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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 ...
Tom Copeland's user avatar
  • 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 ...
Daniel Moskovich'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
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 ...
Victoria Flat's user avatar
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 ...
Gro-Tsen's user avatar
  • 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 ...
Zhen Lin's user avatar
  • 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 ...
dhagbert's user avatar
  • 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 ...
ComplexesOfSheaves's user avatar
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 ...
Edward Hughes's user avatar
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 ...
Mohammad Farajzadeh-Tehrani's user avatar
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 ...
HaroldF's user avatar
  • 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,...
Peter Crooks's user avatar
  • 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 ...
Gabriel's user avatar
  • 721
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 ...
Arshak Aivazian's user avatar
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 ...
Saal Hardali's user avatar
  • 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 ...
Jan Weidner's user avatar
  • 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 ...
S. Carnahan's user avatar
  • 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 ...
user avatar
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 ...
roger123's user avatar
  • 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 ...
David E Speyer's user avatar
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 ...
B. Bischof's user avatar
  • 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 ...
YOURS's user avatar
  • 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 ...
C. Dubussy's user avatar
  • 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 ...
David Corwin's user avatar
  • 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, ...
Maxime Ramzi's user avatar
  • 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 ...
Dmitri Panov's user avatar
  • 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 ...
Justin Curry's user avatar
  • 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 ...
Theo Johnson-Freyd's user avatar
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 ...
Kevin Buzzard's user avatar
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 ...
Dylan Wilson's user avatar
  • 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 ...
Qiaochu Yuan's user avatar
18 votes
2 answers
619 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 ...
Gro-Tsen's user avatar
  • 32.5k
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 ...
J Verma's user avatar
  • 3,218
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, ...
Alexander Körschgen's user avatar
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 ...
Amit Patel's user avatar
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 ...
Dedalus's user avatar
  • 1,071
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 ...
wonderich's user avatar
  • 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 ...
Mike Shulman's user avatar
  • 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 ...
Gabriel's user avatar
  • 721
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 $\{\...
Gro-Tsen's user avatar
  • 32.5k
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 ...
Yuhao Huang's user avatar
  • 5,052
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 ...
Harrison Smith's user avatar
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 ...
Victoria Flat's user avatar
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} \...
Agustí Roig's user avatar
  • 1,975
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 : ...
Agustí Roig's user avatar
  • 1,975
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_\...
Ariyan Javanpeykar's user avatar
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{...
Qfwfq's user avatar
  • 23.3k
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: ...
Hiro's user avatar
  • 945
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....
K Shao's user avatar
  • 623
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 ...
algori's user avatar
  • 23.5k