Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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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 ...
Daniel Moskovich's user avatar
97 votes
10 answers
13k 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 ...
Victoria Flat's user avatar
90 votes
3 answers
10k 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 ...
Saal Hardali's user avatar
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86 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 ...
Jan Weidner's user avatar
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63 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 ...
S. Carnahan's user avatar
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60 votes
8 answers
13k 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 ...
user avatar
56 votes
3 answers
9k views

What are the benefits of viewing a sheaf from the "espace étalé" perspective?

I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. ...
Simon Rose's user avatar
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53 votes
3 answers
10k views

Sheaves and bundles in differential geometry

Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
Harry Gindi's user avatar
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51 votes
7 answers
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What does a projective resolution mean geometrically?

For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
Justin DeVries's user avatar
48 votes
4 answers
7k views

Sheaf-theoretic approach to forcing

Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI. A general ...
Peter Scholze's user avatar
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 ...
Pete L. Clark's user avatar
43 votes
8 answers
13k 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 ...
roger123's user avatar
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39 votes
6 answers
8k views

What is the inverse image sheaf necessary for in algebraic geometry?

Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf $$U \mapsto \...
Charles Staats's user avatar
36 votes
6 answers
8k views

What is the Zariski topology good/bad for?

In a comment to this question the quotation "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry" appears. Well, so some spontaneous questions arise: 1) What is Zariski topology ...
35 votes
5 answers
3k 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 ...
B. Bischof's user avatar
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33 votes
4 answers
6k views

What (if anything) happened to Intersection Homology?

In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...
Alon Amit's user avatar
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33 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 ...
David E Speyer's user avatar
31 votes
4 answers
13k 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 ...
YOURS's user avatar
  • 533
29 votes
3 answers
3k views

Sheaf description of $G$-bundles

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, ...
Charles Siegel's user avatar
28 votes
3 answers
4k views

How do I compare the different notions of Fourier transform for sheaves?

There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...
David Treumann's user avatar
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 ...
David Corwin's user avatar
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26 votes
4 answers
3k views

What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?

There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here: http://ncatlab.org/nlab/show/matching+...
user332's user avatar
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26 votes
1 answer
4k views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
urelement's user avatar
  • 363
26 votes
2 answers
2k views

Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare. Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
Chris Schommer-Pries's user avatar
26 votes
1 answer
1k views

Why there is a Quot-scheme, not a Sub-scheme?

Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
evgeny's user avatar
  • 1,980
26 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 ...
C. Dubussy's user avatar
26 votes
1 answer
1k views

Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
Chris Schommer-Pries's user avatar
25 votes
3 answers
5k views

Stacks and sheaves

I'm a bit confused by the double role which sheaves play in the theory of stacks. On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
Andrea Ferretti's user avatar
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, ...
Maxime Ramzi's user avatar
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24 votes
5 answers
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Applications of the "other" definition of sheaves

In most literature, when you try to look for the definition of sheaves you will see the usual definition for presheaves as a functor from a topological space (or from a Grothendieck topology) to some ...
Jose Capco's user avatar
  • 2,175
24 votes
4 answers
5k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
Qiaochu Yuan's user avatar
24 votes
0 answers
908 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on $...
David Spivak's user avatar
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23 votes
5 answers
1k views

Sheafification via hypercovers

The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...
Mike Shulman's user avatar
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 ...
Dmitri Panov's user avatar
  • 28.8k
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 ...
Justin Curry's user avatar
  • 2,684
22 votes
3 answers
2k 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 ...
dhagbert's user avatar
  • 671
22 votes
1 answer
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 ...
Kevin Buzzard's user avatar
21 votes
2 answers
10k views

Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...
Jesus Martinez Garcia's user avatar
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 ...
Theo Johnson-Freyd's user avatar
21 votes
0 answers
563 views

Is there a useful theory of D-modules on smooth (non-analytic) manifolds?

D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
ಠ_ಠ's user avatar
  • 5,933
20 votes
5 answers
2k views

Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways: (Ordered): ...
David Zureick-Brown's user avatar
20 votes
2 answers
2k 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 ...
ComplexesOfSheaves's user avatar
20 votes
1 answer
2k views

Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...
Qiaochu Yuan's user avatar
20 votes
4 answers
4k views

The tangent bundle to an infinite-dimensional manifold

Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
Theo Johnson-Freyd's user avatar
20 votes
1 answer
1k views

Functorial characterization of open subschemes?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
Manny Reyes's user avatar
  • 5,102
20 votes
3 answers
867 views

Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
Mike Shulman's user avatar
19 votes
6 answers
3k views

Understanding Adjointness of Sheaves in Algebraic Geometry

Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of ...
LMN's user avatar
  • 3,525
19 votes
2 answers
369 views

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 ...
Arshak Aivazian's user avatar
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 ...
Qiaochu Yuan's user avatar
18 votes
2 answers
1k views

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 ...
zzz's user avatar
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