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I recently started studying topos theory, and I am puzzled by the Grothendieck's claim that topos is a "metamorphosis" of the concept of space. Can somebody explain what he means by this?

Thanks, Alexander

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Can you give a reference to where Grothendieck makes this claim? – Jonathan Wise Feb 7 '10 at 2:45
@Jonathan Wise: it was state4d in the Allyn Jackson articles in the AMS Notices, and it can be found in Reaping and Sowing, Grothendieck's memoir. – Alex Feb 7 '10 at 4:15

If you have a space, you can consider the category of sheaves of sets on the space; the latter is a topos (the archetypal example thereof). Since sheaves are (a) very flexible; and (b) highly attuned to the topology of the underlying space, the topos remembers a lot of information about the space. Thus, forgetting the space but remembering the topos, while being perhaps a radical change in perspective, is not really abandoning the idea of the space, but is exactly just changing ones perspective on what a space is.

Thus, passing to the study of topoi from the study of spaces is just one more step in a (very) long mathematical tradition of studying the nature of shape and space.

(Somewhat more bluntly, one might argue that every question about a space that one wants to study is encapsulated in some way sheaf-theoretically, and so remembering the topos precisely remembers everything interesting about the space; hence one is metamorphising the concept of space in such a way as to remember precisly what is interesting, and eliminate from consideration everything that is extraneous.)

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In fact, for a very large class of spaces (the "sober" ones, including all Hausdorff spaces), the topos of sheaves of sets remembers everything about the space, in the sense that the space itself can be reconstructed from the topos up to homeomorphism. – Mike Shulman Feb 7 '10 at 3:30
Can we identify a point in a Hausdorff space by the ultrafilter of opens that converges to it? That's pretty cool actually. So that's why Hausdorff spaces are algebraic (in the mondic sense). They're just a special type of poset. – Harry Gindi Feb 7 '10 at 3:56
Thanks Mr. Emerton. That was a a great (and non-technical) explanation--just what I hoped for. – Alex Feb 7 '10 at 4:17
The article A mad day's work : from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry (traduit du français [77] par Roger Cooke). Bull. Amer. Math. Soc. (N.S.), 38 (2001), 389-408, may be of interest in this respect. – Tim Porter Feb 7 '10 at 8:20
Sorry I forgot to say it was by Pierre Cartier. – Tim Porter Feb 7 '10 at 8:21

In sense, a sheaf over a space is 'representation' of the space. Somewhat akin to a module being a representation of a ring. So, a the catergory (topos) of sheaves over a space plays the same role the category of modules play over a ring. Just as two nonisomorphic rings can have equivalent module categories, two non-homemorphic spaces can have equivalent toposes of sheaves. (As above, in case of 'sober' spaces, such is not the case.)

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The answers already provided are very good and informative, so I just wish to add something concerning the "metamorphosis" of the very notion of space of which Grothendieck speaks in Semailles.

Every space has its associated topos, but there are topoi which are NOT spatial. You can define categorically the notion of point of a topos, and this definition corresponds to the usual notion of points when one restricts to spatial topoi.

Now, the fact that there are plenty of topoi with no points basically means that one can do topology in a pointless world: you can still formally define notions of compactness, coverings, and well as most of the standard topological (and even homotopical) machinery, directly in a given topos, regardless of its having points or not.

As it turns out, the passage from point-set to pointless topology is not just an idle game: for instance in physics at the Planck level you may still want to talk of topological and geometric properties of space-time, and yet you have no well-defined points.

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I agree with two answers already given. I provide some more detail in an answer to a similar question here: What is a topos?

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