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In one of the very first sentences in Hovey's "Model Categories", Ist chapter, we read that

One can always invert these "weak equivalences" formally , but there is a foundational problem with doing so, since the class of maps between two objects in the localized category may not be a set.

I thought that in a model category fibrations and cofibrations provide an explicit description of weak equivalences and how to invert them - but it seems that there is more stuff going on. So can someone provide me an example with a category with weak equivalences, such that the class of maps between two objects in the category with formally inverted weak equivalences is not a set?

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  • $\begingroup$ FWIW, I think Hovey is wrong that there is a foundational problem, as that gives the impression that e.g. some construction might not be possible in ZFC. In fact the only problem is that the resulting category might not be locally small. Granted, many people define a "category" so that it must be locally small, but in this case the problem is still not "foundational" but rather an intrinsic fact about one's choice of definitions: even in a stronger foundational system the same problem would occur. $\endgroup$
    – Mike Shulman
    Jun 26, 2018 at 16:58

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When Hovey says that we "can always invert these 'weak equivalences' formally", the weak equivalences he's talking about aren't necessarily weak equivalences in a model category. He's introducing and motivating model categories by saying that they solve a problem: sometimes we'd like to formally invert some class of arrows, but there's a risk that we can't for foundational reasons. However, if these weak equivalences are part of a model structure then we can use the model structure to show that the localization exists after all.

So if we want an example of a category of weak equivalences whose localization has the foundational problems Hovey's referring to then the weak equivalences can't be part of a model structure. One example along these lines is Example 4.15 of Krause's "Localization theory for triangulated categories". This is an example of a category with weak equivalences whose localization isn't locally small.

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