Questions tagged [sheaf-theory]
For questions about sheaves on a topological space.
931
questions
10
votes
1
answer
506
views
Polylogarithm sheaves
In many different places, I could find the notion on ''(poly)logarithm sheaves''. As is indicated in the name of it, I guess that it should have something to do with (poly)logarithm function: $\mathrm{...
10
votes
1
answer
484
views
What is the total space of a stack after all?
From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
10
votes
1
answer
580
views
How do the direct and inverse image sheaf functors interact with homotopy?
This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$)...
10
votes
0
answers
494
views
Isbell duality between algebras and sheaves
nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...
10
votes
0
answers
361
views
How to model (affine) schemes with a large sketch?
Guitart states in "Toute theorie est algebrique et topologique" as Proposition 17 that the category $\mathbf{Sch}$ of schemes is the category of models of a large mixed sketch. Presumably, ...
10
votes
0
answers
825
views
intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
10
votes
0
answers
731
views
Differential Forms in Infinite Dimensions
In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of ...
10
votes
0
answers
473
views
Applications of sheaf theory to the computation of invariants of LS-category type
I would like to know if sheaf theory can be applied to a particular class of questions in topology.
The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
9
votes
3
answers
2k
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?
9
votes
2
answers
4k
views
Inverse Image as the left adjoint to pushforward
This is a repost of a question on Math stackexchange. No one is biting at it there, so I guess it is harder than I thought.
Assume $X$ and $Y$ are topological spaces, $f : X \to Y$ is a continuous ...
9
votes
4
answers
2k
views
Relative version of sheaf cohomology?
Is there a relative version of sheaf cohomology?
EDIT: I rather mean the cohomology of pairs.
9
votes
2
answers
726
views
Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?
I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning.
Let $M$ be a manifold, and consider the presheaf $C^*(-,...
9
votes
2
answers
1k
views
Geometric interpretation of sheaf cohomology
Please forgive me for the informal and naïve nature of my question, as I am a beginner in algebraic geometry.
In the famous book by Hartshorne, sheaf cohomology is defined as a certain derived functor....
9
votes
1
answer
3k
views
Motivation for equivariant sheaves?
Hello everyone;
i'm looking for a motivation for equivariant sheaves (see http://ncatlab.org/nlab/show/equivariant+sheaf) ~ Why are we interested in them?
More explicitely: Can I think of G-...
9
votes
1
answer
716
views
Why is there no stack of $\ell$-adic sheaves on a curve?
One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...
9
votes
2
answers
338
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 ...
9
votes
1
answer
384
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 $\mathbb{C}$ ...
9
votes
3
answers
2k
views
Representable Presheaf
I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
9
votes
2
answers
1k
views
Is there a description of sheaf cohomology in algebraic-topological terms?
Is there a description of of sheaf cohomology for the sheaf of sections of a continuous function in terms of common constructions in Algebraic Topology?
In more detail: Any sheaf on a space X can be ...
9
votes
1
answer
762
views
Sites which are stacks over themselves
A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
9
votes
1
answer
410
views
Stable homotopy category of complexes of sheaves
Let $X$ be an Hausdorff space, which is locally compact and locally connected. Let $R$ be a commutative and unitary ring and let $D(X)$ be the derived category of unbounded complexes of sheaves of $R$-...
9
votes
1
answer
771
views
Is the analytification functor part of a geometric morphism of topoi?
Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...
9
votes
2
answers
1k
views
Atlas of a manifold as a Sheaf
--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...
9
votes
1
answer
393
views
Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?
Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...
9
votes
1
answer
425
views
Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds
Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:
Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
I understand that this category $\text{...
9
votes
1
answer
376
views
Automorphisms of constant sheaves
Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where $\...
9
votes
1
answer
3k
views
surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
9
votes
1
answer
164
views
W-types and inverse image functor
All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...
9
votes
1
answer
321
views
When are free modules on sheaves of sets quasicoherent?
This question was previously asked over at math.SE.
Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over ...
9
votes
1
answer
411
views
How to view $\textbf{Sh}(\textbf{CartSp})/X$ as "space" in its own right, étale machinery from abstract nonsense perspective for smooth manifolds
Let $\textbf{CartSp}$ be the category of spaces of the from $\mathbb{R}^n$ with smooth maps between them. This is a site with respect to (differentially) good open covers, so consider the Grothendieck ...
9
votes
0
answers
438
views
Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
9
votes
0
answers
278
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
9
votes
0
answers
195
views
Does the category of $G$-equivariant sheaves have enough injectives?
The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...
9
votes
0
answers
502
views
Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
9
votes
0
answers
533
views
In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
9
votes
0
answers
367
views
Is there any notion of "smoothification" from $\mathbb{R}$-schemes to generalized smooth spaces?
I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
9
votes
0
answers
245
views
How does the machinery of left-exact comonads generalize from sheaves to stacks?
Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
9
votes
0
answers
624
views
Colimits of quasi-coherent sheaves on a ringed space
Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...
9
votes
0
answers
359
views
Topologies (and sheaves) on Cat and CAT
I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...
8
votes
2
answers
4k
views
Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
8
votes
2
answers
670
views
Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?
Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\...
8
votes
2
answers
3k
views
Sheaf cohomology question
For a topological space $X$ and a sheaf of abelian groups $F$ on it, sheaf cohomology $H^n(X,F)$ is
defined.
Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally ...
8
votes
2
answers
952
views
Example of a Sheaf (on the site of smooth manifolds) with Nontrivial Cohomology on $\mathbb{R}^n$?
Does such a sheaf of abelian groups exist? If not, is there a reference or a proof? Does such a sheaf of non-abelian groups exist?
I realized recently that while I've taken it for granted that ...
8
votes
1
answer
885
views
What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
8
votes
2
answers
525
views
A very elementary question on the definition of sheaf on a site
I'm now studying the etale cohomology with the book 'Introduction to Etale Cohomology' by Tamme.
In the page 26 of the book, 'a family of effective epimorphisms' is introduced.
'A family $\{ U_{i} \...
8
votes
1
answer
1k
views
Splitting of exact triangles in derived category
Let $A\to B\to C\to A[1]$ be a distinguished triangle in a (bounded below) derived category of an abelian category.
Is there a necessary and sufficient condition that it splits, namely $B\simeq A\...
8
votes
1
answer
1k
views
Kozsul resolution of $\mathcal{O}_X$
Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...
8
votes
1
answer
2k
views
Sheaves of $\mathbb Z$-modules = sheaves of abelian groups
In his "Algebraic Geometry", Hartshorne proves that for any ringed spaces $(X,\mathcal O_X)$, category $Mod(X)$ of sheaves of $\mathcal O_X$-modules has enough injectives. If we take $\...
8
votes
2
answers
2k
views
Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?
In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, ...
8
votes
1
answer
1k
views
Fpqc sheafification and localisation
I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...