Timeline for When do we study maps into an object or from the object to another object?
Current License: CC BY-SA 2.5
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| Nov 28, 2010 at 12:27 | comment | added | David Corfield | Session 6 of Conceptual Mathematics: "The point of view about maps indicated by the terms 'naming,' 'listing,' 'exemplifying,' and 'parameterizing' is to be considered as 'opposite' to the point of view indicated by the words 'sorting,' 'stacking,' 'fibering,' and 'partitioning'." (p. 83) lawvere and Schanuel then go on to explain this 'opposition' philosophically. | |
| Nov 28, 2010 at 5:42 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Nov 28, 2010 at 3:40 | comment | added | Brian | Thanks! I also started to think about the local vs. global like you. In the "local case", we do actually learn something about the local property when we look at "maps in," like the map $I\to M$ (in differentiable manifolds) and $\mathrm{Spec}k[x]/(x^2) \to X$ in Algebraic geometry. I am wondering if there is any other example along these lines. | |
| Nov 28, 2010 at 1:11 | comment | added | David Roberts♦ | Indeed, one often chooses a category of nice objects depending on the niceness of the 'maps out'. For example, one chooses locally convex vector spaces because they have enough maps to the base field. One considers completely regular spaces because they have enough maps to the unit interval to separate subspaces. Compact Hausdorff spaces are particularly nice in some respects because the ring of complex functions is a unital $C^\ast$ algebra. | |
| Nov 28, 2010 at 1:10 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
edited body; added 393 characters in body
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| Nov 28, 2010 at 1:10 | comment | added | Qiaochu Yuan | Oops. Yes, I did. | |
| Nov 28, 2010 at 1:04 | comment | added | Dylan Wilson | Do you mean morse function? | |
| Nov 28, 2010 at 0:27 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |