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Questions tagged [pullback]

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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,...,...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 $\phi^*(*\alpha)$ ...
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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}\...
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0answers
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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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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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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 $f^\#: \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. ...
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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 $\Omega(M)$ ...
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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}^...
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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 (...
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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} &...
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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, $(...
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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 ...
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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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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 $$ (M\circ L)^{\star} = L^{\star}\circ M^{\star}$$ ...
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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 ...
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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 ...
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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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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 $1\in R$ ...
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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 @>&...
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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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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_{...
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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
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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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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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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}...
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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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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 $f:T^*M \rightarrow T^*M$. ...
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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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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,\...
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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 ...
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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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0answers
31 views

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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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_{...
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1answer
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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{...
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2answers
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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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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 $df_{F(p)} \in T_{F(p)}^*N$.
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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 ...
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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 ...
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1answer
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$\Pi_f$ 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 ...
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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\...
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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 ...
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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\...
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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 ...