By browsing the list of open problems in Mark Hovey's book "Model categories", I came across the following one (Problem 8.3): find a model structure on topological spaces, with the same weak equivalences as usual, in which every object is cofibrant, or else prove that this is impossible. I don't see this problem in Mark Hovey's webpage. Does it mean that someone has proved that such a model structure does not exist ?
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3$\begingroup$ At one point in grad school I made a list of the answers to as many of Hovey's open problems as possible. As far as I know, this question has not been answered in the generality Hovey wanted. However, if you are content to work with Delta-generated spaces then a beautiful result of Ching and Riehl lets you restrict to the "algebraically cofibrant objects" and proves that it's a Quillen equivalent model. So there is a model category of spaces (but a severe subclass of spaces) where all objects are cofibrant and the weak equivalences are the usual ones. See arxiv.org/abs/1403.5303 $\endgroup$– David WhiteCommented Mar 7, 2016 at 14:32
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$\begingroup$ Are the morphisms the same? $\endgroup$– Zhen LinCommented Mar 7, 2016 at 17:04
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$\begingroup$ A related theorem due to Cole is the "mixed model structure" which is a model structure on spaces with the usual weak equivalences, Hurewicz fibrations, and the cofibrant objects are those spaces having the homotopy type of a CW complex. $\endgroup$– Dylan WilsonCommented Mar 9, 2016 at 1:06
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