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Singular cohomology as a sheaf of $\infty$-categories

In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
Henry Badhead's user avatar
4 votes
0 answers
102 views

Topos as a totally cocomplete object in a 2-category CART

In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...
Ilk's user avatar
  • 1,347
3 votes
1 answer
260 views

Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme. Let $R^1f_*\mathbb{Q}...
kindasorta's user avatar
  • 2,907
3 votes
1 answer
466 views

Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?

Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
Paul Cusson's user avatar
  • 1,763
2 votes
0 answers
253 views

The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds

$\def\sO{\mathcal{O}} \def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...
Elías Guisado Villalgordo's user avatar
5 votes
2 answers
233 views

References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?

Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category? I know that one can define "categorical ...
Tanny Sieben's user avatar
14 votes
3 answers
2k views

Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology

I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...
Ofek Aman's user avatar
  • 141
10 votes
0 answers
533 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) ...
Ilk's user avatar
  • 1,347
2 votes
0 answers
128 views

On the generalization of a Cech-to-sheaf type spectral sequence

Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
FPV's user avatar
  • 541
6 votes
1 answer
441 views

Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
Jakob Werner's user avatar
  • 1,153
1 vote
1 answer
235 views

Is there a description of cellular automata in form of sheaves?

Cellular automata are defined through rules in a local neighborhood and sheaves, as far as I understand, can be used to glue local data to global data. Has there been any effort to bring those two ...
mathoverflowUser's user avatar
11 votes
1 answer
1k views

Are groups determined by their morphisms from solvable groups?

$\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h_G:=\text{hom}...
Chris H's user avatar
  • 1,949
1 vote
0 answers
106 views

Invariance of categories of sheaves (on simplicial presheaves) under (local) weak equivalence

Let $\mathcal{C}$ be a Grothendieck site (secretly in my head I am thinking of Hausdorff topological spaces with open covers; if I am daring I might be thinking of the big etale site on complex ...
rvk's user avatar
  • 563
6 votes
1 answer
327 views

Topology on cohomology of a sheaf of topological groups

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question: Is there a natural way to introduce topology on $H^i(X,...
 V. Rogov's user avatar
  • 1,170
9 votes
0 answers
308 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
1 vote
0 answers
131 views

Schur's lemma for sheaves with different reduced Hilbert polynomials

Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement: Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
alg_et_geom's user avatar
3 votes
0 answers
83 views

Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
Anna Montaruli's user avatar
4 votes
0 answers
201 views

Infinity-categorical exceptional push-forward

Classically, if $f:X\to Y$ is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor $f_!:Sh(X;k)\to Sh(Y;k)$ among $k$-valued sheaves for, say, a ...
S. carmeli's user avatar
  • 4,189
3 votes
0 answers
375 views

Equivariant sheafs and $G$ actions on modules

I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf. The set up ...
Louis Jaburi's user avatar
2 votes
1 answer
776 views

Mathematical background required to learn about sheaves

Due to my interest in type theory (and higher type theory), I have found that learning about sheaves might be useful (for, e.g., sheaf models of type theories). There is Kashiwara and Schapira's ...
xuq01's user avatar
  • 1,094
9 votes
0 answers
545 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 ...
Morgan Rogers's user avatar
6 votes
1 answer
359 views

Homotopy cosheaf?

Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces. Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every ...
Piotr Achinger's user avatar
2 votes
0 answers
143 views

Computing derived functor of a complex with non-acyclic terms

Let $A^\bullet =(\dots\to A^i\to A^{i+1}\to\dots)$ be a bounded below complex in an abelian category $\mathcal{A}$ with sufficiently many injectives. Let $F\colon \mathcal{A}\to \mathcal{B}$ be an ...
asv's user avatar
  • 21.8k
4 votes
0 answers
432 views

Reference request: sheaf-theoretic operations in the classical topology?

Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
Student's user avatar
  • 273
2 votes
0 answers
83 views

Sheaf of R-modules and modules over compactly supported functions

I'm looking for a reference for the following result: Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over ...
Simon Henry's user avatar
  • 42.4k
5 votes
0 answers
189 views

Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
asv's user avatar
  • 21.8k
4 votes
2 answers
315 views

Equivalence of different cohomology groups

Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups: (1) singular cohomology $H_{sing}^*(X,A)$;...
asv's user avatar
  • 21.8k
11 votes
2 answers
413 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
asv's user avatar
  • 21.8k
0 votes
1 answer
164 views

Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf?

Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact. Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $...
Zhaoting Wei's user avatar
  • 9,019
5 votes
0 answers
377 views

Push forward of the constant sheaf for a Serre's fibration

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
asv's user avatar
  • 21.8k
11 votes
3 answers
935 views

Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book http://www.amazon.com/Introduction-...
David Lucien's user avatar
3 votes
2 answers
488 views

Application of sheaves theory in ring theory

Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
Alex's user avatar
  • 49
8 votes
0 answers
355 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
  • 8,723
3 votes
0 answers
422 views

What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?

It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
Zhaoting Wei's user avatar
  • 9,019
2 votes
2 answers
2k views

Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...
Asten Matshink's user avatar
14 votes
2 answers
924 views

A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ? I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
Mathieu ANEL's user avatar
6 votes
2 answers
580 views

Associating a principal bundle to a torsor

In Introduction to the language of stacks and gerbes, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $...
user40276's user avatar
  • 2,227
8 votes
3 answers
643 views

Is there a name for a "rigid" sheaf?

Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty? In other words, this ...
Bruno Joyal's user avatar
  • 3,910
4 votes
0 answers
261 views

Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
Mikhail Bondarko's user avatar
12 votes
1 answer
2k views

Reference request: sheaves on closed sets

I am faced with a context in which the most natural notion of a sheaf $\mathcal F$ is as a functor on the category of compact subsets of a (locally compact Hausdorff) space $X$. Specifically, I say a ...
John Pardon's user avatar
  • 18.7k
14 votes
1 answer
2k views

Hypercohomology of a complex via Cech cohomology

Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
none's user avatar
  • 141
9 votes
0 answers
672 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 ...
Martin Brandenburg's user avatar
6 votes
1 answer
1k views

Tensor product of sheaves separated?

Let $(X, O_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an ...
LMN's user avatar
  • 3,555
12 votes
3 answers
845 views

Can one characterize those sheaves which have Hausdorff etale spaces?

Given a sheaf of sets $F$ on a space $X,$ under the equivalence of categories between etale spaces over $X$ and sheaves over $X,$ $F$ is associated to a local homeomorphism $$E\left(F\right) \to X$$ ...
David Carchedi's user avatar
10 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?
Louis A's user avatar
  • 360
5 votes
0 answers
336 views

Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?

For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
Mikhail Bondarko's user avatar
1 vote
0 answers
249 views

On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
Mikhail Bondarko's user avatar
1 vote
1 answer
414 views

Restriction of a sheaf to an infinitely small neighbourhood of a closed submanifold: how to work with this ind-sheaf?

Let $X$ be a manifold, $i: Z\to X$ is a closed embedding. For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z_\varepsilon$ the set of points of $X$ ...
2 votes
0 answers
358 views

What are the easiest cases of base change (for sheaves on sites)?

I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
Mikhail Bondarko's user avatar
2 votes
0 answers
414 views

Do inverse images respect flabby sheaves?

Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
Mikhail Bondarko's user avatar