nLab
Hurewicz fibration

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

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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 ${\sigma }_0:X\cong X\times \{0\}\hookrightarrow X\times I$ 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 $X\times I$ 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 $F=p\circ \tilde{F}$:

$$\begin{array}{ccc}X& \stackrel{f}{\to }& E\\ {\downarrow }^{{\sigma }_0}& {}^{\tilde{F}}\nearrow & {\downarrow }^p\\ X\times I& \stackrel{F}{\to }& B\end{array}.$$\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…).