Timeline for Is the category of diffeological spaces a full subcategory of locally ringed spaces?
Current License: CC BY-SA 4.0
11 events
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| Mar 29, 2022 at 13:49 | comment | added | Arshak Aivazian | @SeverinBunk Thank you very much for the link to this book! I have long been interested in synthetic differential geometry - I think now that this is the best approach. But unfortunately it doesn't help me now -- see this comment of mine (correct me if I'm wrong) | |
| Mar 29, 2022 at 7:45 | comment | added | Severin Bunk | I am not aware of a "nice" embedding functor of diffeological spaces into locally ringed spaces. However, maybe you find the framework of $C^\infty$-rings and locally $C^\infty$-ringed spaces helpful; see, for instance, the book "Smooth infinitesimal analysis" by Moerdijk and Reyes, as well as Dominic Joyce's arxiv.org/abs/1001.0023v7. | |
| Mar 28, 2022 at 14:02 | comment | added | Simon Henry | Essentially yes. I wouldn't say it "shows" because the question didn't really specify how you build a locally ringed space out of a diffeological spaces - but at least it shows that Naive constructions using smooth function don't distinguish between the different non-commutative torus (or Irrational if you prefer). It is different from the discrete space though : the discrete space has plenty of non-constant smooth functions on it. | |
| Mar 28, 2022 at 11:59 | comment | added | Konrad Waldorf | @SimonHenry: Ok, now I see your point. The irrational torus has only constant smooth functions and has the discrete D-topology. Hence, it can not be distinguished from the discrete torus as locally ringed spaces. However, it is not diffeomorphic to the discrete torus. This shows that the functor is not full. Correct? | |
| Mar 28, 2022 at 11:39 | comment | added | Konrad Waldorf | And did you mean the "irrational" torus instead of "non-commutative"? Because the D-topology of the irrational torus is discrete, so it seems to fit very well that it doesn't have non-constant smooth functions. | |
| Mar 28, 2022 at 11:32 | comment | added | Simon Henry | I mean't no non-constant smooth functions to $\mathbb{R}$. | |
| Mar 28, 2022 at 11:24 | comment | added | Konrad Waldorf | @SimonHenry: what do you mean by "have no smooth function defined on them"? Every diffeological space has smooth functions, for example all constant ones. | |
| Mar 26, 2022 at 23:12 | comment | added | Arshak Aivazian | @SimonHenry Thanks! In fact, I'm looking for a small extension of the category of smooth manifolds to include the important naturally occurring infinite-dimensional manifolds (such as spaces of smooth functions), but still being a full subcategory of locally ringed spaces. Apparently, diffiological spaces are not suitable for my purposes. Now I asked this question separately. | |
| Mar 26, 2022 at 20:11 | comment | added | Simon Henry | Almost certainly not. Diffelofical strucutre can capture very singular spaces, like the non-commutative torus, that have no smooth function defined on them. I really don't see a way to attach to them a locally ringed space in a way that capture any interesting information | |
| Mar 26, 2022 at 19:34 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Mar 26, 2022 at 19:21 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |