# nLab Hurewicz fibration

### Context

#### Topology

topology

algebraic topology

## Examples

Serre fibration $\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration $\Leftarrow$ shrinkable map

# Contents

## Definition

###### Definition

A continuous map $p \;\colon\; E\longrightarrow B$ of topological space is called a Hurewicz fibration it it satisfies the right lifting property with respect to maps of the form $\sigma_0 \;\colon\; X\cong X\times\{0\}\hookrightarrow X\times I$ for all topological spaces $X$.

###### Remark

This right lifting property is in this context called the homotopy lifting property, because the maps from $X\times I$ are understood as homotopies. In more detail, for every space $X$, any homotopy $F:X\times I\to B$, and a continuous map $f:X\to E$, there is a homotopy $\tilde{F}:X\times I\to E$ such that $f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0$ and $F=p\circ\tilde{F}$:

$\array{ X &\stackrel{f}\to& E \\ \downarrow^{\sigma_0} &{}^{\tilde{F}}\nearrow& \downarrow^p \\ X\times I &\stackrel{F}{\to}& B } \,.$
###### Remark

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.

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 $Top/B_0$ where $B_0$ is a fixed base.

## Abstract Hurewicz fibrations

The concept of Hurewicz fibrations makes sense also more generally in the presence of a (well behaved) interval object, see for instance the early example of such in (Kamps 72) and see (Williamson 13) for review and further developments. Discussion with a view towards homotopy type theory is in (Warren 08).

## 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.

A decent review of Hurewicz fibrations, Hurewicz connections and related issues is in

• James Eells, Jr., Fibring spaces of maps, in Richard Anderson (ed.) Symposium on infinite-dimensional topology

A textbook account of the homotopy lifting property is for instance in

• R. Schwänzl, R. Vogt, Strong cofibrations and fibrations in enriched categories, 2002.

• the textbooks on algebraic topology by Whitehead and Spanier.

Abstract analogues of Hurewicz fibrations can be found in

• K.H.Kamps, Kan-Bedingungen und abstrakte Homotopietheorie, Math. Z. 124,1972, 215 -236

summarised in

and further developed in

Discussion with an eye towards homotopy type theory is in

Revised on July 29, 2015 15:04:32 by Urs Schreiber (82.113.98.28)