Timeline for Why is the concept of topos a "metamorphosis" of the concept of space?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Jun 13, 2011 at 7:21 | comment | added | Andrej Bauer | @Harry: the spaces whose points can be recovered from topology are called "sober". $T_1$-spaces are sober. I was not aware that Hausdorff spaces are algebraic. Perhaps you meant compact Hausdorff spaces? | |
| Apr 4, 2010 at 12:54 | comment | added | David Carchedi | @fpqc Perhaps I misunderstand you, but if you are claiming that Hausdorff spaces are monadic, then I don't believe you. Algebras for the ultafilter monad are COMPACT Hausdorff spaces. Anyway, it's still a really neat fact. | |
| Feb 7, 2010 at 8:21 | comment | added | Tim Porter | Sorry I forgot to say it was by Pierre Cartier. | |
| Feb 7, 2010 at 8:20 | comment | added | Tim Porter | 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. | |
| Feb 7, 2010 at 4:17 | comment | added | Alex | Thanks Mr. Emerton. That was a a great (and non-technical) explanation--just what I hoped for. | |
| Feb 7, 2010 at 3:56 | comment | added | Harry Gindi | 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. | |
| Feb 7, 2010 at 3:30 | comment | added | Mike Shulman | 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. | |
| Feb 7, 2010 at 2:55 | history | answered | Emerton | CC BY-SA 2.5 |