Talk:Sheaf (mathematics)

Comments

Isn't the skyscraper sheaf only defined on Kolmogorov (T0) spaces? Shouldn't that be stated explicitly? 140.247.128.41 (talk) 05:55, 12 November 2009 (UTC)

Why would it only be defined on T₀ spaces? Ozob (talk) 12:27, 12 November 2009 (UTC)

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)

Sheaves are an advanced topic: they build on may advanced ideas in mathematics. I am a postdoctral researcher, and sheaves are the kind of things that we worry about. They are not for an undergraduate. The article may, for some, be hard to read; that is because this is a hard topic. I had heared of sheaves, but did not know exactly what one was. I found this article interesting and informative; with a lot of explanation. I now feel comfortable with these objects. Dharma6662000 (talk) 16:56, 21 July 2008 (UTC)

They are taught in the undergraduate intro Topology class at our university. It should be possible to make this exposition a bit clearer. 140.247.128.41 (talk) 06:04, 12 November 2009 (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)

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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)

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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)

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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)

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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)

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)

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)

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)

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

This is probably a silly thing to add but if you are new to math, you may not be completely aware that it's not enough to review the material; you should also do problems. If you want to learn math, you can't do it with a summary like Wikipedia. It's like condensing an entire textbook into a page. There's a reason why a textbook is 500 pages. Find a book on sheaf theory and do the exercises. You might want to read something like "Global Calculus" by Ramanan, but first you should know basic multivariable calculus, which can be fun too. —Preceding unsigned comment added by 132.206.124.135 (talk) 19:13, 8 March 2010 (UTC)

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

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)

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.

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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)

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)

Some visualization

Visualising a sheaf in terms of stalks over each point of the space

Particular sections (associated to a germ of the stalk) corresponding to each of the red and green open sets of the space.

Gluing together two sections to create a section of the larger open set that they provide a cover of.

Despite the fact that the article is too long already (some if this really should be more properly farmed out to sub-pages) I feel that some visual representation of the basic ideas (necessarily in a simplistic setting) might go a long way to making the concepts involved clearer for new-comers. Of course picking exactly what to illustrate is difficult given the many different uses of sheaves, and how to iluustrate anything is complicated by the many different ways that each field that uses sheaves tends to think of them. In the end I decided that simple and easily representable was best, so following the "A detailed example: A constant sheaf on a two point space" section of the article I went with a constant sheaf over a discrete 2 point space; this does provide a means to illustrate stalks, sections, and gluing in a reasonable clear visual way. For now I just have something preliminary, and would appreciate comments and or suggestions for potential improvements. You can see the three basic images on the right. Hopefully with a little work we can polish these (and possibly one or two more, ideally working from the etale bundle/sheaf bijection) into something that can be efficiently worked into the text and provide a more visual conception of sheaves for those arriving at this article with little prior expertise. -- Leland McInnes (talk) 16:56, 29 June 2008 (UTC)

Fantastic picture, thank you Leland McInnes! You're asking for criticism but I think these three are already quite good and illuminating. Crasshopper (talk) 18:28, 8 August 2014 (UTC)

Hmm. My first thought upon looking at the pictures is, "What's going on?" For the first picture, I'm thinking that the large blue oval at the bottom is the total space, and the red and green ovals inside of it are open sets. The black dots inside the red and green ovals are points. The red dots and green dots represent sections over the stalks at the black dots. The gray dashed ovals and lines are just for visualization's sake. But then I get to the second picture, which says that it's looking at "Particular sections (associated to a germ of the stalk)" and these particular sections are indicated by red ovals and green ovals floating above the base. And now the red and green dots at the height of those ovals are a little darker and more elongated. So it seems like you're representing sections by both dots and floating ovals? And I think that the darkness and elongation of the two distinguished dots is just for visualization's sake, but I'm not sure. And now I get to the third picture, where there's now a blue shape floating over the blue oval. It's intended to represent the glued sections, and it does seem like it would restrict to give the red and green sections over the red and green ovals in the base, but you've also chosen values for the section away from the red and green ovals.

I'm guessing that the problem here is that you're trying to represent a two point topological space, but you'd like the picture to show up well so you're representing the points as red and green ovals rather than as dots. But as I said, my first reaction is to interpret them as open sets rather than as points. I think that would be my reaction even if you took out all the dots and replaced them with stacks of red and green ovals. So my impulse is to say that you should take out all the ovals and make everything dots. Rather than providing dashed gray outlines to distinguish the sections from the space, you could label the sections with the word "Sections" and label the base with the words "Topological space". Or maybe it would be better if every dot were labeled: The sections would be labeled with the numbers ..., −2, −1, 0, 1, 2, ... and the points in the space could be labeled p and q like they are in the text of the article. That frees dashed gray outlines to designate sections—I think outlines won't suggest extra points to the space like I think the blue section now does.

It would be good to illustrate a non-constant sheaf somehow. The only sheaves that can we can even come close to visualizing are pretty small (after all, the étale space of the stalk of a sheaf like C^∞(M) has one point for every germ of a smooth function!). But it would be possible to illustrate the sheaf of sections of a covering space map.

Isn't there a sheaf of orientations on a manifold? The sheaf of top degree differential forms ω_M has a subsheaf of nowhere vanishing forms, and the subsheaf admits an action by R₊, the multiplicative group of positive reals. Taking the quotient should give the orientation sheaf. More geometrically, over an open set U, the orientation sheaf is either Z/2, if U is orientable, or ∅, if U is not. This is something that could conceivably be drawn: Take a circle, split it into two open sets, draw the sections of the orientation sheaf over those two open sets; then do the same for the Möbius strip, and note that while you can glue the orientation sheaf of the circle to produce a global section, you can't do that for the Möbius strip. I bet that would be instructive. Ozob (talk) 18:25, 29 June 2008 (UTC)

Hmm, I admit I should perhaps of taken some time to actually explain what the different elements of the picture represent; having seen similar representations before I had assumed it would hopefully be clear (and apprently it was reasonably possible to guess at). The black dots represent the two points in the base space; the red and green ovals at the bottom are intended to represent the open sets around each point, while the blue represents the total space (as a further open set enclosing the smaller two). The vertical elements are the stalks, following from the rather agricultiral language used we have a stalk sticking up from each point, containing a set of germs (the germs are represented by the individual red and green dots). The difficulty now comes in the tension between discrete and continuous representations: because it is best to think of a section as a homeomorphic image of its corresponding open set, I felt that it was best to represent sections as, literally, a copy of the open set from the base space raised up so that it cuts through germs in the stalks. Thus we would have a section associated to each germ for the red and green opens. The image depicts one section of the red, and one section of the green. The idea, then, is to depict the gluing by depicting the (unique) blue section (i.e. a homeomorphic image of the blue oval) that stitches together (and hence restricts to) the individual red and green sections. I agree some labels would be good, though ultimately they are mostly of use when associated to some particular text, so I've left them off for now.

As to depecting other sheaves: I think ultimately sheaves over discrete spaces are probably all that is easily depictable with the approach I have taken; though perhaps a sheaf over a continuous space with just a few points and stalks picked out is feasible (though would look much like what I have drawn). Certainly non-constant sheaves are easy enough in such a situation.

Finally, your orientation sheaf idea is interesting, and indeed might provide some instructive illustration (if nothing else because it provides a different way of viewing sheaf data than the interpretation I have chosen to use, and more viewpoints are probably helpful). I'll see if I can find some time and work on something along those ideas. Thanks for your thoughts, they are definitely appreciated. I would really like to see this article get some illustrations. -- Leland McInnes (talk) 19:54, 29 June 2008 (UTC)

OK, so it sounds like I was looking at it right. But I disagree that it's best to think of a section as a homeomorphic image of the base in the general case. When the sections of the sheaf are naturally discrete (like they are for covering maps or constant sheaves), then the étale space is a very vivid and instructive picture, and I agree that it's the right way to think about things. But if you want to use it to describe all sheaves, then you need to do it for things where the data aren't naturally discrete: Differential forms, distributions, differential operators, D-modules (modules over the sheaf of differential operators), regular functions on a scheme, and so on. (Someone is about to point out that the structure sheaf of a scheme is not naturally a topological ring, but the stalks naturally have the m-adic topology.) I think that treating a section of a sheaf as a section of the projection from the étale space is wrong in these cases; not formally wrong, but philosophically wrong, because the interesting thing about these sheaves—namely, the gluing of sections—seems more apparent to me when you treat the sections of these sheaves as the geometric objects that you started with rather than as sections of the étale space. That is, I find it easier to glue differential forms than I do sections of the étale space of the sheaf of differential forms.

I think you're right about sheaves over continuous spaces looking a lot like what you've drawn; as I said, that's what it looked like to me at first. I really don't know how one would draw a sheaf like C^∞(M), but since that sort of geometric sheaf is one of the most important kinds it would be nice if one of us could think of a way to draw it. Ozob (talk) 21:24, 29 June 2008 (UTC)

The etale bundle view is always valid, it's just that, in some situations, the etale bundle is particularly hard to visualise. How you visualise a sheaf is, ultimately, going to depend on exactly how you need to interpret it. This comes back to my original comment about "the many different ways that each field that uses sheaves tends to think of them". There are certainly plenty of cases where etale bundles are perhaps not the ideal viewpoint (my own angle on this, coming from topos theory, has a very different conceptual view again). What I am interested in, however, is providing some visualisation of some of the basic concepts of sheaves (stalks, germs, sections, gluing) to help give new-comers something to grasp. Sure, it's not going to cover all the possible cases, but with sheaves nothing ever could. It seems what I have communicates the basics reasonably. When I get some more time I'll see to labelling them better, and try and fit them in with some introductory explanatory text. -- Leland McInnes (talk) 14:19, 1 July 2008 (UTC)

Since you mentioned topoi, I'm going to nitpick and point out that the etale space is only equivalent when the topos has enough points. This pointless correction fills me with childish glee. :-)

I agree that we need some visualization—I know that would have helped me when I was trying to learn about sheaves for the first time. As I said, I wish I knew how to draw C^∞(M); but even then, we'd need pictures like yours to illustrate other sheaves. I'm looking forward to seeing your next version. Ozob (talk) 16:40, 1 July 2008 (UTC)

Well, the étale space viewpoint is in fact correct and equivalent whether or not the topos has enough points: one considers the topos as a site with the canonical topology, and with that the representable functors are precisely the sheaves of the site. In other words, the sheaf is its étale space when one takes the topos viewpoint. Hence the notation F(U) = Hom(U,F) for two objects F and U in a topos. If your topos is Top(X) for a topological space X, then this notation obviously agrees with the "usual" one when U is an open set in X (i.e., the sheaf having étale space U = sheaf represented by U). Stca74 (talk) 08:09, 22 July 2008 (UTC)

You're taking the étale space to mean the existence of a representing object, whereas I'm taking the étale space to be a topological space constructed in a certain fashion. (One takes the disjoint union of all the stalks...) I agree with your argument, given your definition, though I'm not sure why such a definition is useful. (If there are good uses, I'd be curious to know.) Ozob (talk) 19:55, 22 July 2008 (UTC)

Rather than "definition" the idea of treating all sheaves of a topos (when convenient) as their espace étalés is deeply embedded in how one usually wants to work with objects / sheaves in a topos. One simple example is the induced topos: if E is a topos and X and object, then E_/X is naturally a topos together with a morphism of topos into E. In case E=Top(T) for a topological space, it makes sense to view X as an étalé space p: X →T; thenE_/X is étalé spaces Y over T equipped with an T-morphism to X. But these are just étalé spaces over X, and thus E_/X turns out to be equivalent to Top(X) with the associated morphism of toposes being just Top(p). Gets more interesting with the large topos TOP(T), and the viewpoint continues to work when dealing with toposes that do not come from topological spaces in any way. Continuing this way constructions such as pulling a sheaf back to its own étalé space become natural ways to interpret standard operations - see SGA4 Exposé IV for series of examples of how this works.

As for étalé space as a topological space, I do not think having enough points in a topos is sufficient to give you an interpretation of sheaves in the way you describe it. Indeed, the topos (Set) = Top(pt) has enough points. Hence the classifying topos of a discrete group has enough points. However, if the group is not trivial, the only subobjects of the terminal object e of the classifying topos are e itself and the empty sheaf. Thus the classifying topos does not have a generating family of subobjects of the terminal object and fails to be in any way equivalent to a topological space. Interpreting objects in the classifying topos as concrete étalé spaces (mapping to what?) is not clear to me. Stca74 (talk) 15:53, 23 July 2008 (UTC)

Hmm. I think part of our issue is miscommunication: I haven't seen the term "espace étalé" used to describe a representing object for a sheaf. I don't object to it, and for all I know it may be very common (you're better read in topos theory than I am), it's just that it's new to me.

As far as finding topological spaces: That's a nice counterexample, though it's not quite what I had in mind. The precise statement that I thought was true is: If E is a topos with enough points, then there exists a topological space X such that E embeds in Top(X). For instance, if E = Set, then X is a point and the embedding is the obvious one; and if E = B_Z/2, then again X is a point and now E is embedded as all sheaves (over a point) with Z/2-action. It's not as strong as the statement that you disproved (which is the claim that E = Top(X)). Do you know if this is true? Ozob (talk) 23:01, 25 July 2008 (UTC)

Agree that "espace étalé" is not typically used as terminology when dealing with general toposes. Rather, as I've said its more a useful and pervasive viewpoint to treat sheaves often "as" étalé spaces when working with toposes.

Now as for the use of topological spaces, here's some more detail (which should probably be moved to a more topos-related talk page). First, there is indeed a canonical way to associate a topological space to a topos E: one takes the isom classes of points, and the topology is not too difficult to specify (SGA4 Exp IV 7.8) using the "open objects" (subobjects of the terminal object) of the topos. On the other hand, the set of opens in a topos is naturally a category, which can be made into a site Ouv(E). The corresponding topos Ouv(E)~ is the target of a canonical topos morphism E → Ouv(E)~. Now if the topos has sufficiently points, Ouv(E)~is equivalent to Top(Points(E)) (modulo issues with universes, one needs small conservative subset of points in E, see loc.cit.). Thus for toposes E with enough points, there is indeed a universal morphism of toposes from E to the topos Top(Points(E)) associated to the space of points of E.

However, in the example we've discussed above, this morphism fails to be an embedding in a reasonable way. Indeed, the reasonable definition of en embedding for toposes (ibid. 9.1) is that the push-forward functor is fully faithful. For toposes of the form Top(X) for topological spaces this works out as follows: (Notice that an embedding is simply an equivalence followed by the inclusion of a sub-topos.) For Hausdorff X the sub-toposes of Top(X) with enough points correspond precisely to the subsets of X itself (ibid. 9.1.8. e). This applies to (Set) = Top(pt): the only sub-toposes are (Set) itself and the empty topos. Thus only embeddings to (Set) are equivalences to one or the other, and the classifying topos BG for any non-trivial group G cannot embed into (Set). More generally, by ibid. 9.1.9. b, if f: E → F is an embedding, and F is generated by open objects (as Top(X) for any space X), then so is E, again showing that classifying toposes of non-trivial groups cannot embed into any topos of the form Top(X).

Furthermore, the functor you seem to have in mind as the "embedding" of E=B_Z/(2) into (Set) goes in the wrong direction: it is the pull-back of the (essentially unique) point of E. Indeed, an exercise with the functorialities and adjoint properties of sets with group actions shows that the morphisms of toposes f: (Set) → E and g: E → (Set) are the following: f^* is the functor forgetting the G-action (the functor you seem to consider above), f_*(X) = Hom_(Set)(G,X) (with G-action from right action of G on itself), while for g we have g^*(X) = X with trivial G-action and g_* takes a G-set to its subset of G-invariant elements. In addition, both f^* and g^* have also left adjoints f_! and g_!, neither of which is exact if G in not trivial (e.g., f_!(X) = G x X, which fails to take terminal object to terminal object). This shows that neither can be the pull-back of a morphism of toposes,hence neither of f^* and g^* can be push-forward of a topos morphism, and thus the forgetful functor is associated unambiguously to a morphism with target E, not source E. Notice that the compositions of the pull-backs and push-forwards reduce to identities for (Set), as they should.

Finally, it needs to be mentioned that one can view the classifying topos of a (discrete) group G in the following way: realise G as the fundamental group of a topological space X (can be done always). Then BG is the category of covering spaces (or locally constant sheaves) on X. The (usual geometric) fibre functor for any point x in X gives a point in BG. The category of points is a groupoid, equivalent to the fundamental groupoid of X. Notice the obvious ambiguity in term of choosing X. And notice also that the space of points of BG is the one-point space, as there is only one isom. class of points of BG. Stca74 (talk) 11:34, 27 July 2008 (UTC)

Ah, this is very informative. Thank you! To clear up one thing: The precise embedding that I had in mind was to send a set S with a Z/2-action to the graph of Z/2 → Aut(S). (Which, from a set-theoretic point of view, "is" a set with Z/2-action.) And this is a perfectly good embedding of categories; but I didn't notice that it's not a morphism of topoi! Ozob (talk) 18:22, 27 July 2008 (UTC)

Article needs streamlining

The well-intended additions over the past 12 months have unfortunately rendered the article too long and lacking structure. In addition, there are still quite important omissions in the text. My view is that a few improvements are needed to restore the article to B+ status.

• The Introduction comes too close to the formal definition, leading to undue repetition of material
• The examples in the introduction are too detailed: the verification of the axioms (which should come after the Formal Definition anyway) burdens the text too much
• The choice of examples in the introduction could be improved:
  • Being real-valued is irrelevant in the first example
  • The most common function sheaves from geometry (esp. smooth functions, holomorphic functions) should make an early appearance
  • Presheaves that fail to be sheaves should make an appearance early (bounded functions is a good choice)
  • The example on solutions of diff equations may be too specialised here (should be shorter / later if included).
  • Should include the sheaf of sections ofany vector bundle, not just the tangent bundle (vector fields). This is indeed the motivating example for sheaves!
• The detailed example of the constant sheaf on a two-point space is way too detailed. It uses an enormous amount of space for what is a tedious but entirely straightforward exercise. And the example itself is too trivial for treatment in an encyclopaedia article.
• Sheaves of abelian groups require an own section, with exact sequences discussed briefly and a pointer to Abelian categories
• Sheaves of modules over a sheaf of rings need some space; good place to mention that vector bundle = locally free sheaf (over whatever structure sheaf of ones geometric structures) - further discussion belongs elsewhere
• The section on sheaf cohomology is of appropriate length (short) but not balanced: while injective resolutions do not lead to manageable computations, by far the most common way of computing cohomology groups besides the Čech method is using other acyclic resolutions (de Rham cohomology, Dolbeault cohomology, singular cohomology!). Pointing instead to Borel-Bott-Weil theorem is out of place here.

Stca74 (talk) 08:58, 22 July 2008 (UTC)

Some of this I agree with; other parts I'm not sure. If I were to make one comment on your comments, it's that the present article is more pedagogical than you seem to prefer. While the article may be too didactic at some points, it must be comprehensible to someone who is an expert mathematician who has never worked with sheaves before (e.g. certain analysts), and I think it succeeds in that regard.

• I agree that the introduction comes too close to a definition, but I don't know how to improve it.
• I think the examples in the introduction have some strong advantages:
  • Sheaves of functions are the fundamental example. I don't claim that they're the primary motivation for introducing sheaves; I do claim that there is no better first example. It illustrates all the fundamental features of sheaves, and in case that, while trivial, might actually be used in practice. Plus, some of the terminology we use—"restriction" and "gluing"—is immediately apparent from this case. The fact that the functions are real-valued is merely a convenience; it needs to take values somewhere, and it's easier to say R than to introduce another topological space, especially when using R facilitates the differential equation example.
  • Solutions to differential equations have three advantages. One is that when the differential equation is linear, they are a D-module, and D-modules are at the heart of some beautiful and deep stuff. The second is that when the equation is non-linear, they provide an example of a sheaf of sets which is not naturally a sheaf of groups. A lot of sheaves that you naturally encounter are either sheaves of groups or rings or they appear in a purely formal way (e.g., h_X), so it might not be apparent to the reader that sheaves of sets are everywhere. This example shows that they are. Finally, in combination with the first example, they provide a concrete example of a subsheaf.
  • The tangent bundle is much easier to grasp than an arbitrary vector bundle. Just as it's convenient to use R-valued functions in the first example, so is it convenient to use the tangent bundle in the third example. The primary reason why one might want to use the tangent bundle instead of some other well-known bundle (such as the Hopf bundle) is that vector fields are well-known objects with an obvious geometric meaning. The interested reader will read far enough to learn that sections of any vector bundle (indeed, any map) form a sheaf.
• The detailed example may be too long, but is not too trivial. It is exactly the entire notion of gluing which is so confusing to the inexperienced, and this is the simplest possible case of gluing. The section may be too long, but the example is so illustrative that I would strongly object to removing it.
• I agree that sheaves of abelian groups and sheaves of modules over sheaves of rings need their own brief sections and a pointer to a longer article. I'm not convinced that it would be useful to discuss exact sequences here, since it's the standard notion of exactness inherited from an abelian category. But it would be good to mention the correspondence between locally free sheaves and vector bundles.
• Yes, there should be more discussion of other resolutions. The Borel-Bott-Weil theorem is a very particular topic, but when you can't compute your cohomology using a resolution, it's one of the few tools you have left.
  • Actually, I think this section is woefully incomplete because it entirely fails to mention that cohomology with coefficients in a constant sheaf is related to other cohomology theories! Surely we need to mention that on well-behaved spaces, sheaf cohomology agrees with singular cohomology and de Rham cohomology!
  • There should probably be something to the effect of, "In many cases there is a duality theory for sheaves which generalizes Poincaré duality. See Grothendieck duality and Verdier duality.")

Ozob (talk) 20:38, 22 July 2008 (UTC)

Yes, I think your summary of the high-level difference in opinion is that in my mind the current article comes too close to text book, with even a detailed worked-out exercise. So one objection is WP:NOTTEXTBOOK — applies in particular to the long example on two-point space ("step-by-step problem solutions as examples" specifically mentioned in the current policy). Some more specific comments:

• Introduction should aim to explain the "global data (uniquely) specified by compatible local data" property of sheaves without technical notation, as the full definition comes below anyway. That explanation can be mixed with discussion of examples; this way the examples also avoid becomig verifications of semi-technically expressed axioms. Some more motivation would help too: one important piece being the fact that cokernels of morphisms of vector bundles can fail to be bundles (e.g., skyscraper sheaves), leading to a need to enlarge the category one works with.
• I also agree that sheaves of functions are the best introductory example. However, I would do two things: (i) use continuous functions ("such as continuous real-valued functions") as the first example to make it clear that being real-valued is not the essence here; and (ii) mention up-front the few critically important examples of function sheaves (smooth, holomorphic) contrasting them with a non-example (such as bounded functions).
• As for vector fields, I would present the example with a general vector bundle (and not another specific one!) as that is arguably the most important example. Adding a note that reader may wish to think of the tangent bundle and its sections would be fine in case it helps someone to grasp the meaning.
• I'm in favour of keeping D-modules in the article, but would argue that the example may be a bit heavy early in the article. Perhaps the sheaf D of linear differential operators itself would be more straightforward if the introduction needs more examples than the ones above? In any case the full discussion on solution complexes linking D-modules to solutions of diff equations is most naturally done in the derived category context and is clearly beyond the scope of this article.
• In addition to my policy-based objection to the long worked-out exercise on constant sheaf on a two-point space, I do think that at least an example on topologically less trivial space (S¹?) would have been more useful pedagogically. But in my view this material should at least be moved to a sub-page.
• Exactness: it would be god to mention that this is the key property that can be checked on stalks. One sentence enough.
• Indeed, my point about non-injective acyclic resolutions aims at the same as yours: cohomology with constants coefficients. In fact, it is not hard to see that a sheaf version of the singular cochain complex is an acyclic resolution of the constant sheaf and hence singular cohomology agrees with sheaf cohomology. And of course Poincaré's lemma and de Rham cohomology come to the same. This should not be expanded too much here, though, but added to the seriously incomplete article on sheaf cohomology.
• Agree on one sentence on dualities.

Stca74 (talk) 08:24, 23 July 2008 (UTC)

I would like to inject some of my own thoughts on this, since I had been vaguely contemplating similar streamlining ideas. Before I start however, I should make clear my own biases here: I would like to see the article made more accessible to a general audience. Yes, sheaves tend to be embedded in rather deep mathematics, but the fundamental ideas are simple enough that it should be possible to include material that would allow even a high school student to take something useful from the article. The key (as with all attempts to make advanced mathematics articles more accessible) is to provide a reasonable amount of easily graspable material up front in the lede and Introduction, while retaining the more difficult material further down in the article. In general an article should follow a gradient of difficulty, allowing readers to continue through the article according to their ability.

• I agree that the current description in the lede isn't great; it either needs to be less formal (thus removing duplication, and more importantly making the article more accessible), or more succinct. Ideally it should be both -- a decent non-formal description, followed by a very succinct summary of the formal definition (proper full formal details can be handled further down the article).
• I dislike the series of long examples given in the "Introduction". Certainly examples are very useful, and can only really be properly drawn from higher level math, but making the vast majority of the introduction such detailed examples makes the reader wade through far too much material (we haven't even gotten to the formal definition yet). I would definitely prefer to trim the examples down to a few sentences each (and requiring no (sub)headings), and fill out the "Introduction" with more general material.
• As to the choice of examples; they seem not unreasonable to me, though somewhat arbitrary, and not necessarily "natural" -- that is, they seem to be chosen to demonstrate particular pedagogical points, rather than being natural clear examples of sheaves (in this I mostly refer to diff. eqns example, and the specificity (why not sections of general vector bundles) of the other examples).
• I would like to see the detailed example of a constant sheaf on a two point space either cut or completely reworked. The example is both simultaneously too simple and too complex: working with a two point space makes the example necessarily very simple, and not particularly representative -- ultimately it only serves to illustrate the normalization and gluing axioms; it is too complex in that to illustrate an example of how these axioms work (particularly with such a simple case) does not require the (arbitrary) detail given -- it should be entirely possible to give an example that illustrates the same ideas that is far more accessible to a general audience, and this, to my mind, should be done much earlier in the introduction. It is precisely toward such a goal that my illustrations were targetted.
• I'm not actually convinced that sheaves of Abelian groups deserve their own section, but I am not averse either.
• Sheaves of modules over a sheaf of rings is certainly worth giving some space as an example.
• I think the cohomology issue has been largely resolved (although the cohomology section is starting to get long).

A few other points I would like to comment on:

1. The "Examples" section is badly organised, and is made worse by the fact that we already had a series of long worked examples in the introduction (though hopefully that can be fixed). It also seems poorly placed in terms of the overall outline of the article.
2. The history section would be far more usefully turned into proper paragraphs, tightened up (I really don't think we need to hist all the bullet points currently listed), and hoisted up to be much earlier in the article -- it provides a great opportunity to provide some motivation for sheaves that is much more broadly accessible than the examples given in the introduction.
3. In general the article needs streamlining and reorganisation; it currently seems a little bloated and unfocussed.

Obviously a great deal of work has gone into this article, and it is really very good. My comments are directed at the hope that we can put some polish on it and hopefully eventually bring it up to GA status or better. -- Leland McInnes (talk) 15:33, 27 July 2008 (UTC)

It seems like we have consensus on the following points:

• The introduction should be less formal when describing sheaves. It should not nearly define them as it does now.
• The examples in the introduction are too long and too detailed.
• The examples in the introduction should be more representative.
• The example of a two-point space is too long.
• Sheaves of modules over a sheaf of rings need a brief but not extensive description, as well as their own article.

I'm going to trim the two-point space example. The rest of you are welcome to try your hand at the other bullets.

On the other hand, we don't have consensus on these:

• What examples should go in the introduction?
  • In particular, general vector bundles versus the tangent bundle.
• Do abelian sheaves need their own section? (If we have a section on sheaves of modules, I'd say no.)
• What do we do with the example of a two-point space?

It seems that the important point of contention here is over examples: Which ones, where, and how detailed. The only consensus seem to be "Less!"

I still believe that as a first example, it is more useful to use vector fields rather than sections of a general vector bundle. Plenty of beginning graduate students encounter sheaves without being overly familiar with vector bundles and even without being able to define a vector bundle. But they are much more likely to know what a vector field is, because everybody gets some idea of what those are when they take vector calculus. And in a beginning manifolds class, I'd bet they're more likely to see a vector field defined by charts than as a section of the tangent bundle. Now, if this example were later in the article, I'd agree that there would be no reason to single out the tangent bundle. But right now, with the example up front, the simplicity and geometric clarity of the tangent bundle is a strong argument for keeping it the way it is.

Regarding the two-point space, it sounds like we have three points of view:

1. The two-point space is the simplest example of gluing. The presence (though not the length) of this example is appropriate. (This is my view.)
2. The two-point space is too simple to be an interesting example. It should be replaced by a space which has some interesting topology.
3. The two-point space is too simple to be given so much prominence. It should be replaced by a less detailed description.

Is that correct? The one thing that we do agree on is that it's too big. (It used to be even bigger!) I'll try trimming it, and then we can discuss it some more. Ozob (talk) 19:00, 27 July 2008 (UTC)

I should perhaps clarify my view on the two point space example, as I believe (3) was intended to encapsulate my view, but isn't quite what I want to say (it's very close though). I actually agree with (1) that a 2 point space is the simplest example of gluing, and thus is worth discussing. I don't feel that it is worth discussing as an independent example however. I would much rather see such discussion folded into an "Introduction" section that discussed the basic ideas of sheaves in terms graspable by a broad audience. It is this last part that has me objecting to the detail of the description -- much of it is superfluous and arbitrary: do we care that the set chosen for the constant presheaf is the integers? Do we need one example to illustrate both normalization (which is easy to understand) and gluing? Do we need the (pedagogical) explicit verifications at each and every step? If our goal is simply to illustrate relevant ideas (such s normalization and gluing) then I don't think we need any of that. We can simply depict, and explain in general terms, how normalization and gluing work; a two point base space works fine for that, but we don't even need to even (explicitly) define the sheaf to depict and explain these ideas. In this sense I think "less detailed description" doesn't really capture what I am aiming for. I hope this helps clarify my view. -- Leland McInnes (talk) 14:27, 28 July 2008 (UTC)

First of all thanks for Leland for the comments on 27 July. I have the same view that the current state of the article with too much detail makes the content seem unnecessarily difficult. At the same time I'm entirely in favour of making it as accessible to broad mathematical audience.

Then more specifically, regarding the worked-out example on the two-point space, I also think that none of the three options presented represent my viewpoint, which is that the example should either be deleted altogether, or at least farmed out to a subpage (or to the page on gluing axiom, which needs heavy editing, but that's another matter). As I've noted above, I think this type of detailed example is not suitable for an encyclopaedia article (and here I can also cite WP:NOTTEXTBOOK). A more proper way to deal with glueing is in the introduction, while constant sheaves should appear as a (concise) example, which could briefly point at what happens on a disjoint union of open sets. I think these remarks still apply to the new condenced version of the example too. Stca74 (talk) 08:37, 31 July 2008 (UTC)

You seem to have very specific ideas on where you'd like this article to go. Could you put them in the article? Or if you'd rather not put them in the article yet, could you put a sketch on a subpage of your user page, for instance at User:Stca74/Sheaf? Ozob (talk) 17:05, 31 July 2008 (UTC)

I will do what time permits over the weekend - at least with the detailed example, lead and intro. Stca74 (talk) 07:52, 2 August 2008 (UTC)

Managed to have some work done, working with the separate page User:Stca74/Sheaf. More precisely, I have

1. Expanded the lead considerably to cover also some applications. Still not an FA lead, sure, but an improvement I hope.
2. Rewritten the Introduction: it now goes through the definition almost without any technical notation, explaining the definition and logic behind it. I would still like to have it shorter, and it is not free from repeating definitions that follow. But at least it does not go through the motions of detailed axiom verification for several examples. An obvious question: is such "gentle" introduction necessary at all?
3. Restructured the Formal definition: split the sheaf axiom into three (more common in literature, I think), brought in the equalizer definition and the definition of morphisms
4. Moved the detailed example of a two-point space to a subpage with link from the formal definition.
5. Reduced TeX markup and did some other copyediting up to the end of Formal definition.

What I think should still be done (these do not need to amount to too much text, rather pointers elsewhere):

• Abelian sheaves
• Sheaves of modules (in the formal definition section? later?)
• Discuss constant sheaf, locally constant sheaves (after image functors?)
• Incorporate the remaining examples (not used in the lead & intro) into a new Applications section, which should include brief statements about:
  • Structure sheaves as a way to define geometric structure (of a diff mfld, cplx mfld, scheme)
  • Vector bundles, differential forms, divisors etc in such context (see new lead)
  • Varying coefficients in cohomology in the algebraic topology context (local systems as coefficients, Leray spectral sequence...)
  • Links between topological data (cohomology with "discrete" coefficients) and geometric data (cohomology with "continuous" coefficients) (perhaps the example of the exponential sequence and its relation to classification of holom. line bundles used as an example?)

Furthermore, since there should be quite real potential to take the article to GA level:

• Still further copyediting and cleaning once main content is in
• Addition of explicit references.

Do take a look at the existing new material on User:Stca74/Sheaf. Unless objected to, I will start adding material from there to the actual article page in a few days. Stca74 (talk) 19:37, 3 August 2008 (UTC)

I like it. A few comments:

• There's a typo in the third paragraph of the introduction: "whenever two sections over U coincide when restricted to each V_i of , the two sections are identical." I think I saw another one somewhere else, too, but it escapes me at the moment.
• The wikilink map in the lead links to the article on the geographic kind of map.
• At some point it seems that the consensus was that the equalizer formation of the gluing axiom was too abstract. (I would have disagreed had I been here, but I wasn't.) This is why the gluing axiom article was written. It might be good to link that article in the definition section as well as the present link in the sheaving section.
• (As a side note, the gluing axiom page contains a paragraph on sheaves defined on a basis of open sets. That might be good to include on the main sheaf page since that technique is so common for schemes. And the comparison lemma leads into sites and topoi in a very natural way.)
• I'd bring the footnote into the main text. There are a few other things I'd copyedit, too, but nothing worth mentioning.
• Subpages of mainspace pages are disabled (see WP:SP), so User:Stca74/Sheaf/Example would have to be made into an article of its own. Furthermore, as a side effect of the present organization, the article says that it will discuss constant sheaves "below" and then doesn't (because the material is on a subpage).
  • All of a sudden I'm getting a really good idea. Why not have a constant sheaf article? (At present it redirects to locally constant function.) Put the detailed example there (and include a pointer in the sheaf article) and spend the rest of the article talking about the manifold (ha ha) uses of the constant sheaf. Or maybe it would be even better to have a locally constant sheaf article, or to expand the present local system, a short article with big possibilities.
• I think it's a good idea to keep the sentence about F coming from the French faisceau. When I was first learning about sheaves, the only thing it seemed to stand for was a four-letter obscenity.

To answer your question about the existence of the gentle introduction: I think it's an artifact of the historic organization of the article. The present lead is extremely short and barely describes what sheaves are. So there's good reason for a more expansive description, which is what the current introduction tries to be. With a more expanded lead, the introduction section is less necessary. Even better, I think, would be to put the history up here. The timeline is a bad idea, but the article has had it for years and years because nobody's wanted to take the time to write a proper history section. But that would be the right thing to have at that point in the article, because it would tell the reader why sheaves were interesting and give a vague picture of what they were. Then they'd be ready to look at the definition. (If anyone is interested, I think Dieudonne's books on the history of algebraic topology and algebraic geometry might be helpful, and I also think that MacLane and Moerdijk discuss a bit of history in a few places. There might even be a historical article about sheaves somewhere. And the Grothendieck-Serre correspondence might be useful.) Ozob (talk) 22:35, 3 August 2008 (UTC)

(unindent) Thanks for the comments and finding the broken links and typos, Ozob. I have now carried out the changes proposed above — more precisely:

• Added the new expanded lead
• Replaced the old introduction with the new one (leaving for the time being open the issue discussed above whether such section is needed at all)
• Restructured the formal definitions as announced above
• Moved the detailed example to new Constant sheaf (there was an old redirect page to less-than-relevant article). I think it merits its own page, the example indeed fits there if somewhere, and locally constant sheaves / local systems are sufficiently different topic to have a separate page. Constant sheaf is now a stub — one I'm afraid I will not have time to develop for some time.
• Removed the remaining link to Gluing axiom: indeed, whatever the history was, that page does not add to what have (should have?) here. In fact, a main-link to that article from sheafification was rather confusing. My proposal would be to list Gluing axiom for deletion.
• Included the topic of sheaves on sets of a basis
• Found a place to insert a few words on notation and the F-letter
• I kept the footnote as such: alone it is arguably a bit out of place, but working this towards GA will require much more footnotes in any case, so I think its good to have the Notes section there ready.

As I wrote above, I'm afraid I won't have much time for editing in foreseeable future (just back from vacation today...), so I will have to leave the clean-up and expansion of the second half of the article to other enthusiastic editors. Stca74 (talk) 19:38, 4 August 2008 (UTC)

Almost forgot: a great source for the history section is the short article by C. Houzel in Kashiwara's and Schapira's book Sheaves on Manifolds, ISBN 3540518614. The full Houzel article is even available in Google books. Stca74 (talk) 19:48, 4 August 2008 (UTC)

I think an informal introduction needs to exist to provide information for those who are interested, but not as technically inclined. I think the history section should be folded in, and I would actually make the introduction even more informal and low level (possibly renaming the section to "Informal introduction" or similar. It is my view that high school student should be able to read the article and write something intelligible on the subject for a school report. I fully believe that the basic ideas of sheaves (locally defined data that can be stitched together to give global data). One can even give examples that don't require much technical maths to at least communicate the sorts of ideas that sheaves may capture. If I get enugh spare time I will attempt to write some of this up. It will undoubtedly end up being too long, but hopefully with your help we can prune it back into something suitable. -- Leland McInnes (talk) 14:09, 6 August 2008 (UTC)

Two remarks:

1) In the definition of a presheaf (or, at least, in the definition of a sheaf: I'm not sure it does not vary with the author) there should be the condition that the set of sections over the empty set is a one-point set (or, with values in an abstract category, a final object).

2) In the same spirit, the example with the two-point set is bogus: if really the set of sections over the empty set is empty, there can be no restriction map to the set of sections over the empty set! 193.50.42.3 (talk) 13:33, 14 January 2009 (UTC)Alain Gen

Glueing vs gluing

The first is used 3 times, the second 4 times. GeometryGirl (talk) 12:29, 8 January 2009 (UTC)

Unfinished sentence

In the Sheaves subsection of the Formal Definitions section, the sentence "Now a presheaf F is a sheaf precisely when...where the first arrow is an equalizer," is not complete. It should be either finished or deleted. Ixionid (talk) 19:09, 16 May 2009 (UTC)

Sheaf axiom as a limit

Regarding this section:

In general, construct a category J whose objects are the sets U_i and the intersections U_i ∩ U_j and whose morphisms are the inclusions of U_i ∩ U_j in U_i and U_j. The sheaf axiom is that the limit of the functor F restricted to the category J must be isomorphic to F(U).

Please give a reference or explain. Thanks Quiet photon (talk) 11:10, 18 October 2010 (UTC)

See, e.g., Mac Lane and Moerdijk, Sheaves in Geometry and Logic, II.1. Ozob (talk) 00:13, 21 October 2010 (UTC)

I think the phrasing here is ambiguous; I believe the statement is only true if the isomorphism lies in the cone category. That is, the isomorphism has to properly respect the inclusion maps. If I'm right, then it might as well be reformulated as saying that F(U) is a limit of the functor. —Preceding unsigned comment added by 64.26.155.153 (talk) 16:33, 21 April 2011 (UTC)

I am not quite sure what you're saying here, but it sounds questionable to me. If F(U) is a limit of the functor, then any object isomorphic to F(U) is also a limit of the functor because it has the same universal property. Am I misunderstanding you? Ozob (talk) 10:32, 22 April 2011 (UTC)

In your category J, you have a terminal object U, which, in J^op is an initial object, so a limit for any functor F:J^op-->C always exists and its FU — Preceding unsigned comment added by Lhrrwcc (talk • contribs) 20:11, 22 December 2013 (UTC)

J consists of the sets in the open covering but not (necessarily) the set being covered. I've clarified the article. Ozob (talk) 05:34, 29 December 2013 (UTC)

Confused

For the Morphism section, u is a subset of v instead of v a subset of u like in the presheave section, and the picture shows the contravariant res_V,U as F(v)->F(u). Shouldn't v be a subset of u and the picture show F(u)->F(v) and G(u)->G(v)? —Preceding unsigned comment added by 75.93.51.158 (talk) 10:34, 1 December 2010 (UTC)

Yes, the notation is inconsistent. I agree with you that this is bad, and I think that the diagram should be changed. However, I've never figured out how to make a diagram and upload it, so I can't really help. Ozob (talk) 11:47, 1 December 2010 (UTC)

I have changed the diagram so V is a subset of U, and the restriction morphisms go F(U) -> F(V), as they were defined. I changed the sentence including the diagram to say "open subset V of an open set U," but I didn't change any other text. I left the old diagram as SheafMorphism-01.png, because the article in other languages will need to be edited to reflect the new diagram SheafMorphism-01a.png before it can be used. --kundor (talk) 22:14, 29 May 2011 (UTC)

group structure

I am not sure about pre-sheaves, but for sheaves, the objects F(U) need to have a group structure associated with it. For example, polynomials on U, C^\infinity functions on U, etc. This is not clearly stated in the article. Moreover, I believe, the definition has been made too complicated by some unnecessary details. More intuitive definition and some simple examples to start with may be helpful. I will try to add a "simplified definition" with simple examples at the beginning in a day or two. Subh83 (talk) 19:15, 3 March 2011 (UTC)

This is not true. Sheaves of sets are fundamental objects. See for example Grothendieck topos. Ozob (talk) 00:28, 4 March 2011 (UTC)

Exactness

What in the world does it mean for a sequence in a general category with products to be exact? (Referring to the sequence described in "Formal Definitions -> Sheaves.")

Exactness of the first point in the equalizer sequence can be interpreted in any category as injectivity, but the meaning of exactness of the second point is unclear to me. The same terminology is used in a set of lecture notes I'm currently studying, so I came here for clarification and found the same notation. Clearly the notion of "exactness" is not here the same one described in exact sequence, since that one is defined only for categories with kernels and cokernels. 128.100.216.53 (talk) 18:23, 10 August 2012 (UTC)

Exactness here means that the first arrow in the sequence is an equalizer. This is stronger than just being injective; it's analogous to being a kernel and usually is replaces that notion in a general category. In an abelian category, k : K → A being a kernel of f : A → B can be rephrased as k being an equalizer of f and the zero map A → B. Ozob (talk) 22:24, 10 August 2012 (UTC)

Thank you very much! That was very helpful. However, I notice that there is no mention of this more general notion of exactness in terms of equalizers in the article exact sequence. It would appear that this should be changed, but I'm certainly not well-equipped to do so. 142.151.184.236 (talk) 02:48, 12 August 2012 (UTC)

Actually, I retract my statement. This still doesn't seem right: your definition references the zero morphism, but we are working in a general category with products. For instance, the category of sets does not have zero morphisms. 142.151.184.236 (talk) —Preceding undated comment added 03:51, 12 August 2012 (UTC)

Well, the "unreduced version" of Mayer-Vietoris sequence is maybe a better illustration, as it makes explicit use of an equalizer when doing the set intersections. ... Err, well that article should, but doesn't do that either; its hidden and not remarked on: its the (i*, j*) bit, which should be drawn as two parallel arrows, as done here. The Seifert-van Kampen theorem does show this, but its a harder read. This is somehow not at all clear in the exact sequence article. linas (talk) 21:44, 18 August 2012 (UTC)

After digging around, I found that the article coequalizer actually makes remarks about how it generalizes a cokernel via a zero morphism; it would be good if these statements were carefully transposed into the equalizer article, which doesn't seem to actually make this point. linas (talk) 22:55, 18 August 2012 (UTC)

non-sheaf examples!?

Its nice that there are two examples of pre-sheaves that are not sheaves; it would be nice if their sheaf-ification was then given. linas (talk) 21:44, 18 August 2012 (UTC)

I think what I just added should be a good example. -- Taku (talk) 03:50, 31 December 2014 (UTC)

Finiteness conditions for sheaves of modules

I find this materials to be too technical and thus distracting. The definition of "coherent sheaf" is unfortunate and I don't think this is the best place for the discussion. Coherent sheaf gives a fairly good treatment, if intimidating; (the first thing we give is a definition directly from EGA, really?) I propose we merge this section into coherent sheaf. -- Taku (talk) 03:54, 31 December 2014 (UTC)

sheafs on (finite) graphs (as in graph-theory graphs)

I have an application where I'm applying the sheaf concepts and definitions to what is more-or-less-ish a kind-of graph, as in a graph-theory graph, viz a finite number of vertexes and edges. This works in the "obvious" way, because any given vertex can be taken to be a germ, and the edges attaching to it identify the other nearest-neighbor vertexes, and the whole mess can be given a kind-of-like open-set topology. I say "kind-of-ish", because there's more to it than that; in particular, I can have sections and stalks, which don't show up in "ordinary" graph theory (you'd need to take quotients over graphs). I'm sort of making this up as I go along, because its relatively straight-foreward, but was wondering if there are any references for this kind of stuff? To make the question a little more concrete, imagine having topoi viz sheaves applied to logic, but now just replace each logical expression by (labelled) vertexes and (labelled) edges, viz only the locations, not the symbols at the locations. viz "locus solum". Is there any low-brow, undergrad-level treatment for this? 58.153.25.62 (talk) 13:35, 24 September 2017 (UTC)

Is this article "too technical?"

This article has a cleanup tag because it is supposedly "too technical" for non-expert readers to understand. Is it possible to make this article comprehensible to non-expert readers? Jarble (talk) 18:13, 10 October 2019 (UTC)

Confused - is something wrong?

I'm trying to understand the definition of a sheaf from this article, but I'm really confused and suspect something is not correct.

We start by saying, in the definition of presheaf, that if V ⊆ U then there is a function (morphism in Set) $\operatorname {res} _{V,U}\colon F(U)\rightarrow F(V)$.

Then we say res_V,U(s) is denoted s|_V. So the notation leaves the U implicit for some reason, and s|_V is a member of $F(V)$. Whenever we write s|_V there must be some other open set U, a superset of V, such that s is a member of $F(U)$.

Then, in the definition of a sheaf, we have an open covering made up of sets U_i. So $U\subseteq \bigcup _{i}U_{i}$ and there is a function $\operatorname {res} _{U,\,\bigcup _{i}\!U_{i}}\colon F(\bigcup _{i}U_{i})\rightarrow F(U).$ So far so good.

But then we suddenly start talking about s|_Ui. What could that possibly mean? It can only be an element of F(U_i) given by $\operatorname {res} _{U_{i},X}(s)$, for s in some set $X\supseteq U_{i}$ that has been left implicit in the notation. But no such set was ever mentioned in the definition, and hence I can't understand what it's trying to say.

I don't know if this is a problem with the article or with my understanding of it. The only motivation for the unfamiliar s|_V notation is "by analogy with restriction of functions", which means nothing to me, and I guess I might just be reading it wrong somehow. But either way it would be great if the article could be fixed, either by correcting it or by making things more explicit so that it's possible for someone to follow who doesn't already know the material. In either case, it'd be great if I could be pinged when it's done.

Nathaniel Virgo (talk) 16:06, 15 April 2020 (UTC)

Edited to add: ah, maybe the "open covering" is supposed to be $U=\bigcup _{i}U_{i}$ instead of $U\subseteq \bigcup _{i}U_{i}$. Then it makes sense - the missing set X is U itself. But this needs to be stated explicitly, because the linked page on open covers defines them with $\subseteq$ and doesn't seem to mention any 'strict' version of the definition. Nathaniel Virgo (talk) 17:12, 15 April 2020 (UTC)

You are correct. $U$ must exactly equal $\bigcup _{i}U_{i}$, otherwise the restriction $s|_{U_{i}}$ is not well-defined. Equivalently, we can require $U_{i}\subseteq U$ for all $i$. I'm making a (sourced) edit to the page to reflect this, as I too had to bash my head against this definition for a while to see what was wrong. Winthrop23 (talk) 09:15, 29 July 2023 (UTC)

Confusion in opening paragraph

" The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one."

Do we _really_ mean "smaller open sets" - and if so what metric of 'smallness' is in play here? What makes one open set 'smaller than' another open set? Or, as I expect, do we really mean "smaller collections of open sets" where we are referring instead to the _cardinality of_ the collection of open sets. 71.139.124.132 (talk) 13:51, 30 June 2020 (UTC)

"Smaller" here is to be understood in the sense of inclusion : a set $A$ is said to be smaller than a set $B$ if and only if $A\subset B$. --93.25.93.82 (talk) 19:09, 4 July 2020 (UTC)

Not happy with recent edits

I'm not sure the recent edit by Wundzer are an improvement to the article. The concept of a sheaf can be explained, in my mind, quite easily with the notion of continuous functions. (We do have these examples later in the article, and I believe they should be moved up and upgraded to a more welcoming motivation.) Holomorphic functions as a door-keeper to the "motivation" of this article don't serve their purpose that well.

I also wonder whether the historical implied motivation saying that we need sheaves to distinguish between complex manifolds is historically correct. Is there a reference for that?

Any comments on that? Jakob.scholbach (talk) 08:48, 8 October 2020 (UTC)

Hi Jakob.scholbach, thanks for the comments. For your first point, I disagree that using continuous functions is a good motivation for introducing sheaves precisely because they give the illusion that sheaves are just a tool which (initially) makes classical constructions, such as continuous functions on a topological space, a more complicated notion than it has to be. Furthermore, I think the importance of using sheaves, the raison d'etre, can be understood only after looking at the holomorphic and algebrogeomtric categories. In both cases, we want to have a device which is aware of the underlying (holomorphic or algebraic) structure, which gives some direction as to why sheaves are important. For the second point, I recommend looking at Serre's FAC and Cartan's paperCartan's paper Serre cites at the beginning. In addition, there is an excellent math.stackexchange post about the issue. Apparently sheaves caught on historically because of their use in complex analysis/geometry. Apparently Leray originally used them to compute the cohomology of ${\underline {\mathbb {R} }}$ on a smooth manifold, giving the de-Rham cohomology (done while a POW in WW2!). Wundzer (talk) 16:53, 8 October 2020 (UTC).

Here is my comment. We do have the section "Overview" which seems to somehow duplicates the "Motivation" section. But I do get what Wundzer is saying (I think). The article doesn't quite motivate the notion of a sheaf. I think what is missing is the history section. If I remember from what I read like a decade ago :(, Kiyoshi Oka needed what we now call a sheaf to keep track of holomorphic functions on various open subsets. That was really a motivation. For continuous functions, you don't have much need for a sheaf since you can always use a cut-off function to make functions defined on the entire space. In complex analysis, the typical cut-off trick doesn't quite work and so there is this technical need for shaves. Or at least that's what I remember. Unless I missed, the article doesn't explain this. As for sheaves determining a space, that's really the philosophy of Grothendieck: to know a space, one simply has to keep track of "functions" (I am using quotes purposely here) on the space; in other words, sheaves of "functions". That too seems to deserve a mention. But that can be done, again, as a part of the history. (We don't need nor are allowed to judge Grothendieck is correct or not.) -- Taku (talk) 03:48, 9 October 2020 (UTC)

@TakuyaMurata Awesome, it seems like you've gotten my point. Tbh, when I first learned about sheaves, I found the wiki article unhelpful because it just read as a reiteration of the definition, and included some material you find in the first section of Hartshorne.II. By writing the motivational section, I'm really writing this article for my past self who was confused about why I should care about Sheaves. I agree the "Overview" section contains some duplicate material, but my solution would be to assimilate the other material, such as why presheaves are considered, and add another subsection discussing other ways sheaves are found in nature, such as the sheaf of sections of a bundle. Wundzer (talk) 16:04, 9 October 2020 (UTC)

Also, should there be a redirect from cut-off function to Bump function? Wundzer (talk) 16:09, 9 October 2020 (UTC)

@Wundzer: unless I am missing something, what Serre says there is that Cech cohomology is needed for non-fine sheaves. This is quite different from the motivation that your writing indicates. Can you provide a precise citation which supports your point of view?

That said, we also should be careful in not messing up historical motivation and motivating things for a reader who learns about the topic just now. (Compare to the historical motivation for groups: we would never start explaining what a group is by explaining how arc lengths of ellipses etc., à la Abel, gave rise to abelian groups.) We should give the easiest (meaningful) examples for sheaves and motivate why one looks at them. Of course, the sheaf of continuous functions is very particular in that it is a fine sheaf. I am not opposed to mentioning other sheaves, say the sheaf of holomorphic functions, early on. But "motivating" the entire notion of sheaf by the fact that holomorphic functions on a compact manifold are constant is missing the point, I am more than convinced. Jakob.scholbach (talk) 07:24, 9 October 2020 (UTC)

The mathoverflow post also is not saying anything why sheaves (as opposed to sheaf cohomology) have been invented. Really, once again: the motivation has the purpose of motivating people, not of driving them away from the topic. The current motivation will do the latter to many readers (as is implicitly acknowledged also by the title "Technical problems...") Jakob.scholbach (talk) 07:28, 9 October 2020 (UTC)

@Jakob.scholbach: I'm not sure where the confusion is about Serre's paper. In the introduction, the first paragraph states

We know that the cohomological methods, in particular sheaf theory, play an in-creasing role not only in the theory of several complex variables ([5]), but also in classical algebraic geometry (let me recall the recent works of Kodaira-Spenceron the Riemann-Roch theorem).

and that citation leads to Cartan's paper. If you look at sections 4-8 of Cartan's paper you see he is explicitly motivated by constructing ringed spaces for complex manifolds, and later discusses Stein manifolds and sheaf cohomology on them. It seems as though Serre is leading at sheaf theory is indispensable to cohomological methods, and he views sheaves as the main motivation for considering sheaves.

For the math.stachexchange post, I think it's missing the point to consider sheaves without the context of sheaf cohomology since this is why they were invented. Moreover, sheaves are a solution to a technical problem, which is outlined in the motivational section. For the title "Technical problems...", why do mathematicians invent new structures? It's to encapsulate solutions for certain problems, and the problem outlined gives a clear reason for what sheaves accomplish. I think using a sheaf keeping track of structure which can be globally defined, without having to keep track of local information being glued to global information, is a bit of a red herring because it can leave the impression that adding sheaf theory to the mix will over-complicate the issue. In my experience, when I was introduced to sheaves it was in the context of differential topology and I think that lead to some difficulties in my intuition until I got the picture from the complex analytic and algebro-geometric side of the theory. There sheaf theory really shines because it is a technical requirement for a good theory. I think Taku makes a good point that there's a technical need for sheaves in these contexts, and including this technical need is absolutely instrumental to understanding sheaves in complex and algebraic geometry. Wundzer (talk) 15:57, 9 October 2020 (UTC)

I think your approach is simply not working: you are basically narrowing down the readership of this article (or at least the first section, which should be written the most welcoming way possible) to people who know / care about holomorphic functions. IMO, this is annoyingly unhelpful. Introducing locally ringed space as a motivation for sheaves is just driving away tons of readers. Also "This kind of formalism was found to be extremely powerful and motivates a lot of homological algebra such as sheaf cohomology since an intersection theory can be built using these kinds of sheaves. " -- again, this might be (properly improved) be used as a motivation for sheaf cohomology.

What is helpful, instead, would be to come up with easy examples of sheaves. Note that the concept of a sheaf can be explained to any high-school student (who cares): continuous functions, locally constant functions, easy non-examples such as constant functions or bounded functions. Our job is to allow such people, physicists, computer scientists, to understand what a sheaf is. Our job is not to put a barrage of much-higher-level mathematical allusions in front of them that preventssuch an understanding.

Conveying the historical motivation can be done, but is best deferred to the history section. Or maybe a (very brief) section on applications in complex analysis. As far as I can see, and this is also what I see from Serre's citation above, cohomological methods and sheaf cohomology (which is, again, a different topic!) serve to solve / express such problems. If I were to choose a motivating example from complex analysis, I would also rather use the complex logarithm or complex square root.

Again, I am staunchly opposed to this "introduction". Let's keep it simple (early on), and this article will improve. Jakob.scholbach (talk) 12:35, 10 October 2020 (UTC)

As you see, I am in the middle of restructuring the article. I would like to incorporate bits of the "Technical motivation" section elsewhere. However, I somehow struggle to get the point about the "Reformulating..." section: Wundzer, do you want to say something like "The category of complex manifolds is equivalent to some appropriate subcategory of ringed spaces." ? If so, can you give a reference for such a (precise) statement and / or reference which emphasises this point of view on manifolds? In this case, I suggest we add this bit of text to the section "Ringed spaces".

Hi, Jakob.scholbach thanks for your work so far. The main point of the "reformulation" section is it guides the reader towards understanding sheaves as a basic part of geometry, and gives a tool for incorporating geometric concepts through a sheaf theoretic language. I think it's in bad taste to leave readers with the idea that sheaves are some isolated concept, but are in fact a natural construction which is basically everywhere in geometry. Look at definition 1.23 of page 72 of https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf. This section answers your questions. Also, we could cite Ramanan's book on global analysis - he takes this point of view. Wundzer (talk) 23:39, 17 October 2020 (UTC)

OK, I have added a reference to Ramanan's book. Is Demailly's book published? If yes, feel free to add a reference, and also a reference to a precise section in Ramanan would be good (I don't have the book at hand right now). Jakob.scholbach (talk) 13:15, 18 October 2020 (UTC)

Literally the first chapter is titles "Sheaves and differentiable manifolds" but starting on page 12 he defines smooth manifolds using sheaves. Also, the Demailly book is not physically published as far as I know. Wundzer (talk) 20:26, 18 October 2020 (UTC)

Similarly, I don't quite get the point in "Tracking submanifolds...". Is the point here that every submanifold also has its own structural sheaf? That is also something that might go to the ringed spaces section. Jakob.scholbach (talk) 20:28, 17 October 2020 (UTC)

Hi, Jakob.scholbach. Here's a problem we find with projective manifolds: we cannot construct them as the vanishing locus of some holomorphic function. This contrasts the smooth category (smooth/C^\infty manifolds) where given a manifold $X$ and a submanifold $Y$, $Y$ can be constructed as the vanishing locus of some $f\in C^{\infty }(X)$. Back in the algebraic category (and complex manifolds) the sheaves serve as a categorified replacement for the function $f$ defining the subobject $Y\hookrightarrow X$. Then, once you start to think more algebro-geometrically, the sheaf ${\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{Y_{2}}$ is like considering the ideal $I_{1}+I_{2}$, and the ideals are a generalization of using the vanishing locus of some function. Wundzer (talk) 23:39, 17 October 2020 (UTC)

I agree sheaves of ideals are relevant, and they missing right now. I will add them (I hope soon) to the section on sheaves of modules.

Thanks! Wundzer (talk) 20:15, 18 October 2020 (UTC)

Do you have a reference that frames intersection theory based on ideal sheaves? In my experience, the problems of discussing intersections (e.g., multiplicities) are local problems, where there is less (or no?) need of working with sheaves. Jakob.scholbach (talk) 13:15, 18 October 2020 (UTC)

From what I know, ideal sheaves are not directly used since they are resolved if building intersection theory with K-theory, as in SGA 6. Wundzer (talk) 20:15, 18 October 2020 (UTC)

OK. I have therefore removed the motivation section (the material about global vs. local holomorphic functions does now appear elsewhere, and motivating sheaves with intersection theory therefore does not seem to work well.) Jakob.scholbach (talk) 14:05, 19 October 2020 (UTC)

Jakob.scholbach That's not a good idea. You are completely missing the point of the analogy, which can be rigorously understood. Look at the Support of a module page and note the fact the Serre intersection formula is concentrated in degree 0 for cohen macaulay intersections (look in 3264 and all that). The ideal sheaf is only part of the sequence resolving ${\mathcal {O}}_{Y}$. The Serre intersection formula directly uses the derived intersection ${\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{Y_{2}}$, and this derived intersection is the intersection product used in SGA 6. By thinking of ideal sheaves as the correct generalization you are looking at the wrong object! In addition, if you want the direct generalization of vanishing loci of functions in the topological category, constructible sheaves are a direct categorification of constructible functions, the sheaves ${\mathcal {O}}_{Y}$ are a direct categorification of the vanishing locus of $Y$. Wundzer (talk) 20:27, 19 October 2020 (UTC)

(unindent) I know that Serre's intersection formula is about derived tensor product such as $R/I\otimes _{R}^{L}R/J$. What I don't see is how this formula is working as a motivation for sheaves. After all, what is (IMO) interesting / nontrivial / crucial in this topic is already a local question (as you can see from the title "Algebre locale" of Serre's book. That said, once you understand intersection products in the guise of the above derived tensor products, you understand intersection theory completely. Surely you can (and Grothendieck did) phrase intersection theory for coherent sheaves, but this is IMO not a reasonable motivation for the general notion of sheaves. Is there a reference saying that Grothendieck (or anyone else) was motivated by intersection theory to introduce sheaves?

From a technical point of view of writing this article, I also think it is wise to keep forward references to an absolute minimum. The point that global sections do not distinguish between proper cx. manifolds is already mentioned in the stalks section. Also, the point with the "complexity" H(U) = H(C^n) is not striking me as noteworthy here. Jakob.scholbach (talk) 08:56, 20 October 2020 (UTC)

As a reminder, the whole point of referencing applications is the question: "Why should I care?". I think the centrality of intersection theory gives an important motivation for sheaves, sheaf cohomology, and derived categories. This material could be moved down to the derived categories section. Also, H(U) = H(C^n) is a complexity because you are taking local information which all "looks the same" and creating a tool which takes this information and gives a way to glue together on more non-trivial topologies. This gluing process is what allows you to differentiate between different complex structures. Maybe I should elaborate on this point by writing up details with an elliptic or higher genus curve. If you can give me a better word for describing this non-trivial behavior, I'm happy to use it. Wundzer (talk) 15:47, 20 October 2020 (UTC)

I disagree that intersection theory is central to the topic of this article. (I agree that it is an important theory in algebraic geometry as a whole, but this is not an article about algebraic geometry.)

To tell people "Why should you care" you need to choose topics that these people (as opposed to you as a writer) care about. By "these people" I mean an undergrad math student, or a physicist, say. Someone who does not know what a sheaf is will hardly know about intersection theory and obviously even less so about derived categories.

On top of that general problem with your approach, there is another specific issue: intersection theory can and has been formulated without sheaves at all, so the approach to i.t. via sheaves is, IMO, a side-remark on the formulation of intersection theory, but by no means a motivating introductory explanation for sheaves.

I agree that the local vs. global idea is a better motivation. I attempted to clarify your write-up on this section a bit further below ("By contrast, the global information present ..."). So, we now have this content roughly duplicated. I am not opposed to somehow moving it up, even though I find it better placed like this. In any case this content should be merged, and (IMO) the stuff on intersection theory is better removed (in section 1.4.2). Jakob.scholbach (talk) 10:00, 3 November 2020 (UTC)

Proposal for restructuring the article

Sparked by the above discussion about the motivation section, I thought more about this article. Here is a proposal for a new article structure. Comments?

• Definition and first examples
  • Define presheaves, sheaves. Minimize all mention of categorical notions here. Stick to (pre)sheaves of sets.
  • Examples: sheaves on manifolds etc., sections of continuous maps (including sky-scraper); some non-examples

• Operations on sheaves
  • Morphisms of sheaves
  • Basic functoriality (f_*, f^*), stalks
  • Sheafification (Example: constant presheaf vs. constant sheaf)

• Complements
  • Sheaves of modules, sheaves in other categories, sheaves of groupoids
  • Espace étalé
  • (Locally) ringed spaces. (Quasi-)coherent sheaves
  • Sites, topoi

• Sheaf cohomology
  • Basic examples (complex logarithm, for example)
  • Derived categories
  • Poincaré, Verdier duality, exceptional functors

• History

What is currently in the overview section seems to be completely redundant compared to the rest. What is currently in the technical motivation could go to later sections. (Some of this I would just remove, including the section "Reformulating complex manifolds using sheaves".) Jakob.scholbach (talk) 13:07, 11 October 2020 (UTC)

I agree the article should be reorganized and think your proposed organization is good! But, I disagree with your last comment partially, although I see how the language could be improved. Instead of using the word and derivatives of "reforumulate" we could say complex manifolds embed in the category of ringed spaces. This also would extend to complex analytic spaces, but to locally ringed spaces (since the local rings are relevant for singularity theory). Also, if you look on page 72 of https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf this perspective is taken by the author as well. Wundzer (talk) 17:32, 11 October 2020 (UTC)

OK -- We can revisit this point later, but I think this approach to manifolds / ringed spaces belongs (at best) to one of those articles. It is, IMO, at most of secondary / tertiary importance to this article here. Jakob.scholbach (talk) 19:29, 11 October 2020 (UTC)

I have begun restructuring the article as per the above. For now, I have left motivation and overview untouched, but I intend to merge these in elsewhere soon. If someone wants to help with this or other rough parts, please do so! Jakob.scholbach (talk) 22:41, 11 October 2020 (UTC)

Does "Sheaf of sections of a continuous map" only apply to surjective maps

in order for f \circ s = \operatorname{id}_U\ isn't it necessary for f(Y) to include U, and since this is defined for any U, doesn't that mean f must be surjective onto X?

Perhaps this should say "Sheaf of sections of a continuous surjective map"?

clahey (talk) 22:03, 19 November 2020 (UTC)

No, it is right (or applicable) in this generality. If the map f is not surjective, then F(U) may be empty. Jakob.scholbach (talk) 10:30, 20 November 2020 (UTC)

Missing information on global sections

Several other articles redirect "global sections" to this article, but there isn't actually a definition/description of it anywhere, and none of its usages in this article link anywhere else. A small section defining global sections would be a welcome addition! (I don't think I know enough to add this myself at the moment, although I can certainly try to revisit this to help if/when I have more information.)

Dzackgarza (talk) 19:54, 25 June 2021 (UTC)

mathscr

I'm trying about adding an explanation somewhere on wikipedia about the above typeface (image), but where would be a good place to add ? For example, Coherent Analytic Sheaves (doi:10.1007/978-3-642-69582-7) seem to use the above typeface. SilverMatsu (talk) 02:05, 30 March 2023 (UTC)