## Need M combinatorial for existence of injective model structure on $M^G$?

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 "sheaffiable".

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

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 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. – David White Aug 16 at 0:02 @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. – Zhen Lin Apr 12 at 15:51