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I have started a category:reference entry for the article Mandell, May, Schwede, Shipley: Model categories of diagram spectra. I feel that, maybe with the aid of hindsight, there is now room for a somewhat more concise and streamlined presentation of the material in there, for instance by separating basic category theory from the actual constructions a bit more, and I am in the process of typing that into the entry.
So far I worked on part I.
I’ll polish this a bit more, then I am going to feel inclined to copy this over to relevant sections in the entries on sequential-, symmetric- and orthogonal spectra, respectively, for completeness.
Now I have added to Part II – 5.-10. “Plain spectra” the main statement of the sequence of Quillen equivalences. No proofs yet.
I have been typing out the proof of the stable model structure (on the four kinds of diagram spectra in question) here. Mostly done, but not polished or proof-read yet. Need to interrupt now.
I have written out the proof that stable weak homotopy equivalences are stable equivalences (equivalences on generalized cohomology) here.
This and the entire proof of the model structure only needs that maps of free spectra $\lambda_n \colon F_{n+1} S^1 \longrightarrow F_{n} S^0$ co-representing the adjuncts of the structure maps of spectra exist (which follows just by Yoneda), not their actual form. The actual form of these maps is needed only for the analysis of the converse statement, that stable equivalences are stable weak homotopy equivalences (which of course holds except for the case of symmetric spectra). In the writeup I have tried to disentangle that, for clarity.
But what’s a really good way to see which form these maps $\lambda_n$ have? They are meant to be the (FreeSpectrum$\dashv$Component)-adjuncts of the canonical map $S^1 \to (F_n S^0)_{n+1}$. (Hovey-Shipley-Smith 00, remark 2.2.12, Mandell-May-Schwede-Shipley 00, lemma 8.5). I feel like I am missing why this is obvious.
I have completed typing out the proof (from MMSS 00, trying to streamline a little) of the stable model structure (on all four at once: sequential spectra, symmetric spectra, orthogonal spectra, excisive functors): at Proof of the stable model structure.
I decided to create a subsection titled Free spectra; then I moved all material related to free spectra to that section, to have it all in one place. Then I expanded further:
spelled out in full detail the general formula for the free spectra, and its derivation (of course just a big Yoneda-gymnastics, but since the fine detail of this formula matters crucially in the discussion of the subtleties of the stable equivalences, it seems worthwhile);
spelled out the full proof that those $\lambda_n$-maps between free spectra indeed co-represent the adjuncts of the structure maps of sequential spectra (the statement queried in #4 above). I won’t be surprised if you say you have a much shorter way of writing this down.
I have finally fully spelled out that curious proof that those $\lambda_n$-maps between free spectra are stable weak homotopy equivalences in all cases except that of symmetric spectra, here.
I am now spelling out the proof of the pushout-product axiom for the symmetric monoidal smash products of spectra. So far I have written out some lemmas and then the statement and proof that pushout-smash-product takes two cofibrations to a cofibration (here).
I was hoping I could see a way to condense a little the laborious proof by MMSS of the remaining statement that the pushout-product is acyclic if at least one of the inputs is, but I am not there yet.
For transparency, I have spelled out as an example of the general statement that structured spectra modeled on diagrams “$Dia$” are enriched functors
$\mathbb{S}_{Dia} Mod \simeq [ \mathbb{S}_{Dia} FreeMod, Top^{\ast/} ]$how in the special case of sequential spectra, we have an identification
$\mathbb{S}_{Seq}FreeMod^{op} \simeq StdSpheres$where $StdSpheres(S^{n_1}, S^{n_2}) \coloneqq im( S^{n_2-n_1} \to Top^{\ast}(S^{n_1}, S^{n_2}) )$, so that the general statement reduces to the fact (earlier stated e.g. by Lydakis98, prop. 4.2) that sequential spectra are equivalently enriched functors on $StdSpheres$.
I have fine tuned at Model categories of diagram spectra a bit more. For instance I made more explicit why the strict model structures there are cofibrantly generated by the sets $F I$ and $F J$ (here).
I had underappreciated the subtlety behind the small sentence in the proof of corollary 9.8 of MMSS 00
Since $p$ is acyclic, so is $F \to \ast$.
which is one step in establishing the stable model category of any of the flavors of structured spectra.
This sentience is meant to be true due to the existence (stated as Theorem 8.12 (vi) in MMSS00) of the long exact sequence $\cdots \to [Y,E]_{strict} \to [X,E]_{\strict} \to [F,E]_{strict} \to \cdots$ in the homotopy category of the strict (level) model structure for $E$ an structured $\Omega$-spectrum and $F$ the fiber of some level fibration $X \to Y$.
Now of course if we already had a stable model structure in hand, then this sequence would follow formally from the fibrancy conditions and the triangulation of the homotopy category etc., no problem.
But if we want to establish that model structure in the first place, we need to do something else. The proof offered in MMSS00 instead eventually points to section III.2 of LMS 86. There one finds part of the required argument given for sequential spectra. Unless I am missing something, completing and adapting the argument there to structured spectra (as needed for the argument in MMSS00) requires having constructed a handicrafted stable homotopy category of these structured spectra and having established a fair chunk of the pertinent properties.
I suppose that’s possible, but if this detour is needed to make complete the proof of the model structures in MMSS00, then it means that this proof is not suited as a way to introduce the stable homotopy theory of sequential spectra in the first place. As I had been thinking it would.
So I think I’ll have to abandon this route. Alas.
Interesting. Maybe worth an MO question?
Maybe I should do that, right.
I have added to the would-be proof of the relevant lemma here some more detailed comments as to which issue I see.
I understand that it will be hard for any reader here to jump right into the middle of this, but maybe my comments suffice to highlight the core of what’s going on. And maybe somebody sees how to resolve it?!
I don’t have time right now to think about it myself, but I would like to hear what “experts” have to say. Peter May sometimes answers questions on MO.
I have now checked with one author and with another expert. I am still waiting for reply from the main author, but so far I have this:
There are two opions to handle that step highlighted in #11
The first option is what I mentioned above:
Assume that we already know, by other means, that the homotopy category of our diagram spectra is going to be the localization of the strict model structure at the Omega-spectra, and that this localization is pre-triangulated. This means that there are long exact sequences $\cdots \to [B,E]_{strict} \overset{[p,E]_{strict}}{\to} [X,E]_{strict}\to [F,E]_{strict} \to \cdots$ and since $[p,E]_{strict}$ is an isomorphism by assumption on $p$, then $[F,E]_{strict} \simeq \ast \simeq [\ast,E]$, for all Omega-spectra $E$. Hence $F\to \ast$ is a stable weak equivalence.
The second option is:
Exclude symmetric spectra from the discussion. Then re-define the class of stable equivalences to be exactly the class of stable weak homotopy equivalences . Since for all the models except for symmetric spectra the elements in that generating set $K$ are stable weak equivalences, this does not change the conclusions of the other lemmas that go into the proof of the model structure. But now since filtered colimits of abelian groups are exact, it follows for a strict fibration $p$ that all its degreewise long exact sequences of homotopy groups give also degreewise long exact sequences of stable homotopy groups. Hence we have a long exact sequence $\cdots \to \pi_\bullet(F) \to \pi_\bullet(E) \overset{p_\ast}{\to} \pi_\bullet(B) \to \cdots$. Now since $p_\ast$ is an isomorphism, by the (new) assumption that $p$ is a stable weak homotopy equivalence, it follows that $\pi_\bullet(F) = 0$ and hence $F \to \ast$ is a stable weak homotopy equivalence. Which is the desired conclusion under the modified definition.
I have now received a reply from the main author. The idea is indeed to use the argument of lemma 2.3 and theorem 2.4 in chapter II of Lewis-May-Steinberger “Equivariant stable homotopy theory” (parts of the statement there also appears on p. 63 of “A concise course in algebraic topology”). It is claimed that constructing the homotopy commuting diagram there, for all of the flavors of diagram spectra, is still elementary.
Heh. Are you going to try it and see how elementary it is?
I am still waiting for a reply on this. Notice that in LMS there are various explicit continuous functions with explicit homotopies written down by hand for two of the squares. Now pass from sequential spectra to, say, continuous functors on finite CW-complexes. So then these functions and their homotopies need now be suitably equivariant under the full action of the $\infty$-category of finite homotopy types. I don’t see how this may be achieved by hand, without invoking some machinery.
But while I am still interested in what will be the answer on this point, it won’t help my original needs of using the proof as a slick introduction of the stable model category of diagram spectra.
On the other hand, before insisting that the article is meant as written, the main author had suggested a different route to go about it: it should be true, he suggested, that a stable equivalence which is also $K$-injective is already a stable weak homotopy equivalence (“$\pi_\ast$-isomorphism”). I don’t know the proof of this, but if this statement has a nice proof, then that would be my preferred way to proceed. Because in that lemma 9.8 we do have a $K$-injective stable equivalence, and if we may reduce to showing that its fiber has vanishing stable homotopy groups, then we do end up with a nice slick proof.
Oh, okay, I get. So to check that the morphism $\phi \colon Fib(f) \longrightarrow \Omega Cofib(f)$ (as in III.2 of LMS 86) of diagram spectra is a stable weak homotopy equivalence, it is sufficient to check that its image $U(\phi)$ under the forgetful functor to sequential spectra is such. That forgetful functor preserves looping and suspension, so we may apply the argument of LMS verbatim to $U(\phi)$ to obtain the desired conclusion. All right.
Hmm, okay. It’s a bit unsatisfying to have to use facts about sequential spectra to construct other kinds of spectra; it seems like they should each have an elementary independent development.
So there remain two options:
1) Check the suggestion that K-injective stable equivalences are $\pi_\ast$-isos. If true, and if that proof is more self-contained, that would provide an alternative for the step in question.
2) Discard symmetric spectra (and any other potential flavor of diagram spectra with the same problem) and focus just on those flavors for which stable equivalences coincide with $\pi_\ast$-isos. Then modify the definition of the model structure on diagram spectra by defining the weak equivalences to be the $\pi_\ast$ isos right away.
(This is not circular: we do not need to know the model structure beforehand to know whether the stable equivalences coincide with the $\pi_\ast$ isos. What we just need to check is that those morphisms $\lambda_n$ between free diagram spectra (here) are $\pi_\ast$-isos.)
$\,$
On the other hand, maybe reducing to sequential spectra is not too bad. There are pleasant and standard means to set up the model structure on sequential spectra, and with that in hand the handicrafted result of theorem III 2.4 in Lewis-May-Steinberger (that $hofib(f) \to \Omega hocof(f)$ is a $\pi_\ast$-iso) follows by an abstract argument with triangulated structure (here).
But then, maybe I am still stuck with how to completely complete the argument from there. Let me see:
So we consider a morphism $p \colon X \to B$ of diagram spectra, of which we know that it is a stable equivalence, hence that for all $\Omega$-spectra $E$ then $[p,E]_{strict}$ is an isomorphism (homs taken in the “strict”, “levelwise” model structure, the projective model structure on the diagram spectra regarded, indeed, as diagrams). The claim to be shown is: also $hofib(f) \to \ast$ is a stable equivalence.
We are meant to deduce this from observing that we have long exact sequences of the form
$\cdots [\Sigma B,E]_{strict} \overset{\Sigma f^\ast}{\longrightarrow} [\Sigma X,E]_{strict} \longrightarrow [\Sigma hofib(f), E]_{strict} \longrightarrow [B,E]_{strict} \overset{f^\ast}{\longrightarrow} [X,E]_{strict} \,.$It is these sequences that we are to invoke that detour through sequential spectra for: that detour gives stable equivalences $\Sigma hofib(f)\simeq hocof(f)$ and hence reduces the above sequence to an actual cofiber sequence in unstable homotopy theory, which we know is exact.
Okay. But I realize that I still have trouble completing the argument from there. So far this shows that
$[\Sigma hofib(f) \to \ast, E]_{strict}$is an iso for all diagram Omega-spectra $E$. But we need that $[hofib(f) \to \ast,E]_{strict}$ is an iso for all Omega-spectra $E$ (without the $\Sigma$ in there). By adjunction we have of course that
$[hofib(f) \to \ast, \Omega E]_{strict}$is an iso for all $\Omega$-spectra $E$. But to complete the argument (that $[hofib(f) \to \ast, E]_{strict}$ is an iso for all $\Omega$-spectra $E$) we still need to know that every Omega-spectrum is in the image of $\Omega$. Since we don’t know yet that $\Omega$ is invertible, this still needs an argument.
On something else:
Prop. 3.3 in “Model categories of diagram spectra” claims that pullback $\iota^\ast \colon [\mathcal{D},C] \longrightarrow [\mathcal{C},V]$ along a strong monoidal functor $\iota \colon \mathcal{C}\longrightarrow \mathcal{D}$ preserves the unit of the Day convolution product up to isomorphism.
That doesn’t seem true, or am I misreading something? The proof (on p.64) says that it’s an iso because its the adjunct of an iso. But adjuncts of isos need not be isos, of course.
Instead, the Day convolution units are the functors corepresented by the units in the base, and so there is a comparison morphism
$y(1_{\mathcal{C}}) = \mathcal{C}(1_{\mathcal{C}},-) \longrightarrow \mathcal{D}(\iota(1_{\mathcal{C}}) , \iota(-)) \simeq \mathcal{D}(1_{\mathcal{D}} , \iota(-)) \simeq \iota^\ast y(1_{\mathcal{D}}) \,,$but it’s not in general an iso. It’s an iso when $\iota$ is faithful.
I see that it’s an iso when $\iota$ is fully faithful. But if $\iota$ is the free abelian group functor, which is faithful but not fully faithful, then I don’t think the map is an iso: there are more elements in the free group on a set than there are in that set. In general, if $\iota$ has a right adjoint, then this is the map on global elements induced by the unit of the adjunction, which might sometimes be an iso even if the unit itself is not, but in general won’t be.
Right, sorry, that’s what I meant to say, fully faithful.
Do they use that part of the lemma elsewhere?
I don’t think so. And it contradicts a main point of the article. If the lemma were true, it would follow that the restriction of the sphere spectrum in its incarnation as an excisive functor – which is the tensor unit there – to the other diagram spectra would remain the tensor unit. While the whole point is that this is not the case, and that instead after restriction one needs to check that the restricted sphere spectrum is still a commutative monoid and consider its modules.
Coming back to the issue with the proof of the model structure: For the special case of symmetric spectra the claim is that the model structure obtained has as weak equivalences those morphisms $f$ such that $[f,E]_{strict}$ is an iso for every Omega-spectrum $E$. But for the Hovey-Shipley-Smith model structure on symmetric spectra, the analogous definition also requires that $E$ be an injective object in symmetric spectra. There is no such condition in MMSS00. (?)
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