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Are there any books or papers where I can find some general statements and methods for working with Quillen functors that are not equivalences (and not localizations)? In particular, I would like to know which structured are preserved by Quillen functors between stable homotopy categories. I wasn't able to find much in Hovey's book.

P.S. Does there exist something like "Model categories for dummies"?:)

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  • $\begingroup$ I was going to recommend Hovey's book! Why do you say you didn't find much there? What did you expect to find? $\endgroup$
    – Fernando Muro
    Commented Oct 30, 2013 at 21:09
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    $\begingroup$ "model categories don't make it easy to prove that a left Quillen functor preserves finite homotopy limits, but in fact when it's between stable model categories it's automatic (this is in Hovey)." : a comment by Marc Hoyois. Yet I was not able to find anything like that in Hovey. So I wonder whether there exist some more sources on the subject (or possibly somebody could explain me how to read Hovey). $\endgroup$
    – Mikhail Bondarko
    Commented Oct 30, 2013 at 21:53
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    $\begingroup$ Derived left Quillen functors preserve homotopy push outs and finite coproducts, these coincide with homotopy pull backs and finite products by stability, hence all finite homotopy limits are preserved $\endgroup$
    – Fernando Muro
    Commented Oct 31, 2013 at 9:13

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