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In his book "Model Categories", Hovey sets out to write a self-contained introduction to model categories. The final chapter briefly discusses some questions which stayed unresolved.

I have been wondering if there is some type of overview discussing the developments concerning these questions. Have they all been resolved one way or another?

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They have certainly not all been resolved one way or the other. As the other thread points out, Tyler Lawson and others were working on this a few years ago. During my last years of grad school, I started an annotated version documenting which of the problems had been solved and where. If you email me with a specific problem you want to know the answer to, odds are I've at least searched (well, in 2014) to see if it was answered.

Arguably the biggest problem on the whole list was about a 2-model category of model categories, and I think if you keep your eyes on arxiv you will see a lot of interesting work on that in the coming months.

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