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?

foundationalproblem, 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. $\endgroup$