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It is natural to ask if the category of model categories can be endowed with the structure of a model category where the class of weak equivalences is given by the Quillen equivalences knowing that Quillen equivalences satisfy the 2-out-of-3 property, see Hovey's book Corollary 1.3.15.

In the same book Hovey said that we don't know the answer. I would like to know what is the state of art about this question. Does one make progress toward this model structure ?

Thanks

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    $\begingroup$ Julie Bergner would probably be the most natural person to ask. She has studied homotopy (co)limits of model categories in a structure of some sort on the category of model categories. I'll write more later if I can find time, but many others who frequent MO are well-suited to give answers. $\endgroup$
    – David White
    Feb 20, 2015 at 17:23
  • $\begingroup$ Is the category of categories (finitely) (co)complete, as required by the definition of a model category? It appears that a problem might already arise at this level. $\endgroup$
    – Dmitri Pavlov
    Feb 22, 2015 at 18:40
  • $\begingroup$ @Adeel: Sorry, I forgot to insert the second adjective “(co)complete”. What I meant is whether the category of (finitely) (co)complete categories is (finitely) (co)compelte. I already have a problem establishing it for equalizers: if 1 and 2 are indiscrete categories with 1 respectively 2 objects and one morphism between any pair of objects, then what is the equalizer of two different functors 1→2? It can only be the empty category, which is neither complete nor cocomplete. $\endgroup$
    – Dmitri Pavlov
    Feb 25, 2015 at 20:05
  • $\begingroup$ I actually don't think that's the biggest problem. One could still ask for a model structure on the category of model categories (using the terminology of Hovey's book) and this side-steps the issue of (co)completeness. This approach was taken recently in a paper on $S^1$-equivariant homotopy, by Blumberg and Mandell $\endgroup$
    – David White
    Feb 25, 2015 at 20:22
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    $\begingroup$ Actually I didn't thought about the distinction between model structure and model category structure when I wrote the question. Let say that a model structure only will be fine but in this case I'm afraid that the resulting model structure would not be cofibrantly generated. $\endgroup$
    – user2664
    Feb 26, 2015 at 15:58

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