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](http://ncatlab.org/nlab/show/model+structure+on+functors) 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.



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> (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.
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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.



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> (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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Need M combinatorial for existence of injective model structure on $M^G$?
It seems this question has been answered very nicely since I asked it, in the paper [*Left Induced Model Structures and Diagram Categories*](http://arxiv.org/abs/1401.3651) (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](https://arxiv.org/abs/1509.08154v2) (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.


