Re: category of fraction and set-theoretic problem

["Prof. T.Porter" <t.porter@bangor.ac.uk>]

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.

Dear All,

Philippe's question may be answered in part by looking 
at the construction by Baues and Dugundji (Trans Amer Math Soc 140
(1969) 239 - 256). Another point is that in the homotopical applications
it is not that the Ore condition is satisfied but that it is satisfied
up to homotopy that counts. A discussion of this in at least one case is
to be found on pages 90 - 111 of the book by Heiner Kamps and myself.
(see my homepage for the detailed coordinates if you want. The set
theoretic question was looked at by various people including Markus
Pfenniger in an unpublished manuscript in 1989.

Tim  
************************************************************
Timothy Porter                   
Mathematics Division,		  
School of Informatics,                                          
University of Wales Bangor                            
Gwynedd LL57 1UT                  
United Kingdom    
tel direct:        +44 1248 382492   
home page:            http://www.bangor.ac.uk/~mas013  
Mathematics and Knots exhibition: http://www.bangor.ac.uk/ma/CPM/