nLab
Hurewicz cofibration

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

Hurewicz cofibrations are a kind of cofibrations of topological spaces, hence a kind of continuous functions satisfying certain extension properties.

In point-set topology Hurewicz cofibrations are often just called cofibrations. If their image is a closed subspace they are called closed cofibrations.

A continuous function is a Hurewicz cofibration if it satisfies the homotopy extension property for all spaces and with respect to the standard notion of left homotopy of topological spaces given by the standard topological interval object/cylinder object.

More generally, one may speak of morphisms in any category with weak equivalences having the homotopy extension property with respect to a chosen cylinder object, one speaks of h-cofibrations.

Definition

Definition

A continuous function i:AXi \colon A \longrightarrow X is a Hurewicz cofibration if it satisfies the homotopy extension property in that:

  • for any topological space YY,

  • all continuous functions f:AY f \colon A\to Y, f˜:XY\tilde{f}:X\to Y such that f˜i=f\tilde{f}\circ i=f

    A f Y i f˜ X \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }
  • and any left homotopy F:A×IYF \colon A\times I\to Y such that F(,0)=fF(-,0)=f

there is a homotopy F˜:X×IY\tilde{F} \colon X\times I\to Y such that

  • F˜(i×id I)=F\tilde{F}\circ(i\times id_I)=F

    A×I F Y i×id I F˜ X×I \array{ A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X \times I }
  • and F˜(,0)=f˜\tilde{F}(-,0)=\tilde{f}

    A id A×const 0 A×I F Y i×id I i×id I F˜ X id X×const 0 X×IAAA=AAAA f Y i f˜ X \array{ A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X &\underset{id_X \times const_0}{\longrightarrow}& X \times I } \phantom{AAA} = \phantom{AAA} \array{ \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X } }
Definition

A Hurewicz cofibration i:AXi \colon A\to X (def. 1) is called a closed cofibration if the image i(A)i(A) is a closed subspace in XX.

If AXA\subset X is closed and the inclusion is a cofibration, then the pair (X,A)(X,A) is called an NDR-pair.

There is also a version of the definition for pointed spaces.

Properties

Subspace inclusions

Proposition

A topological subspace inclusion AXA \hookrightarrow X is a Hurewicz cofibration precisely if A×IX×{0}A \times I \cup X \times \{0\} is a retract of X×IX\times I.

Corollary

A subcomplex inclusion into a CW-complex is a closed Hurewicz cofibration.

e.g. Bredon Topology and Geometry, p. 431

More generally, every retract of a relative cell complex inclusion is a closed Hurewicz cofibration.

This is part of the statement of the Quillen adjunction between then classical model structure on topological spaces and the Strøm model structure (see below)

Closedness

Every Hurewicz cofibration ii is an injective map and if the image i(A)i(A) is closed then it is a homeomorphism onto its image. In the category of weakly Hausdorff compactly generated spaces, i(A)i(A) is always closed (the same in the category of all Hausdorff spaces), but in the category Top of all topological spaces there are pathological counterexamples.

The simplest example (see the classical monograph Dieck, Kamps, Puppe, Homotopietheorie, LNM 157) is the following: let A={a}A =\{a\} and X={a,b}X=\{a,b\} be the one and two element sets, both with the codiscrete topology (only XX and \emptyset are open in XX), and i:AXi:A\hookrightarrow X is the inclusion aaa\mapsto a. Then ii is a non-closed cofibration (useful exercise!).

Strøm’s model structure

The collections

make one of the standard Quillen model category structures on the category Top of all topological spaces Strøm's model category.

The identity functor id:TopTopid \colon Top \to Top is left Quillen from the classical model structure on topological spaces (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.

Top StromididTop Quillen. Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \,.

This means in particular that any retract of a relative cell complex inclusion is a closed Hurewicz cofibration.

Interaction with pullbacks

Theorem

Let

X 0 X p 0 p B 0 B E 0 E \array{ X_0 &\hookrightarrow & X \\ {}^{\mathllap{p_0}}\downarrow && \downarrow^{\mathrlap{p}} \\ B_0 &\hookrightarrow& B \\ \uparrow && \uparrow \\ E_0 &\hookrightarrow& E }

be a commuting diagram of topological spaces such that

  • the horizontal morphisms are closed cofibrations;

  • the morphisms p 0p_0 and pp are Hurewicz fibrations.

Then the induced morphism on pullbacks is also a closed cofibration

X 0× B 0E 0X× BE. X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E \,.

This is stated and proven in (Kieboom).

Corollary

The product of two closed cofibrations is a closed cofibration.

References

  • Dieter Puppe, Bemerkungen über die Erweiterung von Homotopien, Arch. Math. (Basel) 18 1967 81–88; MR0206954 (34 #6770) doi

  • Arne Strøm, Note on cofibrations, Math. Scand. 19 1966 11–14 file MR0211403 (35 #2284); Note on cofibrations II, Math. Scand. 22 1968 130–142 (1969) file MR0243525 (39 #4846)

The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in

Revised on July 12, 2017 17:15:01 by Urs Schreiber (46.183.103.17)