I am reading Gelfand-Manin, and am a little confused about their proof that the equivalence relationship between roofs in the localization of a category $B$ at a localizing class of morphisms.

In particular, in proving transitivity, it seems that one can just put a roof over the bottom diagram on page 149, and then check all of the commutativity stuff (which I did, I think). But their argument is more complicated than this - in particular, it uses the third axiom of a localizing system ($ft = gt$ is equivalent to $sf = gs$ for some s). What am I missing?

(I'm not sure how to tex up these diagrams, sorry!)


I think you may be assuming (using the notation from the passage of Gefand & Manin that you refer to) that $p\in S$, otherwise I'm not sure how you want to "put a roof over the bottom diagram".

But we only have that $tp\in S$, not necessarily $p\in S$.

The description of the equivalence relation on roofs just before this is a little ambiguous: by the "third roof" they mean $(sr,gh)$, not $(r,h)$. I.e., they don't require that $r\in S$, only that $sr\in S$.

  • $\begingroup$ Yeah, I think that was it. Thanks! $\endgroup$ – Lorenzo Najt Dec 23 '15 at 18:25
  • $\begingroup$ I'm confused. For the case of quasi-isomorphisms I think it is the same, because if $s$ and $sr$ are quasi-isomorphisms is the same as $s$ and $r$ are quasi-isomorphisms. $\endgroup$ – Lorenzo Najt Dec 23 '15 at 19:02
  • $\begingroup$ And the same for any class of morphisms compatible with a cohomological functor (F is a cohomological functor, I mean all of the s with F(s) an isomorphism.) Is there some reason why Gelfand-Manin's definition is better in some more general context? $\endgroup$ – Lorenzo Najt Dec 23 '15 at 19:04

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