category of fraction and set-theoretic problem
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.
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.
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/
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.
participants (4)
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Jiri Rosicky -
Michael Barr -
Philippe Gaucher -
Prof. T.Porter