If S is the class of weak equivalences in a Quillen model structure then C[S^{-1}] is always locally small. See, e.g., M. Hovey, Model categories, AMS 1999, Jiri Rosicky On Thu, 30 Nov 2000, Philippe Gaucher wrote:
Bonjour,
I have a general question about localizations.
I know that for any category C, if S is a set of morphisms, then C[S^-1] exists. And moreover if C is small, then C[S^{-1}] is small as well (as in the Borceux's book Handbook of categorical algebra I)
If S is not small, and if we suppose that all sets are in some universe U, then the previous construction gives a solution as a V-small category for some universe V with U \in V (the objects are the same but the homsets need not to be U-small). So it does not work if one wants to get U-small homsets.
Another way is to have a calculus of fractions (left or right) and if S is locally small as defined in Weibel's book "Introduction to homological algebra".
But in my case, the Ore condition is not satisfied. Hence the question : is there other constructions for C[S^{-1}] ?
Thanks in advance. pg.