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added to simplicial model category a handful of theorems that state when and how a model category is Quillen equivalent to a simplicial model category.
My motivation for filling this in was actually that I was reading van den Berg/Garner types are weak omega-groupoids and my impression was that the main theorem there is morally the usual simplicial resolution technique in model categories, only that instead of simplicial objects they use globular objects.
The other main statement in there I hope we can isolate in some other entry (and it may go back to other authors?): that the context categories of certain type theoreies with identity types naturally carry the structure of somthing close to a category with fibrant objects.
noticed that a bit of information was missing and somewhat hastily added some in at simplicial model category: a section on Properties, a warning on terminology, a section on combinatorial simplicial model categories.
Expanded the Properties-section at simplicial model category by more discussion of Enrichment, tensoring, and cotensoring
I am trying to brush-up and expand simplicial model category for a seminar talk.
There is a uniqueness theorem for model structures on simplicial objects cited here, being theorem 3.1 in the article by Rezk-Schwede-Shipley mentioned there. Am I missing something: isn’t this uniqueness a trivial consequence of the fact that model structures are determined by their cofibrations and fibrant objects? Is the observation of that general fact younger than that theorem?
okay, I have added more details in the section Simplicial Quillen equivalent model structures.
I would speculate that you are correct, i.e. that the general fact (due I think to Joyal?) was not known to RSS at the time of writing that paper.
that the general fact (due I think to Joyal?)
At least I know it from his notes on quasi-categories.
was not known to RSS at the time of writing that paper.
Okay, I see. Seems to be kind of surprising, given that it is a simple elementary argument. But then, that’s how it goes.
The fact surprised me a bit when I heard it. It’s obvious from the definition of model category that any two of the three classes of maps determine the third, and after you get used to that idea it may also seem “obvious” that just knowing the fibrant objects probably wouldn’t be enough, that you’d need to know all the fibrations. Of course what’s surprising to one person may be obvious to another. (-:
With the fact in hand, it’s hard to imagine how hard it is to guess it. But at least phrasing the proof in words as follows makes it seem non-surprising and non-mysterious:
“Given the cofibrations, fibrant objects are sufficient for computing derived homs into fixed objects, and these in turn are sufficient for determining the weak equivalences.”
Of course the answer to every riddle is obvious once one has it. But at least this one seems to be straightforward and not involve any tricks.
I have added the proof of three of the items in the big theorem in the section on resolutions by simplicial model structures.
Will have to quit now, hope to further expand and polish tomorrow.
I have written out here essentially all of the proof of
on how a model category may be replaced by a Quillen equivalent simplicial one.
I am feeling a bit dumb, but there is one step which seems to have a gap to me:
in theorem 5.2 b) the claim is that the usual Quillen adjunction $(colim \dashv const) : C \to [\Delta^{op}, C]_{proj}$ descends to the localization of $[\Delta^{op}, C]_{proj}$ that makes it have as weak equivalences the maps that become weakly equivalent under hocolim.
It says in the proof: “$const$ clearly preserves fibrations and trivial fibrations”. Is this clear? For the unlocalized structure of course, but for the local structure?. What is clear from item c) of the same theorem is that $const : C \to [\Delta^{op}, C]_{proj,localized}$ preserves fibrant objects, and from a) that it preserves weak equivalences. But fibrations?
Perhaps one needs to insert a reference to Hirschhorn, 3.3.18? If an adjunction $M\rightleftarrows N$ is Quillen and we localize $M$ at some class of maps whose derived images become weak equivalences in $N$, then we still have a Quillen adjunction.
Ah, thanks Mike. I was missing that. I was actively aware of the similar statement for simplicial left proper model categories which is currently prop. 4 at Quillen equivalence. This would imply this result… were it not for the fact that in the present context simplicialness is the very thing to be shown.
Okay, thanks, so I added that prop. 3.3.18 at Quillen adjunction- behaviour under localization and have patched the discussion at simplicial model category with a pointer to this.
3.3.18 makes use of a statement (8.5.4) similar to Prop. 4, that for a right adjoint to be a right Quillen functor it suffices for it to preserve (1) acyclic fibrations and (2) fibrations between fibrant objects. I haven’t looked at the proofs of either, but is there any chance that Prop. 4 only uses the simplicial structure as a crutch and would work just as well in the non-simplicial case?
Prop. 3 at Quillen adjunction also seems to me like it really shouldn’t depend on the model categories being simplicial.
That could well be, yes. I should go through all these statements and generalize them where possible.
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