nLab
Hurewicz fibration

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

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


Contents

Definition

Definition

A continuous map p:EBp \;\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 σ 0:XX×{0}X×I\sigma_0 \;\colon\; X\cong X\times\{0\}\hookrightarrow X\times I for all topological spaces XX.

Remark

This right lifting property is in this context called the homotopy lifting property, because the maps from X×IX\times I are understood as homotopies. In more detail, for every space XX, any homotopy F:X×IBF:X\times I\to B, and a continuous map f:XEf:X\to E, there is a homotopy F˜:X×IE\tilde{F}:X\times I\to E such that f=F˜σ 0:=F˜ 0f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0 and F=pF˜F=p\circ\tilde{F}:

X f E σ 0 F˜ p X×I F B. \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 0Top/B_0 where B 0B_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

See also

  • 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)