Timeline for The "right" topological spaces
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2018 at 20:31 | comment | added | Denis Nardin | @Qfwfq Actually, let me be even a bit more radical. For me the model of the homotopy type is not the topological space: it is the Kan complex itself. When I have a nice topological space $X$, its homotopy type is represented by the Kan complex $\mathrm{Sing}_*X$ (and I see as an irrelevant accident that you can indeed model homotopy types with topological spaces) | |
| Oct 12, 2018 at 20:16 | comment | added | Denis Nardin | @Qfwfq This seems a fair way of putting it. Essentially the same machinery that produces the étale homotopy type would work for differentiable manifolds (and produces the "correct" homotopy type) but in this case it is not necessary and taking the homotopy type associated to the underlying topological space works as well. | |
| Oct 12, 2018 at 19:38 | comment | added | Qfwfq | I re-bumped into this question, so even though years later I take the occasion to ask @Denis Nardin: so probably we can say (a bit tautologically) the following? In studying the (étale or whatever) homotopy type of an algebraic variety the underlying topological space does not happen to serve as a model; one still has a nice CG Hausdorff model for the homotopy type though, which happens to be the geometric realization of a Kan complex. For differentiable manifolds (resp. classical homotopy type of $X(\mathbb{C})$) the underlying topological space (resp. the analytic topol.) is already enough. | |
| Sep 21, 2017 at 20:26 | vote | accept | coudy | ||
| Sep 21, 2017 at 20:26 | vote | accept | coudy | ||
| Sep 21, 2017 at 20:26 | |||||
| Oct 9, 2016 at 2:30 | answer | added | Tim Campion | timeline score: 14 | |
| Oct 5, 2016 at 18:37 | answer | added | John Rognes | timeline score: 14 | |
| Sep 1, 2016 at 15:02 | answer | added | user51223 | timeline score: 1 | |
| Sep 1, 2016 at 12:26 | comment | added | Philippe Gaucher | Concerning the interest of weak Hausdorff spaces, see my answer http://mathoverflow.net/a/204627/24563. | |
| Sep 1, 2016 at 11:28 | comment | added | Denis Nardin | @Qfwfq Usually people do not study the underlying topological space of algebraic varieties, but rather more exoteric objects like the étale homotopy type which is a (pro)homotopy type in the sense I described (that is a (pro)Kan complex) | |
| Sep 1, 2016 at 6:58 | answer | added | Ronnie Brown | timeline score: 4 | |
| Sep 1, 2016 at 2:41 | comment | added | David Roberts♦ | Actually one can be happy with CG Weak Hausdorff spaces. A Qfwfq points out, the underlying topological space of the ringed space underlying an algebraic variety is not Hausdorff, but IMHO classical point-set topology is not the right thing to study varieties with, since one should probably be at least looking at étale topology or other Grothendieck topologies. | |
| Sep 1, 2016 at 1:00 | comment | added | Qfwfq | Algebraic varieties are not Hausdorff... | |
| S Aug 31, 2016 at 23:41 | history | suggested | Martin Sleziak |
corrected tags: (gen.general-topology) -> (gn.general-topology)
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| Aug 31, 2016 at 23:30 | review | Suggested edits | |||
| S Aug 31, 2016 at 23:41 | |||||
| Aug 30, 2016 at 17:51 | comment | added | Dylan Wilson | (As a sanity check, it is indeed true that if you kill the real line in the above homotopy theory, you recover the homotopy theory of spaces as we know it.) In any case, topological spaces work just fine, and are probably not going anywhere any time soon. Though people are getting good at avoiding them when they're unnecessary. | |
| Aug 30, 2016 at 17:49 | comment | added | Dylan Wilson | I feel like 'spaces' arise for two completely distinct reasons. There are manifolds, and there are $\infty$-groupoids. The latter are combinatorial and can be modeled by simplicial sets, and the former are very intuitive and behave how we want them to behave (no pathological point-set nonsense). Sometimes we want to do space-y things with objects that aren't quite manifolds, but they're close enough... I think most of the examples are captured if we copy motivic folk and look at sheaves (of $\infty$-groupoids) on the site of manifolds. | |
| Aug 30, 2016 at 16:02 | comment | added | Justin Campbell | As I understand it, in homotopy type theory space (synonymous with $\infty$-groupoid) is a primitive notion. Perhaps if these foundations are widely accepted then one will be able to work "invariantly" or "model-independently" in a perfectly rigorous manner. As Denis Nardin points out, this is already common practice. | |
| Aug 30, 2016 at 14:56 | answer | added | tttbase | timeline score: 3 | |
| Aug 30, 2016 at 14:16 | comment | added | Denis Nardin | I think that for the purposes of homotopy theory every convenient category of topological spaces (in the sense of Steenrod) is acceptable (see ncatlab.org/nlab/show/convenient+category+of+topological+spaces for a thorough discussion of the issue). These technicalities are often swept under the rug and considered unimportant (for example many topics in homotopy theory can be done as easily using Kan complexes instead of topological spaces, obtaining equivalent theorems) | |
| Aug 30, 2016 at 14:11 | history | asked | coudy | CC BY-SA 3.0 |