# Questions tagged [fibration]

A branch of topology that deals with the notion of a fiber bundle.

276 questions
Filter by
Sorted by
Tagged with
113 views

### What does it mean that terminal object have morphism to other object?

Wiki defines terminal object as the object towards which there is a single morphism from every other object of the category https://en.wikipedia.org/wiki/Initial_and_terminal_objects This definition ...
62 views

50 views

### Geometrically seeing the linking of the fibres in the Hopf fibration?

I'm working through the following document: https://arxiv.org/pdf/0908.1205.pdf I can understand the construction of the Hopf map using the Riemann sphere but I struggle to understand the geometric ...
128 views

### Lifting a map to the total space of a circle bundle

Let $\pi:P \to M$ be a smooth circle bundle, so $S^1$ is the fibre, $f:N \to M$ a smooth map. I would like to know what are the necessary and sufficient conditions for $f$ to lift to a map $g:N \to P$...
32 views

### Seeing that the subobject fibration is a fibration

Let $\mathcal{B}$ be a category with pullbacks. We have that $cod:\mathcal{B}^\rightarrow\to\mathcal{B}$ is a fibration, with the Cartesian arrow over $k:X\to cod(g:Y\to Z)$ given by the pullback ...
83 views

### Is the map $z\mapsto z^2$ a fibration?

Let $f:\mathbb C \to \mathbb C$ be defined as $f(z) = z^2$. Is this a fibration? Or at least a Serre fibration? I am not sure how to approach this.
80 views

### generalization of fibration

A Hurewicz fibration $E\rightarrow B$ is a map so that for all maps $X\rightarrow E$ and $X\times I\rightarrow B$ making the obvious square commute, there is a lifting $X\times I\rightarrow E$ so that ...
76 views

### What is the essential difference between a sheaf and a fibration with lifting property?

Are there cases where the two notions coincide? By sheaf I mean a pre-sheaf ($\mathcal{C}^{op} \rightarrow \mathcal{Set}$) satisfying the sheaf condition. The sheaf condition says that, for every ...
58 views

### Equivalence of contractible and trivial fibrations

Let $B$ be a connected space. I am wondering, which of the following are equivalent, and under which conditions? $B$ is weakly contractible. $B$ is contractible. All fibrations $E \rightarrow B$ are ...
101 views

### Exact sequence of homotopy groups from a short exact sequence of Lie groups

Let $$1\to G'\to G\to G''\to 1$$ be a short exact sequence of real Lie groups. I assume that $G',\,G$, and $G''$ have finitely many connected components, but are not necessarily connected. I am ...
53 views

### Up to homotopy principal bundle

A principal bundle (say, in the category of topological spaces and topological groups) induces homeomorphisms on the fibers. This works well for groups $G$. I am wondering if there is an analogue for &...
55 views

### What is a (co)fibration in category theory

Coming from a computer science background, I have some working knowledge on category theory but none in algebraic topology. I see many definition involves some homotopy theory which I completely have ...
45 views

### Model categories: $\text{Ho}$ and $\cal C_{cf}/\sim$

I have asked this question about model categories: Why $\text{Ho} \ \cal C$ is $\cal C_{cf}/\sim$ and not $\cal C/\sim$ and I got this answer: take for cofibratiobns Iso, weak equivalences all arrows. ...
### $\pi_2(T \vee \mathbb{C}P^2)$ and action of $\pi_1$ on $\pi_2$
Let $X= T \vee \mathbb{C}P^2,$ where $T$ denotes the 2-dimensional torus. The task is to compute $\pi_2(X)$ and describe the action of $\pi_1(X)$ on $\pi_2(X)$. As for the first part, is there any ...