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Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". History that changes not with the question but only with ever more sophisticated tools.

Can you explain how one (esp. physicists with visual intuition interested in extending into abstract mathematics and algebraic geometry) should think about describing a general space, from a historical perspective?

Something like the below:

You might want to study a certain space by

  1. using algebra (by studying behavior of functions on some field),
  2. using algebraic varieties (by studying solution sets to systems of algebraic equations)
  3. with Schemes
  4. cohomology between objects
  5. with sheaves
  6. Topos (by describing categorically how other objects relate to the one you're trying to study)
  7. Motive theory (unsure but put it here as per Mumford's writings on Grothendieck's work)

Don't take the above list seriously. I am merely regurgitating undigested material without much in-depth understanding. My goal here is to understand how to find a motivation for using more advanced tools in abstract mathematics for studying spaces—usually they are unforgiving in the technical rigor almost from the first paragraph, hence my asking for "historical" perspective.

Thank you in advance!

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    $\begingroup$ I am not sure one can think about a motive as a space. Motive remembers only the cohomological information, for some geometric questions that is not enough. That may be because I don't understand what exactly do you want (what do you mean by describing a space?). What is cohomology between objects? Do you mean higher direct image sheaves or something like relative homology from topology? $\endgroup$ – user143954 Aug 6 at 7:05
  • $\begingroup$ @jpp my elementary understanding is no doubt tripping us up here. As a physicist I'm used to using geometric and algebraic tools as methods for probing some aspects of spaces, say the shape of a magnetic field to interpret properties of particles, or shapes of fermionic fluids in topological insulators. I use the words "shape" and "space" as general property of physical systems. My question is: Since we are interested in describing more esoteric properties, how should I best think about the progression of these tools and the information they carry. Apologies if these questions are too vague $\endgroup$ – i am circle Aug 6 at 8:00
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    $\begingroup$ I don't think it's a progression. You can make a progression from some views of space, but not all. For instance, "space as a datatype whose topology carries information-theoretic content" is not naturally reconcilable with notions of space arising in algebraic topology. $\endgroup$ – Andrej Bauer Aug 6 at 9:25

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