Questions tagged [pullback]
383 questions
21
votes
4 answers
What is a simple definition of the pullback of a section?
I am simply asking for a definition for something everyone uses but nobody defines. Really, this is used in class and in Hartshorne, and I have tried to look for a definition in Hartshorne, Qing Liu, Wikipedia, nothing comes up, so I am wondering…
Evariste
- 2,787
21
votes
3 answers
What is a pullback of a metric, and how does it work?
The term "metric" is familiar, but not the idea of a pullback on it. I have tried to find intuitive, beginner-friendly explanations of this concept without success. Your attempts would be appreciated. Pictures and concrete examples would be…
Tensor McTensorstein
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15
votes
2 answers
Demystifying the Magic Diagram
Vakil calls the following pullback diagram the magic diagram . I have also seen it being called the magic square. It often shows up in fiber product diagram chases such as those associated with separatedness assertions. $\require{AMScd}$ \begin{CD}…
Qi Zhu
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14
votes
1 answer
Interior Product and Pullback Properties
Let $f:M\rightarrow N$ a diffeomorfism between differentiable manifolds. $X$ is a $C^{\infty}$ vector field over N. If $\omega \in \Omega^{k}(N)$. (i.e. $\omega$ is a $k$ - form), prove that $$f^{\ast}(i_X \;\omega)=i_{f^{\ast} X}f^{\ast}\omega $$…
Moe
- 206
12
votes
1 answer
Prove that the pullback $F^* g$ of a Riemannian metric $g$ is a Riemannian metric iff $F$ is a smooth immersion
So I'm trying to prove that $F^*g$ , where $F^*$ is the pullback of $F: M \rightarrow N$ and $g$ is a Riemannian metric on $N$ , is a Riemannian metric on $M$ if and only if $F$ is a smooth immersion. While trying to solve this exercise (Page 329…
user637978
10
votes
2 answers
What is the intuition behind pushouts and pullbacks in category theory?
What is the intuition behind pullbacks and pushouts? For example I know that for terminal objects kind of end a category, they are kind of last is some sense, and that a product is a kind of pair, but what about pullbacks and pushouts what are the…
geckos
- 235
10
votes
1 answer
Is the pull-back of the structure sheaf the structure sheaf?
Maybe this is a stupid question, but I got irritated by it: Suppose $f: X \rightarrow Y$ is a morphism of schemes. That comes with a map of sheaves $f^\#: \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$ . Because $f_*$ and $f^*$ are adjoint to each…
red_trumpet
- 11,587
10
votes
2 answers
Kähler form on the Blow up of a Kähler manifold
Given a Kähler manifold $X$, the blow up is a Kähler manifold aswell (see Hodge theory and CAG by Voision Prop. 3.24). The idea is of course to use the pull-back the Kähler form $\pi^*\omega_X$. It says that this form is not positive , but only…
Notone
- 2,500
10
votes
1 answer
Pullback of a differential form by a local diffeomorphism
Suppose I have to smooth oriented manifolds, $M$ and $N$ and a local diffeomorphism $f : M \rightarrow N$. Let $\omega$ be a differential form of maximum degree on $N$, let's say, $r$. How can I rewrite $$\int_N \omega$$ in terms of the integral of…
user143144
- 377
9
votes
2 answers
Injectivity, surjectivity and pullback diagrams
Consider the following pullback diagram (in any category): $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}…
57Jimmy
- 6,468
9
votes
1 answer
Why doesn't the functor $\bar{\mathcal{P}}\bar{\mathcal{P}}$ preserve pullbacks?
I've tried finding examples on my own but the sizes of the sets is a bit hard to manage. In the litterature I've seen this fact referenced in a few places but they all point to Rutten: Universal coalgebra: a theory of systems which mentions, in the…
Strange Brew
- 455
8
votes
1 answer
Sections of pullback bundle are generated by pullback of sections
Let $M,N$ be smooth manifolds, $\pi:E\to M$ a smooth rank $r$ vector bundle, $f:N\to M$ a smooth map. We define $$ f^*E=\{(p,v)\in N\times E: f(p)=\pi(v)\}, $$ which has a natural structure of a vector bundle over $N$ . Consider a section…
Sha Vuklia
- 4,366
8
votes
2 answers
Why is $\operatorname{colim} F \cong \pi_0\left (\int F\right )$?
Given a small functor $F:\mathsf{C \to Set}$ , I need to prove that $\operatorname{colim} F$ is isomorphic/in bijection with the connected components of the category of elements $\int F$ . It's not the least bit clear to me how to make this…
Mnifldz
- 13,306
8
votes
1 answer
How to transform $(r,s)$ tensor fields?
Let $\mathcal M$ and $\mathcal N$ be smooth manifolds and let $f : \mathcal M \rightarrow \mathcal N$ be a diffeomorphism. How can we transform general $(r,s)$ tensor fields, with mixed convariant and contravariant indices, along that mapping? A…
shuhalo
- 8,186
8
votes
2 answers
Pullback of a pullback square along $f$ is again a pullback square
Awodey's Category Theory is at it again, asking me to do things without fully explaining what any of it means. Part (b) of Problem 2 in Chapter 5 reads as follows: Show that the pullback along an arrow $f:Y\to X$ of a pullback square over $X$,…
D. Brogan
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