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In algebraic topology and homotopy theory, Hurewicz cofibrations are a kind of cofibration of topological spaces, hence a kind of continuous function satisfying certain extension properties.
Specifically, a continuous function is a Hurewicz cofibration (Strøm 1966) if it satisfies the homotopy extension property for all target spaces and with respect to the standard notion of left homotopy of topological spaces given by the standard topological interval object/cylinder object.
A pointed topological space $(X,x)$ for which the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a Hurewicz cofibration is called a well-pointed topological space. A simplicial topological space all whose degeneracy maps are Hurewicz cofibrations is called a good simplicial topological space.
In point-set topology Hurewicz cofibrations are often just called cofibrations,for short. If their image is a closed subspace they are called closed cofibrations.
Beware that there are other relevant classes of cofibrations between topological spaces, notably the Serre cofibrations, and (in homotopy theory and model category-theory, at least) also often just called “cofibrations”. But “closed cofibration” always refers to closed Hurewicz cofibrations.
In fact, the notion of homotopy extension property makes sense in any category with a chosen cylinder object; and in this generality one also speaks of h-cofibrations (following Mandell, May, Schwede & Shipley 2001, p. 16).
(Hurewicz cofibration)
A map (continuous function) $i \colon A \xrightarrow{\;} X$ betwee topological spaces is a Hurewicz cofibration if it satisfies the homotopy extension property for all spaces, hence if
equivalently:
(re-formulation in terms of right homotopies)
In a convenient cartesian closed category $TopSp$ of topological spaces (such as that of compactly generated topological spaces) the product $\dashv$ internal hom-adjunction
allows to pass to adjunct morphisms in Def.
and equivalently express the above diagram of left homotopies as the following (somewhat more transparent) diagram of right homotopies, where $ev_0 \,\colon\, Y^{[0,1]}\to Y$ denotes evaluation at 0 $\gamma \mapsto \gamma(0)$:
In this equivalent formulation, the homotopy extension property is simply the right lifting property against the evaluation map $ev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y$ out of the path space of $Y$.
(e.g. May 1999, p. 43 (51 of 251))
(closed cofibrations)
A Hurewicz cofibration $i \colon A\to X$ (Def. ) is called a closed cofibration if the image $i(A)$ is a closed subspace in $X$. In this case the pair $(A,X)$ is also called an NDR-pair.
Every Hurewicz cofibration $i$ is injective and a homeomorphism onto its image.
(tDKP 1970 (1.17)).
In the category of weakly Hausdorff compactly generated spaces, the image $i(A)$ of a Hurewicz cofibration is always closed. The same holds in the category of all Hausdorff spaces.
(e.g. May 1999, Sec. 6.2, p. 44 (52 of 251))
But in the plain category Top of all topological spaces there are pathological counterexamples:
Let $A =\{a\}$ and $X=\{a,b\}$ be the one and two element sets, both with the codiscrete topology (only $X$ and $\varnothing$ are open subsets of $X$), and $i:A\hookrightarrow X$ is the inclusion $a\mapsto a$. Then $i$ is a non-closed cofibration.
When the image of a Hurewicz cofibration is a closed subspace – which is automatically the case in weakly Hausdorff topological spaces, by Prop. HurewiczCofibrationsInCGWHSpacesAreClosed – then there are a number of equivalent reformulations of the defining homotopy extension property (Def. ).
A closed topological subspace-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if its pushout product with the endpoint inclusion $0 \,\colon\, \ast \xhookrightarrow{\;} [0,1]$ into the topological interval
admits a retraction $r$:
Let $A \xhookrightarrow{i} X$ be a closed topological subspace-inclusion of compactly generated topological spaces which is a Hurewicz cofibration. Then for $Y$ any compactly generated topological space, the k-ified product space-construction $Y \times A \xhookrightarrow{ id_Y \times i } Y \times X$ is itself a Hurewicz cofibration.
Since compactly generated spaces with the k-ified product space construction form a cartesian closed category, the operation $Y \times (-)$ is a left adjoint and hence preserves the pushout in (1). It follows that the retract which exhibits, according to Prop. , the cofibration property of $i$, may be extended as a constant function in $Y$ to yield a retract that exhibits the cofibration property of $Y \times i$.
A closed topological subspace-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if the following condition holds:
There exists:
(1) a neighbourhood $U$ of $A$ in $X$
(2) a left homotopy $\eta$ shrinking this neighbourhood into $A$ relative to $A$, in that it makes the following diagram commute:
(3) another continuous function $\phi \colon X \xrightarrow{\;} [0,1]$ which makes the following diagram commute:
and in fact such that only $A$ is the preimage of zero: $A \,=\, \phi^{-1}(\{0\})$.
(This is due to Strøm 1966, Thm. 2; recalled, e.g., in Bredon 1993, Thm. 1.5 on p. 431)
If $X$ is a normal topological space then any inclusion $A \xhookrightarrow{\;} X$ is an h-cofibration iff its factorization $A \hookrightarrow{\;} U_A$ through any open neighbourhood $U_A \subset X$ of $A$ is an h-cofibration.
The composition of two h-cofibrations is itself an h-cofibration.
If
is a pair of Hurewicz cofibrations, with $i_1(A_1) \underset{clsd}{\subset} X_1$ closed, then the inclusion
into the product space of the codomains is itself a Hurewicz cofibration.
(fiber product of Hurewicz fibrations preserves Hurewicz cofibrations)
Let
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 fiber products
is also a closed cofibration.
(Cartesian product preserves Hurewicz cofibrations)
If
is a pair of Hurewicz cofibrations, then their image under the product functor is, too:
This is the special case of Prop. for $B$ and $B_0$ being the point space
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 $id \colon Top \to Top$ 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.
This means in particular that any retract of a relative cell complex inclusion is a closed Hurewicz cofibration.
Any relative CW complex-inclusion is an h-cofibration.
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).
Any point-inclusion into a (finite-dimensional) locally Euclidean Hausdorff space (e.g. a topological manifold) is an h-cofibration.
By definition of locally Euclidean spaces, any point has a neighbourhood $U$ which is chart, being a Euclidean space that may be identified with a vector space $\mathbb{R}^n$ with the given point being the origin $0 \,\in\, \mathbb{R}^n$:
Now let:
$\eta \colon U \times [0,1]\to X$ the homotopy given by $(\vec x, t) \mapsto (1-t)\cdot \vec x$;
$K \;\coloneqq\; B_{\leq 1}(0)$ the closed ball of unit radius around the origin in $\mathbb{R}^n$ (hence compact by Heine-Borel);
$\phi \colon X \to[0,1]$ be given by:
$x \mapsto min\big( \| x\|, 1 \big)$ on $U$ (the distance from the origin cut off at 1),
$x \mapsto 1$ on the complement $X \setminus K$.
It is manifest that this is a well-defined function and that the restrictions $\phi|_{U}$ and $\phi|_{X\setminus K}$ are continuous functions. Moreover, since compact subspaces of Hausdorff spaces are closed, $X \setminus K$ is open and $\big\{ U ,\, X \setminus K \big\}$ is an open cover of $X$. Therefore (by the sheaf-property of continuous functions), $\phi$ is continuous on all of $X$.
It is immediate to see that this data satisfies the conditions discussed in Prop. . Since Hausdorff spaces are $T_1$, so that all of their points are closed, that proposition applies and implies the claim.
Let $X$ be an absolute neighbourhood retract (ANR) and $A \xhookrightarrow{i} X$ a closed subspace-inclusion. Then $i$ is a Hurewicz cofibration iff $A$ is itself an ANR.
Let $X$ be a paracompact Banach manifold. Then the inclusion $A \hookrightarrow X$ of any closed sub-Banach manifold is a Hurewicz cofibration.
Being a closed subspace of a paracompact space, $A$ is itself paracompact (by this Prop.). But paracompact Banach manifolds are absolute neighbourhood retracts (this Prop.) Therefore the statement follows with Prop. .
(point inclusion into PU(ℋ))
The projective unitary group PU(ℋ) on an infinite-dimensional separable Hilbert space is
a Banach Lie group in its norm topology, and as such well-pointed by Prop. ;
no longer a Banach space in its weak/strong operator topology, but nevertheless still well-pointed in this case, by this Prop..
Named after:
Original articles:
Arne Strøm, Note on cofibrations, Math. Scand. 19 (1966) 11-14 (jstor:24490229, dml:165952, MR0211403)
Arne Strøm, Note on cofibrations II, Math. Scand. 22 (1968) 130–142 (jstor:24489730, dml:166037, MR0243525)
Dieter Puppe, Bemerkungen über die Erweiterung von Homotopien, Arch. Math. (Basel) 18 1967 81–88 (doi:10.1007/BF01899475, MR0206954)
Textbook accounts:
Tammo tom Dieck, Klaus Heiner Kamps, Dieter Puppe, Chapter I of: Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (doi:10.1007/BFb0059721)
Glen Bredon, Section VII.1 of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)
Peter May, Chapter 6 of: A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Section 4.1 in: Algebraic topology from a homotopical viewpoint, Springer (2002) (doi:10.1007/b97586, toc pdf)
Lecture notes:
Exposition:
Akhil Mathew, Cofibrations, 2010 (web)
Akhil Mathew, Examples of cofibrations, 2010 (web)
The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in
The terminology “h-cofibration” is due to: