category of fraction and set-theoretic problem

category of fraction and set-theoretic problem

[Philippe Gaucher]

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.

Re: category of fraction and set-theoretic problem

[Michael Barr]

It is not clear if you are interested in special cases or in general
conditions.  If the latter, I cannot help, but here is an example of a
special case.  But first, I might ask why it matters.  Gabriel-Zisman
ignores the question and I think they are right to.  Every category is
small in another universe.  

Consider the category C of chain complexes from some abelian category. By
this I mean bounded below with a boundary operator of degree -1.  Arrows
are chain maps of degree 0.  Let S denote the class of homotopy
equivalences and T the class of homology isomorphisms.  Then S < T and
there is neither a calculus of right or left fractions for either.  On the
other hand S^{-1}C is equivalent to C/~ in which you have identified
homotopic arrows.  This is locally small because you leave the objects
alone and it is a quotient.  From S < T, it follows that T^{-1}C =
T^{-1}S^{-1}C = T^{-1}(C/~) and the image of T in C/~ does have a calculus
of fractions (both left and right; duality implies that they are
equivalent).  Thus there is a notion of homotopy calculus of fractions in
this case.  I have tried, without success, to find a general condition
of which this would be a special case.

Michael

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.
> 
> 

Re: category of fraction and set-theoretic problem

[Prof. T.Porter]

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/

Re: category of fraction and set-theoretic problem

[Jiri Rosicky]

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.
> 
> 
>