TITLE: Class of maps in localized category may not be a set
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: 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.