# Questions tagged [pullback]

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### How do I think of the Hom functor?

For an object $X$ in a category $C$, there is a functor $C(-\,,X)$ from $C^{\mathrm{op}}$ to Set that assigns to each object $Z$ the set $C(Z,X)$ and to each morphism $f: Y \to Z$ the pullback $f^*$ ...
1 vote
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### Serre fibrations and pullbacks

Let $h:Y \to B$ be a surjective Serre fibration and let the following be a pullback diagram. $$\require{AMScd} \begin{CD} X @>>> E @. \\ @VfVV @VgVV \\ Y @>>h> B @. \end{CD}$$ Then ...
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### Which name receives this "pulled-back" sheaf?

Let $f:X \to Y$ be a continuous map of topological spaces, and let $\mathcal{F}_Y$ be a subsheaf of the sheaf of germs of continuous functions over $Y$, i.e. $\mathcal{F}_Y \subset \mathcal{C}^0_Y$. I'...
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### Pullbacks, Terminal objects and Products: proof of a proposition

I am trying to prove proposition 11.13 in Adamek’s Joy of Cats. The proposition says that if, in a pullback square, the sink object is a terminal object, then the pullback is a product. I tried to ...
67 views

### I get $d\phi^*\alpha = 0$ for any $1$-form $\alpha$ on $\mathbb{R}^2$. This cannot be correct.

I'm trying to get a hold on differential forms; I'm not sure about the following calculation. Let $\alpha$ be a 1-form on $\mathbb{R}^2$, i.e. \alpha(x,y) = a\ dx + b\ dy \end{...
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### Missing minus sign in pullback calculation of a $1$ form on $\mathbb{C}$

Consider $S^2$ with a coordinate chart given by the stereographic projection through the north pole. We identify $\mathbb{C}$ with $\mathbb{R}^2$, then for a point $(\theta,\phi)\in S^2$, the ...
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### Explicit Expression for Pullback Measure via a Surjective Mapping

Suppose we have a measurable, surjective function $f: (X, \Sigma) \to (Y, \Omega)$ and $X,Y$ are two locally compact metric spaces (can even assume $X,Y$ are subsets of $\mathbb{R}^n$). Given a ...
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### pullback of schemes

So we had the following statment in the lecture: I just dont get why this equation is true, even though it probably just follows immediately from the universal property of the pullback. Thanks in ...
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### Squares in additive infinity categories

I need to show that a commutative square in an additive infinity category, where the vertical maps both are split epimorphisms and both admit fibers/kernels, is cartesian if and only if it is ...
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1 vote
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### Computing the pull-back of a simple one-form

Let $(u, v)\in\mathbb{R}^2$ and consider the 1-form $\alpha_{(u,v)} : \mathrm{T}_{(u, v)}\mathbb{R}^2 \to \mathbb{R}$ defined by \begin{align} \alpha_{(u,v)}(w) &= (e^v ~\mathrm{d}u + u ~\mathrm{d}...
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### Gradient of a function defined on a Lie group by pulling back to identity

I am reading this paper dealing with optimization on Lie groups and in the introduction (second full paragraph of the second page) the authors write: ''However, a specialization in Lie group will ...
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1 vote
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1 vote
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### Pullback with $f^*g_N=\alpha\cdot g_M$

I'm new here, so if I need to correct something, let me know. Calculating some things I have noticed the following: If $f:(M,g_M)\to (N,g_N)$ is a differentiable mapping, where $g_N$ is a metric of $N$...
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1 vote
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### Map from continuous functions in $S^1$ to continuous functions in $\mathbb{R}$ injective subset of periodic functions

Let $S^1 = \{ (x,y)\in \mathbb{R}^2 | x^2+y^2=1$. Let $z:\mathbb{R} \to S^1$ given by $z=(cos(\theta),sin(\theta))$. Define the map $z^* : C^0 (S^1) \to C^0 (\mathbb{R})$ by $z^* (f)= f \circ z$. I am ...
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### Understanding $\rho^*\delta_0$ the pullback of the delta distribution by a function

I don't understand the meaning of theorem 6.15 (page 136) from the Hörmander Book "The Analysis of Linear Partial Differential Operators I". It states a connection between the pullback of ...
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### Pull-back between covector fields

https://youtu.be/mJ8ZDdA10GY?t=2184 I was following this lecture and at this point in the lecture the professor starts discussing why it is easier to construct pull-backs between covector fields ...
129 views

### Pullback of a differential form by the zero section

Let $\pi: E \rightarrow M$ be a vector bundle of rank $n$ and let $j: M \rightarrow E$ be its zero section. Let $\beta$ be a closed differential form. I was convinced that $j^* \beta$ is the zero ...
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### When does the Galois cohomology functor $H^1(Gal(K^{sep}/ -), G(K^{sep}))$ preserve pullback diagrams?

Fix a field $K$ and a separable closure $K^{sep}$ of $K$. Consider the category of all separable field extensions $K \subset L \subset K^{sep}$ with field inclusions as morphisms. Consider the first ...
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### Given a vector function which is a gradient, did the pullback of that function still a gradient?

Let $\Omega$ be a bounded open subset of $\mathbb{R}^3$ and $n$ the outward-pointing unit normal to the boundary of $\Omega$. We know that the space $(L^2(\Omega))^3$ of square integrable functions ...
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1 vote
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### Exercise: pullback in the category of R-modules

True or False: (i) It always exist the pullback of two morphisms $f:A \rightarrow C$ and $g:B \rightarrow C$ in the category of R-modules. (ii) if $f: A \rightarrow B$ is a homomorphism of R-modules ...
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### Equivalent definitions of flat morphism

Suppose $\pi : X \to Y$ satisfies that pullback on quasi-coherent sheaves is exact, how do I prove that $\pi$ is flat via the local definition; i.e stalkwise $O_{X,p}$ is a flat $O_{Y,q}$ module ...
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### Definition of differentiable map that preserves orientations

I'm reading Frank W.Warner's "Foundations of Differentiable Manifolds and Lie Groups". In page 139, he defines the differentiable map that preserves orientations: Let M and N be orientable ...
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### Pullback connection from a local diffeomorphism under an integral.

Suppose $(M^n, g)$ and $(N^n,h)$ are Riemannian manifolds, and for each $p\in M$ there is a neighborhood $U\subset M$ and a diffeomorphism $\phi:(U, g|_U) \to (N,h)$ such that $g|_U = \phi^*h$. I ...
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I have to prove the next equality $$\psi^*\left(\frac{xdx+ydy+zdz}{x^2+y^2+z^2}\right)=dt$$ wher $\psi^*$ is the pull-back of $$\psi:S^2\times\mathbb{R}\rightarrow \mathbb{R}^3-\{(0,0,0)\}, \$$ $$... 2 votes 0 answers 48 views ### d : \Omega^r(R^n) \rightarrow Ω^{r+1}(R^n) show that: d ◦ \phi^∗ = \phi^∗ ◦ d Let U, V \subset R^n be two open sets and \phi : V \rightarrow U  a diffeomorphism. For the outer derivative d : \Omega^r(R^n) \rightarrow Ω^{r+1}(R^n) show that:$$d ◦ \phi^∗ = \phi^∗ ◦ d. ...
Let $\epsilon$ be a topos and $P$ be an object of $\mathcal{E}$ such that $(-)^P: \mathcal{E} \rightarrow \mathcal{E}$ preserve epis then the right adjoint to pull back \$\Pi _P : \mathcal{E} /P \...