# Questions tagged [fibration]

For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.

183
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### Closure properties of fibrations

I am trying to prove the above theorem, I think I can do the backward direction. I wanted to be sure about the forward direction:
$(\implies)$ Suppose $F$ is a fibration. Then since fibrations are ...

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### Conduché fibrations

Given the following definition
Defn: Suppose $\mathcal{A}$ is a category with finite limits. An object $Y$ of $\mathcal{A}$ is called exponentiable if the functor $(- \times Y) \colon \mathcal{A} \to \...

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### Reference for the tricategory of elements associated to a trifunctor

The theory of bicategorical fibrations has been relatively well studied, e.g. by Baković and by Buckley. In particular, given a trifunctor $F : \mathcal K \to \mathbf{Bicat}$ from a bicategory $\...

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### Fibrations using adjoints

For ordinary fibrations is it true that:
Given a functor $F \colon C \to D$ with $C$ finitely complete, and a fully faithful functor $U$ such that $F \dashv U$ and F preserving limits. Then F is a ...

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### Pullbacks in Cat in a 2-dimensional sense

$\newcommand\Fib{\mathrm{Fib}}\newcommand\Cat{\mathrm{Cat}}\newcommand\OpF{\mathrm{OpF}}\DeclareMathOperator\cod{cod}$In proving that a codomain functor from the 2-category $\Fib$ to $\Cat$ is a 2-...

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### How is the behaviour of a deformation retract under a fibration? [duplicate]

Let $p:E \rightarrow B$ a fibration and take $A\subset B$ a deformation retract of B. Is it true that $p^{-1}(A)$ is a deformation retract of E?
By deformation retract I mean the weaker definition.
I'...

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### Is this $\mathbb C$-fibration over compact Riemann surface trivial?

I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions:
$p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...

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### Division of fibration by $\Sigma_{n}$ gives Serre fibration

This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...

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### Is the pushforward of an exponentiable fibration along an exponentiable fibration again exponentiable?

Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold:
The pullback functor $p^*\colon \...

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### Are all homotopy equivalences realized by fibrations over [0,1]?

Given two homotopy equivalent spaces $X$ and $Y$, does there always exist a Hurewicz fibration $p: E\rightarrow [0,1]$ with $p^{-1} (0) = X$ and $p^{-1} (1)=Y$?
This issue shows up in the accepted ...

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### Cohomology of the amplitude space of unlabeled quantum networks

I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...

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1
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### A fiber-like method to show equivalence of infinity categories

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...

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### Is the fiberwise suspension of a Serre fibration a Serre fibration?

Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma_f X = X \times [0,1] / {\sim}$,...

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### Fibration of hyperbolic 3-manifold

A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence
$$
1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1
$$
where $N$ is the fiber.
I've heard that, if $M$ ...

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### Reference request - Fibrations between spaces of embeddings

This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...

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### Monodromy action on homogeneous spaces

If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
G/H\to BH\to BG.
$$
By choosing a model for $EG$ we can promote this into a fibre bundle.
My question is about how to understand ...

3
votes

1
answer

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### Looking for examples of non-singular holomorphic foliations with compact leaves

I am looking for examples (or what is known about) of the following kind of object:
X compact Kähler manifold
F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...

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2
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### Associativity of consecutive fibrations

[ I asked the same question on stackexchange but attracted little attention. Besides, I made some progress after I posted it. So I decided to move it here. ]
Consider path-connected CW-complexes $A$, ...

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### (Co)cartesian fibrations and left Kan extensions

Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...

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### Computing the homotopy type of $B\operatorname{Aut}(K(G,1))$ using a fibration sequence: why is $G \to \text{Aut(G)}$ given by conjugation?

$\newcommand{\Aut}{\operatorname{Aut}}$Let $G$ be an abelian group.
It seems to be a well-known fact (for example here) that $B\Aut(K(G,1))$, the classifying space of the topological monoid of (...

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### Fibrations of $n$-groupoids in the folk model structure on $n$-categories

Define a strict $n$-groupoid to be a strict $n$-category all of whose morphisms are weakly invertible.
[For $1\leq k < n$ a $k$-morphism $f:x\to y$ is weakly invertible if there exists $g:y\to x$ ...

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### The $\pi_1(BM)$ action on $H^*(BH,R)$ in a Serre fibration $BH\to BG\to BM$

$R$ is a ring. Applying the Serre spectral sequence to a fibration $F\to E\to B$, to avoid local coefficients, we need to require that $\pi_1(B)$ acts on $H^*(F, R)$ trivially. Consider a fibration of ...

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### When is the Grothendieck / category of elements construction a fibration on geometric realizations?

Suppose we have a simplicial complex / poset / small category without loops $X$ equipped with a functor $F$ into the category of posets / small categories without loops. Suppose further that for each ...

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### Schemes as categories fibered in thin groupoids

Every time I start to read about schemes from a birds-eye view (like in the introduction to The Geometry of Schemes by Eisenbud and Harris) I get really excited; they sound like a categorical approach ...

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### Objects whose representable presheaf is a fibration

Is there any literature on representable presheaves which are fibrations, or categories such that all representable presheaves are fibrations?
A representable presheaf $$\mathcal{C}(-,X):\mathcal{C}^{...

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### Vietoris-Begle type result for differentiable fiber bundle

In Vietoris-Begle Theorem, we consider a closed and surjective map between two paracompact and Hausdorff spaces and we get some relation involving the homologies of the fiber, total space, and the ...

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### A fibration equivalent to having a terminal object

It is well known that the codomain functor $$cod:\mathcal{C}^\to\to\mathcal{C}$$ from the arrow category of a category $\mathcal{C}$ to itself is a fibration iff $\mathcal{C}$ has binary pullbacks.
...

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1
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### Ehresmann's fibration theorem for CW or simplicial complexes

Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ?
Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is ...

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1
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### Complex fibration over complex torus

Let $M$ be a 3-dimensional complex manifold, and $\Lambda$ a discrete lattice in $\mathbb C^2$. Suppose there is a holomorphic submersion $f:M\to\mathbb{C}^2/\Lambda$ with fibers given by 1-...

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### Terminology for an kind-of principal fibration

My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...

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### fibre-preserving homotopy equivalence

Let $p:E\to B$ and $p':E'\to B$ be fibrations. It is well known that if $f:E\to E'$ a fibrewise map that is also a homotopy equivalence, then it is a fibrewise homotopy equivalence.
What about the ...

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### Trivialization of fibration by etale base change

Let $f:Y \to X$ be a smooth fibration over $\mathbb{C}$ in the sense that $X$ is a smooth, quasi-projective, connected variety and $f$ is a smooth, projective (surjective) morphism. Suppose that every ...

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### Fundamental group of the complement of some quadric cones

cross-posting from MathSE
Problem
Consider the domain
$$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$
and the map
$$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...

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### Integrable systems and Lagrangian fibrations

It is known that every integrable system gives rise to a Lagrangian fibration via action-angle variables. My question is how to tell if a given Lagrangian fibration is an integrable system, that is ...

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### Historical proof of Leschetz Hyperplane Theorem

I browse in Phillip Griffiths' Slides
on historical development of
Hodge-theory and these include a sketch of the original approach
with Lefschetz used to study complex surfaces in his famous
...

6
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### Universal property of the codomain fibration

Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...

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1
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### Can we show that a functor is a fibration without choosing a cleavage?

Is there a standard method for showing that a functor $F:\mathcal{C}\to\mathcal{D}$ is a fibration, aside from constructing a cleavage?
In the proof of the Grothendieck construction, the fibration we ...

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2
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### Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma

I have a general question about the motivation behind to definition the smooth morphisms
as we know it from algebraic geometry. The most common
definition of a smooth morphism $: X \to Y$ between two ...

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### Can we recover $\pi_2(S^2)$ from this simplicial set?

Let $S^3 \rightarrow S^2$ be the Hopf fibration. Can we recover $\pi_2(S^2)$ of $S^2$ from the simplicial set $X : \Delta^{op} \rightarrow \text{Set}$,
$$ X(n) = \pi_0 (S^3 \times_{S^2} \cdots \times_{...

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### Homotopy limits of section spaces

Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak ...

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### Free abelian group on a space and fibrations

Let $X$ be a topological space. Endow the free abelian group on $X$, $\mathbb Z[X]$, the quotient topology coming from the surjection $\bigsqcup_n X^n \times \mathbb Z^n \to \mathbb Z[X]$. For $Y$ a ...

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### Action of fundamental group on homotopy fiber

For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi_1\left(B,b_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\...

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1
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### Homotopy equivalent fibers and Fibrations

If a morphism of topological spaces $X\rightarrow Y$ is a fibration, and the target space is connected, then the fibers of the points $y\in Y$ are homotopy equivalent, i.e. for all $y_1,y_2\in Y$ we ...

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### CW structure on infinite-dimensional manifolds

It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they ...

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### Tischler's Theorem on nonvanishing $1$-forms on open manifolds

I have been trying to find a generalized version of the following theorem due to D. Tischler,
Theorem 1. Let $M^n$ be a closed $n$-dimensional manifold. SUppose $M^n$ admits a nonvanishing closed $1$-...

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### A fiber bundle of the Euclidean space over an orbifold

Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...

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### Foliation of $\mathbb R^n$ by connected compact manifolds

Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not ...

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3
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### Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions

Let $A$ be an abelian group and let $n \geq 2$. For any connected CW complex $X$, it is standard that a fibration $f\colon E \rightarrow X$ whose fibers are homotopy equivalent to a $K(A,n)$ is ...

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### A notion of fibration on bisimplicial sets

[I am not trained in this stuff, but have an outside research interest, so sorry if this question is standard.]
I am interested in notions of fibrations, or fibrant objects, in bisimplicial sets. In ...

5
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1
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### Delooping a fibration sequence with loopspace fiber and finite CW complexes

The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $K(G,n)$, and so I don't see the immediate ...