---
tag: PB9JXQ2KC3
kind: concept
name: Serre fibration
msc: [55R05]
sources:
  - url: https://en.wikipedia.org/wiki/Fibration
---

## Summary

A Serre fibration is a continuous map $p \colon E \to B$ that has the [[homotopy lifting property|G4QDM01P6J]] with respect to all cubes $I^n$.
It is the weak form of a fibration used most often in homotopy theory.
Every [[Hurewicz fibration|Y773B7EQA7]] and every [[fiber bundle|CX4M8K61AB]] is a Serre fibration.
For a Serre fibration the homotopy groups of the fiber, the total space and the base fit into the [[long exact sequence of homotopy groups|E0RV5D1GQ0]].

## Explanations

### L1 human

Think of $E$ as lying above $B$, with $p$ pressing each point of $E$ down to $B$.
In a Serre fibration, every deformation in $B$ of a cube-shaped family of points can be followed by a deformation in $E$.
One only needs to choose the starting position in $E$.
The projection of the product $B \times F$ onto $B$ is the simplest example: one keeps the $F$ coordinate fixed and moves the $B$ coordinate.
Covering maps, such as $\R \to S^1$, $t \mapsto e^{2\pi i t}$, are also Serre fibrations.