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My understanding is that one thinks to rational homotopy theory for computational advantage. However, thinking about things in terms of localizations still lacks some amount of intuition for me.

In particular, if $f,g$ are continuous functions and $\gamma$ is localization functor by rational homotopy equivalence and $\gamma(f)=\gamma(g)$, is there something I can say analogous to continuously ("rationally"?) transforming $f$ to $g$".

Thanks!

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I think that you really need to bite the bullet, and think hard about what the localization functor does. –  André Henriques Apr 17 '13 at 19:40
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Wasn't it Von Neumann who said that in mathematics you don't understand things, you just get used to them? –  Angelo Apr 18 '13 at 2:38

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