Timeline for Colimits in the category of smooth manifolds
Current License: CC BY-SA 2.5
9 events
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| May 30, 2018 at 15:26 | comment | added | Tom | I know it's usually not good stile to comment something written 8 years ago, but I just wanted to mention that a "continuous" map from a metric space to a complete space with noncomplete image won't do. (For example the arctan function takes complete R to noncomplete open interval). So, you should take an isometric embedding (or a uniformly continuous embedding, since completeness is really a property of uniform spaces). | |
| Mar 29, 2010 at 12:47 | comment | added | Chris Schommer-Pries | The calculation of the dimension of I^n/I^{n+1} is a standard part of the theory of jets and jet bundles. It is the dimension of the vector space of those functions which vanish up to order (n-1) modulo those which vanish up to order n. I first learned about this from the book "Stable Mappings and Their Singularities" by Golubitsky and Guillemin. I suspect that there are many other references out there, too. | |
| Mar 29, 2010 at 0:35 | comment | added | Reid Barton | In my experience people usually say "the functor Man^op -> Ring preserves limits" and then there is a certain amount of head-scratching to figure out what is actually happening. | |
| Mar 27, 2010 at 12:37 | comment | added | Andrew Stacey | Whoops! Yes. I suppose I could rescue it by considering it as a function in to C^infty-rings^OP. There's probably some technical term for this property as well (a contravariant functor taking colimits to limits or vice-versa). | |
| Mar 27, 2010 at 9:39 | comment | added | user2146 |
By "As the functor $M\mapsto C^\infty(M)$ [...] preserves colimits and so is suitable for the argument to go through." you mean that it takes them to limits, right? Anyhow, +1 and thanks for making me wonder about Fröhlicher spaces.
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| Mar 27, 2010 at 3:02 | comment | added | Andrew Stacey | One of my reasons for recasting Reid's answer was that that step seemed very opaque to me as I don't think well "algebraically" about manifolds! Once I'd worked out what was happening for myself, I figured it worth posting in case anyone else was like me. So I can't give you a reference for that bit - you should ask Reid and Chris for that! | |
| Mar 27, 2010 at 1:28 | vote | accept | Glen M Wilson | ||
| Mar 27, 2010 at 1:28 | comment | added | Glen M Wilson | Andrew, this is a great explanation! Thank you for elaborating on the general method behind showing this; it reinforced my understanding of Reid's solution (which is very intriguing). Your method using Frölicher spaces is very nice. P.S.: I'm still slightly confused as to what goes into computing $dim_{\mathbb{R}} I^{n+1}/I^{n}$, but I think it might just take some elbow grease to get it to work. I hate to belabor the point, but if this computation isn't all that trivial, I'd appreciate a literature reference. If not, just ignore me! =) | |
| Mar 26, 2010 at 22:54 | history | answered | Andrew Stacey | CC BY-SA 2.5 |