# Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{sSet}_*$ is the Quillen model category of pointed simplicial sets.

With this structure one can define suspension and loop functors on $\mathrm{Ho}(\mathcal C)$. The suspension $\Sigma$ is the (derived) smash product with the simplicial $\mathbf S^1$ and the loop functor is its right adjoint. By definition, $\mathcal C$ is called stable iff $\Sigma$ is an autoequivalence of $\mathrm{Ho}(\mathcal C)$.

In case $\mathcal C$ is not necessarily stable, I'm looking for canonical ways to stabilize $\mathcal C$. In case $\mathcal C$ fulfills extra axioms, e.g. it is left proper and combinatorial, one can find by [1] a simplicial model structure on the simplicial objects $\mathrm s \mathcal C$ of $\mathcal C$ which is Quillen equivalent to $\mathcal C$ itself. So for left proper and combinatorial model categories $\mathcal C$, one may assume that they are already simplicial so that the suspension is already defined on the level of the model category. Then one can use [2] or [3] to consider (symmetric) spectrum objects in $\mathcal C$ with respect to the suspension and gets a canonical stable model category, where the model structure is the stable one.

Now my question is as follows: Is there a canonical way to stabilize a model category without going through simplicial enrichments (and which may work with any closed model category)? Bonus questions are: Are there better or more canonical references than the two I gave? Does it make sense just to stabilize the homotopy category $\mathrm{Ho}(\mathcal C)$, e.g. just to consider spectra on the homotopy level?

[1] Dugger: Replacing Model Categories with Simplicial Ones

[2] Hovey: Spectra and symmetric spectra in general model categories

[3] Schwede: Spectra in model categories and applications to the algebraic cotangent complex

• “Does it make sense just to stabilize the homotopy category Ho(C), e.g. just to consider spectra on the homotopy level?”: It should not be very difficult to construct two model categories C and D such that Ho(C) is equivalent to Ho(D), but Ho(Stab(C)) is not equivalent to Ho(Stab(D)). – Dmitri Pavlov Jul 13 '15 at 16:22
• “Is there a canonical way to stabilize a model category without going through simplicial enrichments (and which may work with any closed model category)?”: The answer to this question, as stated, is yes: start by replacing your model category with a Quillen equivalent left proper combinatorial model category. Present the suspension as a left Quillen endofunctor (any choice works) and apply Hovey's machinery (Definition 3.3 in his paper), which does not need a simplicial enrichment. – Dmitri Pavlov Jul 13 '15 at 16:40
• Is every model category Quillen equivalent to a left proper combinatorial one? – Marc Nieper-Wißkirchen Jul 13 '15 at 17:37
• @MarcNieper-Wisskirchen: Certainly not. See my recent MathOverflow question mathoverflow.net/questions/209734 Here are 3 counterexamples: the Strom model structure on Top, the absolute model structure on chain complexes, and pro model structures. – David White Jul 13 '15 at 20:43
• @MarcNieper-Wißkirchen: If you're not concerned with specific examples, then one can simply take the underlying quasicategory and then stabilize it using one of the constructions in Proposition 1.4.2.24 in Higher Algebra. This is guaranteed to produce the correct answer. – Dmitri Pavlov Jul 14 '15 at 9:27

This question is near and dear to my heart. Since it is asked within the context of model categories I will try to answer in that context. I agree with Dmitri that you should not try to stabilize the homotopy category. Your main question seems to be about whether or not $C$ needs to be simplicial. The answer is no. Hovey's machine does not require a simplicial model category, but requires you to have the functor you want to stabilize. Via the machinery of framings (see chapter 5 of Hovey's book) you can write down that functor even when $C$ is only cofibrantly generated. This is the approach taken in chapter 6 of Hovey's book. Next, let's discuss the hypotheses that $C$ be left proper and cellular or combinatorial.

As far as I am aware all stabilization results involve Bousfield localization. However, the version of Bousfield localization which Schwede uses (improved upon in Bousfield's paper on Telescopic localization) does not require combinatoriality or cellularity. Those hypotheses are present in order to actually build the localization functor, which works via some transfinite process you need to know will eventually end. But if you already have the localization functor then you don't need those hypotheses. So I think what you propose can be done, but you will want to be careful to determine which model of spectra you want. As Hovey points out, Schwede's machine and Hovey's machine build different model categories of spectra, and only Hovey's appears to have the property you want. So basically you would want to take Hovey's proof and make it go with a localization functor built "by hand" so as to avoid the need for $C$ to be combinatorial or cellular. Note that without left properness you can still say something, and I have a preprint on that if you would like to email me. It's not yet ready to write about publicly here. I advise working out an example, e.g. take $C$ to be the stable module category and figure out what the Ho(sSet) action does. Use a framing to write down the suspension endofunctor and try to build the localization following Hovey and Schwede. If it works then you have a nice roadmap for what to do in general, and the material in chapters 5-6 of Hovey's book will help fill in the details.

There are also other ways to stabilize, e.g. Biedermann's paper L Stable Functors, which basically defines the stabilization you want, tries to build it, and has a comparison to Hovey's machine. The relevant results are in section 5, $M$ is what you called $C$, and $V$ can be simplicial sets or chain complexes or something else. This uses the version of Bousfield localization I mentioned above, but it (and also Proposition 2.2 of Motivic Functors by Dundas, Rondigs, Ostvaer) technically require locally presentable in order to embed into a presheaf category. Perhaps this can be avoided; I advise asking a category theorist.

Since Dmitri mentioned Higher Algebra, let me just say that I don't understand the proof of 1.4.2.24. It first does the case where $C$ is presentable, which we already knew how to do thanks to the references in the question, and then it says without loss of generality we can assume $C$ is small. I don't follow this step. Perhaps it's one of these arguments where you enlarge the Grothendieck universe but in this context it seems to be getting something for free. In any event, this step is probably not replicable in the context of model categories because once you bump up to a larger Grothendieck universe I don't know what happens. If the model category truly becomes small then it's uninteresting (e.g. just a poset), but I think what really happens is more in line with the discussion at Is the $\infty$-category of presentable $\infty$-categories presentable?, i.e. in this new universe $C$ won't have all limits and colimits, only those of some bounded size. In any event, I don't think you can reduce questions about model categories to dealing with small ones.

• One can perfectly well have small model categories. There are no interesting small combinatorial model categories, however. – Zhen Lin Jul 14 '15 at 21:44
• There is no problem with 1.4.2.24 of Higher Algebra. The condition in the statement requires that $\mathcal{C}$ be pointed and have finite limits, which is a property clearly independent of the universe you're in. – Dylan Wilson Jul 15 '15 at 4:16
• @DavidWhite: I'll go through the references you gave in the second to last paragraph. As to writing down the suspension functor via framings, at appears to me that one would need functorial factorizations to get functorial cosimplicial frames of cofibrant replacements. Without functoriality, I only see how to get a well-defined suspension functor on the homotopy category. So one may have to check how to extends Hovey's machinery to "functors, well-defined only up to homotopy". – Marc Nieper-Wißkirchen Jul 15 '15 at 15:10
• @ZhenLin: I guess we both already thought about this point, and the conclusion was that you need to weaken the model category axioms to only ask for finite limits and colimits, right (i.e. take Quillen's version not Hovey's)? I'm thinking of mathoverflow.net/questions/106924 and mathoverflow.net/questions/108739 – David White Jul 15 '15 at 17:11
• @DylanWilson: I did not say anything was wrong with it, and even if I thought there was something wrong this would have been the wrong forum to bring it up. I just said I can't follow it, and can't make it work for model categories. Am I right that the step reducing it to $C$ small is a universe enlargement, or is there something else going on? You are saying you can read the proof straight through and follow every step? If so I have questions for you. – David White Jul 15 '15 at 17:14