# nLab Hurewicz fibration

### Context

#### Topology

topology

algebraic topology

## Examples

Serre fibration $⇐$ Hurewicz fibration $⇒$ Dold fibration $⇐$ shrinkable map

# Contents

## Definition

A Hurewicz fibration $p:E\to B$ is a continuous map of topological spaces that satisfies the right lifting property with respect to maps ${\sigma }_{0}:X\cong X×\left\{0\right\}↪X×I$ for all topological spaces $X$.

Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient category of spaces one is working in, since frequently this is a convenient category of spaces. In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of all topological spaces, but in practice this usually doesn’t make a whole lot of difference.

This right lifting property is in this context called the homotopy lifting property, because the maps from $X×I$ are understood as homotopies. In more detail, for every space $X$, any homotopy $F:X×I\to B$, and a continuous map $f:X\to E$, there is a homotopy $\stackrel{˜}{F}:X×I\to E$ such that $f=\stackrel{˜}{F}\circ {\sigma }_{0}:={\stackrel{˜}{F}}_{0}$ and $F=p\circ \stackrel{˜}{F}$:

$\begin{array}{ccc}X& \stackrel{f}{\to }& E\\ {↓}^{{\sigma }_{0}}& {}^{\stackrel{˜}{F}}↗& {↓}^{p}\\ X×I& \stackrel{F}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X &\stackrel{f}\to& E \\ \downarrow^{\sigma_0} &{}^{\tilde{F}}\nearrow& \downarrow^p \\ X\times I &\stackrel{F}{\to}& B } \,.

Instead of checking the homotopy lifting property, one can instead solve a universal problem:

###### Theorem

A map is a Hurewicz fibration precisely if it admits a Hurewicz connection. (See there for details.)

## Appearance in a model structure

There is a Quillen model category structure on Top where fibrations are Hurewicz fibrations, cofibrations are closed Hurewicz cofibrations and weak equivalences are homotopy equivalences; see model structure on topological spaces and Strøm's model category. There is a version of Hurewicz fibrations for pointed spaces, as well as in the slice category $\mathrm{Top}/{B}_{0}$ where ${B}_{0}$ is a fixed base.

## References

The historical paper of Hurewicz is

• Witold Hurewicz, On the concept of fiber space, Proc. Nat. Acad. Sci. USA 41 (1955) 956–961; MR0073987 (17,519e) PNAS,pdf.

Hurewicz fibrations are nowadays a standard topic in textbooks of algebraic topology (Whitehead, Spanier, Hatcher…).

Revised on April 7, 2013 19:03:34 by Urs Schreiber (89.204.130.212)