Questions tagged [fibration]
For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
192 questions
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Fibrations of categories with terminal objects
Fix $\mathbf{B}$ for a category with pullbacks.
Proof: $(\Longrightarrow)$ Suppose that $P$ is a fibration of categories with terminal objects. Then for every $I$ in $\mathbf{B}$ there is an object $...
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An internal functor in $\mathbf{Fib(B)}$
I have been thinking about this notion of an internal functor in the category $\mathbf{Fib(B)}$ of fibrations over the same base $\mathbf{B}$. Say $f \colon P \Rightarrow Q$ is an internal functor ...
- 615
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2
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Connectivity of fibers under fibration replacement
Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
- 1,177
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48
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Double analogue of the domain functor
For a category $\mathcal{C}$, the domain functor $\mathbf{dom} \colon \mathcal{C}^2 \to \mathcal{C}$ is a fibration.
Denote $\mathbb{D}_1^2$ for the category of proarrows and cells between them and ...
- 615
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106
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Joyal's Cartesian squares
I am looking at Joyal's definition of cartesian squares Defn 1.1, Is this the same as saying that a commutative square is cartesian iff the induced arrows on the fibers are equivalences?
- 615
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Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
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2
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127
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Closure properties of fibrations (cartesian functor)
Given fibrations $P$ and $Q$ over the same base and a cartesian functor $F$ as shown below.
I know that when $P$ is discrete, one can easily show that $F$ is a fibration. But this is not true in ...
- 615
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81
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Geometric morphisms for double categories
Is such there a notion 'Geometric morphism' for double categories? I have been reading Jean Benabou's lecture notes on fibred categories. He has slides on fibred geometric morphisms for arbitrary ...
- 615
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90
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Equivalence of fibrations
I know that the for fibrations $P \colon E \to B$ and $Q \colon F \to B$ over $B$ there is an equivalence
$$\mathbf{Fib(B)}(R, P \to Q) \cong \mathbf{Fib(B)}(R \times_B P, Q) $$ natural in $R \in \...
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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 ...
- 615
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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 \...
- 615
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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 ...
- 615
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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-...
- 615
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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 ...
- 8,803
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1
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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
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
- 571
4
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A fiber-like method to show equivalence of infinity categories
Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
- 2,152
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Is the fiberwise suspension of a Serre fibration a Serre fibration?
Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma_f\pi \colon \Sigma_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma_f X = X \times [0,1] / {\sim}$,...
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2
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Fibration of hyperbolic 3-manifold
A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence
$$
1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1
$$
where $N$ is the fiber.
I've heard that, if $M$ ...
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315
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Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
- 2,292
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Monodromy action on homogeneous spaces
If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
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 ...
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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 ...
- 280
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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$, ...
- 663
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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 ...
- 1,906
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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 ...
- 1,996
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1
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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}^{...
- 10.1k
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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 ...
- 1,177
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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.
...
- 10.1k
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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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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 ...
- 35.9k
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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+...
- 1,205
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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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221
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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
...
- 5,998
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1
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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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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 ...
- 10.1k
3
votes
2
answers
2k
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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 ...
- 5,998
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votes
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203
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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_{...
user30211
3
votes
0
answers
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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 ...
user126256
4
votes
0
answers
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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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