Questions tagged [pullback]

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Cartesian product is to monoidal product as pullback is to what?

I'm trying to complete the following pattern product : monoidal product : coproduct pullback : ? : pushout That is, if the monoidal product is a ...
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1 vote
2 answers
183 views

Pullback of Lie algebras [closed]

Let $k$ be a field of characteristic 0 and let $\varphi:\mathfrak{g}\rightarrow\mathfrak{f}$ and $\psi:\mathfrak{h}\rightarrow\mathfrak{f}$ be maps of Lie algebras. Is there a reference showing that ...
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3 votes
1 answer
174 views

Homotopy pullback in the category of DG algebras

I was wondering if somebody could tell me the definition of homotopy pullback. More precisely, is there a description of the homotopy pullback in the category of (graded-commutative) DG algebras over ...
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4 votes
0 answers
172 views

Is the restriction of an injective sheaf on a closed subscheme still injective?

Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$. Question. Is $i^*\mathcal{I}$ still an ...
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1 vote
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229 views

Covariant Derivative of sections of a pullback bundle

Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between ...
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2 votes
1 answer
207 views

Pull-back a section of a vector bundle

Let $M$ be a manifold of dimension $n$ and $\mathcal D$ be a distribution of dimension $n-1$. We consider the quotient bundle $TM/\mathcal D = \bigsqcup_{p \in M} T_pM/\mathcal D_p$ with the ...
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  • 268
2 votes
0 answers
63 views

What is, explicitly, a pullback in the category of $L_\infty$ algebras?

I was wondering if the category of $L_\infty$ algebras is complete and in particular I am looking for an explicit construction of the pullback for $\require{AMScd}$ \begin{CD} @. B\\ \phantom V @VV ...
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  • 21
2 votes
1 answer
470 views

Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?

Cross-post from MSE. For a continuous map $f:(M,g)\to (N,h)$, between Riemannian manifolds $(M,g)$ and $(N,h)$ we can pullback $h$ by $f$. Most experts take the trace from this new tensor and work ...
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1 vote
1 answer
259 views

Measure in $\mathbb {C} ^p$ [closed]

If we have a non-constant holomorphic map $ f: \mathbb C ^ p \to X $, where $ X $ is a complex manifold. Let $ \omega $ be a metric on $X$, so $ \omega $ is a positive definite $ (1,1) $-form. Is $ f ...
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  • 19
2 votes
0 answers
127 views

Pull back of a Bounded form

Let $(X, \omega) $ be a complex manifold and let $\alpha $ be a $p$-form $\omega$-bounded on $X$. Let $f:Y\to X$ be a holomorphic function. Is $f^*\alpha$ $f^*(\omega)$-bounded on $Y$ ?
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11 votes
1 answer
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Why do elementary topoi have pullbacks?

In the book of Szabo "Algebra of Proofs", Definition 13.1.9 introduces an elementary topos as a cartesian closed category with a subobject classifier. On the other hand, many other sources including ...
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1 vote
1 answer
165 views

Pullback of boundary divisors under forgetful maps

Let $\overline{\mathbf{M}}_{0,n}$ be the moduli space of stable $n-$pointed smooth rational curve of genus zero and $\overline{\mathbf{U}}_{0,n}$ the universal family described by $\pi_n:\overline{\...
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1 vote
0 answers
183 views

Splitting after pullback under finite flat morphisms

I recently became aware of an interesting result, which claims for smooth projective curve over a finite field, you can pull back any vector bundle, under combinations of Galois covers and Frobenius, ...
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1 vote
1 answer
426 views

Injectivity of the cohomology map associated to the pullback of line bundles

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
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2 votes
0 answers
215 views

Pullback connection and diffeomorphism of the base

Let $p \colon E \to B$ be a vector bundle, $\nabla^E \colon E \to E \otimes \Omega^1_B$ a connection on $E$, and $\phi \colon B \xrightarrow{\sim} B$ a diffeomorphism. Further, let there be a natural (...
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1 vote
0 answers
153 views

Nef divisors on abelian varieties are pullbacks of ample ones

It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
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4 votes
1 answer
433 views

Does Spec functor sends pushouts of rings into pullbacks of sets?

This question was posted here on StackExchange. Let $A$ be a commutative ring and $B,C$ be two commutative $A$-algebras. Consider the pushout square of ring homomorphism $\require{AMScd}$ \begin{CD} ...
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3 votes
1 answer
113 views

Pullback of homogeneous twisted differential operators

Let $X,Y$ be smooth complex varieties and let $G$ be an smooth affine algebraic group acting on $X$ and $Y$ such that $X,Y$ are $G$-homogeneous spaces (the $G$ action is transitive). We also let $f:Y \...
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3 votes
0 answers
127 views

Techniques for computing homotopy pullbacks

I am in the context of simplicial sets. I have a square diagram that I want to show to be homotopy pullback. What I can grasp is that it is a pullback up to some equivalences in the middle of my ...
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1 vote
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71 views

Transverse measures in pseudo-Anosov diffeomorphisms

I've recently begun doing research involving pseudo-Anosov diffeomorphisms, which are diffeomorphisms on surfaces $f : M \to M$ admitting two singular measured foliations $(\mathcal F^s, \nu^s)$ and $(...
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3 votes
1 answer
361 views

Restriction of Ext sheaves on closed subschemes

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
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1 vote
0 answers
192 views

Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...
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0 votes
0 answers
78 views

$H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?

Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$. Is there a constant $C=...
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  • 625
4 votes
0 answers
192 views

Exercise in the book "Lectures on Kähler geometry"

I am currently studying the book "Lectures on Kähler geometry" by Andrei Moroianu and am looking for help concerning Exercise 5.8 (3) which is to prove the following Lemma 5.11 Let $f: M \...
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  • 149
5 votes
0 answers
93 views

Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients: An oriented, (assumed here to be smooth) manifold $M^n$ A finite group $G$ (and a field, chosen to be $\...
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1 vote
1 answer
249 views

Properties of codimension under pull back

If I pull back a cycle of codimension $c$ along a morphism of schemes I can easily see that the codimension can stay the same or the codimension can drop to all the way to $0$. But intuitively it ...
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  • 3,788
15 votes
3 answers
2k views

History of the pullback corner notation

Where/when did the convention originate of marking pullback (and/or pushout) squares by that little right-angle symbol in the corner? The earliest instance I’ve been able to find is in Paul Taylor’s ...
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1 vote
0 answers
166 views

Kernel of Gysin map

Let $X$ be a variety, let $D_1$, $D_2$ be linearly equivalent effective Cartier divisors on $X$ and let $\varphi_i\colon A_{k+1}(X)\to A_{k}(D_i)$ be the Gysin homomorphism. Is $\ker(\varphi_1) = \ker(...
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1 vote
0 answers
502 views

Isomorphism of probability spaces

Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
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2 votes
2 answers
480 views

Does "symmetry" of a pullback connection should be obvious?

$\newcommand{\M}{M}$ $\newcommand{\N}{N}$ $\newcommand{\TM}{TM}$ $\newcommand{\TN}{TN}$ $\newcommand{\TstarM}{T^*M}$ $\newcommand{\Ga}{\Gamma}$ Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a ...
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  • 6,286
3 votes
0 answers
104 views

Smooth submersions - smallest universal subclass of regular epimorphisms?

The smooth category has many problems. One is that pullbacks of regular epimorphisms need not exist. However, pullbacks along submersions always exist. It also seems that submersions are universal (i....
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3 votes
1 answer
112 views

Fiber product of nilmanifolds

Let $M_1$ and $M_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi_i:M_i \to B_i\ \ i=1,2$. Suppose that $B_1=B_2$ and that the fibers are torus of the same dimension and ...
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  • 209
4 votes
1 answer
911 views

Left adjoint of pullback

In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states: Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback ...
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11 votes
2 answers
1k views

Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
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  • 615
1 vote
1 answer
248 views

finite generation of a certain type of subring

Let $k$ be a field, and let $R$ be a finitely generated $k$-algebra. (If it helps, you may assume $R$ is an integral domain.) Let $I$ be an ideal of finite colength. Note that $A:=k+I$ is a subring ...
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  • 1,658
2 votes
0 answers
224 views

Is continuity of a functor stable under pullback?

Let $p:C\rightarrow D$, $i:F\rightarrow D$ be functors of 2-categories, and we form the lax pullback of $p$ along $i$ $$ \bar{p}:C\times_D^{lax} F\rightarrow F$$ Q1: Is it true that if $p$ ...
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  • 670
0 votes
2 answers
2k views

Pullback of a constant sheaf

Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type. Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and $\mathscr{...
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  • 241
1 vote
2 answers
924 views

Definition of subobject classifier in presheaves

I am reading Awodey (Category Theory, 1st edition), p 175, and I have difficulties to understand the paragraph about the subobject classifier of $\mathbf{Sets}^{\mathbf{C^{op}}}$. First let me quote ...
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35 votes
3 answers
9k views

Pullback measures

Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist? It's true that the naive treatment of such a ...
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  • 4,670