most recent 30 from http://mathoverflow.net 2013-06-20T00:08:51Z http://mathoverflow.net/feeds/question/14452 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14452/why-is-the-concept-of-topos-a-metamorphosis-of-the-concept-of-space Alex 2010-02-07T02:31:02Z 2011-06-12T21:10:21Z

<p>Hi,</p> <p>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? </p> <p>Thanks, Alexander </p>

http://mathoverflow.net/questions/14452/why-is-the-concept-of-topos-a-metamorphosis-of-the-concept-of-space/14453#14453 Emerton 2010-02-07T02:55:32Z 2010-02-07T02:55:32Z

<p>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. </p> <p>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. </p> <p>(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.)</p>

http://mathoverflow.net/questions/14452/why-is-the-concept-of-topos-a-metamorphosis-of-the-concept-of-space/20242#20242 Mukul Patel 2010-04-03T14:28:01Z 2010-04-03T14:28:01Z

<p>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.)</p>

http://mathoverflow.net/questions/14452/why-is-the-concept-of-topos-a-metamorphosis-of-the-concept-of-space/20300#20300 David Carchedi 2010-04-04T12:57:35Z 2010-04-04T12:57:35Z

<p>I agree with two answers already given. I provide some more detail in an answer to a similar question here: <a href="http://mathoverflow.net/questions/101/what-is-a-topos/18164#18164" rel="nofollow">http://mathoverflow.net/questions/101/what-is-a-topos/18164#18164</a></p>

http://mathoverflow.net/questions/14452/why-is-the-concept-of-topos-a-metamorphosis-of-the-concept-of-space/67603#67603 Mirco Mannucci 2011-06-12T21:10:21Z 2011-06-12T21:10:21Z

<p>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. </p> <p>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.</p> <p>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. </p> <p>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. </p>