6
votes
1answer
317 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? I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...
18
votes
0answers
542 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 ...
1
vote
0answers
84 views

Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
4
votes
1answer
268 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 ...
8
votes
1answer
515 views

When does the sheaf cohomology of a topological space vanish?

The question is in the title. A more precise formulation is: Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$? The obvious example is a ...
3
votes
1answer
278 views

Restricting a Soft Sheaf to an Open is again Soft?

Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :) Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are ...
9
votes
1answer
379 views

When do sheaves which are constant along the fibers come from the base?

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves ...
0
votes
1answer
367 views

Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question

Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
2
votes
1answer
510 views

Spectral sequences in Hypercohomology of sheaves

Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
1
vote
2answers
384 views

Hypercohomology of a complex of sheaves that might be acyclic (or might not)

Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves ...
5
votes
3answers
1k views

Where can I find a proof of the de Rham-Weil theorem?

Where can I find a proof of the de Rham-Weil theorem? Does anyone know?
58
votes
5answers
3k 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 ...
1
vote
0answers
139 views

Soft sheaves on indiscrete paracompact spaces

Let $X$ be some space, I have basically 2 questions: 1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
1
vote
0answers
471 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 $X$: $$ ...
1
vote
1answer
649 views

How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
3
votes
1answer
381 views

Functoriality of base change

Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism ...
3
votes
1answer
370 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 ...
13
votes
2answers
557 views

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 ...
26
votes
8answers
2k 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 sheaf ...