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I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. We can assume $M$ is cofibrantly generated and left proper, but I'm trying to avoid assuming $M$ is combinatorial. If necessary we can assume $M$ is cellular, but right now I don't see how that can help. It has turned out to be useful to know that $M^G$ has a model structure where the cofibrations are $G$-equivariant maps which are cofibrations in $M$. This is simply the injective model structure on $M^G$, where a map $f$ is a weak equivalences or cofibrations if the underlying map in $M$ is such.

I tend to think of $M^G$ as a special case of a diagram category $M^I$ (where $I$ is small), because I feel like I have a decent grasp on diagram categories. In that setting, I believe one must know that $M$ is combinatorial in order to know the injective model structure exists. However, I don't have a good reference for this other than having seen it mentioned without proof or reference in a number of papers.

(1) Can anyone provide a reference which proves the injective model structure on $M^I$ exists? I'd like to see where the hypothesis that $M$ is combinatorial gets used. I'd also like the reference to prove the injective model structure is cofibrantly generated.

It's worth noting that Proposition A.2.8.2 in Lurie's Higher Topos Theory proves existence of the injective model structure, but I'm not satisfied with that for a couple of reasons. First, the proof is very complicated because Lurie wants it to hold in the setting where $M$ and $I$ are enriched over an excellent model category. My ideal reference would be a simpler proof holding in the non-enriched setting, preferably the first place the injective model structure was defined. Second (and related), because everything in this appendix is about combinatorial model categories, I can't help but wonder if there's a proof which relies on that hypothesis less. Finally, it's almost impossible for me to get my hands on the generating (trivial) cofibrations from that proof. Lurie relies on the very-complicated Lemma A.3.3.3 to get generating cofibrations and on Proposition A.2.6.8, which says basically that if you're in a category which is almost combinatorial (missing only generating trivial cofibrations) then you can get the generating trivial cofibrations for free from the generating cofibrations.

In the special case where $I$ is a group $G$, I can't seem to find anything on the injective model structure. Most of the work I can find on equivariant homotopy theory uses the projective model structure instead of the injective (and this one is known to exist if $M$ is cofibrantly generated). I imagine that with so much structure on $I$ and with so much theory which has been developed out there for equivariant homotopy theory, one should be able to come up with a much better proof in this setting than the one in HTT.

(2) Is the hypothesis that $M$ is combinatorial still necessary to prove existence of the injective model structure on $M^G$? In what ways is this model structure nicer than $M^I$ for a generic $I$?

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It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheafifiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

Edit: Apparently Lurie's construction has been studied further in an abstract context by Makkai and Rosický and Makkai and Rosický and Vokřínek.

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  • $\begingroup$ Hi. Thanks for the answer and references. I'll have to look into them. I'm voting you up, but not accepting the answer yet as I still hope others come along and give more references or ideas about avoiding this A.3.3.3 argument. My hope was that early references wouldn't have that. I suppose I'll find out soon when I read the ones you mention. $\endgroup$
    – David White
    Commented Aug 16, 2012 at 0:02
  • $\begingroup$ @Mike At first I thought "logical approach" meant "straightforward", but having skimmed the paper now, I see you mean "logical" in the literal sense! A very remarkable paper. $\endgroup$
    – Zhen Lin
    Commented Apr 12, 2013 at 15:51
  • $\begingroup$ Mike, concerning your last paragraph, you probably have seen these: Reedy model structures on diagrams indexed by an inverse category. $\endgroup$
    – Fernando Muro
    Commented Jun 16, 2015 at 9:25
  • $\begingroup$ @FernandoMuro, of course, but what I meant was an injective model structure for a general diagram shape. $\endgroup$
    – Mike Shulman
    Commented Jun 17, 2015 at 17:04
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It seems this question has been answered very nicely since I asked it, in the paper Left Induced Model Structures and Diagram Categories (Contemp. Math. 641 (2015) 49-81). They prove in Proposition 4.17 and Theorem 4.19 that if $M$ has a Postnikov presentation and the class of cofibrations coincides with the class of monomorphisms, and if $D$ is a small indexing category, and if maps in $M^D$ can be factored into a trivial cofibration followed by a fibration via the Postnikov presentation on $M$ then $M^D$ admits an injective model structure. In particular, this does not require $M$ to be combinatorial. You can also drop the need for cofibrations to equal monomorphisms if you ask $M^D$ to have both types of factorization. This paper also does a fantastic job spelling out the duality between asking a model category to be cofibrantly generated vs. to have a Postnikov presentation. Since the injective model structure is dual to the projective, I don't think a better answer than this one can be found, but I am glad to know combinatoriality is not needed. For those who enjoy the Bayeh et. al. paper, I also recommend the extension of this paper which can be found in the appendix to Hess and Shipley's Waldhausen K-Theory of Spaces via Comodules.

EDIT: A follow-up paper, A necessary and sufficient condition for induced model structures (Journal of Topology, 2017) gives an answer completely in terms of factorization systems, and again avoids the need for $M$ to be combinatorial. This is the best I could have hoped for when I asked the question way back in 2012.

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  • $\begingroup$ To what new examples does this apply? $\endgroup$
    – Lennart Meier
    Commented Jun 17, 2015 at 7:13
  • $\begingroup$ My gut instinct is that it applies to Top. Since Top is not combinatorial it wasn't an example before. But I think you'd have to carefully work out the Postnikov presentation $\endgroup$
    – David White
    Commented Jun 17, 2015 at 13:03
  • $\begingroup$ Also: part of my reason for posting this answer is that I got a message from another mathematician who presumably had some application in mind and wanted to know if I'd ever delved deeper into this question. I think even in the combinatorial case it's nice to have a second proof that does not pass through the machinery of Smith's theorem, which can be mysterious to new-comers to the field. $\endgroup$
    – David White
    Commented Jun 17, 2015 at 16:45
  • $\begingroup$ I think an even better example is Pro categories, which are not combinatorial but often are fibrantly generated, which is very related to having a Postnikov presentation (I think Postnikov presentations were invented to generalize the notion of fibrantly generated). This paper seems to give injective model structures over pro-categories. $\endgroup$
    – David White
    Commented Jul 27, 2015 at 8:30
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    $\begingroup$ In my paper `Images directes cohomologiques dans les catégories de modèles', there is Theorem 6.16, which says the folowing. If a model category $M$ has small limits, is right proper and has the class of monomorphisms as its class of cofibrations, then the injective model structure exits on $M^D$ for any small $D$. If you look at the proof, you can see that we do not really need the cofibrations to be the monomorphisms, but only that the class of cofibrations is closed under small limits. $\endgroup$
    – D.-C. Cisinski
    Commented Mar 10, 2019 at 21:17
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This is probably not the sort of answer that the original question was going for, since you said you were happy to assume cofibrant generation but not combinatoriality (i.e. not local presentability), but it may be useful to other readers. Namely, if you're willing to keep local presentability and instead relax cofibrant generation to accessibility, then there is now a very general result about the existence of injective model structures in the paper Injective and Projective Model Structures on Enriched Diagram Categories (arXiv:1710.11388) by Lyne Moser.

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  • $\begingroup$ Thanks for sharing! I did not know about that paper. $\endgroup$
    – David White
    Commented Feb 13, 2019 at 11:48

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