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When proving that colimits are universal (stable under pullback), why is it sufficient to prove it for coproducts and coequalizers?

I am trying to understand Borceux's proof that colimits are universal in Set. He opens by saying that it is sufficient to prove this for coproducts and coequalizers. I saw this answer, but I am ...
LandOnWords's user avatar
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Partial Derivative of Pullback of Differential form

I'm new to differential forms and the book I'm reading contains a part I don't understand. It states the following: Let $k\geq 1$ and assume that $D \subset \mathbb{R}^k$ and $U \subset \mathbb{R}^n$ ...
Josef K.'s user avatar
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1 answer
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Projection maps fibred product of same object twice can be chosen to be equal?

Suppose $\mathscr{C}$ is a (locally small)category and $X,Y \in \mathscr{C}$ and $X \rightarrow Y$ is a single morphism, what I am curious about is if the projections $X \times_Y X \rightarrow X$ of ...
Sadim's user avatar
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Vakil's Foundations of Algebraic Geometry, Exercise 1.3.W - Pullback square exists if map is monic

In this exercise, the goal is to show that a morphism $\pi: X \to Y$ is a monomorphism iff the fibered product $X \times_Y X$ exists and that the induced diagonal morphism $\delta_\pi: X \to X \...
Corlio's user avatar
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Wedge Product and Differential Forms, example

Let $x=id_{\mathbb{R}^4}$, $\alpha=dx^1+x_2dx^2\in \Omega^1\mathbb{R}^4$, $\beta=\sin(x_2)dx^1\wedge dx^3+\cos(x_3)dx^2\wedge dx^4\in \Omega^2\mathbb{R}^4$, $h(x_1, x_2, x_3, x_4)=(x_1, x_2, x_3x_4, ...
Lu1998's user avatar
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2 votes
1 answer
49 views

Pullback of k-algebras

I am trying to show that for 3 reduced and finitely generated ("of finite type") k-algebras R,S and T and maps $\varphi_1: R\to S$ and $\varphi_2: T\to S$, the Pullback $R\times_S T$ exists. ...
Fix's user avatar
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Interaction of pullback and the Fourier transform

I'd like to understand how the spectra of functions on a given domain are affected by (different kinds of) maps of that domain. Specifically, consider the Schwartz space $S(\mathbb R^n)$ of test ...
Yaque's user avatar
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1 vote
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33 views

If the pullback of $\phi_1(x)=\exp_x(X_x)$, preserves the volume of the Riemannian manifod $M$, then $\operatorname{div}(X)=0$?

Let $M$ be an orientable compact Rimeannian manifold whithout boundary with volume form $\operatorname{vol}_M$. Take $X\in \mathfrak{X}(M)$ such that if $\phi_t(x)$ is the flow of $X$ and $\phi_1:M\to ...
Gomes93's user avatar
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What is the induced smooth covering map $F_0^*E\times I\rightarrow H^*E$?

Let $F_0,F_1:M\rightarrow N$, be smoothly homotopic maps, and $E\rightarrow N$ a smooth vector bundle over $N$. Then by the homotopy lifting property there exists a bundle morphism $\tilde{H}:F_0^*E\...
Chris's user avatar
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Diagonal metric with induced metric being diagonal as well

A nondegenerate cyclic surface in $\Bbb L^3$ (Lorentz-Minkowski space) with constant (Gaussian curvature) $K \ne 0$ is a surface of revolution. Take a surface of revolution $S\subset \Bbb L^3$ with ...
zeta space's user avatar
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33 views

Inverse of pullback metric

Suppose we have the inclusion $\iota: X \hookrightarrow \mathbb{P}^n$ which is injective, but not a diffeomorphism. Given the standard metric $g_{\mu \overline{\nu}} dz^{\mu} \otimes d\overline{z}^{\...
Eweler's user avatar
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Two definitons of a pullback of a differential form

Let $f: X \to Y$ be a morphism of spaces with admitting differential forms (e.g. real manifold, complex manifold, smooth algebraic variety, schemes). Let $\Omega^n_Y$ denote the sheaf of $n$-forms on $...
CJ Dowd's user avatar
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2 votes
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Question about pullback of differential form

I'm reading Spivak's Calculus on Manifolds (I'm on page 89). For a differentiable function $f : \mathbb{R}^n \to \mathbb{R}^m$, $Df(p): \mathbb{R}^n \to \mathbb{R}^m$ is a linear transformation (on ...
Alan Chung's user avatar
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Pullback metric on sphere

I am learning differential geometry, and wanted to see a calculation for the round (induced) metric on the sphere $S^n$. To do this, I wanted to consider the immersion $\iota:S^n \rightarrow \mathbb{R}...
Ari Krishna's user avatar
3 votes
1 answer
69 views

Elements of $\infty$-cats Corollary 4.1.3

I am currently studying the book Elements of $\infty$-Cats and stumbled across the following Corollary (this is Corollary 4.1.3 on page 88 in the book): I am a bit confused on how to get the induced ...
h3fr43nd's user avatar
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1 answer
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Trouble showing kernel-preserving additive contravariant functors are left exact

I'm reading Advanced Modern Algebra: Part 1 by Rotman and I'm having trouble verifying the functor in the following statement is left exact: kernels of R-maps are pullbacks. Thus, kernels are inverse ...
ctk's user avatar
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Is the pullback functor $q^*$ essentially injective when $q$ is a regular epimorphism?

Let $q : W \rightarrow V$ be a regular epimorphism in a category $\mathcal{C}$, and consider the pullback functor $q^* : \mathcal{C}/V \rightarrow \mathcal{C}/W$. Is $q^*$ necessarily essentially ...
Sambo's user avatar
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Isometries of a submanifold with induced metric

I need help to find isometries of a submanifold of a semi-Riemannian manifold. To be crystal clear, let me start with what I mean by an isometry. $\textbf{Definition:}$ Let $(M,g)$ be a semi-...
lolabol's user avatar
  • 21
-1 votes
2 answers
87 views

Is the sentence "the diagram is a pullback" wrong? [closed]

In the context of category theory I've seen both sentences "the diagram is a pullback" and "the diagram is pullback" being used in the literature to talk about pullback squares. ...
Francisco's user avatar
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When is a Riemannian metric the pullback of the Euclidean metric?

Let $F:M \to (N,\bar{g})$ be a smooth map between two smooth manifolds $M \subset \mathbb{R}^2$ and $N \subset \mathbb{R}^2$ of dimension 2, with $\bar{g}$ the Euclidean metric. From what I ...
arthur's user avatar
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2 votes
1 answer
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Prime ideals of pullbacks of commutative rings [closed]

Let $A,B,C$ be commutative rings with given ring homomorphisms $f:A\rightarrow C $ and $g: B \rightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ be their pullback (with the subring ...
user1401's user avatar
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0 answers
107 views

Open immersion of schemes and pullbacks

Assume $f:X\to Y$ is a morphism of schemes and take $U_1,U_2$ open subsets of $X$ s.t. $f(U_i)\subset V$ for some open subset $V\subset Y$. I would like to show the induced map $U_1\times_V U_2\to X\...
t_kln's user avatar
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2 votes
1 answer
56 views

Is the normal 1-form to an embedded surface always not fully determined?

I'm a theoretical physicist working on general relativity, I am familiar with differential geometry but there's something I've never understood. Let there be an embedding $$\phi:S\rightarrow M$$ where ...
P. C. Spaniel's user avatar
4 votes
1 answer
69 views

Show that $\lambda:G \to \mathbb{R}^\ast$ where $R_h^\ast \mu = \lambda (h) \mu$ is smooth

Let $G$ be a Lie group of dimension $n$, and let $\mu$ be a left-invariant $n$-form. I've already shown that the pullback under a right translation, $R_h^\ast \mu$ is also a left-invariant form and ...
ImHackingXD's user avatar
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1 vote
1 answer
89 views

Showing that Pullback of Zero is Zero

Recall that, given $f:\mathbb{R}^n \to \mathbb{R}^m$ a differentiable function, then we define the pullback function $f^* : \Lambda ^k \left(\mathbb{R_{f(p)}} ^m\right) \to \Lambda^k \left(\mathbb{R_{...
HtmlProg's user avatar
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If two pullback bundles are equivalent, must the maps be homotopic?

Let $E \to M$ be a fiber bundle and $f,g \colon N \to M$. We know (e.g. from Steerod) that if $f$ and $g$ are homotopic, then the pullback bundles are equivalent bundles over $N$, $f^\ast E \cong g^\...
Joe's user avatar
  • 189
2 votes
0 answers
29 views

2-pullbacks in 2-functor 2-categories

Let $\mathcal{C}$ and $\mathcal{D}$ be 2-categories, and let $[\mathcal{C}, \mathcal{D}]$ be the 2-category of 2-functors from $\mathcal{C}$ to $\mathcal{D}$. If $\mathcal{D}$ has 2-pullbacks, does it ...
User7819's user avatar
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2 votes
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Brouwer's fixed-point theorem

I'm trying to understand the cohomological proof of this theorem, but I'm stuck at a point, which brings me to ask a more general question: let $i:Z\rightarrow X$ be the inclusion of a closed set into ...
CarloReed's user avatar
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0 answers
47 views

Transition functions for the cotangent bundle

In R.W.R Darling, on differential geometry, an exercise is to construct the cotangent bundle $T^{*}M$ from transition functions $g_{\gamma\alpha}(p)$. A quick analysis suggest using equivalence ...
Bin's user avatar
  • 3
1 vote
1 answer
116 views

Pullback of a Partition of Unity

I'm trying to prove that the collection of function $\{F^*\rho_\alpha\}$ is a partition of unity on $N$ subordinate to the open cover $\{F^{-1}(U_\alpha)\}$ of $N$. Here $\{\rho_\alpha\}$ is a ...
一団和気's user avatar
1 vote
0 answers
177 views

Rigorously proving that the pullback connection of the Levi-Civita connection via an isometric immersion gives the Levi Civita connection

Suppose $(M, \left\langle \cdot, \cdot \right\rangle_M), (\tilde M, \left\langle \cdot, \cdot \right\rangle_{\tilde M})$ are Riemannian manifolds and $f: M \rightarrow \tilde M$ is an isometric ...
rosecabbage's user avatar
  • 1,697
0 votes
1 answer
71 views

Basic pullback property

Let $f: M \to K$ be a local diffeomorphism between manifolds. We define the pullback of a vector field $\xi$, denoted as $f^{*}\xi : M \to TM$, where $f^{*}\xi = (T_{x}f)^{-1}\xi(f(x)) \in T_{x}M.$ ...
Thomas Petit's user avatar
1 vote
0 answers
30 views

Contraction of metric with derivatives of 1-form components

Let $(M,G)$ be a Riemannian metric and $V$ a $1$-form on $M$. Is there a coordinates-free, geometrical interpretation of the scalar $$G^{ij} \partial_i V_j$$ with summation over repeated indices ...
DavideL's user avatar
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1 vote
0 answers
26 views

Pullbacks in D(CW/X)

It seems like homotopy pullback is pullback for the case of D(CW) (derived category of CW-complexes). Is this true? In general, I want to say that homotopy pullback satisfies the universal property of ...
user avatar
0 votes
0 answers
166 views

Pull back of the laplacian using differential form.

Given a smooth manifold $\mathcal{M}$ a way to express of expressing the Laplacian operator $\Delta$ is through the combination of differentials and hodge star operator $\star$ $$ \Delta f = \star d \...
user8469759's user avatar
  • 5,317
0 votes
0 answers
185 views

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(...
LeonardoOiler's user avatar
2 votes
0 answers
40 views

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 ...
Dog_69's user avatar
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2 votes
2 answers
96 views

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 \...
tony-c's user avatar
  • 59
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0 answers
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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 ...
William Gibson's user avatar
2 votes
1 answer
118 views

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 ...
Seth's user avatar
  • 3,695
1 vote
0 answers
106 views

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 ...
Seth's user avatar
  • 3,695
1 vote
1 answer
81 views

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)...
user avatar
1 vote
1 answer
147 views

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 \...
Ben123's user avatar
  • 1,307
0 votes
1 answer
82 views

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 $\mathscr{I}/\mathscr{I}^2$, ...
aaa acb's user avatar
  • 207
0 votes
1 answer
52 views

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 " $S^*\xi$ &...
psl2Z's user avatar
  • 2,813
2 votes
1 answer
122 views

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: ...
Генивалдо's user avatar
2 votes
2 answers
58 views

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 $v_2=(1,0,1)$. ...
Tutusaus's user avatar
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0 answers
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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' ...
Jos van Nieuwman's user avatar
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0 answers
73 views

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 ...
Jorge A. Mateos's user avatar
4 votes
2 answers
256 views

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\...
Samuel Adrian Antz's user avatar

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