Dear Carlos
I am looking for references giving lots of details on the constructions of localization in Gabriel-Zisman.
Horst Schubert "Categories" (translated from German by Eva Gray, Springer-Verlag, 1972) Chapter 19 is pretty good. In some sense, for the general construction, you don't need much detail. The category Cat_0 of small categories is a reflective full subcategory of the category Simp of simplicial sets. There are many reasons for this: but in any case the left adjoint to the inclusion is easy to construct explicitly; that too is in Gabriel-Zisman. Features are that the inclusion preserves exponentiation so that the left adjoint (reflection) preserves finite products. Colimits in Cat_0 are constructed by forming them in Simp and applying the reflection. So Cat_0 is complete and cocomplete. The existence of the arrow category ("lax path") construction a |--> a^2 and the "lax cylinder" construction a |--> 2 x a (where here 2 is the category with two objects 0 and 1 and only one non-identity arrow which goes 0-->1) implies that the limits and colimits in Cat_0 become 2-limits and 2-colimits (in the strict enriched category sense) in the 2-category Cat. Let s be a set of arrows in a category a. Let 2 be as before and let i be the category consisting of an isomorphism 0 --> 1. Regard s as a discrete category so that there is an obvious natural transformation sigma from dom : s --> a to cod : s --> a. Then the category of fractions a[s^{-1}] is the pushout of the functor 2 x s --> a (corresponding to sigma) along 2 x s --> i x s. (In our terminology, this is the inverter of the 2-cell sigma.) For the calculus of fractions case, I suggest Schubert. If your categories all have pullbacks and functors preserve them, things work better as documented in: Jean Bénabou, Some remarks on $2$-categorical algebra. I. Actes du Colloque en l'Honneur du Soixantième Anniversaire de René Lavendhomme (Louvain-la-Neuve, 1989). Bull. Soc. Math. Belg. Sér. A 41 (1989), no. 2, 127--194. Best regards, Ross 21-Feb-2005 15:45:38 -0400,1193;000000000000-00000000