Timeline for Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?
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23 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2022 at 4:18 | comment | added | Z. M | @DmitriPavlov Thanks. If it is lawful, maybe also add a link in ncatlab.org/nlab/show/Fr%C3%A9chet+manifold#Losik94 and wikipedia page (there is a footnote for Losik's paper) en.wikipedia.org/wiki/Diffeology | |
| Oct 4, 2022 at 0:31 | comment | added | Dmitri Pavlov | @Z.M: Losik's paper is available here: dmitripavlov.org/scans/… | |
| Oct 1, 2022 at 18:27 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Oct 1, 2022 at 5:03 | comment | added | Alec Rhea | I agree that abandoning excluded middle would be ambitious for a 10th grade class, to say the least haha -- the texts I had in mind were Synthetic Geometry of Manifolds and Synthetic Differential Geometry by Anders Kock. | |
| Sep 30, 2022 at 21:55 | comment | added | Dmitri Pavlov | @Z.M: A translation does exist, but it appears that Springer's online archive only covers years from 2007 on. I ordered it through my library. | |
| Sep 30, 2022 at 21:06 | comment | added | Z. M | @DmitriPavlov Do you have any English material which summarizes that proof? There seems no English translation of that paper. I only find an article by the same author which summarizes the theorems but not the proofs. | |
| Sep 30, 2022 at 20:30 | comment | added | Arshak Aivazian | @AlecRhea The fact is that I give my students a course on differential geometry and, in parallel, a course on the theory of sheaves (more precisely, sheaves on sites and Grothendieck topoi). I would like to use the powerful mechanism of sheaves to construct all the missing spaces, but I would not want the only place where the differential geometry lives for them is only non-classical topoi. I consider constructive mathematics fundamentally more natural and important, but nevertheless it seems inappropriate (today) to force students in the 10th grade to abandon the law of the excluded middle. | |
| Sep 30, 2022 at 20:30 | comment | added | Arshak Aivazian | @AlecRhea Thank you, I like synthetic differential geometry and I will definitely study it! I have several texts, but please send yours - perhaps I will find new and interesting ones among them. However, it doesn't work for me right now. | |
| Sep 30, 2022 at 19:42 | comment | added | Dmitri Pavlov | @Z.M: The Banach or Fréchet structure can be canonically recovered from the sheaf structure (on finite-dimensional manifolds), see the paper of Losik “Fréchet manifolds as diffeological spaces”. It proves that the category of Fréchet manifolds embeds fully faithfully in the category of diffeological spaces. | |
| Sep 30, 2022 at 19:33 | vote | accept | Arshak Aivazian | ||
| Sep 30, 2022 at 19:23 | comment | added | Z. M | @DmitriPavlov Thanks. But I am not sure whether this recovers the "infinite-dimensional manifold" structure, or even the topological structure. For example, if we take some form of "geometric realization" of a sheaf in the category of topological spaces, it is a co-end, thus a colimit of finite dimensional manifolds, so there is no Banach structure (on the tangent spaces) or something like this. | |
| Sep 30, 2022 at 18:48 | comment | added | Dmitri Pavlov | @Z.M: The category of sheaves of sets (or simplicial sets) on the site of smooth manifolds is a cartesian closed category. In particular, the internal hom Hom(M,N) between two smooth manifolds M and N exists and has the expected properties. For example, the tangent space at any point can be computed as relative vector fields along a smooth map, etc. Likewise, the Lie algebra of the (infinite-dimensional) group of diffeomorphisms M→M can be computed as the Lie algebra of vector fields on M. The book by Iglesias-Zemmour explains all this. | |
| Sep 30, 2022 at 18:28 | history | became hot network question | |||
| Sep 30, 2022 at 18:22 | comment | added | Alec Rhea | Is synthetic differential geometry appealing to you? The general setting is a topos (sometimes Grothendieck, sometimes just well-pointed or even less than a topos works) — I have a few good texts in mind if that sounds interesting. | |
| Sep 30, 2022 at 17:14 | comment | added | Z. M | I wonder the relation of the first sentence (about infinite-dimensional manifolds) and the later. If I understand correctly, smooth spaces / stacks are finitary in nature. | |
| Sep 30, 2022 at 17:00 | comment | added | Andy Putman | I agree with @ChrisGerig here in the sense that this is not how mainstream people working in differential geometry conceive of their subject. This is not to say that it might not be useful for whatever it is you're trying to do, just that any such book won't help you read the differential geometry literature. | |
| Sep 30, 2022 at 15:52 | answer | added | Dmitri Pavlov | timeline score: 12 | |
| Sep 30, 2022 at 15:12 | comment | added | Arshak Aivazian | @ChrisGerig I don't understand what you mean. | |
| Sep 30, 2022 at 14:45 | comment | added | Chris Gerig | I'd go as far to say you're not really doing differential geometry anymore... | |
| Sep 30, 2022 at 12:01 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Sep 30, 2022 at 10:46 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Sep 30, 2022 at 10:35 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Sep 30, 2022 at 10:27 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |