a Quillen equivalence between the stable model structure for ordinary sequential spectra with structure maps $S^1 \wedge X_n \to X_{n+1}$ and that of sequential spectra on “even graded sequences” with structure maps $S^2 \wedge X'n \to X'_{n+1}$.

(I have been following in outline the discussion in section 10.4 *Local homotopy theory*, trying to strip away all that is not necessary for just this statement. And I may be wrong but right now I feel like I also filled in some small things. For instance it seems to me right now that prop. 10.53 there shouldn’t claim a stable equivalence, but a zig-zag of those. In any case, the “proof” text after 10.53 seems to have been chopped off or left over from a copy-and-pasting operation. Anyway, staring at it it becomes clear what the idea is.)

I have spelled out a proof (here) that the “fake suspension” on sequential spectra descends to an equivalence of the stable homotopy category with itself.

]]>I have created an entry *model structure on topological sequential spectra*.

In parts this directly parallels the entry *Bousfield-Friedlander model structure*.

But now I have spelled out full proof of the model structure and its cofibrant generation: here

I did this by taking the more general proof that I had earlier spelled out at *Model categories of diagram spectra*, and specializing it to the case of sequential spectra.

The effect of that is that those tedious technical lemmas about the maps of free spectra collapse to something simple, with the result that the actual proof may start right away with less preliminaries, which makes the writeup a bit more transparent. On the other hand, the neat thing is that apart from that analysis of the free spectra the proof is verbatim the same now for all cases (sequential, symmetric, orthogonal spectra and pre-excisive functors), so in the other entries it’ll be possible to turn this around and say: “after this analysis of the free symmetric/orthogonal spectra the proof of their model structure now follows verbatim as at model structure for topological sequential spectra”.

As far as exposition and writeup goes, the only remaining “gap” I left is that at one point the proof invokes that $Top_{Quillen}^{\ast/}$ and hence $[StdSpheres, Top_{Quillen}^{\ast/}]_{proj}$ is a topological model structure (this is used in the proof of this lemma ). I plan to spell that out, too. But not tonight.

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