Hello everyone, This is primarily a reference request, but I would also welcome corrections or indications of "better ways to approach this". I am not a category-theorist by training! My current PhD student and I have been looking at the localization of a certain subcategory of Grp, with respect to a class of morphisms that satisfy some but not all of the conditions for a left/right calculus of fractions. The key feature is that in the localization, every abelian group becomes isomorphic to the trivial group. We then wanted to show that every abelian group behaves as a zero object in the localized category (initial and terminal), and verified this in an ad hoc way. However, inspecting our arguments, they seem to work in much greater generality, leading to: CLAIM: let C be any category with a terminal object 1, and let W be any class of morphisms in C. Then 1 is a terminal object in the localization C[W^{-1}]. Initially this seemed too good to be true, since localization "adds more morphisms", but at the time of writing I can't find a mistake in our argument. Q1. Does anyone know a reference for this result/claim? The sources that I have consulted so far mostly discuss localization in contexts where C or W satisfy additional conditions, and say very little about general properties of the localisation functor when W is arbitrary. In the general setting I see no reason for the localisation functor to admit a left adjoint, which would be the cheapest way to show that it preserves terminal objects. Q2. Assuming we haven't made an error: is there a "conceptual" reason or explanation for this result? Our current argument relies on the concrete model of C[W^{-1}] in terms of equivalence classes of zigzags (a la Gabriel-Zisman), and shows that any zigzag starting at an object A_0 and ending at 1 is equivalent to the unique C-morphism A_0 \to 1. The proof is "hacky": it works by performing an induction on the length of the zigzag, and requires a small but mildly annoying case-by-case analysis. I have often been told that a "true" categorical perspective should just use the appropriate universal property of localization (perhaps in a 2-categorical context), but at the moment I can't see a way to achieve this. Remark: essentially the same reasoning seems to show that localization preserves initial objects, so both Q1 and Q2 apply to that result/claim. Regards Yemon -- Dr. Y. Choi School of Mathematical Sciences Lancaster University Bailrigg, Lancaster Lancashire LA1 4YF