Questions tagged [pullback]
The pullback tag has no usage guidance.
0
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Prove that $(\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_*$ and $(\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^*$
Let $\mathbf f:\mathbf R^n\rightarrow\mathbf R^m$ and $\mathbf g:\mathbf R^m\rightarrow\mathbf R^k$. I figured out the pull-back part by finding
$$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...
2
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1answer
48 views
Monomorphisms preserved in a pullback
The question is answered, e.g, here.
Suppose $(P, p_1, p_2)$ is a pullback for $f:A\to C$ and $g:B\to C$ with $fp_1 = gp_2$.
if $f$ is a monomorphism, show $p_2$ is a monomorphism.
What we do is ...
2
votes
1answer
78 views
Pullback of Fubini-Study form on $\mathbb {CP}^1$
Question: Let $\varphi_S: S^2\setminus \{N\} \to \mathbb C$ be given by $\varphi_S (x_1, x_2, x_3) = \left(\frac{x_1}{1 - x_3}, \frac{x_2}{1 - x_3}\right)$, i.e., the stereographic projection. Also ...
1
vote
1answer
34 views
Pushforward of covariant and contravariant tensor.
Le $F : M \rightarrow N$ be a map between manifolds.
What is the pushforward of a covariant or contravariant tensor?
I think that for a covariant tensor $T : T_pM \times...\times T_pM \rightarrow \...
2
votes
1answer
43 views
Are all split epimorphisms effective? [duplicate]
An epimorphism is called split if it has a section (a right inverse).
An epimorphism is called effective if it has a kernel pair and is the coequalizer of its kernel pair.
The ncatlab implies here ...
0
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1answer
18 views
Regular categories and composite regular epis.
I'll put the question and my work in a picture because mathjax does not recognize tikzcd.
If there are morphims $a_2: Q \to P$ and $a_1: P \to A$ such that the diagram commutes, we are done because ...
0
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1answer
80 views
hodge star and pull back
Let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be an orthogonal linear map. Prove that $\phi^*(*\alpha) = *\phi^*(\alpha)$ for all $k$-forms $\alpha$ on $\mathbb{R}^n$.
I tried to write out $\phi^*(*\alpha)$ ...
4
votes
2answers
69 views
Injectivity, surjectivity and pullback diagrams
Consider the following pullback diagram (in any category):
$$
\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
1
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0answers
44 views
Scheme-theoretic Preimage and Fibered Product of Schemes
Following Eisenbud-Harris The Geometry of Schemes, and I'm having trouble understanding a specific part of their proof that fibered products exist in the category of schemes.
The affine case is okay,...
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0answers
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Categories Fibered in Groupoids and Yoneda
My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category $\{X_0/X_1\}$ providing a functor $p: \{X_0/...
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0answers
106 views
Is the pull-back of the structure sheaf the structure sheaf?
Maybe this is a stupid question, but I got irritated by it: Suppose $f: X \rightarrow Y$ is a morphism of schemes. That comes with a map of sheaves $f^\#: \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. ...
0
votes
2answers
36 views
pull back of smooth covering space is injective
I need to prove this but I don't really know where to start:
Let $p:M\to N$ be a smooth covering space between smooth manifolds. Show that $p^*:\Omega(N)\to\Omega(M)$ is injective.
Where $\Omega(M)$ ...
0
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1answer
32 views
Involution action on $H^1(S^1\times S^2)$
I am studying about an action $I^*$ on a de Rham cohomology group $H^1(S^1\times S^2)$ induced from an action $I\cdot (z,x)=(\overline{z},-x) $ where $S^1\times S^2\subset \mathbb{C}\times \mathbb{R}^...
0
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1answer
29 views
(wrong) Proof of stability of homotopy equivalences under pullback
In studying the category Top localized by homotopies, I asked me this question:
"Is homotopy equivalence stable by pullback (base change)?"
I know that it is necessary to have a further condition (...
3
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1answer
27 views
Pullback of a single map
We work in a category where pullbacks exist. Given a map $f:Y\to Z$, there is a pull-back diagram
$$\begin{array}& Y \times_Z Y & \stackrel{g_1}{\longrightarrow} & Y \\ \downarrow{g_2} &...
1
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1answer
147 views
Is pullback of inverse metric an inverse of the pullback metric?
Let $M$ be a manifold, $(N,g)$ be a Riemannian manifold and $f:M \to N$ be a diffeomorphism.
Does the following equallity holds?
$$(f^*g^{-1}) = (f^*g)^{-1} $$
(here $g^{-1}$ - inverse metric to g, $(...
4
votes
1answer
185 views
Pullback of differential forms and determinant
I'm studying differential geometry using the book "Godinho Natàrio - An introduction to Riemannian Geometry". These are the definitions and theorems I'm working with:
Definition 1 (Pullback of a ...
0
votes
1answer
24 views
morphism between pullback bundles
Let $\alpha_1=(E_1,\rho_1,M)$, $\alpha_2=(E_2,\rho_2,M)$ two smooth vector bundles over M and $f:N\longrightarrow M$ a smooth surjective map. If $f^*\alpha_1$ and $f^*\alpha_2$ (the pullback bundles) ...
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1answer
232 views
The universal property of polynomial rings and universal identities
To verify some identity involving say two variables $x, y \in R$ for any commutative ring $R$ it suffices to verify this identity in $\mathbb{Z}[x,y]$. This is just the sort of trick that should be ...
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0answers
65 views
Pull-back of composition
I just started a course on differential topolgy, but I'm having some trouble proving easy identities.
For example, I'm trying to prove this identity
$$ (M\circ L)^{\star} = L^{\star}\circ M^{\star}$$
...
0
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1answer
81 views
Definition of pushforward and pullback for modules
Let $R$ be a commutative ring with unit and $\sigma:R\longrightarrow R$ an endomorphism of the ring $R$. Let $M$ be an $R$-module.
What are the definitions of $\sigma^*M$ and $\sigma_* M$?
How do ...
3
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0answers
73 views
The power and the use of homotopy pullback?
A homotopy commutative diagram
$\require{AMScd}$
\begin{CD}
W @>\varphi_y>> Y\\
@V\varphi_xV V @VgVV\\
X @>f>> Z,
\end{CD}
is called a homotopy pullback, if there ...
1
vote
0answers
75 views
Universal property of homotopy pullbacks
I am working in a model category $\mathcal C$. Given a fibration $p: Y \to B$ and a map $u : A\to B$ where $A$ and $B$ (and thus $Y$ also) are fibrant, it is know that the usual pullback $A\times_B Y$ ...
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0answers
73 views
Is pullback of non-commutative rings well defined?
I know that pullback is defined for commutative ring, but what about non-commutative case?
Let's consider the following diagram, where $R_i,\bar{R}$ are rings and $R$ is the pullback:
Then $1\in R$ ...
0
votes
1answer
38 views
Why Bousfield localization preserves homotopy pull-backs?
Studying chromatic homotopy theory I encountered the chromatic fracture square
$\require{AMScd}$
\begin{CD}
L_{K(n) \vee K(m)}X @>>> L_{K(m)}X\\
@V V V @VV V\\
L_{K(n)}X @>&...
1
vote
2answers
690 views
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. ...
2
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1answer
49 views
What equivalence relation is being used to define the category of partial maps?
Here's what Awodey says in his Category Theory.
For any category $\mathbf{C}$ with pullbacks, define the category $\mathbf{Par}(\mathbf{C})$ of partial maps in $\mathbf{C}$ as follows: the objects ...
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0answers
64 views
Pullback of $\mathcal{O}(1)$ via the degree $n$ map $\mathbb P^1 \to \mathbb P^1$
Let $k$ be a field and consider the degree $n$ map $f: \mathbb P^1_k \to \mathbb P^1_k$ given by $(x:y) \mapsto (x^n: y^n)$. I want to show that $f^*(\mathcal O_{\mathbb P^1_k}(1)) \simeq \mathcal O_{...
3
votes
2answers
137 views
How to recover the covariant derivative from the pull back from that on the principal bundle
I am watching these lecture series by Fredric Schuller.
Covariant derivatives - Lec 25 - Frederic Schuller @minute 01:10:11
When we arrive at the covariant derivative from the principal bundle $P$ ...
3
votes
2answers
188 views
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 ...
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votes
2answers
103 views
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 ...
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0answers
33 views
Correspondence of the symplectic structure induced by a Lagrangian on TM and the canonical symplectic structure on T*M
I am trying to prove in a coordinate-independent way (see Fecko's "Differential Geometry and Lie Groups for Physicists, problem (18.3.2)) that
$$\theta_L := S(dL) = \hat{L}^* \theta \tag{1}\label{eq1}...
0
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1answer
42 views
Pullback of divisor under the map $z\mapsto z^p$
I'm a little confused right but I think this question can easily be answered.
Let $X\subset \mathbb C^3$ be the (affine) surface defined by $z=x^ay^b$, where $(x,y,z)$ are the coordinates on $\mathbb ...
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0answers
60 views
Typo in Spivak’s Mechanics?
In the last string of equalities, shouldn’t $ f^*(dq + \tau dp)\wedge f^*dp $ instead read $(dq + \tau dp)\wedge dp$?
$Q, P, q, p$ are coordinate functions on $T^*M$, and $f:T^*M \rightarrow T^*M$. ...
3
votes
1answer
210 views
Why are epis stable under pullback in an abelian category ?
I think the title says it all.
My question is partly motivated by the fact that this makes "element"-style reasoning with generalized elements possible; but also motivated by the result in itself.
...
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0answers
77 views
Push-forward and Category theory
There obviously seems to be a connection between the push-forward and the pull-back of a smooth function $f:M \to N$ between smooth manifolds, and the Hom-maps from category theory $f^*=Mor_C(f,\...
3
votes
1answer
80 views
What is a Presheaf-category enriched pullback?
I have a question about presheaf-enriched categories, like sSet for example that I think is pretty basic, but I don't know how to go about.
So I have a category $C$, like $\Delta^\text{op}$, that is ...
2
votes
0answers
49 views
Given Pullback of $R$-modules, show that if $f$ is injective, then $\alpha$ is injective
Given the following diagram, I want to show that if $f$ is injective, then $\alpha$ is injective, and then if $f$ is surjective, $\alpha$ is surjective.
$\begin{array}
XX & \stackrel{\alpha}{\...
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Proof of a lemma concerning pull-backs of bundles
One has the following definitions:
Let $f: M \rightarrow N$ be a smooth map between manifolds and let $\pi: P \rightarrow N$ be a $G$-principal bundle over $N$. The $pull-back$ of $P$ by $f$ is ...
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0answers
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Prove pullback of $f$ on $T^{k}(V)$ is a linear transformation
Given:
$V$ is a vector space.
$f: V \to V$ is a linear transformation
The pullback on $f$ on $T^{k}(V)$ is the map:
$f^{*}: T^{k}(V) \to T^{k}(V)$
$f^{*}(T)(\vec{v_{1}}, ...,\vec{v_{k}})= T(f\vec{v_{...
0
votes
1answer
82 views
Why the rightmost square is a pullback?
I am currently stuck in the problem below, which is somehow a converse to a previous question I asked.
Suppose that the following commutative diagram in an abelian category $$\require{AMScd} \begin{...
2
votes
2answers
74 views
Why $f'$ is an isomorphism if the rightmost square is a pullback?
Here is a really baffling problem for me, which states as follows:
In the following row-exact commutative diagram in an abelian category $$\require{AMScd} \begin{CD}
0 @>>> X'@>>>...
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0answers
40 views
Equality Proof of Pushforward and Pullback
Trying to prove $df_{F(p)}(F_*(v_p)) = F^*(df_{F(p)})(v_p)$. It is given that F is a smooth function between manifolds M and N, p $\in$ M, $v_p \in T_pM$ and $df_{F(p)} \in T_{F(p)}^*N$.
0
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1answer
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Suppose that $X$ is a sub affine variety of $Y$ , and let $φ : X \to Y$ be the inclusion. Prove that $φ^*$ is surjective…
where $\varphi^*$ is the pullback homomorphism $k[Y] \to k[X]$
If you could also give an explanation for as to what the pullback morphism is, I'd really appreciate it; my understanding is somewhat ...
2
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0answers
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Find a non vanishing differential form on the torus
Inspired by this question
I attempted to pull back the 2-form $dt\wedge ds \in \bigwedge^2(\Bbb R^2)$ using the product of the stereographic projections. I'd like to have my calculations checked, for ...
3
votes
1answer
57 views
$\Pi_f$ for a morphism $f$ between simplicial sets
This
nLab article says presheaf categories (including $\mathsf{sSet}$,
the category of simplicial sets) are locally cartesian closed.
For presheaf categories, it can be proven
by use of the category ...
0
votes
1answer
113 views
Pullbacks and differential forms, require deep explanation + algebra rules
Can somebody help me understand this. Let $\omega$ be a closed two-form on $\mathbb{R}^3$ and $\eta$ a one-form such that $\omega=d\eta$. $M$ is an orientable manifold with boundary $\partial M$. $i:M\...
3
votes
1answer
43 views
Pullback square with two identical sides
I'm wondering if any known property holds for a pullback square of the following form:
$$
\require{AMScd}
\begin{CD}
Y @>{g}>> X \\
@V{h}VV @V{f}VV \\
Y @>{g}>> X
\end{CD}
$$
In ...
1
vote
0answers
152 views
Is a conformal transformation also a general coordinate transformation?
As far as I understand, a general coordinate transformation is induced by a diffeomorphism $f:M\rightarrow M$ where $M$ is a manifold (which can locally be described with coordinates). So if $x:M\...
2
votes
1answer
162 views
Pullbacks and pushouts with surjective functions and quotient sets?
I'm in the category $Set$:
A pullback of two insertion maps $f,g$ with the same codomain is the intersection of the two domains.
A pushout of two insertion maps $f,g$ with the same domain (which ...
