Timeline for Presheaves are locally sheaves?
Current License: CC BY-SA 2.5
20 events
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| Jan 14, 2011 at 2:21 | comment | added | Sándor Kovács | Dear Georges, I totally agree and my comment was more to those who read yours than to you. (See p.p.s above). What I meant to point out is that for sheaves this is not an issue = yet another reason why sheaves are good and presheaves are less so. | |
| Jan 14, 2011 at 1:24 | comment | added | Georges Elencwajg | (continuation) So I feel we should help beginners in that field by not only telling them what is true but also by drawing their attention to potential misconceptions and (this is more subjective) ambiguities. Anyway, thanks a lot for your expert and relevant comments. | |
| Jan 14, 2011 at 1:16 | comment | added | Georges Elencwajg | Dear Sándor, I agree with everything you say and I am sure we are in complete agreement on everything concerning the mathematical substance of (pre)sheaves. And, yes, my sentence on ambiguity was probably ambiguous! It's just that sheaves are pretty subtle the first time you meet them : all vector bundles give rise to sheaves that are locally but not globally isomorphic, etc. (to be continued) | |
| Jan 14, 2011 at 0:43 | comment | added | Sándor Kovács | p.p.s: And I agree that it is ambiguous to say that a presheaf is locally isomorphic to a sheaf but I don't think this is widely used. | |
| Jan 14, 2011 at 0:42 | comment | added | Sándor Kovács | p.s.: I am sure I am not saying anything new to you. :) | |
| Jan 14, 2011 at 0:40 | comment | added | Sándor Kovács | (cont'd from above) The point is that in your example the objects are not sheaves but pre-sheaves. So, it is perfectly all right to say that "Two sheaves are locally isomorphic if their stalks are isomorphic" because it is equivalent to saying that the ambient space has an open cover such that for any of the open sets in this cover the restrictions of the two sheaves are isomorphic. In some sense, this is the point of studying sheaves, instead of pre-sheaves, they can be re-built from their stalks, or one could say they are locally determined. | |
| Jan 14, 2011 at 0:37 | comment | added | Sándor Kovács | @Georges: Just a comment on your sentence "I find it ambiguous, as proved by this very question, to call a morphism of sheaves a "local isomorphism" if it is an isomorphism on the stalks.": Actually, it is not at all ambiguous, but there are different issues here: First of all, if you have a morphism of sheaves which is an isomorphism on stalks, then it is an isomorphism (don't need the "local"!). I think what you may have had in mind is to say that: Two sheaves are locally isomorphic if their stalks are isomorphic.This might seem ambiguous in light of your example, but it is not! (cont'd) | |
| Jan 14, 2011 at 0:32 | comment | added | Sándor Kovács | @Harry: it seems that your view on what "local isomorphism" means keeps coming up and it seems that no one else thinks that it should mean what you are suggesting. Requiring the existence of a morphism for saying that two objects are "locally isomorphic" is an overkill. In particular, in most cases, this will imply that those objects are actually isomorphic. Think of sheaves; if there is a morphism $\mathcal F\to \mathcal G$ that's an isomorphism on the stalks, then it is an isomorphism. | |
| Jan 13, 2011 at 21:13 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Added "An answer to Roy's question"
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| Jan 12, 2011 at 17:29 | comment | added | Georges Elencwajg | Dear Harry, since our discussion hinges on terminology, I have nothing to add: we just seem to have different definitions. Thank you for your interest in this post and for sharing your point of view. | |
| Jan 12, 2011 at 15:21 | comment | added | Harry Gindi | Dear Georges, two manifolds are locally diffeomorphic iff there exists a local diffeomorphism from one to the other (whence globally morphic). | |
| Jan 12, 2011 at 14:18 | comment | added | Georges Elencwajg | Harry, all manifolds of the same dimension are locally diffeomorphic, independently of any morphisms betwen them. On the other hand a submersion of a manifold onto another one of lower dimension has local sections about each point but the manifolds are certainly not locally diffeomorphic. Finally I have never heard the notion of objects being globally morphic . | |
| Jan 12, 2011 at 13:36 | comment | added | Harry Gindi | Also, Georges, I find it natural to require two objects to be globally morphic before we can talk about whether or not they're locally isomorphic. The OP's condition regarding restriction sheaves should instead be called "local satisfaction of the sheaf conditions". For instance, we say two manifolds are locally diffeomorphic if there exists a morphism admitting sections locally about each point. | |
| Jan 12, 2011 at 13:14 | comment | added | Harry Gindi | That's not why it's called a local isomorphism. It's called a local isomorphism because it becomes an isomorphism of "local objects". This is pretty standard terminology from as far back as Gabriel-Zisman. | |
| Jan 12, 2011 at 0:14 | vote | accept | Daniel Barter | ||
| Jan 11, 2011 at 17:38 | comment | added | Kevin Ventullo | An isomorphism on the level of stalks should probably be called an "infinitesimal isomorphism." | |
| Jan 11, 2011 at 9:39 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Changed last line of introduction. Added explanations in definition of presheaf $\mathcal F$
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| Jan 11, 2011 at 9:17 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
added last line "On the other hand..."
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| Jan 11, 2011 at 8:54 | history | edited | Georges Elencwajg | CC BY-SA 2.5 |
Added sentence in the introduction
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| Jan 11, 2011 at 8:48 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |