Timeline for Why sheaves are important and why do we care about them? [closed]
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2018 at 16:10 | vote | accept | Lolman | ||
| Apr 9, 2023 at 17:53 | |||||
| Jun 18, 2018 at 1:35 | history | closed |
Mike Shulman Stefan Kohl♦ R.P. abx Pace Nielsen |
Needs details or clarity | |
| Jun 17, 2018 at 13:53 | answer | added | fosco | timeline score: 12 | |
| Jun 17, 2018 at 12:15 | comment | added | Lolman | @Yemon all mathematician try to relate their fields with others. CT gives a general framework, sometimes unneded,sometimes useful, sometimes insightful. Having a new framework or insight is better than not having it usually. My point was that too many things fit the idea of a sheaf. If you are interested I will send you my thoughts on stochastic processes one day. | |
| Jun 17, 2018 at 12:13 | comment | added | Lolman | @Fosco to me the answer seems easy now. Most constructs in analysis are around the reals, a poset. So at 1categorical level it they are quite poor. But everything changes at 2categorical level. E.G. the limit in metric spaces is a weighted limit. | |
| Jun 17, 2018 at 12:07 | comment | added | Lolman | It seems that somehow my question seems to have been forgotten. Maybe I'll formulate it better in a future time. | |
| Jun 17, 2018 at 10:43 | comment | added | fosco | @YemonChoi Allow for an OT before the thread is closed (I hope it won't be); please don't take it as an insult. Since I met category theory eleven (!) years ago, I question myself at least once a day about what is precisely the intrinsic feature of mathematical analysis preventing it to be described by category theory. I've gained no satisfactory answer since then. I'd be happy to hear your opinion on that, but of course this needs a different thread. (see also Freyd, "I've always disliked analysis"). | |
| Jun 17, 2018 at 2:50 | review | Close votes | |||
| Jun 18, 2018 at 1:35 | |||||
| Jun 17, 2018 at 0:37 | comment | added | Yemon Choi | "Maybe we should consider rings on processes or gluing of processes. Or maybe something already is a sheaf without us knowing." On the other hand, how about actually studying stochastic processes, reading works of e.g. Meyer, Dellacherie, Doob ... and considering the hypothesis that not everything in mathematics fits into your own personal view of what happens in mathematics ? (I speak as an analyst who's actually very sympathetic towards categorical methods/perspectives in analysis, bu thinks derived functors don't help characterize the endomorphisms of the Banach algebra $A_+({\bf T})$.) | |
| Jun 16, 2018 at 23:12 | comment | added | Lolman | @Yemon I edited the question. Reading wiki for stochastic process I would say that it is more of a morphism than anything else. And ao a lax limit woould capture the category about them. But if they are the object they themself are not interesting. I ahold dwell into them more to be able to say something, but they do not appear to satisfy any equation as yo be sheaf-like. Maybe we should consider rings on processes or gluing of processes. Or maybe something already is a sheaf without us knowing. Just because a manifold is a sheaf it doesn't mean that everyone calls it that. | |
| Jun 16, 2018 at 23:04 | history | edited | Lolman | CC BY-SA 4.0 |
added 676 characters in body
|
| Jun 16, 2018 at 17:49 | comment | added | Yemon Choi | Who is "we" meant to be in this context? I mean, I care about sheaves for analytic/topological reasons, people not so far from my interests might care about them because of e.g. Cartan's Theorems A and B. Also, "Why it seems that mathematics is made of sheaves" would be news to people doing e.g. stochastic processes - or is that somehow not mathematics? | |
| Jun 16, 2018 at 13:02 | comment | added | Phil Tosteson | I think the best motivations are not purely categorical: sheaves are important because local to global principles are important in geometry and topology. However for categorical motivation, given any category, you can try to present it as a category of functors to set via the yoneda lemma. Various classes of categories correspond to various classes of functor. The book "Locally presentable and accessible categories" might be helpful. | |
| Jun 16, 2018 at 12:36 | comment | added | fosco | (as an aside comment, $Set$ has a universal property, it's the terminal object of a suitable category of toposes; we choose $Set$ because it's the simplest we can go, but I feel a lot of topos theory is "over a generic base", i.e. you fix a topos $\cal S$ and study the category ${\bf Topoi}/{\cal S}$; of course ${\bf Topoi}/Set\cong{\bf Topoi}$) | |
| Jun 16, 2018 at 12:34 | comment | added | fosco | > Why it seems that mathematics is made of sheaves? Maybe because every category of sheaves allows you to instantiate known mathematical structures inside it, thanks to its internal language? | |
| Jun 16, 2018 at 11:52 | comment | added | Wojowu | That was meant to answer the first question "why do we prefer $Set$ above all else?". We simply don't and in many contexts sheaves on $Set$ are next to useless. | |
| Jun 16, 2018 at 11:49 | comment | added | Lolman | Saying that sheaves pop out in some algebraic geometry theory isn't an answer. Why it seems that mathematics is made of sheaves? | |
| Jun 16, 2018 at 11:48 | comment | added | Wojowu | @Lolman My point was that sheaves on $Set$ are not central in algebraic geometry. | |
| Jun 16, 2018 at 11:45 | comment | added | Lolman | @Wojowu you aren't really helping. An affine scheme is a particular ring object in a topos. So by itself already a sheaf. A scheme is a glueing of affine schemes, again a sheaf. An O(X)-module? A sheaf of abelian groups with an action of a scheme, that is another sheaf. | |
| Jun 16, 2018 at 11:29 | comment | added | Wojowu | If you read up on algebraic geometry, you will see that sheaves of rings play the central role there. | |
| Jun 16, 2018 at 11:24 | comment | added | Lolman | @Will The question is why sheaves not on $Set$. Because "why sheaves" is unclear. I edited, should be better. | |
| Jun 16, 2018 at 11:22 | history | edited | Lolman | CC BY-SA 4.0 |
added 132 characters in body
|
| Jun 16, 2018 at 11:09 | comment | added | Will Sawin | So your two questions are why we only use sheaves of sets, and why we also use sheaves of groups, rings, and modules? Don't these contradict each other? | |
| Jun 16, 2018 at 10:59 | history | asked | Lolman | CC BY-SA 4.0 |