Cofibration and homotopy extension property

Intuition [Beginner]

A cofibration is an inclusion $i:A\to X$ that is well-behaved with respect to homotopies: any homotopy defined on the subspace $A$ that starts compatibly with a map on the whole of $X$ can be extended to a homotopy on all of $X$. Picture a movie playing on $A$ — frames changing over time — together with a still photograph on $X$ whose initial frame matches the start of the movie. Cofibration means the movie on $A$ can be extended to a full movie on $X$ that agrees with the still photograph at time zero.

This extension property is the homotopy extension property (HEP), and it captures exactly what it means for $A$ to sit inside $X$ "nicely" from a homotopy-theoretic viewpoint. A stupid embedding — one where $A$ is wedged into $X$ in some pathological way — fails HEP; the cofibration condition rules out such pathologies.

The standard examples are friendly. The boundary circle of a disc is a cofibration of the disc. Every closed inclusion of a CW complex into a larger CW complex is a cofibration. The mapping cylinder construction turns any continuous map into a cofibration without changing its homotopy type. These three families cover essentially all the inclusions you will meet in algebraic topology.

Visual [Beginner]

Two cylinders are drawn side by side. On the left, a thin strip $A$ inside a wider space $X$, with a homotopy on $A$ shown as a vertical sweep through time. On the right, the homotopy has been extended to a sweep on all of $X$, agreeing with the original at the bottom slice $X\times \{0\}$ and matching the strip data along $A\times [0,1]$.

The point of the picture is that the bottom face plus the side strip together determine the entire cylinder above $X$ when $A\hookrightarrow X$ is a cofibration.

Worked example [Beginner]

Take $A=\{0,1\}$ and $X=[0,1]$, the two endpoints sitting inside the unit interval. The inclusion $A\hookrightarrow X$ is a cofibration. Given a map $f:[0,1]\to \mathbb{R}$ with $f(0)=0$ and $f(1)=0$, and a homotopy $H:\{0,1\}\times [0,1]\to \mathbb{R}$ given by $H(0,t)=t$ and $H(1,t)=2t$, find an extension $\tilde{H}:[0,1]\times [0,1]\to \mathbb{R}$ with $\tilde{H}(\cdot ,0)=f$ and $\tilde{H}{|}_{\{0,1\}\times [0,1]}=H$.

A linear interpolation works. Set $\tilde{H}(x,t)=f(x)+t\cdot ((1-x)\cdot 1+x\cdot 2)=f(x)+t(1+x)$. Check the boundary conditions: $\tilde{H}(x,0)=f(x)$ matches; $\tilde{H}(0,t)=0+t=t$ matches $H(0,t)$; $\tilde{H}(1,t)=0+2t=2t$ matches $H(1,t)$. The takeaway: HEP is a concrete extension condition, and for tame inclusions the extension can be written down explicitly.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $i:A\to X$ be a continuous map of topological spaces.

Definition (homotopy extension property, HEP). $i$ has the HEP with respect to a space $Z$ if for every continuous map $f:X\to Z$ and every homotopy $H:A\times [0,1]\to Z$ with $H(a,0)=f(i(a))$ for all $a\in A$, there exists a homotopy $\tilde{H}:X\times [0,1]\to Z$ with $\tilde{H}(x,0)=f(x)$ for all $x\in X$ and $\tilde{H}(i(a),t)=H(a,t)$ for all $(a,t)\in A\times [0,1]$.

Definition (cofibration). $i:A\to X$ is a cofibration if it has the HEP with respect to every topological space $Z$.

The condition is most often used when $i$ is a closed embedding — and in fact, for cofibrations into Hausdorff spaces, the embedding is automatically closed. We adopt the convention that $A$ is regarded as a subspace of $X$ via $i$.

Equivalent characterisation (mapping-cylinder retract). $i:A\to X$ is a cofibration if and only if the inclusion $$ X \times {0} ,\cup_{A \times {0}}, A \times [0, 1] ;\hookrightarrow; X \times [0, 1] $$ admits a continuous retraction. The "L-shaped" subspace on the left collects the data of $f:X\to Z$ together with $H:A\times [0,1]\to Z$; a retraction $r:X\times [0,1]\to X\times \{0\}\cup A\times [0,1]$ lets one extend the L-shape to all of $X\times [0,1]$ by composing with $r$.

Standard examples.

• CW pair inclusion. For a CW complex $X$ with subcomplex $A$, the inclusion $A\hookrightarrow X$ is a cofibration. In particular, the skeletal inclusion $X^{(n-1)}\hookrightarrow X^{(n)}$ is a cofibration for every $n$.
• Mapping cylinder. For any continuous $f:X\to Y$, the inclusion of $X$ into the mapping cylinder $M_f=(X\times [0,1])\bigsqcup Y/(x,1)\sim f(x)$ as the bottom slice is a cofibration.
• Pushout product. The inclusion of the boundary $\partial D^n\hookrightarrow D^n$ is the prototypical cofibration; products and pushouts of these generate the cofibrations of CW topology.

A non-example: the inclusion $\mathbb{Q}\hookrightarrow \mathbb{R}$ is not a cofibration. Density of $\mathbb{Q}$ obstructs the existence of the retract demanded by the equivalent characterisation.

Key theorem with proof [Intermediate+]

Theorem (mapping-cylinder factorisation). Every continuous map $f:X\to Y$ factors as $$ X \xrightarrow{;j;} M_f \xrightarrow{;r;} Y $$ where $j:X\hookrightarrow M_f$ is a cofibration and $r:M_f\to Y$ is a homotopy equivalence with homotopy inverse the inclusion $Y\hookrightarrow M_f$ as the top slice.

Proof. Recall $M_f=(X\times [0,1])\bigsqcup Y/(x,1)\sim f(x)$, with the quotient topology ^{[Hatcher §0]}. Define $j(x)=[(x,0)]$ and $r$ by $r([(x,t)])=f(x)$ on the cylinder part and $r([y])=y$ on the $Y$ part; the relation $(x,1)\sim f(x)$ makes $r$ well-defined and continuous.

Step 1: $j$ is a cofibration. Construct an explicit retraction $$ \rho: M_f \times [0, 1] ;\to; M_f \times {0} ,\cup, X \times [0, 1] $$ where the right side identifies $X$ with $j(X)\subset M_f$. Send $(([(x,s)],t))\mapsto ([(x,\min (s,1-t))],\max (0,s+t-1))$ on the cylinder portion of $M_f\times [0,1]$, and $(([y],t))\mapsto ([y],0)$ on the $Y$ portion. This map is continuous because the two formulas agree on $X\times \{1\}\times [0,1]$ via the identification $(x,1)\sim f(x)$. It is a retraction because it fixes $M_f\times \{0\}$ and $j(X)\times [0,1]$. By the equivalent characterisation, $j$ has HEP, hence is a cofibration.

Step 2: $r$ is a homotopy equivalence. The composite $r\circ {\iota }_Y:Y\to Y$ where ${\iota }_Y(y)=[y]$ equals the identity. The composite ${\iota }_Y\circ r:M_f\to M_f$ sends $[(x,s)]\mapsto [f(x)]=[(x,1)]$ on the cylinder and is the identity on $Y$. The straight-line homotopy $H_u([(x,s)])=[(x,s+u(1-s))]$ for $u\in [0,1]$ deforms the identity of $M_f$ to ${\iota }_Y\circ r$, with $H_0={\mathrm{id}}_{M_f}$ and $H_1={\iota }_Y\circ r$, while restricting to the constant homotopy on $Y$. Hence ${\iota }_Y\circ r\simeq {\mathrm{id}}_{M_f}$, completing the equivalence. $\square$

Bridge. The mapping-cylinder factorisation builds toward the model-category structure on topological spaces: every map factors as a cofibration followed by a homotopy equivalence (or, dually, as a homotopy equivalence followed by a fibration). The foundational reason this works is exactly the symmetry that the equivalent characterisation makes visible — cofibrations are retract pairs, fibrations are lift pairs, and the two factorisations are dual constructions. The construction also appears again in 02.01.07 (fibration), where the path-space replacement plays the same role on the dual side: there the bridge is the fact that $\mathbf{Top}$ has both a left and a right factorisation system, which putting these together identifies $\mathbf{Top}$ with a model category in the sense of Quillen. The central insight is that cofibrations and fibrations are dual halves of the homotopy theory of $\mathbf{Top}$, and this is exactly what generalises to abstract homotopy theory in any model category. The same mapping-cylinder construction further generalises to the path-space construction in spectra, in chain complexes, and in any pointed model category.

Exercises [Intermediate+]

Advanced results [Master]

The HEP-based formulation of cofibration is the modern descendant of Borsuk's 1931 theory of absolute neighbourhood retracts. The key fact making the theory algebraically powerful is the duality with fibrations: cofibrations and fibrations are the two halves of the model-category structure on $\mathbf{Top}$, and every theorem on one side has a mirror on the other.

Theorem (Strom 1968). The following are equivalent for a closed inclusion $i:A\hookrightarrow X$ in a Hausdorff space $X$:

1. $i$ is a cofibration.
2. The pair $(X,A)$ has the homotopy extension property with respect to every space.
3. The L-shaped subspace $X\times \{0\}\cup A\times [0,1]\subseteq X\times [0,1]$ is a retract of $X\times [0,1]$.
4. There exist a continuous function $u:X\to [0,1]$ with $A\subseteq u^{-1}(0)$ and a homotopy $H:X\times [0,1]\to X$ with $H(\cdot ,0)={\mathrm{id}}_X$ and $H(a,t)=a$ for $a\in A$, satisfying $H(x,t)\in A$ whenever $u(x)<t$.

The fourth condition makes the cofibration structure visible as a "halo" function $u$ together with a homotopy that pushes the halo into $A$; it is the form most useful for explicit constructions on CW pairs.

Theorem (cofibration-fibration duality). Every continuous $f:X\to Y$ admits two factorisations:

$$X\stackrel{\text{cofibration}}{\to }{M}_f\stackrel{\text{homotopy equivalence}}{\to }Y$$ $$X\stackrel{\text{homotopy equivalence}}{\to }{P}_f\stackrel{\text{fibration}}{\to }Y$$

via the mapping cylinder $M_f$ and the path-space $P_f=\{(x,\gamma )\in X\times Y^{[0,1]}:\gamma (0)=f(x)\}$.

These two factorisations are the source of the model-category structure on $\mathbf{Top}$ in the sense of Quillen 1967. The cofibrant-fibrant replacement $X\to M_f\to Y\to P_{M_f\to Y}$ then becomes the standard derived-functor machinery for homotopy theory.

Theorem (cofibre sequence and Puppe). Let $i:A\hookrightarrow X$ be a cofibration. The mapping cone $C_i=X{\cup }_{i}CA$ is naturally homotopy equivalent to $X/A$, and the Puppe sequence $$ A \xrightarrow{i} X \to X/A \to \Sigma A \to \Sigma X \to \Sigma(X/A) \to \Sigma^2 A \to \cdots $$ induces a long exact sequence on $[-, Z]_$foranypointedspace$Z$.Incohomology,applying$H^(-; G)$producesthestandardlongexactsequenceofthepair$(X, A)$.

Generalisations.

• Closed model categories (Quillen 1967): cofibrations are abstractly defined by lifting properties dual to fibrations, and the topological case is one example among many. Chain complexes, simplicial sets, and spectra all carry analogous model structures.
• Cellular cofibrations (Hovey Model Categories 1999): the cofibrations of any cofibrantly-generated model category are retracts of transfinite compositions of pushouts of generating cofibrations. For $\mathbf{Top}$ the generators are the boundary inclusions $\partial D^n\hookrightarrow D^n$.
• $\infty$-categorical lift: in any presentable $\infty$-category, the homotopy-coherent generalisation of cofibrations is given by left lifting properties against acyclic fibrations, and the cofibre sequence becomes the cofibre construction in any pointed stable $\infty$-category.

Synthesis. Cofibrations and fibrations are dual halves of one structural device for organising the homotopy theory of $\mathbf{Top}$. The central insight is exactly the duality: every statement about HEP has a mirror about HLP, every cofibration factorisation has a fibration factorisation, every cofibre sequence has a fibre sequence. The mapping cylinder is dual to the path space; the suspension $\Sigma$ is dual to the loop space $\Omega$; the cofibre sequence is dual to the fibre sequence — the bridge is exactly Eckmann-Hilton. The foundational reason every map factors through a cofibration is that the mapping cylinder is the universal solution to that factorisation problem in $\mathbf{Top}$, and putting these together identifies the homotopy category of $\mathbf{Top}$ with the localisation at the weak equivalences of the Quillen model structure generated by the cofibration-fibration pair. This generalises directly to chain complexes, simplicial sets, and spectra; the central insight that a homotopy theory is exactly a model category appears again in every modern formulation of derived algebraic geometry and stable homotopy theory.

Full proof set [Master]

Proposition (HEP equivalent to retract). An inclusion $i:A\hookrightarrow X$ has the homotopy extension property with respect to every space $Z$ if and only if $X\times \{0\}\cup A\times [0,1]$ is a retract of $X\times [0,1]$.

Proof. ($\Rightarrow$) Apply HEP with $Z=X\times \{0\}\cup A\times [0,1]$, $f:X\to Z$ defined by $f(x)=(x,0)$, and $H:A\times [0,1]\to Z$ defined by $H(a,t)=(a,t)$. The hypothesis $H(a,0)=(a,0)=f(i(a))$ holds. HEP supplies $\tilde{H}:X\times [0,1]\to Z$ extending $H$ and starting at $f$; the map $\tilde{H}$ is a continuous retraction $X\times [0,1]\to Z$ because $\tilde{H}$ restricted to $Z\subseteq X\times [0,1]$ is the identity.

($\Leftarrow$) Given a retraction $r:X\times [0,1]\to X\times \{0\}\cup A\times [0,1]$ and the data $f:X\to Z$, $H:A\times [0,1]\to Z$ with $H(\cdot ,0)=f\circ i$, define $\tilde{H}:X\times [0,1]\to Z$ by composing with $r$: combine $f$ and $H$ into a single continuous map $g:X\times \{0\}\cup A\times [0,1]\to Z$ ($g(x,0)=f(x)$, $g(a,t)=H(a,t)$, well-defined because the two formulas agree at $A\times \{0\}$ via the hypothesis), then set $\tilde{H}=g\circ r$. This $\tilde{H}$ extends $f$ at $t=0$ and extends $H$ on $A\times [0,1]$. $\square$

Proposition (cofibrations are closed inclusions in Hausdorff spaces). Let $i:A\to X$ be a cofibration with $X$ Hausdorff. Then $i$ is a closed embedding.

Proof. By the equivalent characterisation, there is a retraction $r:X\times [0,1]\to X\times \{0\}\cup i(A)\times [0,1]$. The set $i(A)=\{x\in X:r(x,1)\in i(A)\times \{1\}\}$ is the preimage under the continuous map $x\mapsto r(x,1)$ of the closed set $i(A)\times \{1\}$ (closed because $X$ is Hausdorff and the diagonal-style construction lands in a closed slice). Hence $i(A)$ is closed in $X$. Injectivity of $i$ on $A$ follows because the retraction distinguishes points of $A$ via their distinct $t=1$ images. $\square$

Proposition (CW pair inclusion is a cofibration). If $X$ is a CW complex and $A\subseteq X$ is a subcomplex, then $A\hookrightarrow X$ is a cofibration.

Proof sketch. Induct on the cells of $X\setminus A$ in increasing dimension. The base case is $X=A$, where the inclusion is the identity. For the inductive step, attach a single $n$-cell along $\phi :\partial D^n\to A$. The inclusion $\partial D^n\hookrightarrow D^n$ is a cofibration (Exercise 1), so its pushout along $\phi$, which is $A\hookrightarrow A{\cup }_{\phi }{D}^n$, is a cofibration by stability of cofibrations under pushout. Iterating over all cells and passing to the colimit (which preserves cofibrations in the CW topology) yields the claim. $\square$

Theorem (mapping cone of a cofibration is the strict quotient up to homotopy). If $i:A\hookrightarrow X$ is a cofibration, the natural map $C_i=X{\cup }_{i}CA\to X/A$ collapsing the cone is a homotopy equivalence.

Proof sketch. The cone $CA$ is contractible to its apex $a_{\ast }$. The collapse map sends $CA$ to a point, identifying $C_i\to X/A$. By Exercise 3 applied to the cofibration $A\hookrightarrow C_i$ with $A$ contractible inside $CA$, the collapse $C_i\to C_i/CA\cong X/A$ is a homotopy equivalence. $\square$

Connections [Master]

• Fibration 02.01.07. The Eckmann-Hilton dual of cofibration. Where cofibrations are characterised by the homotopy extension property and arise as left halves of factorisations, fibrations are characterised by the homotopy lifting property and arise as right halves. The two together give the Quillen model structure on $\mathbf{Top}$.

• Quotient and identification topology 02.01.06. The mapping cylinder, mapping cone, and quotient $X/A$ are all identification-topology constructions, and the proof that $i:A\hookrightarrow X$ being a cofibration makes $X/A$ homotopy-equivalent to the mapping cone uses the universal property of the quotient.

• Homotopy and homotopy group 03.12.01. The HEP definition uses homotopies directly, and cofibrations are exactly the inclusions for which the long exact sequence of the pair $(X,A)$ extends to a Puppe sequence on homotopy groups via the suspension.

• Covering space 03.12.02. Covering maps are fibrations with discrete fibre, dual to the cellular cofibrations $\partial D^n\hookrightarrow D^n$ in the model-categorical picture; the lifting properties on the two sides match up under Eckmann-Hilton.

• Eilenberg-MacLane space 03.12.05. The Postnikov-tower construction uses cofibration-fibration factorisations at every stage to convert a space into its tower of $K({\pi }_n,n)$ pieces; the cofibre sequence $K({\pi }_n,n)\to X^{(n)}\to X^{(n-1)}$ is the dual of the fibration sequence visible there.

Historical & philosophical context [Master]

The theory of cofibrations grew out of Karol Borsuk's 1931 paper Sur les rétractes (Fund. Math. 17, 152--170) ^{[Borsuk 1931]}, which introduced the absolute neighbourhood retracts (ANRs) and proved the first homotopy-extension theorems for compact metric spaces. Borsuk's framing was geometric: an ANR is a space that is a retract of every nice neighbourhood of an embedding into Euclidean space. Norman Steenrod's The Topology of Fibre Bundles (Princeton 1951) ^{[Steenrod 1951]} codified the cofibre-side of the theory in the bundle-theoretic language of the day.

The modern HEP-based axiomatisation crystallised in the 1960s. Steenrod's 1967 paper A convenient category of topological spaces ^{[Steenrod 1967]} gave the cartesian-closed framework in which HEP behaves cleanly with function spaces. Arne Strom's 1968 paper Note on cofibrations II (Math. Scand. 22, 130--142) ^{[Strom 1968]} proved the equivalence of the HEP characterisation, the L-shaped retract characterisation, and the halo-function characterisation that bears his name. J. Peter May's A Concise Course in Algebraic Topology (1999) ^{[May 1999]} is the canonical pedagogical exposition.

The unification with fibrations under Eckmann-Hilton duality was made explicit by Daniel Quillen's 1967 Homotopical Algebra (Lecture Notes in Mathematics 43) ^{[Quillen 1967]}, which axiomatised the cofibration-fibration pair as a model category and proved that the homotopy category of any model category is the universal localisation at the weak equivalences. The $\infty$-categorical incarnation in the work of Joyal, Lurie, and others through the 2000s shows the same cofibration-fibration pair structuring every presentable $\infty$-category.

Bibliography [Master]

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