Timeline for Is there a model category describing shape theory?
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Apr 26, 2020 at 16:03 | history | bounty ended | CommunityBot | ||
| S Apr 26, 2020 at 16:03 | history | notice removed | CommunityBot | ||
| Apr 24, 2020 at 15:14 | history | edited | Sebastian Goette | CC BY-SA 4.0 |
added 208 characters in body
|
| S Apr 18, 2020 at 14:51 | history | bounty started | Sebastian Goette | ||
| S Apr 18, 2020 at 14:51 | history | notice added | Sebastian Goette | Draw attention | |
| Nov 8, 2019 at 15:34 | comment | added | Sebastian Goette | @DenisNardin Thanks for the suggestion. I see the point now and did the edit. | |
| Nov 8, 2019 at 15:33 | history | edited | Sebastian Goette | CC BY-SA 4.0 |
Removed the algebraic varieties as suggested by Denis Nardin
|
| Nov 8, 2019 at 12:26 | comment | added | Denis Nardin | @SebastianGoette Well, it is sort of the same definition, only you need to generalize it to a notion of the shape of a topos rather than just of topological spaces (you might have heard it under the name "étale homotopy type").The point is that the underlying space of an algebraic variety does not have much useful information (e.g. all curves over an alg. closed field are homeomorphic!),and you need more refined objects (e.g the étale topos)to recover the information present in the topology for manifolds. Still, your question is interesting, I would only remove the reference to alg. varieties. | |
| Nov 8, 2019 at 11:29 | comment | added | Sebastian Goette | @DenisNardin I was really thinking of the shape as a topological space, as in the wikipedia article. I was not even aware that there is a separate definition in algebraic geometry. | |
| Nov 7, 2019 at 17:02 | comment | added | Denis Nardin | @SimonHenry It's not idempotent, but it becomes one if you consider the profinite shape instead (see the appendix to SAG and the Barwick-Glasman-Haine work on the stratified shape) | |
| Nov 7, 2019 at 15:54 | comment | added | Simon Henry | In Lurie's perspective shape theory is an adjunction between the $\infty$-category of $\infty$-toposes and the $\infty$-category of pro-spaces. I always had the feeling that this adjunction is idempotent ( not sure if this is really true though, is there someone who know ?) if it is the case "shapes" are a localization of the category of pro-spaces, as such it should be possible to construct a model structure representing them (maybe on pro-simplicial sets...). If this works one can start thinking about whether this transfer to a model structure on some category of topological spaces... | |
| Nov 7, 2019 at 13:52 | comment | added | Denis Nardin | @SebastianGoette I'll have to admit I am a bit confused. What do you mean with the shape of an algebraic variety then? The notion I know is not encoded in the topology. | |
| Nov 7, 2019 at 12:23 | comment | added | Sebastian Goette | @DenisNardin I don't expect to see the 'algebraic' cohomology (in whatever sense) here, because the structure sheaf seems to be an extra datum that the topological space does not have. But maybe apart from constant sheaves, we also want to (maybe have to) consider constructible sheaves. But then all information we need is still encoded in the topology. | |
| Nov 7, 2019 at 12:16 | comment | added | Sebastian Goette | @SimonHenry I must admit that I have no idea. The correct notion of equivalence seems to be part of the problem. Also I wonder whether it is possible to work on topological spaces, or whether one needs topoi. | |
| Nov 6, 2019 at 18:24 | comment | added | Simon Henry | What is the precise definition of 'shape equivalences' (it is not in the wikipedia link) ? I'm not expert in shape theory but I believe there has been several non-equivalent definition... is it equivalent to the definition Lurie gave for infinity toposes ? | |
| Nov 6, 2019 at 17:56 | history | edited | Sebastian Goette | CC BY-SA 4.0 |
Minor edits
|
| Nov 6, 2019 at 17:36 | comment | added | Denis Nardin | Hmm... the shape of algebraic varieties does not depend on the underlying topological space, unless I'm mistaken (rather, it is the shape of the étale topos), so I doubt a naive approach would work there. | |
| Nov 6, 2019 at 16:52 | history | asked | Sebastian Goette | CC BY-SA 4.0 |