Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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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 $...
David Spivak's user avatar
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21 votes
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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 ...
ಠ_ಠ's user avatar
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18 votes
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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, ...
Sebastian Goette's user avatar
18 votes
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529 views

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 ...
Benighted's user avatar
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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 ...
user7316's user avatar
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17 votes
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633 views

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 ...
Saal Hardali's user avatar
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16 votes
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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 ...
Akhil Mathew's user avatar
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13 votes
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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 $\{\...
Gro-Tsen's user avatar
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12 votes
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494 views

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 $...
Ekaterina Bogdanova's user avatar
12 votes
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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 ...
Harrison Smith's user avatar
10 votes
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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) ...
Ilk's user avatar
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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, ...
Martin Brandenburg's user avatar
10 votes
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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 ...
Amos Kaminski's user avatar
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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 ...
Matthias Ludewig's user avatar
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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 ...
Mark Grant's user avatar
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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 ...
Markus Zetto's user avatar
9 votes
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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 ...
Konrad Waldorf's user avatar
9 votes
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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 ...
Zhaoting Wei's user avatar
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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 ...
Morgan Rogers's user avatar
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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 ...
ಠ_ಠ's user avatar
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9 votes
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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{...
Saal Hardali's user avatar
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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 ...
pnips's user avatar
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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 ...
Martin Brandenburg's user avatar
9 votes
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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 ...
David Roberts's user avatar
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8 votes
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387 views

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{...
Qi Zhu's user avatar
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320 views

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. ...
user234212323's user avatar
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572 views

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 ...
curious math guy's user avatar
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264 views

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 \...
S. carmeli's user avatar
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8 votes
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335 views

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 ...
Dmitri Pavlov's user avatar
8 votes
0 answers
244 views

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'...
John Pardon's user avatar
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8 votes
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433 views

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 ...
John Pardon's user avatar
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8 votes
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300 views

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) ...
მამუკა ჯიბლაძე's user avatar
8 votes
0 answers
566 views

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 \...
ಠ_ಠ's user avatar
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8 votes
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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 ...
Exterior's user avatar
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8 votes
0 answers
345 views

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 ...
Vivek Shende's user avatar
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8 votes
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331 views

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?
Vivek Shende's user avatar
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8 votes
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368 views

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 ...
Hanno's user avatar
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7 votes
0 answers
618 views

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 ...
EJAS's user avatar
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7 votes
0 answers
154 views

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 ...
John Rached's user avatar
7 votes
0 answers
544 views

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). ...
ಠ_ಠ's user avatar
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7 votes
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401 views

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 $X$ ...
Saal Hardali's user avatar
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7 votes
0 answers
196 views

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 \...
ಠ_ಠ's user avatar
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7 votes
0 answers
370 views

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-...
Jonas Gomes's user avatar
7 votes
0 answers
205 views

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 ...
Rafael Mrden's user avatar
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7 votes
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203 views

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 ...
paul Monsky's user avatar
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7 votes
0 answers
758 views

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 ...
Kestutis Cesnavicius's user avatar
6 votes
0 answers
208 views

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 ...
Arrow's user avatar
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6 votes
0 answers
181 views

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, ...
Misha Verbitsky's user avatar
6 votes
0 answers
161 views

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 ...
Arshak Aivazian's user avatar
6 votes
0 answers
148 views

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 $X$,...
Lukas Miaskiwskyi's user avatar

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