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?



Class of maps in localized category may not be a set
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"](https://arxiv.org/abs/0806.1324). This is an example of a category with weak equivalences whose localization isn't locally small.


