# Questions tagged [pullback]

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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 ...
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1 vote
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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 $U \subset \mathbb{R}^n$ ...
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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 ...
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### 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. ...
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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 ...
1 vote
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 $f^{*}\xi = (T_{x}f)^{-1}\xi(f(x)) \in T_{x}M.$ ...
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1 vote
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### 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 ...
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### 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 ...
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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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