Questions tagged [sheaf-theory]
For questions about sheaves on a topological space.
334
questions with no upvoted or accepted answers
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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 $...
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563
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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 ...
18
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440
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Is there a model category describing shape theory?
Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, ...
18
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529
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Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
18
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Idea of presheaf cohomology vs. sheaf cohomology
Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...
17
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633
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Is there an Infinite dimensional sheaf theory for analysis on manifolds?
I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the ...
16
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Sheaf cohomology and inverse limits
In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13:
Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
13
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270
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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 $\{\...
12
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Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?
Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
12
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446
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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 ...
10
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494
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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) ...
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361
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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
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825
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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
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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
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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
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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
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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
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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
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502
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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
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533
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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 ...
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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
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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
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624
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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
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359
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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
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387
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Sheaf of compact Hausdorff spaces but not a condensed anima
Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
8
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320
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Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
8
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572
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Stalks of limit sheaves
Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map
$$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow ...
8
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265
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Grothendieck axioms and sheaf categories
An abelian category, $A$, is said to satisfy axiom AB6 if for every family of filtered diagrams $I_j$ indexed by a set $J$, the canonical map $\mathrm{colim}_{\prod_{I_j}}(\prod_j M_{ij}) \to \prod_J \...
8
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What is a morphism of ∞-sites?
Recall that a morphism of sites
is a covering-flat functor
that preserves covering families.
Morphisms of sites can be identified with those
geometric morphisms of induced toposes
for which the ...
8
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244
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Why mu-stratifications?
In the microlocal theory of sheaves developed by Kashiwara and Schapira, there is the notion of a $\mu$-stratification, which is a stratification satisfying a stronger property ("$\mu$") than Whitney'...
8
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433
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Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
8
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300
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Examples of locally-but-not-globally (pre)sheaf toposes?
Re-reading my own recently posted question What is the total space of a stack after all? I realized that I don't know something more simple and presumably more basic.
Are there (bounded if you like) ...
8
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566
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Can we use sheaf cohomology to say anything interesting for vector bundles with non-flat connections?
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \...
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Grothendieck - A group as a sheaf over simplicial complexes
In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
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Why do Kashiwara and Schapira require a base ring of finite global dimension?
In the book "Sheaves on Manifolds" by Kashiwara and Schapira, they work always with sheaves of $R$-modules, where $R$ is a ring of finite global dimension.
Why do they do this, what care ...
8
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331
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What's the Hochschild homology of the category of constructible sheaves?
Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?
8
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368
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Dualizing complex of the product of two locally compact spaces
Hello!
In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
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What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
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Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?
Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
7
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544
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What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?
The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
...
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Generalities on sheaves - Where can I find the technology that can make this "proof" of Atiyah duality precise?
Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct).
Let $X$ ...
7
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196
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Category of all sheaves on all spaces and their (co)morphisms
In Bredon's Sheaf Theory text, he defines a category of all sheaves on all spaces, whose arrows, called cohomomorphisms, are pairs $(\psi, f): \mathscr{F} \to \mathscr{E}$, where $\mathscr{E} \in \...
7
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Sheafs in O-minimal Structures
Let $\mathcal{N} = (N, <, \ldots)$ be an o-minimal structure and let $X \subset N^m$ be a definable set. Following the procedure stablished by Edmundo, Jones and Peatfield in "Sheaf cohomology in o-...
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Correspondence between Verma module morphisms and invariant differential operators - is it exact?
For a (complex, connected, simply connected, simple) Lie group $G$, a parabolic subgroup $P \subseteq G$, and a $\mathfrak g$-integral $\mathfrak p$-dominant weight $\lambda$, we can construct ...
7
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203
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sheaves on thickened nodal cubics
Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...
7
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758
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A question on a proof that fine sheaves are soft
Let's open R.O.Wells "Differential Analysis on Complex Manifolds" p. 53 and have a look at the Proposition 3.5 stating that all fine sheaves are soft (over a paracompact Hausdorff $X$). In the proof ...
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Is the right adjoint to presheaf direct image interesting?
Let $X\overset{f}{\to}Y$
be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
6
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181
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G-sheaves on spaces with a free G-action
Let $X$ be a topological space
equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined
"$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space,
...
6
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161
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Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
6
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148
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Cech cohomology on product covers & Fréchet sheaves
My question is about the paper [Ka67].
Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...