Re: category of fraction and set-theoretic problem

[Jiri Rosicky <rosicky@mathstat.yorku.ca>]

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.
> 
> 
>