Questions tagged [pullback]
The pullback tag has no usage guidance.
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Pullback of Differential Forms
In exercise 47 from Gauge Fields, Knots and Gravity by Baez and Munain, we want to show that if $\phi:M\to N$ is a map of smooth manifolds, then there is a unique pullback map on forms $$\phi^*:\Omega(...
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Find a subgroup $K$ to complete the pullback diagram $G/g_1Hg_1^{-1}\leftarrow G/H\to G/g_2Hg_2^{-1}$.
EDIT: I have realised I made a mistake when decompsoing the morphisms of $\mathscr B_G$.
Nevertheless, the question seems to be interesting on its own, so i will leave it. I would also like to cite ...
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Tensor fields and scalar function pullbacks
For convenience, $(p,q)$ tensor fields on a differentiable manifold $M$ is defined to be the entire scalar field.
On the other hand, in my textbook, the pullback of the $(p,q)$ tensor $T(x)$ at $x \...
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Pullback of a translation map of a divisor in Birkenhake-Lange's book "Complex Abelian Varieties"
I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement:
'Another observation, which will prove ...
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Showing that the Kernel in $\textbf{Vect}$ is given by a pullback diagram
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes, "Advanced Modern Algebra" 2nd edition by Rotman, and
"Representation ...
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Showing that the CoKernel in $\textbf{Vect}$ is given by a pushout diagram
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes, "Advanced Modern Algebra" 2nd edition by Rotman, and
"Representation ...
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Doubt about splitting principle
I have a doubt about splitting principle.
We know that if $E\to M$ is a complex vector bundle of rank $k$, then there exist a manifold $T=T(E)$ and a proper smooth map $f:T\to M$ such that
$f^*:H^*(M)...
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Orientation preserving diffeomorphism $\iff$ positive determinant
I am stuck on a step in a proof, so I will write out the statement and the proof (it is not too long). I would like someone to explain the last 2 steps of the proof, if possible.
Statement: Let $U,V \...
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Is conormal sheaf a pullback?
Apologize if this is a newbie question.
Let $i: Z\hookrightarrow X$ be a closed immersion with ideal sheaf $\mathscr{I}$. The conormal sheaf of $Z$ in $X$ is defined as $\mathscr{I}/\mathscr{I}^2$, ...
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Pullback along a family of maps
Is there a construction to handle pullbacks along many maps simultaneously? Something like this: Given a subset $S \subseteq C(B',B)$ and a bundle $\xi = (E,\pi,B)$. Is there a bundle " $S^*\xi$ &...
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Pullback of a Lorentzian metric is non-degenerate
Background: Recently I've read this post of one person trying to prove that the pullback $F^*g$ of a riemannian metric $g$ is a Riemannian metric iff $F: N \to (M,g)$ is a smooth immersion.
Question: ...
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Pullback of lorentzian metric is degenerate at the boundary of embedded submanifold
Let $\Sigma = \mathbb{R} \times [0,1]$ be a smooth manifold with boundary $\partial \Sigma = \mathbb{R} \times \{0,1\}$.Let $M = R^{1,D-1}$ be the $D$-dimensional Minkowski space with scalar product (...
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Computing the pullback of an exterior differential of a vector field at a point given two linear independent vectors
Given the $1$-differential form $\omega=xzdx+xdy+xdz$ and the vector field $F(x,y,z)=(e^y,e^z,e^x)$, I must compute $$(F^{*}d\omega)_{p}(v_1,v_2)$$ where $p=(1,-1,0), v_1=(1,1,0)$ and $v_2=(1,0,1)$. ...
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Pullback of the identity is 'basically' the identity?
I encountered the titular statement in a discussion on subobject classifiers (in particular: in the proof that the domain of such a subobject classifier is terminal).
However, I think his 'basically' ...
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Existence of a right adjoint functor of the inverse image via a morphism of schemes between the categories of quasi-coherent modules
Let $f:X\to Y$ a morphism of schemes. It induces a covariant functor:
$$f^*:Qcoh(Y)\to Qcoh(X)$$
Which happens to be the inverse image. Now, fixing any quasi-coherent $O_X$-module N we can define ...
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Is there a cojoin, the dual construction of the join of topological spaces?
The join $X\star Y$ of topological spaces $X$ and $Y$ can alternatively be written as a homotopy pushout of the canonical diagram $X\leftarrow X\times Y\rightarrow Y$:
\begin{equation}
X\star Y
=X\...
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Determinant of pullback of metric tensor
Consider a pseudo-Riemannian manifold $(M,g)$ with $\dim M=n$.
Let $\Sigma$ be an embedded submanifold with $\dim\Sigma=k$ and inclusion map $\iota:\Sigma\rightarrow M$. We then define the induced ...
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Understanding Pullback of Cartier Divisors
Let $g\colon X\to Y$ be a morphism of schemes, and let $D=(U_i,f_i)$ be a Cartier Divisor on $Y$. I've seen the following definition of the pullback of $D$: $g^*D\colon=(g^{-1}(U_i),f_i\circ g)$, as ...
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Pullback and pushforward of vector fields
I am studying the article On the Lagrange-Dirichlet converse in dimension three.
Lemma 2.15. With respect to the coordinate chart corresponding to $θ_1 \neq 0$ and defining $ξ_1 = 1$ for notational ...
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Pullbacks respect order of forms
Consider a holomorphic map $f:M\to N$ between two complex manifolds. This gives rise to a pullback $f^*:\Omega^k(N)\to\Omega^k(M)$, that is, a complex linear map between the complex $k$-forms on $N$ ...
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What is the pullback of the adjoint representation on a Lie algebra?
On a Lie Algebra, we often work with the adjoint representation
\begin{align*}
\text{ad}_u: \mathfrak{g}&\to\mathfrak{g}\\
v&\mapsto [u,v].
\end{align*}
Provided we have some symmetric, ...
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Pullback seen as universal cone, arrow wanted
I am trying to learn some concepts of category theory and there is something I don't understand about categorical limits. I found these notes online with the following definitions (please see the ...
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Description of $\operatorname{Spec}(A\times_C B)$ in terms of the spectra for $A,B,C$
I assume rings are commutative. Given surjective ring morphisms $f:A\to C$, $g:B\to C$, I wonder if it's possible to determine the prime ideals of the pullback $A\times_C B$ in terms of the prime ...
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Trying to understand that $Imm(\Bbb S^1,\Bbb R^d)\simeq Map(\Bbb S^1,\Bbb S^{d-1})$
I'm looking at various seminars that Pascal Lambrechts has given about manifold calculus. One of the lectures I've found is in the following slides of one of his lectures, but I have problems ...
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The pullback and derivative on GL(n)
$GL(n)$ is the space of invertible matrices.
Let $\phi^i_j: GL(n) \rightarrow R$
be the function where $\phi^i_j(g)$ is the element in the $i$-th row and $j$-th column of $g$.
Let $d\phi$ be the ...
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Clarifying Calculation of Induced Mappings (Pullback) in Coordinates
I am reading Mathematical Analysis by Andrew Browder and am confused in his calculating the meaning of $\mathbf{f^\ast},$ the "induced mapping" or pullback of $\mathbf{f^\ast}.$ Here's his ...
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Intuition for the commutation of the pullback and the exterior derivative
I know that there are already several questions to this topic but I haven‘t seen a satisfactory answer. Deriving that the pullback and the exterior derivative commute is no problem but I want to know ...
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Pullback of tangent bundle of codomain induced by a submersion is a quotient of tangent bundle of domain
Let $M,N$ be smooth manifolds and $f: M \to N$ a smooth map. Then, we have the following commutative diagram
Here, $\pi_1, \pi_2$ are the bundle projections from the tangent bundles, $p_1, p_2$ are ...
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Example of a finite category without pushouts or pullbacks
I'm trying to find an example of a finite category in which there are no pushouts or pullbacks. By finite category, I mean a category with a finite amount of objects and morphisms.
The concepts of ...
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Pull-Push homomorphism of group convolution algebra
Given three finite groups $H, G, K$ with homomorphisms $\displaystyle H\xleftarrow{\pi_H}{} G \xrightarrow{\pi_K}{} K,$ we have two homorphisms of group (convolution) algebras, the first going from ...
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Clarification needed for the definition of Pushout and Pushout diagram from definition of Pullback and Pullback diagram
The following is taken from $\textit{Categories}$ by Horst Schubert
Definition for Pullback
Let $f:A\rightarrow C, g:B\rightarrow C$ be two morhisms with the same codomain. A $\textit{pullback}$ (...
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Residue of the pullback
Suppose $p\colon Y \rightarrow X$ is a holomorphic mapping of Riemann surfaces, $a\in X$, $b\in p^{-1}(a)$ and $k$ is the multiplicity of $p$ at $b$.
Then I am trying to show the claim:
Given any ...
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Equivalence problem and Maurer-Cartan structure equations
I am reading a proposition. The proposition is as follows:
Let $M$ and $\bar{M}$ be two manifold, with $\{\mu_i\}$ and $\{\bar{\mu}_i\}$ co-frame, respectively. And there is an diffeomorphism s.t. $\...
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Quick further question about the notion of "pullback/pushout".
Previously, I asked a question about the notion of "pullback" in category theory here, I just have one further quick question about the concept of "pullback/pushout", that is, does ...
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When is the induced map in a pullback diagram a fibration?
Let $X$, $Y$ and $Z$ be topological spaces and let $f:X \to Z$ and $g:Y \to Z$ be continuous. Let $P$ be the pullback of $X\stackrel{f}{\to}Z\stackrel{g}{\leftarrow} Y$. Let $E$ be another space and ...
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Pullback of a 2-tensor exercise
Let $(e_1,e_2,e_3)$ be a basis of $V$ and $(\varepsilon^1,\varepsilon^2, \varepsilon^3)$ be the dual basis. Let $(f_1,f_2)$ be a basis of $W$ and $(\phi^1,\phi^2)$ the dual basis. Let $L: V \...
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When is a pair of morphisms is a pullback of some pair of morphisms?
Given $p$ and $q$ morphisms on a category (preferably arbitrary, but it could be interesting to know on some other class of categories too), are there any useful necessary and/or sufficient conditions ...
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Does subclass of morphisms closed under wide pullbacks necessarily consist of only monomorphisms?
For category $\mathcal{C}$, does subclass $M \subseteq \operatorname{Mor}(\mathcal{C})$ closed under even largely wide pullbacks necessarily consist of only monomorphisms? The assumption being more ...
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Can we pull back a measure with additional information on one coordinate?
Let $\mu$ be a finite measure on a product sigma algebra $(\Sigma_1\times\Sigma_2, \mathcal A_1\otimes\mathcal A_2)$, let $f:\Sigma'_2\to\Sigma_2$ be a measurable function from another measurable ...
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Torsion on a flat connection (Geometric Intuition).
Start with the 2-sphere $\mathcal{S}^2$ with the standard $(\theta, \phi)$ chart ($\theta = 0$ is the North pole etc) and metric:
$$ds^2 = d\theta^2 + \sin^2 \theta d\phi^2$$
Remove the two poles from ...
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Sections of pullback bundle are generated by pullback of sections
Let $M,N$ be smooth manifolds, $\pi:E\to M$ a smooth rank $r$ vector bundle, $f:N\to M$ a smooth map. We define
$$
f^*E=\{(p,v)\in N\times E: f(p)=\pi(v)\},
$$
which has a natural structure of a ...
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Compute $\int_𝛷^{ }dx_1\wedge dy_1 + dy_1\wedge dx_2 +dx_2 \wedge dy_2 \,\mathrm .$
Let $z_j=x_j+iy_j $ $(j=1, 2, ... n)$ be the coordinates in $ℂ^n$.
We can identify $ℂ^n$ with $ℝ^ {2n}$ by writing $(z_1, ... z_n) =(x_1,y_1, ... x_n,y_n) $
Let $D⊂ℂ$ be the square of the points $z=x+...
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Relation between the $t$-derivative of the pullback of a form $\omega$ by the flow $f_t$ and the Lie derivative of $\omega$ along the generator
at the moment I am self-studying differential geometry using the book by Guillemin and Haine. In this book, Exercise 2.6.ix asks me to prove the following:
Let $U \subset \mathbb{R}^n$ open, $f_t: U \...
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Prove a square is a pullback
Let $C$ an abelian category, and consider the following diagram:
$$
\require{AMScd}
\begin{CD}
P
@> \beta_1 >>
A_1
\\
@V \beta_2 VV
@VV \alpha_1 V
\\
A_2
...
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Pullback of an epimorphism in the category of Hausdorff spaces
Can anyone give me an example of a pullback of an epimorphism which is not an epimorphism, in the category of Hausdorff spaces?
I've been thinking about it but I have no idea.
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Constructing finite limit from terminal object and pullbacks
I'm trying to understand, why in Proposition 3.1 in Finite Limit in ncatlab (3) implies (1). My intuition tells me to use induction to show that the limit for any diagram having $n$ vertices (objects) ...
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Pullback notation in Arapura's book "Algebraic Geometry over the Complex Numbers"
In Arapura's Algebraic Geometry over the Complex Numbers, exercise 2.1.16. reads:
Let $F:X\to Y$ be a surjective continuous map. Suppose that $\mathscr{P}$ is a sheaf of $T$-valued functions on $X$. ...
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Pullback to spherical coordinates and intuition on pullbacks in general
Pretty much what the question says. Right. So, to remind everyone, spherical coordinates are:
\begin{align}
x & = \rho \sin(\phi)\cos(\theta) \\
y & = \rho \sin(\phi)\sin(\theta) \\
z & ...
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Pullback of projection
I am trying to understand what an axial map $f \colon \mathbb{R}P^n \times \mathbb{R}P^m \to \mathbb{R}P^k$ does on cohomology, where a map is called axial, if the restrictions $* \times \mathbb{R}P^...
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Showing that a specific set is really a limit. (Tom Leinster "Basic Category Theory" Ex. 5.1.37)
Here is the question I want a concrete example of it so that I can feel that the given set is really a limit:
Show that the set $(5.16)$ in Example 5.1.22 really is a limit of $D.$
Here is Example 5.1....
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