Timeline for Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?
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| Oct 1, 2022 at 18:27 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Oct 1, 2022 at 16:52 | comment | added | Dmitri Pavlov | @Z.M: Yes, so (concrete) sheaves of sets on the site of complex manifolds are equivalent to (concrete) sheaves of sets on the site of Stein manifolds, which are equivalent to (concrete) sheaves of sets on the site of polydisks. These are the precise analogues of diffeological spaces in the complex analytic setting. See ncatlab.org/nlab/show/complex+analytic+infinity-groupoid on the nLab. | |
| Oct 1, 2022 at 16:34 | comment | added | Z. M | No, I am asking an analytic analogue of diffeologies. Complex analytic spaces are of finite type, in this sense it is not an analogue of diffeologies (there are other differences: in some sense the site might be somehow polydiscs instead of Stein's, I guess). | |
| Oct 1, 2022 at 15:16 | comment | added | Dmitri Pavlov | @Z.M: Yes, the functor of points approach can be used to define complex analytic spaces as certain sheaves of sets on the site of Stein spaces. This is completely analogous to how schemes can be defined as certain sheaves of sets on the site of affine schemes, i.e., the opposite category of commutative rings. | |
| Oct 1, 2022 at 10:46 | comment | added | Z. M | Is there any analytic version of this? An analytic version might also have a p-adic analogue (aka. locally analytic geometry). | |
| Sep 30, 2022 at 21:58 | comment | added | Dmitri Pavlov | @AivazianArshak: I would not characterize the cited work of Christensen and Wu as “finicky”, in fact, it gives a simple and conceptual treatment of tangent bundles of diffeological spaces. | |
| Sep 30, 2022 at 21:31 | comment | added | Arshak Aivazian | Now I looked through the book and see that the tangent space is defined just as well as for stacks, and judging by the things that you mention and the content of the book, there are no special obstacles for the development of differential geometry in this context (and in the article mentioned in the comment, it was about problems with other versions of definitions). It will be interesting to see what stops working or becomes more complicate when the condition of concreteness of the sheaves is abandoned. | |
| Sep 30, 2022 at 21:31 | comment | added | Arshak Aivazian | Thank you! I asked a related question earlier, and from this comment it seemed to me that diffiological spaces have unpleasant problems at the level of defining a tangent space, so I stopped being interested in them. | |
| Sep 30, 2022 at 19:33 | vote | accept | Arshak Aivazian | ||
| Sep 30, 2022 at 18:44 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Sep 30, 2022 at 15:52 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |