All Questions
Tagged with sheaves or sheaf-theory
979 questions
19
votes
6
answers
2k
views
Is the dual notion of a presheaf useful?
It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...
- 13.5k
19
votes
2
answers
393
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Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?
A friend of mine had the following question while reading the section "C2.2 The topos of sheaves" in "Sketches of an Elephant".
Let $G$ be a group (considered as a category with ...
- 6,962
19
votes
1
answer
1k
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A sheaf is a presheaf that preserves small limits
There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.
However when reading ...
19
votes
2
answers
3k
views
Explaining Mukai-Fourier transforms physically
A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...
- 10.5k
18
votes
9
answers
2k
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What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 118k
18
votes
2
answers
1k
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Measuring a presheaf's failure to be a sheaf?
Apologies for the vagueness of question.
Background
this thread has some nice examples of presheaves failing to be sheaves.
Question
Is there a generic way to measure "how badly" a presheaf fails ...
- 928
18
votes
4
answers
6k
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Derived categories of coherent sheaves: suggested references?
I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
- 3,218
18
votes
2
answers
2k
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What are the points (and generalized points) of the topos of condensed sets?
What are the topos points of $\mathrm{CondSet}$, i.e. the geometric morphisms $\mathrm{Set} \to \mathrm{CondSet}$?
More generally, is there a concise description of the geometric morphisms $\mathcal{E}...
- 1,110
18
votes
2
answers
4k
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Locally constant sheaves for the étale topology, lack of intuition about "étale-localness"
I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
- 291
18
votes
3
answers
2k
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Can $\mathcal O_X$ be recognized abstract-nonsensically?
This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.
In the ...
- 18.4k
18
votes
1
answer
1k
views
Why do rigid spaces have "not enough points"?
In Brian Conrad's notes
here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
- 585
18
votes
4
answers
1k
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Cohomology of a sheaf of functions locally constant along a foliation
Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...
- 28.9k
18
votes
2
answers
618
views
In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?
For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
- 32.5k
18
votes
0
answers
462
views
Is there a model category describing shape theory?
Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, ...
- 6,813
18
votes
0
answers
548
views
Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
- 1,701
17
votes
4
answers
1k
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Can one glue De Rham cohomology classes on a differential manifolds?
Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
- 47.5k
17
votes
3
answers
3k
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Sheafification - Why does twice suffice?
Hi,
I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve.
It is known that ...
- 1,071
17
votes
2
answers
517
views
Characterize constant objects in the internal language of a topos?
A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $Q$. We say a sheaf $X\colon Q^{\text{op}}\to\mathsf{Set}$ is constant if it is the sheafification of a ...
- 8,659
17
votes
2
answers
1k
views
Cosheafification
Hello all. I have a pre-cosheaf in the category of vector spaces. How do I cosheafify? I've failed to find literature on this topic.
I'll be more specific. Let $\mathbb{X}$ be a topological ...
- 171
17
votes
1
answer
1k
views
Is a direct sum of flabby sheaves flabby?
Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal ...
- 47.5k
17
votes
2
answers
1k
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Etale spaces using Kan extensions
Mac Lane - Moerdijk's "Sheaves" gives this cryptic hint in page 91 that the equivalence between etale spaces and sheaves on a space $X$ can be cooked up using formal methods.
More precisely, we are ...
- 13.6k
17
votes
2
answers
1k
views
Is the site of (smooth) manifolds hypercomplete?
By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
17
votes
1
answer
442
views
Examples of statements that are valid in every spatial topos
I am looking for statements¹ that, when interpreted in the internal language of a topos, are valid in all spatial toposes (i.e., the topos of sheaves of any topological space) that are not valid in ...
- 32.5k
17
votes
2
answers
1k
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What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos)....
- 9,408
17
votes
0
answers
648
views
Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
- 7,789
16
votes
3
answers
5k
views
Do we have non-abelian sheaf cohomology?
Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative ...
16
votes
2
answers
2k
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Understanding the definition of stacks
First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
- 1,093
16
votes
4
answers
2k
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Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 827
16
votes
2
answers
4k
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Under what circumstances do morphisms on the stalks of a sheaf induce a sheaf morphism
It is very well known that if $\alpha : \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves, then it induces homomorphisms on the stalks. I have been wondering for a while if given a collection of ...
- 3,830
16
votes
2
answers
9k
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Canonical Sheaf of Projective Space
I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow \mathcal{O}_{\mathbb{P}^...
- 953
16
votes
6
answers
14k
views
What books should I read before beginning Masaki Kashiwara and Pierre Schapira's "Sheaves on Manifolds"
I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara and Pierre Schapira's book "Sheaves on Manifolds", but I find it is not easy for me to understand.
What ...
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.5k
16
votes
3
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5k
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Stalks of sheaf-hom
Let $F$ and $G$ be sheaves on $X$. Under what conditions is the natural map from the stalk at $p$ of $\mathcal{H}\kern{-1pt}\mathit{om}(F,G)$ to $\mathrm{Hom}(F_p, G_p)$ an isomorphism?
- 161
16
votes
1
answer
448
views
Zorn's lemma for Grothendieck sites
In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
- 163
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2
answers
1k
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Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections?
I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every ...
- 118k
16
votes
0
answers
4k
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Sheaf cohomology and inverse limits
In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13:
Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
- 25.6k
15
votes
1
answer
2k
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How to motivate constructible sheaves
I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
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15
votes
2
answers
616
views
Is it true that $\operatorname{2-colim}_U \textsf{QCoh}(U) = \textsf{Vect}(K_X)$, as $U$ shrinks to the generic point?
Let $X$ be an integral scheme with function field $K$. If $U\subset X$ is an open subscheme, we may consider the restriction functor
$$\textsf{QCoh}(X) \to \textsf{QCoh}(U).$$
I don't know much about ...
- 711
15
votes
3
answers
900
views
What information is lost in $X \to \mathrm{Sh}(X)$?
Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure?
Thanks, Adrian.
- 151
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2
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2k
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Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 321
15
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1
answer
926
views
Sheaf-theoretically characterize a Riemannian structure?
A smooth structure on a topological manifold can be characterized as a sheaf of local rings, see for example the discussion here.
Q: Is there a way to characterize a Riemannian structure on a smooth ...
- 928
15
votes
1
answer
1k
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The real numbers object in Sh(Top)
If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...
- 66.8k
15
votes
1
answer
1k
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Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
- 10.5k
15
votes
2
answers
2k
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Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
- 5,548
14
votes
2
answers
904
views
What's the easiest example of a morphism of topoi that is not from that of a site?
A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of ...
- 5,052
14
votes
3
answers
2k
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Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology
I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...
- 141
14
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3
answers
1k
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Counterexamples to gluing complexes of sheaves
Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
- 937
14
votes
1
answer
702
views
Donaldson and DT invariants
Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can ...
- 587
14
votes
1
answer
2k
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Hypercohomology of a complex via Cech cohomology
Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
- 141
14
votes
1
answer
457
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References regarding a connection between recursion theory and sheaves
In Manin's A Course in Mathematical Logic for Mathematicians, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by:
$\mathcal{E}$ is the set of all ...
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