nLab
Hurewicz fibration

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

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Definition

Definition

A continuous map $p \;\colon\; E\longrightarrow B$ of topological space is called a Hurewicz fibration if it satisfies the right lifting property against all continuous functions

$$\sigma_0 \;\colon\; X \cong X \times\{0\} \hookrightarrow X \times I$$

including any topological space $X$ as one end of the cyclinder? over it (where $I = [0,1]$ denotes the topological interval).

Remark

In this context this right lifting property is 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 $F=p\circ\tilde{F}$:

$$\array{ X &\stackrel{f}\longrightarrow& E \\ {}^{\mathllap{\sigma_0}} \big\downarrow &{}^{\tilde{F}}\nearrow& \big\downarrow {}^{\mathrlap{p}} \\ X\times I &\stackrel{F}{\to}& B } \,.$$

Remark

Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient category of TopologicalSpaces 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.)

Properties

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.

There is also a model category whose fibrations are the Hurewicz fibrations and whose weak equivalences are the weak homotopy equivalences, obtained by mixing the above model structure with the classical model structure on topological spaces.

Relation to Serre fibrations

Every Hurewicz fibration is a Serre fibration. Conversely, a Serre fibration between CW-complexes is a Hurewicz fibration.

Local recognition over numerable open covers

The following proposition says that being a Hurewicz fibration is a local property with respect to the codomain space, at least as long as the local open cover used is numerable.

Proposition

(Local recognition of Hurewicz fibrations over numerable open covers)

Let $p \colon E \to B$ be a continuous function such that there exists a numerable open cover $U \overset{\phi}{\longrightarrow} B$ over which $E$ is a Hurewicz fibration, hence such that the pullback $\phi^\ast p$ is a Hurewicz fibration.

Then $p$ is a Hurewicz fibration

A proof may be found spelled out in e.g. May 99, Sec. 7.4

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).

Examples

Empty bundles

Example

(empty bundles are Hurewicz fibrations)
All empty bundles $\varnothing \longrightarrow B$ are Hurewicz fibrations, because none of the commuting squares that one would have to non-trivially lift in actually exist:

$$\array{ X &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ X \times I &\longrightarrow& B \mathrlap{\,,} }$$

since the empty topological space is a strict initial object: There is no morphism to it from any inhabited space $X$

(If $X$ itself is empty, then so is $X \times I$ and then a square only exists if $B$ in turn is empty. This square, consisting entirely of empty spaces, does have a unique lift, trivially, and hence also the empty bundle over the empty space is a Hurewicz fibration – for what it’s worth.)

Covering spaces

Example

(covering spaces are Hurewicz fibrations)

Every covering space projection is a Hurewicz fibration, by this prop..

Fiber bundles

Proposition

(numerable fiber bundles are Hurewicz fibrations)

Every numerable fiber bundle, hence in particular every fiber bundle over a paracompact topological space, is a Serre fibration.

Proof

By definition of local triviality, for a numerable fiber bundle $p \colon E \to B$ – such as a fiber bundle over a paracompact topological space – there exists a numerable open cover $\phi \colon U \to B$ such that the pullback bundle $\phi^\ast p$ is trivial, i.e. is the Cartesian product projection

$$\phi^\ast p \;\colon\; U \times F \longrightarrow U$$

(for $F$ the typical fiber). Since such a projection is clearly a Hurewicz fibration the claim follows by the local recognition of Hurewicz fibrations over numerable open covers, from Prop. .

Notice that the example of empty bundles (Example ) is a special case of a fiber bundle: With typical fiber the empty topological space.

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

• Alan Hatcher, from p. 61 on in Algebraic Topology (web)

See also

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

• the textbooks on algebraic topology by Whitehead and Spanier.

• Peter May, A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)

• Tyrone Cutler, Fibrations III – Locally trivial maps and bundles, 2020 (pdf, pdf)

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

• K. H. Kamps, Tim Porter, Abstract Homotopy and Simple Homotopy Theory (GoogleBooks)

and further developed in

• Richard Williamson, Cylindrical model structures (arXiv:1304.0867, web)

Discussion with an eye towards homotopy type theory is in

• Michael Warren, Homotopy theoretic aspects of constructive type theory, 2008 (PDF)