Contents

# Contents

## Idea

Hurewicz cofibrations are a kind of cofibrations of topological spaces, hence a kind of continuous function 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 $i \colon A \longrightarrow X$ is a Hurewicz cofibration if it satisfies the homotopy extension property in that:

• for any topological space $Y$,

• all continuous functions $f \colon A\to Y$, there exists $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$

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

there is a homotopy $\tilde{F} \colon X\times I\to Y$ such that

• $\tilde{F}\circ(i\times id_I)=F$

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

$\array{ A &\overset{id_A \times const_0}{\longrightarrow}& A \times I &\overset{F}{\longrightarrow}& Y \\ {}^{\mathllap{i \times id_I}}\downarrow && {}^{\mathllap{i \times id_I}}\downarrow & \nearrow_{\mathrlap{\tilde F}} \\ X &\underset{id_X \times const_0}{\longrightarrow}& X \times I } \phantom{AAA} = \phantom{AAA} \array{ \array{ A &\stackrel{f}{\to}& Y \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X } }$
###### Definition

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

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

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

## Properties

### Subspace inclusions

###### Proposition

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

###### 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 $i$ injective and a homeomorphism onto its image (Homotopietheorie (1.17)). In the category of weakly Hausdorff compactly generated spaces, $i(A)$ 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 Homotopietheorie) is the following: let $A =\{a\}$ and $X=\{a,b\}$ be the one and two element sets, both with the codiscrete topology (only $X$ and $\emptyset$ are open in $X$), and $i:A\hookrightarrow X$ is the inclusion $a\mapsto a$. Then $i$ is a non-closed cofibration (useful exercise!).

### Strøm’s model structure

The collections

• closed Hurewicz cofibrations (def. , def. ),

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

The identity functor $id \colon Top \to Top$ 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.

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

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

### Interaction with pullbacks

###### Theorem

Let

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

be a commuting diagram of topological spaces such that

• the horizontal morphisms are closed cofibrations;

• the morphisms $p_0$ and $p$ are Hurewicz fibrations.

Then the induced morphism on pullbacks is also a closed cofibration

$X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E \,.$

This is stated and proven in (Kieboom).

###### Corollary

The product of two closed cofibrations is a closed cofibration.

• 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)

• Tammo tom Dieck, Klaus Heiner Kamps, and Dieter Puppe. Homotopietheorie. Vol. 157. Springer-Verlag, 1970.

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