Timeline for Is an injective smooth map an immersion?
Current License: CC BY-SA 4.0
16 events
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| S Sep 29, 2022 at 18:37 | history | edited | Glorfindel | CC BY-SA 4.0 |
Edited to use MathJax
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| S Sep 29, 2022 at 18:37 | history | suggested | Crypton | CC BY-SA 4.0 |
Edited to use MathJax
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| Sep 29, 2022 at 18:27 | review | Suggested edits | |||
| S Sep 29, 2022 at 18:37 | |||||
| Feb 15, 2010 at 10:17 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
added 245 characters in body
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| Feb 15, 2010 at 9:40 | comment | added | Harry Gindi | Yeah, exactly. The reason that the functors of points are especially nice is that we can characterize our maps as actual injective local diffeomorphisms without really referring to the underlying spaces by specifying everything locally. That's what I really like about sheaves in general. | |
| Feb 15, 2010 at 9:34 | comment | added | Ryan Budney | They're local diffeos, sure. But a covering map is a local diffeo without a right or left inverse, in general. | |
| Feb 15, 2010 at 9:33 | comment | added | Harry Gindi | When I said "morphisms," I probably should have said "structure morphisms", which are denoted by P. They are a distinguished class of morphisms that determine which sheaves are geometric varieties (generalization of schemes, manifolds in that they are "covered by" open immersions of representable functors). | |
| Feb 15, 2010 at 9:30 | comment | added | Harry Gindi | Check out cours 2 page 4 example 4. In fact, the open immersions are characterized precisely by being injective. Like any good functor of points theory, the underlying topological space is not important. Precisely, open immersions are injective (at the sheaf level) local diffeomorphisms. | |
| Feb 15, 2010 at 9:21 | comment | added | Ryan Budney | Open immersions generally aren't injective, either. This isn't a "failure" of the theory you're seeing. Your expectations are too narrow. | |
| Feb 15, 2010 at 8:56 | comment | added | Harry Gindi | It's interesting if you think about this in terms of Toen's "Geometric Contexts". I see it as a failure of the classical theory that injective morphisms do not correspond to immersions. Rather, if we replace the classical notions with functors of points, a lot of these "degeneracies", if they could be called that, disappear to some extent. This is because injective morphisms (our choice of morphisms depends on the geometric context, of course) do in fact become "open" immersions. | |
| Feb 15, 2010 at 6:20 | comment | added | Daniel Barter | thanks for the help! The cubic example was just what i was looking for. i feel a bit silly now though | |
| Feb 15, 2010 at 6:07 | comment | added | Kevin H. Lin | algori: I fixed it. And yeah that works too. | |
| Feb 15, 2010 at 6:05 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
added 2 characters in body
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| Feb 15, 2010 at 6:03 | comment | added | algori | Kevin -- this is not continuous;) On a more serious level: $t\mapsto t^3$ would do, no multivariate calculus necessary. | |
| Feb 15, 2010 at 6:03 | comment | added | Ryan Budney | I give a homework problem to my manifolds students ,asking them to construct a smooth injection (an immersion at all but one point)from $\mathbb R$ and onto the "corner space" $(x,y)$ such that $x \geq 0$ and $y=0$ union $x=0$ and $y \geq 0$. | |
| Feb 15, 2010 at 5:55 | history | answered | Kevin H. Lin | CC BY-SA 2.5 |