Questions tagged [fibration]
For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
16 questions from the last 365 days
3
votes
1
answer
197
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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 $...
- 615
1
vote
0
answers
46
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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
7
votes
2
answers
383
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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
2
votes
0
answers
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
2
votes
0
answers
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
9
votes
0
answers
120
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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,...
- 371
1
vote
2
answers
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
4
votes
0
answers
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
2
votes
0
answers
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 \...
- 615
3
votes
1
answer
156
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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
4
votes
0
answers
191
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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
9
votes
0
answers
85
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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 $\...
- 10.6k
6
votes
1
answer
243
views
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
2
votes
0
answers
69
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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
0
votes
0
answers
45
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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'...
6
votes
1
answer
285
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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 $\mathbb{C}$ ...
- 661