# Isomorphisms in the localization of a category

Let $\mathcal{C}$ be a (small) category, and $S \subset \mathcal{C}$ a class of morphisms in $\mathcal{C}$. Suppose $f$ is a morphism in $\mathcal{C}$ that becomes an isomorphism in the localization $S^{-1}\mathcal{C}$. Suppose moreover that $S$ satisfies the two-out-of-three property (i.e. in a composition, if two of the terms belong to $S$, then so does the third) and contains all isomorphisms in $\mathcal{C}$. When can we conclude that $f \in S$ itself?

In the special case that I'm considering, $S$ is the class of weak equivalences in a model category $\mathcal{C}$. In this case it is true (and follows from the alternative description of the homotopy category) that an isomorphism in the homotopy category is a weak equivalence, but the proof involves some manipulations. I am curious if a simpler approach exists.

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I guess you don't want to have a description of the localization via a calculus of fractions (which as far as I know doesn't work for model categories) but in which stability of $S$ under weak pull-backs or push-outs is sufficient. There is a relatively recent preprint by Sebastian Thomas on various conditions on $S$ that you can impose, but I wouldn't call that simple and some manipulations would be a euphemism. The same can be said about the Dwyer-Hirschhorn-Kan-Smith approach, of which you certainly are aware and on which Thomas's paper is based. – t.b. May 24 '11 at 1:22
@Theo: Dear Theo, thanks! I wasn't aware of any of this, though it does indeed appear somewhat difficult. – Akhil Mathew May 24 '11 at 2:20
The calculus of fractions is due to Gabriel-Zisman and can be found (at least outlined) in Ch. 10 of Weibel. It is patterned after Ore's localization theory for non-commutative rings and gives you a handy description of the localized category. But the hypotheses are a bit too strong to apply to model categories. Nevertheless I'd have a look at that. It is treated nicely in an old book on categories by Schubert (Ch 19 of the German ed.). DHKS propose a localization approach for model categories based purely on weak equivalences and Thomas made a systematic study of all these theories. – t.b. May 24 '11 at 2:35
@Theo: Thanks for the references! I hadn't considered that there was an analogy between Ore and G-Z localization; interesting. I'll take a look at ch. 10 of Weibel. – Akhil Mathew May 24 '11 at 14:32
One last remark that may or may not be helpful: While checking that everything works fine in the category defined by fractions, I found pages 300-302 of Lam's Lectures on modules and rings very useful for orientation. It is very easy to get lost in all those diagrams (e.g. while checking transitivity of the equivalence relation or associativity of the composition) and Lam's exposition can be easily read to apply to categories, not only rings. – t.b. May 24 '11 at 15:32