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I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written:

[...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and understanding vector bundles (a type of “space over M”) is the same as understanding functions

Can someone make this insight more obvious (manifest)? Maybe through some illuminating example or picture.

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    $\begingroup$ I guess, one way to interpret this is via the Gelfand-Neumark Theorem. Let $X$ be a locally-compact Hausdorff space, then X can be identified with the spectrum of the C* algebra A=C(X) and the vector bundles over X can be identified with finitely generated projective modules over A. $\endgroup$
    – user473423
    Commented May 28, 2022 at 13:48
  • $\begingroup$ @Echo The GN theorem seems to be in a rather different context, though, since C(X) lacks certain properties one would usually want for coordinate rings of varieties, and is algebraically very "rigid" $\endgroup$
    – Yemon Choi
    Commented May 28, 2022 at 22:56

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I think the most famous example of this is the Gabriel–Rosenberg reconstruction theorem, which says that we can recover a (nice enough) scheme from its category of quasi-coherent sheaves. Indeed, the Wikipedia page itself — though very short — contains the sentence:

[...] the theorem says (in a sense) working with stuff on a space is equivalent to working with the space itself.

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