All Questions
Tagged with sheaf-theory dg.differential-geometry
50 questions
113
votes
4
answers
13k
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 ...
- 22.1k
54
votes
3
answers
11k
views
Sheaves and bundles in differential geometry
Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" ...
- 19.6k
26
votes
2
answers
2k
views
Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
24
votes
4
answers
6k
views
What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
- 118k
24
votes
1
answer
837
views
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 ...
- 6,025
23
votes
4
answers
5k
views
De Rham decomposition theorem, generalisations and good references
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
- 28.9k
20
votes
4
answers
4k
views
The tangent bundle to an infinite-dimensional manifold
Suppose that $A,B$ are smooth ($\mathrm C^\infty$) manifolds, and denote by $\hom(A,B)$ the set of $\mathrm C^\infty$-maps $A \to B$. It is a perfectly well-defined set, but often one wants more. ...
- 54.6k
17
votes
0
answers
648
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 ...
- 7,789
16
votes
3
answers
3k
views
Physical interpretations/meanings of the notion of a sheaf?
I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
- 10.5k
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 ...
- 141
12
votes
4
answers
3k
views
Is there an easy way to describe the sheaf of smooth functions on a product manifold?
A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic (...
- 6,069
11
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. ...
- 2,074
11
votes
1
answer
866
views
Serre spectral sequence for de Rham cohomology
Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...
- 2,305
10
votes
2
answers
1k
views
Differential forms as a sheaf on the site of all manifolds vs. sheaf on an individual manifold
The de Rham complex can be viewed as a sheaf $\Omega$ on the entire site $\mathsf{Man}$ of smooth manifolds via the usual pullback of differential forms $(f: M \to N) \mapsto (f^*: \Omega(N) \to \...
- 6,025
9
votes
1
answer
447
views
Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds
Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:
Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
I understand that this category $\text{...
- 6,023
9
votes
0
answers
570
views
In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...
- 6,025
9
votes
0
answers
378
views
Is there any notion of "smoothification" from $\mathbb{R}$-schemes to generalized smooth spaces?
I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...
- 7,789
8
votes
1
answer
1k
views
Relative version of de Rham cohomology with local coefficients
Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\...
- 6,025
8
votes
0
answers
588
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 \...
- 6,025
7
votes
3
answers
1k
views
Encounters with partitions of unity
Not sure how this would be received here. This question is about smooth partitions of unity.
Let $M$ be a manifold. Consider an open cover $\{U_\alpha\}_{\alpha\in \Lambda}$ of $M$. A collection of ...
- 6,023
7
votes
2
answers
518
views
Do pseudo-differential operators form a sheaf of algebras?
Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
- 7,789
7
votes
2
answers
1k
views
Natural operators in differential geometry - why are they natural?
I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...
- 10.5k
6
votes
1
answer
752
views
Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user82886
6
votes
0
answers
179
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 ...
- 6,962
5
votes
1
answer
351
views
Exercises around Diffeological Spaces or a Diffeologic Atlas Theory
Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
- 2,074
5
votes
1
answer
286
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 6,038
5
votes
1
answer
331
views
Is the sheaf associated to a differential structure of a specific type?
On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
5
votes
0
answers
154
views
Sheaf-like reconstruction of a continuous function
Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
- 5,405
5
votes
0
answers
346
views
Properties of microlocalization
Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
- 303
5
votes
0
answers
380
views
Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
- 6,025
4
votes
0
answers
278
views
Are manifolds "naturally" ringed or locally ringed spaces?
My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.
On the one hand, it's reasonable to ...
- 711
3
votes
1
answer
747
views
Are cohomology functors sheaves?
Question is the following:
Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
More generally, are cohomology functors sheaves in ...
- 6,023
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 ...
- 1,763
3
votes
1
answer
89
views
The sheaf propagation is open in the zero section
Let $X$ a smooth manifold and $F$ a sheaf (let's say of abelian groups) on $X$. We will say that $F$ propagates at $x\in X$ in the (co)-direction $p \in T_x^*X$ if for all $C^1$-function $\phi$ ...
- 1,017
3
votes
0
answers
99
views
Cohomology of differentiable stacks: should the sheaf be fine?
I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fifth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
- 105
3
votes
0
answers
64
views
Canonical map in the direct image of $\mathscr{D}_X$
Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...
- 193
2
votes
2
answers
451
views
Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?
Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...
- 54.6k
2
votes
1
answer
326
views
Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
- 129
2
votes
1
answer
177
views
Are vector bundles acyclic for $\Gamma_c$?
Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
- 711
2
votes
1
answer
399
views
Locally free sheaves and vector bundles over smooth connected projective curve
Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
- 395
2
votes
0
answers
75
views
Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 301
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 ...
2
votes
0
answers
111
views
Canonicity in split sequence in cotangent spaces
Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...
- 167
2
votes
0
answers
35
views
If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
2
votes
0
answers
524
views
What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
1
vote
1
answer
443
views
Is this Sequences of Complexes of Sheaves Exact?
So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ \underline{...
- 1,611
1
vote
0
answers
100
views
Site structure on smooth fibered manifolds
Let $\mathsf{FB}_{s}$ be the category whose objects are smooth fibered manifolds, and whose morphisms are smooth strong projectable maps. Recall that given fibered manifolds $(\pi:Y\rightarrow X)$ and ...
- 3,307
1
vote
0
answers
101
views
NSR superstring as a map of supermanifolds
On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
- 11
1
vote
0
answers
395
views
the push forward of the differential idea of sheaf
This is about the differential idea of a sheaf as a vector bundle $E$ (not necessarily locally trivial) on a real manifold $M$ with a zero curvature connection $\nabla$. The zeroth cohomology is then ...
- 1,143
1
vote
0
answers
91
views
Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle
Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow k(...
- 251