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Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What is the most natural choice to this question: what should the classses of fibrations, cofibrations and weak equivalencies be?

user122424
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  • To define a category, you have to be sure that for every object $X,Y$ $Hom(X,Y)$ is a set. Are you sure that the set of functors between two categories endowed witha Quillen model is a set ? – Tsemo Aristide Dec 01 '18 at 15:41
  • Well, we may encounter an illegitimate category $\mathbf{CAT}$. – user122424 Dec 02 '18 at 16:03
  • I've edited my question by inserting the word "small". Everything goes smoothly now ? – user122424 Dec 02 '18 at 16:42
  • Well, very few model categories of interest are small...Anyway, that's not really the point. A model category structure gives, in particular, homotopy limits and colimits, and it's extremely difficult to construct even these for model categories. Julie Bergner is one author who's written papers on special cases like homotopy fiber products. Much easier is to study similar questions $\infty$-categorically. – Kevin Carlson Dec 02 '18 at 21:13
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    @KevinCarlson "Very few model categories of interest are small". Maybe that's true in your universe. I'm in the next one up, and I see lots of interesting small model categories! – Alex Kruckman Dec 03 '18 at 16:30
  • @AlexKruckman Of course, and as I said, this isn't really the main issue. It's interesting to note, though, that even without changing universe, the category of left Quillen functors out of any combinatorial model category, which covers all the standard examples up to Quillen equivalence, is essentially small. – Kevin Carlson Dec 04 '18 at 03:11
  • To ask more concretely, what is the trivial model structure on the category of all small model categories? – user122424 Dec 04 '18 at 19:08
  • You have a forgetful functor from Model Categories to Cat, and that Cat is a model category with equivalences as weak equivalencies, isofibrations as vibrations, and injective on objects as cofibrations. If we could show that this possess a left adjoint, and that Cat is cofibrantly generated, then we would have a model structure by transferring along the adjunction. This would be such that weak quivalences are equivalences of underlying categories, and fibratios are jsofibration of underlying categories. But this is not so interesting in my opinion. Can we find a MS with Quillen equivalences? – Andrea Marino Feb 20 '19 at 22:05
  • Wait... Model Categories is not cocomplete. Indeed, the union of two cocomplete categories is not cocomplete. As an example, take two posets closed for sup. If you make the disjoint union, it is no longer closed for sup. So there is no model structure. – Andrea Marino Feb 20 '19 at 22:08

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