Hi,
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
Hi, 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 


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 sheaftheoretically, 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.) 


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 nonhomemorphic spaces can have equivalent toposes of sheaves. (As above, in case of 'sober' spaces, such is not the case.) 


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 pointset 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 spacetime, and yet you have no welldefined points. 


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

