# Questions tagged [pullback]

269 questions

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### 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

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### 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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### 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…

user143144

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### 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

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### 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

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### What is the intuition behind pushouts and pullbacks in category theory?

What is the intuition behind pushbacks and pullouts? 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

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### 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 $F$ is and only if $F$ is a smooth immersion. Among trying to prove this exercise (Page 329 Lee…

Frog will do

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### 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

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### 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}\ \…

57Jimmy

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### 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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### 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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### Why isn't this homotopy pullback a point?

Consider any point in a topological space $X$ and consider the homotopy pullback of the diagram $* \to X \leftarrow *$. Why is said pullback the loop space of $X$ instead of just a point?
If we have any other space $P$ then we have precisely one…

user571594

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### 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

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### Any volume form on a smooth $n$-dimensional manifold is locally a pullback of the standard volume form $dx_1\wedge ...\wedge dx_n$ on $\mathbb R^n$?

Let $M$ be an $n$-dimensional smooth manifold equipped with a volume form $\omega$ , and let $\omega_0:=dx_1\wedge ...\wedge dx_n$ be the standard volume form on $\mathbb R^n$ , then is it true that for every $a \in M$ , there exists an open set $U$…

user228169

**6**

votes

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### Pullback in the category of graphs

Consider the category of (undirected) multigraphs (possibly with loops) and multigraph homomorphisms.
What are pullbacks in such a category? Is there an informal, colloquial and intuitive way to describe them?
According to the definition of…

Taroccoesbrocco

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