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.