Talk:Sheaf (mathematics)

From Wikipedia, the free encyclopedia

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	A-BB+ Class	Top Priority	Field: Topology
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Nice work, but lead needs to be expanded as an overview of the article, and some diagrams might help convey the abstract concepts. Geometry guy 22:55, 9 June 2007 (UTC)

[edit] Comments

I don't know if I'm the only one, but I know what sheaves are, and I thought this page wasn't easy to grok. Loisel 07:56 30 Jun 2003 (UTC)

I don't know what sheaves are, but I have studied category theory and other advanced mathematics, and I found this page completely impenetrable. Dominus 01:27, 11 Aug 2003 (UTC)

The article reads better now. Loisel 20:53, 5 Sep 2003 (UTC)

Just like to say it is a jolly good article now, infomal introduction, technical description, history. Could do with a list of references, more along the lines of 'further reading' than historical sources? Billlion 20:40, 14 Dec 2004 (UTC)

Someone has added sheaves of groups etc.: this of course only works for sheaves with values in some concrete category - not, as is implied, any category.

Charles Matthews 07:36, 10 Sep 2003 (UTC)

Why does one need a concrete category in general anyway? It seems to me that the reason is to be able to state the sheaf axiom, which is about elements of the F(U). Is that why? Can one state the sheaf axiom entirely arrow-theoretically using, say, universal properties? -- Miguel 18:30 Nov 15 2003 (UTC)

Yes, we use concrete categories for the sheaf axiom. It can be stated abstractly as follows: if U_i are open subsets of X with union U, then we get three maps:

• r : F(U) → Π_i F(U_i), induced from the restrictions of F(U) to the sets U_i
• s : Π_i F(U_i) → Π_k,l F(U_k ∩U_l), induced from the restrictions of F(U_i) to the sets U_i∩U_l
• t : Π_i F(U_i) → Π_k,l F(U_k ∩U_l), induced from the restrictions of F(U_i) to the sets U_k∩U_i

The sheaf axiom then says that r is the equalizer of s and t. Obviously, for an arbitrary category C we would in addition have to assume that all occurring products exist in C.

It's probably overkill to put this in the article. However, I don't entirely like the "concrete category" assumption either. In order to discuss stalks, we need that our category contains all direct limits, and not all concrete categories do. I don't know what to do about it right now. AxelBoldt 15:39, 22 Nov 2003 (UTC)

Yes, it's kind of the wrong approach, in that a sheaf of abelian groups is (morally) an abelian group in a sheaf category, rather than something in a functor category to abelian groups; at least I think that explains better what limits you need where. But given just that one case to think about (which is fundamental for applications), it's not really harmful to say the things here, I believe.

Charles Matthews 16:14, 26 Nov 2003 (UTC)

Correct me if I'm wrong, but isn't the stalk at a point defined as the direct limit resp. colimit rather than (inverse) limit as the article states? Rvollmert 19:19, 26 Oct 2003 (UTC)

You are correct. I found this concrete example helpful, which comes from the Zariski topology. Let R = the ring of integers. One special case of the question is this: with R₍₂₎ the ring of fractions with odd denominators (the stalk at the point 2R of Spec(R)), and S^-1R running over the subrings corresponding to finite sets of permitted prime numbers in the denominators. Then R₍₂₎ is the direct limit of the S^-1R.

Perhaps this could be adapted to an example for the article, or the Zariski topology article.

Charles Matthews 09:48, 28 Oct 2003 (UTC)

It has been suggested that this article needs (a) history and (b) motivation. That's quite true, naturally. There is an extensive history of sheaf theory in the article by Houzel in the Kashiwara-Schapira book on sheaves on real manifolds. It might be hard to do that justice. The motivations from algebraic geometry can be cited. They actually came after those from several complex variables - about which there is zilch at WP. (There is a passing reference to the Cousin problem in the Whitehead problem article - which is a good sheaf theory matter, though you'd never guess. The algebraic topology roots are more obscure to me - Cech cohomology, anyone?

My point: a great deal to do before this could responsibly be put in place. As it is, the technical side of the article isn't yet right. Local homeomorphism needs seeing to. I was trying to avoid the word étale throughout - éspace étalé as in Godement has been added by someone else. The point on terminology as I'd see it is to use local homeomorphism consistently for étale here: so that the étale cohomology page doesn't have to start by saying that étale needs disambiguation.

Charles Matthews 11:49, 6 Nov 2003 (UTC)

We also need to mention sheaves of O_X modules on a ringed space, and go on to coherent sheaves. AxelBoldt 00:57, 15 Nov 2003 (UTC)

These are mentioned on Scheme_(mathematics); I've added a link. Rvollmert 17:59, 17 Nov 2003 (UTC)

Hello,

My comment is in regards to the definition of "res". In going from the section "Definition of a presheaf" to "The gluing axiom" it seems that the definition of "res" has been reversed. Am I wrong? I.e. when we say res U, V when mean that V is a "subobject" of U in the fixed topology (I am abusing subobject a bit I know .. I mean that V "subset of" U, i..e in the Heyting algebra (in simpler terms poset) on the fixed topology).

Regards, Bill

Yes, you are right about the inconsistent notation: I've now changed it round to match the previous usage.

Some of the category theory in that part seems to be getting a bit out of hand.

Charles Matthews 07:19, 28 Apr 2004 (UTC)

I feel like moving the gluing axiom general discussion (values in a category with finite limits) to its own page. It is rather advanced for a first reading, anyway. An additional reason is that there is now mention of the Mayer-Vietoris axiom in homotopy theory elsewhere; and it would be a useful connection to make.

There are probably further things to do, to smooth out the treatment here. No article yet on local systems of coefficients, which were an important case in the formulation. We ought to bite the bullet and write down various things about resolutions of sheaves.

Charles Matthews 16:35, 2 Nov 2004 (UTC)

I think the section added on Sheaves on a basis of open sets would actually fit better on the gluing axiom page. Charles Matthews 06:08, 14 Dec 2004 (UTC)

[edit] in my opinion

i think the detailed example of a sheaf on a two point set is rather silly. 1. in an encyclopedic article is such a section sensful? 2. if yes, shouldn't we give a better example, i mean non-trivial and interesting? something in the direction of de rham theory, or even better serre duality? —Preceding unsigned comment added by 77.1.51.58 (talk) 20:03, 25 December 2007 (UTC)

[edit] restrictions, revisited

I think there is another bug concerning "res" and associated terminology: First it is written :

   * ... called the "restriction from U to V"... write it as resU,V.

And in the next paragraph:

   * ...i.e., the restriction of F(U) to U is the identity.

I think this should read "restriction from U to U" according to what precedes, or "restriction from F(U) to F(U)" according to later usage, e.g. in the next point :

   * ... i.e. the restriction of F(U) to F(V) and then to F(W) is ...

Also, I tentatively added a subsection title for my post, shouldn't this be done for each one, to make selective editing easier, and capture more easily the subject of the post ? (Please feel free to remove it, if you feel its not appropriate, or add one elsewhere if you approve.) MFH 12:56, 9 Mar 2005 (UTC)

[edit] Counter-examples/fiber bundle

It would be nice to include a brief statement of how sheaves differ from fiber bundles, with simple counter-example, a sheaf that's not a fiber bundle, e.g. because fibers aren't isomorphic or don't have some natural isomorphism, or whatever. linas 03:44, 9 May 2005 (UTC)

Any fibre bundle has a sheaf of sections. Almost no fibre bundle is a sheaf as we define it: it's a kind of space, and a sheaf is a kind of functor on the poset of open sets of a space. I'm not exactly sure what you are asking for. The concepts are different, though of course they are historically related. Charles Matthews 11:06, 9 May 2005 (UTC)

Hmm. Well, put yourself in a mindset of someone coming across this for the first time. If the sections of a fiber bundle form a sheaf, then what about the converse? Can every sheaf be expressed as a set of sections of a fiber bundle? If not, why not? Naively (i.e. sticking to concrete sets, not categories), one wants to equate the idea of a stalk to a fiber. A sheaf of analytic functions on the complex plane seems naively equivalent to the analytic sections of the trivial bundle C x C (C==complex plane), and naively one thinks one might be able to show some sort of isomorphism between a stalk and a fiber. i.e. the set of germs at a point x seems to be a preimage π^-1(x). Is that not possible? What are the fallacies one might trip over as one generalized? linas 00:46, 10 May 2005 (UTC)

OK, the most helpful thing to say is probably this: the sheaf concept is (just about) equivalent to the concept fibre bundle with _discrete fibre_ that can vary from point to point. I.e. variable F, but always a discrete space. The trouble is that the F can be fantastically complicated, for the simplest bundles. For example I take the bundle R x R, F at a point P will be the ring of all germs of continuous functions at P (see local ring). People really only 'got' the sheaf concept through the complex analysis case, where the sheaf is a disjoint union of total analytic continuations stacked up 'above' a domain D in C. Charles Matthews 10:35, 10 May 2005 (UTC)

Hmm. Thanks. I sort of get it ... I guess I wanted to say that the article sort of lulls one into thinking there's more similarity than there is (the cocylce condition, etc. seem so familiar, and then there is the very first example of sections) but that's probably my fault for just skimming in a half-awake state. linas 04:54, 11 May 2005 (UTC)

[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as sheaves, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

What kind of sources/reference are you using for your study? Not just wikipedia hopefully! :-) Dmharvey 23:04, 26 January 2006 (UTC)

Beno, as Schopenhauer said, concepts are derived from perceptions. That is why you are desperate to visualize a sheaf. As concepts become more abstract and removed from the originating perceptions, they become less easier to understand. Examples of the original perceptions are needed. From these perceptions, properties have been successively removed or abstracted until only the bare symbolism of signs remains. Anyone who claims to understand sheaves is either referring to their own private mental picture, which no one else can share, or is referring to the manipulation of mathematical symbols through various conventional operations. One must go back to the original perceptions, then to the concepts that were derived from them. From those basic concepts, more abstract concepts were devised until a vast complexity of concepts, signified by symbols, resulted. It is impossible to understand sheaves by starting at the wrong end, that is, by trying to understand the vast complexity instead of the original basic concepts.Lestrade 15:04, 11 February 2006 (UTC)Lestrade

[edit] Original perceptions

And, Lestrade, what are those original perceptions for sheaves? I´m very interested in sheaves theory related to fiber bundle classification.

The analogy of sheaves is modeled after a bundle of sticks, fagots, strands, cords, fibers, or filaments that are tied together. See fascine and fascis. The concept sheaf is simply an alternative way of saying, in common language, bundle, cluster, collection, set, or group. Any concept, including mathematical concepts like "sheaf," must be ultimately based on some kind of perception. Otherwise, it is an empty concept, that is, a meaningless symbol or sign. . Lestrade 16:15, 17 May 2007 (UTC)Lestrade

[edit] The term "étalé space"

The article states:

The [epace étalé] should not be referred to as an "etale space", as the word "etale" has other mathematical meanings.

What's wrong with the term "étalé space"? I don't know of any conflicting meanings (whether or not one forgets the accent marks), and other serious resources on the Net (like PlanetMath [1]) use it. (Although the books that I read seem to prefer to keep it all in French.) I suppose that it could be confused with the term "étale map" (note accents) and its relatives, but in fact these concepts are closely related: an étale map is the generalisation to algebraic geometry of the topological notion of local homeomorphism that is referred to in this article. --Toby Bartels 03:38, 30 May 2006 (UTC)

For notes on spelling, see Wikipedia talk:WikiProject Mathematics/Archive11#french spelling. --Toby Bartels 21:55, 20 September 2006 (UTC)

[edit] about the introduction

" restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one"

we are not restricting open sets, we are not gluing open sets. we are restricting and gluing sections of the sheaf.

For some reasons, in the section "Direct and inverse images" the minus sign in the superscript -1 is sometimes missing. I checked the code, and it is there. Can someone with more knowledge about this than me have a look, please? 131.188.103.41 15:33, 21 November 2006 (UTC)

I checked again, and it is a font issue. The horizontal line in f is running into the minus sign rendering it invisible. I added some spaces to fix this. 131.188.103.41 15:37, 21 November 2006 (UTC)

[edit] too long

The article is currently very long. Tightening and especially moving the things to appropriate subpages would be good. Jakob.scholbach (talk) 15:07, 15 February 2008 (UTC)