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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 $\...
psl2Z's user avatar
  • 301
4 votes
1 answer
437 views

Delta distribution on manifolds

Let $M$ be a manifold. There is the sheaf of distributions $\mathcal{D}'$ and the sheaf of distribution densities $\mathcal{D}(\cdot)'$ on $M$. A delta distribution density $\delta_p \in \mathcal{D}(M)...
psl2Z's user avatar
  • 301
4 votes
1 answer
541 views

Clarification on smooth de Rham theorem

I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology: Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex $$\mathbb{R}...
locally trivial's user avatar
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,\...
ABIM's user avatar
  • 5,405
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{...
Praphulla Koushik's user avatar
5 votes
1 answer
512 views

Residue of the canonical sheaf along subvariety

Let $S$ be a smooth projective surface over an algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K_S)$ ...
user267839's user avatar
  • 6,038
2 votes
0 answers
66 views

For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?

Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces. Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...
Arrow's user avatar
  • 10.5k
2 votes
1 answer
170 views

On sections into Banach bundles over a compact manifold

Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...
H1ghfiv3's user avatar
  • 1,255
5 votes
1 answer
404 views

Cohomology of sheaf of Schwartz distributions with support in a submanifold

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...
asv's user avatar
  • 21.8k
4 votes
0 answers
195 views

Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
asv's user avatar
  • 21.8k
10 votes
0 answers
762 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
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 ...
Saal Hardali's user avatar
  • 7,789
9 votes
1 answer
426 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 ...
KeD's user avatar
  • 221
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 ...
Arrow's user avatar
  • 10.5k
3 votes
2 answers
566 views

Vanishing of sheaf cohomology with compact support

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions. 1) Under what sufficient conditions on $F$ for any compact subset $K\...
asv's user avatar
  • 21.8k
0 votes
1 answer
252 views

Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R. Is it possible to extend this function to a smooth function on R²? ...
Dmitri Pavlov's user avatar
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. ...
Mark.Neuhaus's user avatar
  • 2,074
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 ...
Mark.Neuhaus's user avatar
  • 2,074
1 vote
0 answers
178 views

Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
John's user avatar
  • 111
2 votes
1 answer
407 views

smooth manifold vs. exceptional inverse image

A well-known theorem in topology says that for a smooth manifold $M$ of dimension $n$ the map $f: M \rightarrow point$ satisfies $$f^! \mathbf R = \mathbf R[n]$$ Here $\mathbf R$ is the constant sheaf....
Jakob's user avatar
  • 2,040
7 votes
2 answers
7k views

On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement. ...
agt's user avatar
  • 4,306