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 $f^*$ ...
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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 ...
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2 votes
1 answer
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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. \begin{equation} \alpha(x,y) = a\ dx + b\ dy \end{...
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  • 3,108
2 votes
1 answer
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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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Pullback of a density by the exponential map

In this lectures notes Geometric wave equation by Christian Bär at page 17 he has Definition 1.2.27. Let $\Omega$ be a starshaped with respect to $x$. We define the smooth positive function $\mu_{x}: ...
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Pullback Connection/Riccati-Equation

I'am currently trying to understand the paper: https://epub.uni-regensburg.de/23578/1/MP171.pdf The point where I'am stucked at is in the proof of Theorem 4.6. You dont need to read the whole article, ...
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34 views

Pullback of Lie derivative acting on $k-$ forms

I have to prove the following. Let $M$ be a differentiable smooth manifold and let $\chi \in \Gamma(TM)$ a smooth vector field on $M$. Denote by $\mathcal{L}_{\chi}$ the Lie derivative along $\chi$ ...
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1 answer
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Why Pushout in Set are not simple unions? [closed]

Why do we use disjoint unions instead of simple unions to have pushout in Set?
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Smooth functions and pullback map

This question was asked in my assignment on manifolds and I am struck on it. Question: For a 1-form w= f(x) dx on $\mathbb{R}$ , define $\int_{a}^{b} w = \int_{a}^{b} f(x) dx $ for $a, b \in \...
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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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Proof of the push-pull formula $V(F^*f)=DF(V)(f)$

Let $M_1, M_2$ two smooth manifolds, $F: M_1 \rightarrow M_2$ a smooth map and $f:M_2 \rightarrow \mathbb{R}$ a smooth function. I'm trying to prove that for any vector field $V$ on $M_1$, we have $V(...
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2 answers
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Calculate the pullback of a 2-form

Hi folks I've been trying to tackle the following problem that I found in the book of Introduction to Manifolds by Loring Tu but I haven't really made any progress. Any help would be appreciated. On ...
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1 vote
1 answer
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Pullback of tautological line bundle of $\Bbb{CP}^n$

Now let $\pi:\Bbb{C}^{n+1}\backslash\{0\}\to\Bbb{CP}^n$ be the natural projection, and $\mathcal{O}(-1)$ be its tautological line bundle, defined by $\{([x],l)\in\Bbb{CP}^n\times\Bbb{C}^{n+1}\mid l\in ...
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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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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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Domain and codomain of pushforward and pullback

Let$F:M \to N$ be a smooth map of manifolds. The pointwise pushforward is a linear map $$ F_{*,p} : T_p M \to T_{F(p)} N; \qquad v \mapsto F_{*,p}v: C^\infty(N) \ni \phi \mapsto v (\phi \circ F) \in \...
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2 votes
2 answers
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Pullback of a set is its preimage

(posting this as a question-answer combo, as figuring out this relation helped me understand the terms, and I hope it might for others) I have heard that the preimage of a function is sometimes ...
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Definition of the pullback of $L=Im(dS)$ and related questions on defintions

Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function ...
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  • 179
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Is the pullback of a contractive split again contractive?

If I have a pullback of $C^*$-algebras and I consider the underlying banach spaces Banach, in particular all maps in the pullback square are contractive, and I have a contractive split $s$ for one map,...
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1 vote
1 answer
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Equivalence between sections of the pullback bundle and lifts in the corresponding commutative diagram

Let $\pi: P \to B$ denote a principal $G$-bundle over base $B$, and let $f: B' \to B$ be a continuous map from another space $B'$ to $B$. I've been reading Stephen Mitchell's Notes on Principal ...
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1 vote
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What is the definition of the pullback of this map

Let $M$ be a smooth manifold on which acts a compact Lie group $G$. Let's suppose we have a diffeomorphism on $M$ $$f : M \rightarrow M $$ If $\Phi : M \rightarrow \mathfrak{g}^* $ is a map between $M$...
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Pullback of a line bundle on a stack amounts to a rational character

Let $m,q$ be positive integers. Let $\mathcal M$ be a stack and $\mathcal L\in \operatorname{Pic}(\mathcal M)\otimes \Bbb Q$. Let $B\Bbb G_m^q:=[\operatorname{Spec}\Bbb Z/\Bbb G_m^q]$ over a field $k$ ...
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  • 2,740
0 votes
1 answer
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Flat morphism preserves associated points

Let $f:X\to Y$ be a flat scheme morphism, let $\mathcal F\in \operatorname{QCoh}(Y)$. Let $\operatorname{Ass}_Y(\mathcal F)$ be the set of associated points to $\mathcal F$. Is it true that $Ass_X(f^{*...
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1 vote
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Inverse orthogonal projection of smooth functions and 1-forms on submanifold

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\s}{\mathring{\Delta}^n} \newcommand{\topspace}{\R^{n+1}_{>0}} \newcommand{\P}{\mathcal{P}} \newcommand{\O}{\mathcal{O}} $$ Quotient Let $M$ be a smooth ...
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Metric tensor via inclusion

Consider the Riemannian manifold $(\mathrm S^3_+, i^*g)$, where $i : \rm S^3 \rightarrow\Bbb R^4$ is the inclusion, $g = g_{\Bbb R^4}$ is the euclidean metric and $\rm S^3_+ = \{ (x,y,z,t)^T : t >0\...
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0 votes
1 answer
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Pullback induces isomorphism on the fibers

Suppose we are given a pullback diagram of topological spaces $\require{AMScd}$ \begin{CD} A @>>> B\\ @VVV @VVV\\ C @>>>D \end{CD} and we want to look at the fibres of the maps $A\to ...
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Pullbacks and isomorphisms of the form $id \times_Z f$ in Set

Let $X \xrightarrow{x} Z \xleftarrow{y} Y$ be a cospan of sets, and consider the pullback $X \times_Z Y$. Given $f\colon Y \to Y'$, I would like to show that $\newcommand{\id}{\operatorname{id}}\id_X \...
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1 answer
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Why is the pullback of a constant map zero?

Im confused about the proof of Poincaré's Lemma given in the book Geometry, Topology and Physics by M. Nakahara. He states that for a closed $r$-form $\omega $ on a contractible chart U the composite ...
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Pullback in local coordinates

Let $M,N$-smooth manifolds and $$\chi:M\rightarrow N$$ is a map between them. Then if we have a map $$f:N\rightarrow R$$Then the pullback of $f$ is the smooth function $\chi^*f$ defined by $$(\chi^*f)(...
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1 vote
1 answer
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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
1 answer
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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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0 votes
1 answer
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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 ...
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0 votes
1 answer
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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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0 answers
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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
1 answer
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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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  • 501
2 votes
1 answer
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Descent of holomorphism also holomorphic - hyperelliptic Riemann surfaces

My question is about how to finish a proof of Lemma 1.9 from Rick Miranda's Algebraic Curves and Riemann Surfaces. Essentially, a hyperelliptic Riemann surface is the solution set of the equation $y^2=...
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1 answer
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Pullback exists in the category of modules

I was reading the pullback definition on https://en.wikipedia.org/wiki/Pullback_(category_theory)#Commutative_rings and one of the examples was the existence of pullbacks in the category of rings. ...
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Pullback of ring localizations.

In what follow let $R$ be a commutative ring with unit and let $S_1$ and $S_2$ two saturated multiplicative subsets (by saturated I mean that in the localization morphism $i_{S_i} \colon R \to S_i^{-1}...
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2 votes
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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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2 votes
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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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2 votes
1 answer
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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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1 vote
1 answer
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Specific example Pull-back equality in manifolds

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 $$ \psi:S^2\times\mathbb{R}\rightarrow \mathbb{R}^3-\{(0,0,0)\}, \ $$ $$ ...
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2 votes
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$d : \Omega^r(R^n) \rightarrow Ω^{r+1}(R^n)$ 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: $$d ◦ \phi^∗ = \phi^∗ ◦ d.$$ ...
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3 votes
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Internally projective object gives a commuting square

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 \...
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