Timeline for When do we study maps into an object or from the object to another object?
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| Nov 28, 2010 at 6:58 | comment | added | Theo Johnson-Freyd | @Qiaochu: Funny, I would have thought the existence of math.SE would mean we could be close more questions, rather than fewer? OTOH, I'm also less eager to close than may I used to be --- I certainly don't vote to close either this question of Martin's old one. I did vote down this question because I think it's "unclear or not useful". | |
| Nov 28, 2010 at 5:42 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Nov 28, 2010 at 4:26 | history | edited | Brian | CC BY-SA 2.5 |
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| Nov 28, 2010 at 3:58 | comment | added | Qiaochu Yuan | @Martin: at least for me, my closing policy was stricter before the advent of math.SE than it is now. I think norms have just shifted in the intervening months. | |
| Nov 28, 2010 at 3:55 | comment | added | Theo Johnson-Freyd | -1. I disagree with the premise of the question. | |
| Nov 28, 2010 at 3:43 | history | edited | Brian | CC BY-SA 2.5 |
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| Nov 28, 2010 at 1:30 | answer | added | Akhil Mathew | timeline score: 3 | |
| Nov 28, 2010 at 0:27 | answer | added | Qiaochu Yuan | timeline score: 10 | |
| Nov 27, 2010 at 23:22 | comment | added | Ryan Budney | IMO the premise of the question is mis-informed. In particular the differential geometry example is largely missing the point of the subject -- the study of the properties of smooth maps $M \to \mathbb R$ is part of differential topology, and even then it fits into the study of smooth maps $M \to N$, and in that setting the directionality of the map becomes rather irrelevant. | |
| Nov 27, 2010 at 22:56 | comment | added | Martin Brandenburg | Yes Dan, I wonder why my question was closed and this here gets good answers ... | |
| Nov 27, 2010 at 22:11 | answer | added | André Henriques | timeline score: 8 | |
| Nov 27, 2010 at 20:38 | answer | added | Terry Tao | timeline score: 19 | |
| Nov 27, 2010 at 20:33 | comment | added | Terry Tao | In dynamics (or ODE, or ergodic theory) we study orbits and trajectories, i.e. maps from Z or R into the space of interest. (One also studies factor maps from the space into simpler spaces, so it goes both ways in this case.) And in virtually any subject of mathematics, we study elements of a space, i.e. maps from a point into the space... | |
| Nov 27, 2010 at 19:59 | comment | added | Kevin H. Lin | @Brian: That only works for affine schemes. | |
| Nov 27, 2010 at 19:52 | comment | added | Dan Petersen | See also this old (closed) question: mathoverflow.net/questions/12326/co-objects-are-better-closed | |
| Nov 27, 2010 at 19:52 | comment | added | Brian | @Kevin Lin: At least in Algebraic Geometry, the surprising fact is that $\mathrm{hom}(\mathrm{Spec}(A), \mathrm{Spec}(\mathbb{Z}[x]))$ carries all the information of $\mathrm{Spec}(A)$. So, in this case, we only need to consider ONE set of maps from our object to $\mathrm{Spec}(\mathbb{Z}[x])$. | |
| Nov 27, 2010 at 19:45 | history | edited | Brian | CC BY-SA 2.5 |
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| Nov 27, 2010 at 19:40 | history | edited | Brian | CC BY-SA 2.5 |
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| Nov 27, 2010 at 19:29 | answer | added | Pete L. Clark | timeline score: 11 | |
| Nov 27, 2010 at 19:25 | comment | added | Kevin H. Lin | In algebraic topology, we study maps from simplices to spaces -- homology. We also study maps from spheres to spaces -- homotopy groups. As Karl says, in algebraic geometry the study of maps of curves into varieties has been very fruitful. This has also been fruitful in symplectic geometry -- look up Gromov-Witten theory. It should not be surprising that maps to an object are interesting, because of Yoneda's lemma, which says that maps to an object contain essentially all information about that object. See mathoverflow.net/questions/3184/… | |
| Nov 27, 2010 at 19:24 | comment | added | Brian | @José Figueroa-O'Farrill: The question is not intended to be misleading. If I had known everything about it, I wouldn't ask the question anyway. | |
| Nov 27, 2010 at 19:18 | comment | added | José Figueroa-O'Farrill | The question is a little misleading. There are plenty of cases where one learns about an object by mapping into it. For example, in differential geometry, the study of geodesics, minimal surfaces,... can shed a lot of light on the target manifold, as can the theory of harmonic maps (a.k.a. sigma models in the Physics literature). | |
| Nov 27, 2010 at 19:16 | comment | added | José Figueroa-O'Farrill | There is a related question: mathoverflow.net/questions/17325/… | |
| Nov 27, 2010 at 19:07 | comment | added | Karl Schwede | In algebraic geometry, people certainly study maps from curves (especially $\mathbb{P}^1$) to various varieties. The study of rational curves (ie, exactly such maps) on a given variety is an area of very active research. Higher dimensional generalizations have also begun to be explored. | |
| Nov 27, 2010 at 19:00 | history | asked | Brian | CC BY-SA 2.5 |