Questions tagged [pullback]
The pullback tag has no usage guidance.
270
questions
2
votes
2
answers
143
views
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
0
answers
32
views
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 ...
0
votes
0
answers
27
views
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'...
0
votes
1
answer
46
views
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 ...
2
votes
1
answer
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.
\begin{equation}
\alpha(x,y) = a\ dx + b\ dy
\end{...
2
votes
1
answer
29
views
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 ...
3
votes
1
answer
51
views
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}: ...
1
vote
0
answers
28
views
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, ...
0
votes
0
answers
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 $\chi$ ...
-3
votes
1
answer
50
views
Why Pushout in Set are not simple unions? [closed]
Why do we use disjoint unions instead of simple unions to have pushout in Set?
0
votes
0
answers
16
views
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 \...
2
votes
1
answer
30
views
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 ...
0
votes
1
answer
53
views
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(...
4
votes
2
answers
220
views
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 ...
1
vote
1
answer
36
views
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 ...
0
votes
1
answer
35
views
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 ...
0
votes
0
answers
32
views
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 ...
1
vote
0
answers
46
views
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}...
0
votes
0
answers
37
views
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 ...
1
vote
0
answers
45
views
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
$$
F_{*,p} : T_p M \to T_{F(p)} N; \qquad v \mapsto F_{*,p}v: C^\infty(N) \ni \phi \mapsto v (\phi \circ F) \in \...
2
votes
2
answers
116
views
Pullback of a set is its preimage
(posting this as a question-answer combo, as figuring out this relation helped me understand the terms, and I hope it might for others)
I have heard that the preimage of a function is sometimes ...
0
votes
1
answer
32
views
Definition of the pullback of $L=Im(dS)$ and related questions on defintions
Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function ...
0
votes
1
answer
15
views
Is the pullback of a contractive split again contractive?
If I have a pullback of $C^*$-algebras and I consider the underlying banach spaces Banach, in particular all maps in the pullback square are contractive, and I have a contractive split $s$ for one map,...
1
vote
1
answer
48
views
Equivalence between sections of the pullback bundle and lifts in the corresponding commutative diagram
Let $\pi: P \to B$ denote a principal $G$-bundle over base $B$, and let $f: B' \to B$ be a continuous map from another space $B'$ to $B$.
I've been reading Stephen Mitchell's Notes on Principal ...
1
vote
0
answers
51
views
What is the definition of the pullback of this map
Let $M$ be a smooth manifold on which acts a compact Lie group $G$.
Let's suppose we have a diffeomorphism on $M$ $$f : M \rightarrow M $$
If $\Phi : M \rightarrow \mathfrak{g}^* $ is a map between $M$...
1
vote
0
answers
32
views
Pullback of a line bundle on a stack amounts to a rational character
Let $m,q$ be positive integers.
Let $\mathcal M$ be a stack and $\mathcal L\in \operatorname{Pic}(\mathcal M)\otimes \Bbb Q$.
Let $B\Bbb G_m^q:=[\operatorname{Spec}\Bbb Z/\Bbb G_m^q]$ over a field $k$ ...
0
votes
1
answer
38
views
Flat morphism preserves associated points
Let $f:X\to Y$ be a flat scheme morphism, let $\mathcal F\in \operatorname{QCoh}(Y)$.
Let $\operatorname{Ass}_Y(\mathcal F)$ be the set of associated points to $\mathcal F$.
Is it true that $Ass_X(f^{*...
1
vote
0
answers
30
views
Inverse orthogonal projection of smooth functions and 1-forms on submanifold
$$
\newcommand{\R}{\mathbb{R}}
\newcommand{\s}{\mathring{\Delta}^n}
\newcommand{\topspace}{\R^{n+1}_{>0}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\O}{\mathcal{O}}
$$
Quotient
Let $M$ be a smooth ...
0
votes
0
answers
47
views
Metric tensor via inclusion
Consider the Riemannian manifold $(\mathrm S^3_+, i^*g)$, where $i : \rm S^3 \rightarrow\Bbb R^4$ is the inclusion, $g = g_{\Bbb R^4}$ is the euclidean metric and $\rm S^3_+ = \{ (x,y,z,t)^T : t >0\...
0
votes
1
answer
22
views
Pullback induces isomorphism on the fibers
Suppose we are given a pullback diagram of topological spaces
$\require{AMScd}$
\begin{CD}
A @>>> B\\
@VVV @VVV\\
C @>>>D
\end{CD}
and we want to look at the fibres of the maps $A\to ...
0
votes
0
answers
56
views
Pullbacks and isomorphisms of the form $id \times_Z f$ in Set
Let $X \xrightarrow{x} Z \xleftarrow{y} Y$ be a cospan of sets, and consider the pullback $X \times_Z Y$.
Given $f\colon Y \to Y'$, I would like to show that $\newcommand{\id}{\operatorname{id}}\id_X \...
0
votes
1
answer
135
views
Why is the pullback of a constant map zero?
Im confused about the proof of Poincaré's Lemma given in the book Geometry, Topology and Physics by M. Nakahara. He states that for a closed $r$-form $\omega $ on a contractible chart U the composite ...
0
votes
0
answers
68
views
Pullback in local coordinates
Let $M,N$-smooth manifolds and $$\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)(...
1
vote
1
answer
65
views
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 $N$...
1
vote
1
answer
42
views
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 ...
0
votes
1
answer
77
views
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 ...
0
votes
1
answer
62
views
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 ...
0
votes
1
answer
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 ...
4
votes
0
answers
48
views
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 ...
0
votes
0
answers
25
views
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 ...
1
vote
1
answer
81
views
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 ...
2
votes
1
answer
82
views
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=...
0
votes
1
answer
139
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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. ...
2
votes
0
answers
37
views
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}...
2
votes
0
answers
83
views
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 ...
2
votes
0
answers
69
views
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 ...
2
votes
1
answer
88
views
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 ...
1
vote
1
answer
59
views
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
$$
\psi:S^2\times\mathbb{R}\rightarrow \mathbb{R}^3-\{(0,0,0)\}, \
$$
$$
...
2
votes
0
answers
48
views
$d : \Omega^r(R^n) \rightarrow Ω^{r+1}(R^n)$ show that: $d ◦ \phi^∗ = \phi^∗ ◦ d$
Let $U, V \subset R^n$ be two open sets and $\phi : V \rightarrow U $ a diffeomorphism. For the outer derivative $d : \Omega^r(R^n) \rightarrow Ω^{r+1}(R^n)$ show that:
$$d ◦ \phi^∗ = \phi^∗ ◦ d.$$
...
3
votes
0
answers
49
views
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 \...