All Questions
13 questions
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
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
12
votes
1
answer
2k
views
Reference request: Book of topology from "Topos" point of view
Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
- 545
9
votes
1
answer
400
views
The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
- 23.3k
6
votes
1
answer
494
views
Prove category of constructible sheaves is abelian
Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of ...
- 1,701
5
votes
1
answer
723
views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
- 7,442
5
votes
1
answer
350
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 6,038
4
votes
1
answer
390
views
Is every soft sheaf of countable $\mathbb Q$-vector spaces a direct sum of skyscraper sheaves?
Let $X$ be a finite-dimensional compact metrizable space (these properties might partially be irrelevant; on the other hand, the case $X=[0,1]$ is already interesting to me).
Let $\mathcal F$ be a ...
- 3,184
3
votes
0
answers
877
views
The "pullback presheaf" and the proper base change theorem in topology
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
V\...
- 7,609
2
votes
1
answer
177
views
Is the restriction of a simple sheaf of modules simple?
Let $X$ be a topological space, $A$ a sheaf of (unital and associative but not necessarily commutative) rings on $X$. Suppose $M$ is a simple quasicoherent $A$-module and $U$ an open subset of $X$. Is ...
- 3,079
2
votes
0
answers
126
views
Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
- 3,079
2
votes
0
answers
35
views
If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
1
vote
0
answers
200
views
Question regarding affine fibre bundles
Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...
- 585