Timeline for What's the "correct" smooth structure on the category of manifolds?
Current License: CC BY-SA 2.5
18 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 3, 2014 at 0:56 | vote | accept | Theo Johnson-Freyd | ||
| Apr 10, 2010 at 20:43 | comment | added | Theo Johnson-Freyd | @DY: Fair enough. Incidentally, around here "bordism" has replaced the older word "cobordism", because it isn't "co" to anything. A "bordism" is a manifold with prescribed "bord"ers. | |
| Apr 1, 2010 at 1:01 | comment | added | Deane Yang | I still don't understand what you're after. You seem to want to define a cobordism (what's a "bordism"?) as a smooth curve in the space of manifolds, but I don't see why that's a reasonable expectation. I am sympathetic to wanting to put things like this into a clean abstract category theoretic framework, but, as others have said, it's rather difficult to find the "right path" without knowing what you are going to use this for. What new insight or theorem are you hoping to get this way? Or is there an existing insight or theorem that you're trying to restate in a more category-theoretic form? | |
| Mar 31, 2010 at 21:06 | answer | added | Andrew Stacey | timeline score: 5 | |
| Mar 31, 2010 at 19:32 | comment | added | Theo Johnson-Freyd | @JJV: every fiber bundle is trivializable over a contractible space. But certainly they are not all trivialized. This is a big difference from the category perspective. The point is what I do in the later paragraphs, by asking "What is Hom(M,Man)?" The answer is that a functor M -> Man is smooth iff it restricts to a curve along every curve \R -> M. So if Hom(\R,Man) are arbitrary bundles, then I think Hom(M,Man) are arbitrary bundles. But if Hom(\R,Man) are trivialized bundles, then Hom(M,Man) are trivialized bundles. | |
| Mar 31, 2010 at 19:29 | comment | added | Theo Johnson-Freyd | @TL: Thanks. I've never really known whether to say something is "soft". I wouldn't have asked the question if I didn't think it had real mathematical content --- I'm don't think that MO is the right place for "advice" questions, for example, but I could imagine that "soft question" means "devoid of actual math". | |
| Mar 31, 2010 at 19:06 | comment | added | Tom Leinster | Theo, I think the "soft question" tag is a bit modest (or something). I guess I can see why you put it on, but it's a question with real mathematical content, and I don't think it's soft at all. | |
| Mar 31, 2010 at 16:16 | history | edited | Theo Johnson-Freyd | CC BY-SA 2.5 |
added some motivation
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| Mar 31, 2010 at 10:44 | answer | added | Jeffrey Giansiracusa | timeline score: 2 | |
| Mar 31, 2010 at 8:20 | comment | added | Jan Jitse Venselaar | I agree with the above comments that some motivation would help to understand the question better. As for Definition 1, it feels like you lose too much by just considering $P \times \mathbb{R}$, or am I missing something and are your fiber bundles not trivial over contractible spaces? | |
| Mar 31, 2010 at 7:56 | answer | added | Konrad Waldorf | timeline score: 2 | |
| Mar 31, 2010 at 5:34 | comment | added | BCnrd | @Bo: what do you mean by "useful"? | |
| Mar 31, 2010 at 5:29 | comment | added | Ryan Budney | I agree with Carnahan and Yang, there is no correct definition until you have a purpose for the device you want to construct, be it a category or whatever. | |
| Mar 31, 2010 at 4:48 | comment | added | Bo Peng | I like your category. Feels like something potentially useful. | |
| Mar 31, 2010 at 3:36 | comment | added | Deane Yang | But what are you going to do with this? Are you really going to be working in a setting where the difference between the two really matters? (I mean these to be naive questions from someone uses manifolds only in the simplest ways and is still quite uneducated on why category theory is useful for studying differential geometry) | |
| Mar 31, 2010 at 3:30 | comment | added | S. Carnahan♦ | How do you plan to use this category? A potential application would give your question a good point of focus. | |
| Mar 31, 2010 at 2:57 | history | asked | Theo Johnson-Freyd | CC BY-SA 2.5 |