Talk:Sheaf (mathematics)

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

text: 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)

text: 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)

text: 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 Ui are open subsets of X with union U, then we get three maps:
* r : F(U) → Πi F(Ui), induced from the restrictions of F(U) to the sets Ui
* s : Πi F(Ui) → Πk,l F(Uk ∩Ul), induced from the restrictions of F(Ui) to the sets Ui∩Ul
* t : Πi F(Ui) → Πk,l F(Uk ∩Ul), induced from the restrictions of F(Ui) to the sets Uk∩Ui
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)
thread_title: Comments

message: 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)
thread_title:  in my opinion 

message: 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)
thread_title:  restrictions, revisited 

message: 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)
thread_title: Counter-examples/fiber bundle

message: 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)
replies: text: 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)

text: 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)
thread_title: New To Advanced Math

message: 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)
thread_title:  Original perceptions 

message: 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 ) 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)
thread_title:  The term "étalé space" 

message: " restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one"
we are not restricting open sets, we are not gluing open sets. we are restricting and gluing sections of the sheaf.
----
For some reasons, in the section "Direct and inverse images" the minus sign in the superscript -1 is sometimes missing. I checked
the code, and it is there. Can someone with more knowledge about this than me have a look, please?
131.188.103.41 15:33, 21 November 2006 (UTC)
thread_title:  about the introduction 

message: 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)
thread_title:  too long 

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

message: 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)
thread_title:  Unfinished sentence 

message: Regarding this section: 
:In general, construct a category J whose objects are the sets Ui and the intersections  and whose morphisms are the inclusions of  in Ui and Uj. 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)
replies: text: See, e.g., Mac Lane and Moerdijk, Sheaves in Geometry and Logic, II.1. Ozob (talk) 00:13, 21 October 2010 (UTC)

text: 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)

text: 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)
replies: text: 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)
thread_title:  Sheaf axiom as a limit 

message: 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 resV,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)
replies: text: 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)

text: 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)
thread_title:  Confused 

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

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Exactness 

message: 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)
replies: text: I think what I just added should be a good example. -- Taku (talk) 03:50, 31 December 2014 (UTC)
thread_title: non-sheaf examples!?

message: 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)
thread_title:  Finiteness conditions for sheaves of modules 

message: 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)
thread_title:  sheafs on (finite) graphs (as in graph-theory graphs) 

message: 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)
thread_title:  Is this article "too technical?" 

message: 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  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 Ui. 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(Ui) 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)
replies: text: You are correct.  U  must exactly equal  \bigcup_i U_i , otherwise the restriction s|_  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)
thread_title:  Confused - is something wrong? 

message: " 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)
replies: text: "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)
thread_title:  Confusion in opening paragraph 

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Proposal for restructuring the article 

message: 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)
replies: text: 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)
thread_title:  Does "Sheaf of sections of a continuous map" only apply to surjective maps 

message: 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)
thread_title:  Missing information on global sections 

message: 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 () seem to use the above typeface. SilverMatsu (talk) 02:05, 30 March 2023 (UTC)
thread_title:  mathscr