All Questions
Tagged with or sheaf-theory sheaf-theory
89 questions
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
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
48
votes
8
answers
8k
views
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 65.4k
98
votes
10
answers
14k
views
equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Given a topological space $X$, we can define the sheaf cohomology of $X$ in
I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$)
or
II. the Čech ...
- 1,871
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
29
votes
3
answers
4k
views
Is there a good general definition of "sheaves with values in a category"?
Let $\mathcal{A}$ be a category.
There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
- 15.9k
22
votes
3
answers
3k
views
Necessity of hypercovers for sheaf condition for simplicial sheaves
I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.
For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that ...
- 661
21
votes
2
answers
3k
views
Sheaves of complexes and complexes of sheaves
Let A be an abelian category, and X a topological space.
There are two ways one could try to construct some oo-category of sheaves on X from this data:
Consider the category $Sh(X,A)$ of sheaves on ...
16
votes
4
answers
2k
views
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
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 ...
13
votes
1
answer
1k
views
How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
- 433
13
votes
1
answer
1k
views
Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude,...
- 4,920
11
votes
2
answers
1k
views
Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
- 711
9
votes
2
answers
377
views
Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
- 6,962
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
89
votes
5
answers
18k
views
What is sheaf cohomology intuitively?
What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...
- 13.2k
66
votes
4
answers
11k
views
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
- 45.7k
62
votes
8
answers
14k
views
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
45
votes
8
answers
14k
views
How should one think about sheafification and the difference between a sheaf and a presheaf
The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
- 2,782
36
votes
3
answers
4k
views
What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 156k
35
votes
5
answers
4k
views
Heuristic explanation of why we lose projectives in sheaves.
We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...
- 4,842
34
votes
4
answers
15k
views
When will the pushforward of a structure sheaf still be a structure sheaf?
Let $f:X\rightarrow Y$ be a morphism of schemes.
When $PicY\rightarrow PicX$ is an embedding and $f_{*}\mathscr{O}_{X}$ is invertible, it is the structure sheaf of $Y$.
In the proof of Zariski's Main ...
- 563
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
27
votes
1
answer
2k
views
Galois Group as a Sheaf
I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
- 15.4k
25
votes
2
answers
3k
views
Are there (enough) injectives in condensed abelian groups?
The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact, ...
- 15.9k
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
22
votes
5
answers
6k
views
Cohomology of Structure Sheaves: Algebraic, Constructible and more
I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
- 2,684
21
votes
3
answers
2k
views
Is there a "categorical" description of Grothendieck's algebra of differential operators?
First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due ...
- 54.6k
21
votes
2
answers
2k
views
Naive question about constructing constructible sheaves.
In algebraic geometry, an etale sheaf on a Noetherian scheme is called constructible if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each ...
- 41.4k
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
18
votes
9
answers
2k
views
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
4
answers
6k
views
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
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
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
3
answers
3k
views
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
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, ...
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
15
votes
1
answer
1k
views
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
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
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
0
answers
297
views
Ordinal-valued sheaves as internal ordinals
Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
- 32.5k
13
votes
0
answers
481
views
Making the conceptual leap from locales to Grothendieck topologies?
I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
- 373
12
votes
2
answers
818
views
global fibrations of simplicial sheaves
I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : ...
- 1,975
12
votes
1
answer
1k
views
Mitchell's embedding theorem
Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} \...
- 1,975
12
votes
6
answers
1k
views
Assumptions on the category C for sheafification of C-valued presheaves
For any category C and topological space X we have the notion of a C-valued presheaf on X.
What assumptions must be made about C in order that we have the notion of such a presheaf being a 'sheaf'? I ...
- 1,871
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
11
votes
5
answers
8k
views
When is the push-forward of the structure sheaf locally free
Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then $f_\...
- 9,497
11
votes
2
answers
4k
views
Is Sheafification Functor Exact?
I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.
My question is:
...
- 945
11
votes
1
answer
892
views
Are all Grothendieck topologies on Set equivalent?
The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X_i\stackrel{\phi_i}{\hookrightarrow} X\}\in\mathrm{...
- 23.3k
11
votes
1
answer
1k
views
Etalé space construction for presheaves on a Grothendieck site
As it is described for example in [Mac Lane-Moerdijk, Sheaves in Geometry and Logic, II.6.], one can construct the sheafification functor very lucidly by associating to a presheaf a certain bundle (cf....
- 623