---
tag: Y773B7EQA7
kind: concept
name: Hurewicz fibration
msc: [55R05]
sources:
  - url: https://en.wikipedia.org/wiki/Fibration
---

## Summary

A Hurewicz fibration is a continuous map $p \colon E \to B$ that has the [[homotopy lifting property|G4QDM01P6J]] with respect to all topological spaces.
Fibrations behave in homotopy theory like fibre bundles.
The fibres over points of a path-connected base are homotopy equivalent, and there is a long exact sequence of homotopy groups.
A [[Serre fibration|PB9JXQ2KC3]] is the weaker notion that asks for the lifting property only for cubes, or equivalently for CW complexes.

## Explanations

### L1 human

Think of $E$ as a space lying over $B$, with a fibre over each point.
Suppose a family of points moves in $B$ and you know where it starts in $E$.
The map is a fibration if you can always lift the whole motion to $E$ continuously.
Product projections, covering maps and fibre bundles over nice bases are fibrations.
The path space fibration, sending a path in $B$ to its end point, is a central example in homotopy theory.