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?
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$\begingroup$ 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 ? $\endgroup$– Tsemo AristideCommented Dec 1, 2018 at 15:41
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$\begingroup$ Well, we may encounter an illegitimate category $\mathbf{CAT}$. $\endgroup$– user122424Commented Dec 2, 2018 at 16:03
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$\begingroup$ I've edited my question by inserting the word "small". Everything goes smoothly now ? $\endgroup$– user122424Commented Dec 2, 2018 at 16:42
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$\begingroup$ 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. $\endgroup$– Kevin CarlsonCommented Dec 2, 2018 at 21:13
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2$\begingroup$ @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! $\endgroup$– Alex KruckmanCommented Dec 3, 2018 at 16:30
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