Cofibration

In mathematics, in particular homotopy theory, a continuous mapping

${\displaystyle i:A\to X}$,

where ${\displaystyle A}$ and ${\displaystyle X}$ are topological spaces, is a cofibration if it lets homotopy classes of maps ${\displaystyle [A,S]}$ be extended to homotopy classes of maps ${\displaystyle [X,S]}$ whenever a map ${\displaystyle f\in {\text{Hom}}_{\textbf {Top}}(A,S)}$ can be extended to a map ${\displaystyle f'\in {\text{Hom}}_{\textbf {Top}}(X,S)}$ where ${\displaystyle f'\circ i=f}$, hence their associated homotopy classes are equal ${\displaystyle [f]=[f'\circ i]}$.

This type of structure can be encoded with the technical condition of having the homotopy extension property with respect to all spaces ${\displaystyle S}$. This definition is dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces. This duality is informally referred to as Eckmann–Hilton duality. Because of the generality this technical condition is stated, it can be used in model categories.

Definition[]

Homotopy theory[]

In what follows, let ${\displaystyle I=[0,1]}$ denote the unit interval.

A map ${\displaystyle i\colon A\to X}$ of topological spaces is called a cofibration^{[1]pg 51} if for any map ${\displaystyle f:A\to S}$ such that there is an extension to ${\displaystyle X}$, meaning there is a map ${\displaystyle f':X\to S}$ such that ${\displaystyle f'\circ i=f}$, we can extend a homotopy of maps ${\displaystyle H:A\times I\to S}$ to a homotopy of maps ${\displaystyle H':X\times I\to S}$, where

${\displaystyle {\begin{aligned}H(a,0)&=f(a)\\H'(x,0)&=f'(x)\end{aligned}}}$

We can encode this condition in the following commutative diagram

where ${\displaystyle S^{I}={\text{Hom}}_{\textbf {Top}}(I,S)}$ is the path space of ${\displaystyle S}$.

Cofibrant objects[]

For a model category ${\displaystyle {\mathcal {M}}}$, such as for pointed topological spaces, an object ${\displaystyle X}$ is called cofibrant if the map ${\displaystyle *\to X}$ is a cofibration. Note that in the category of pointed topological spaces, the notion of cofibration coincides with the previous definition assuming the maps are pointed maps of topological spaces.

Examples[]

In topology[]

Cofibrations are an awkward class of maps from a computational perspective because they are more easily seen as a formal technical tool which enables one to "do" homotopy theoretic constructions with topological spaces. Fortunately, for any map

${\displaystyle f:X\to Y}$

of topological spaces, there is an associated cofibration to a space ${\displaystyle Mf}$ called the mapping cylinder (where ${\displaystyle Y}$ is a deformation retract of, hence homotopy equivalent to it) which has an induced cofibration called replacing a map with a cofibration

${\displaystyle i:X\to Mf}$

and a map ${\displaystyle Mf\to Y}$ through which ${\displaystyle f}$ factors through, meaning there is a commutative diagram

where ${\displaystyle r}$ is a homotopy equivalence.

In addition to this class of examples, there are

• A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if ${\displaystyle (X,A)}$ is a CW pair, then ${\displaystyle A\to X}$ is a cofibration). This follows from the previous fact since ${\displaystyle S^{n-1}\to D^{n}}$ is a cofibration for every ${\displaystyle n}$, and pushouts are the gluing maps to the ${\displaystyle n-1}$ skeleton.
• Cofibrations are preserved under pushouts and composition, which is stated precisely below.

In chain complexes[]

If we let ${\displaystyle C_{+}({\mathcal {A}})}$ be the category of chain complexes which are ${\displaystyle 0}$ in degrees ${\displaystyle q<<0}$, then there is a model category structure^{[2]pg 1.2} where the weak equivalences are quasi-isomorphisms, so maps of chain complexes which are isomorphisms after taking cohomology, fibrations are just epimorphisms, and cofibrations are given by maps

${\displaystyle i:C_{\bullet }\to D_{\bullet }}$

which are injective and the cokernel complex ${\displaystyle {\text{Coker}}(i)_{\bullet }}$ is a complex of projective objects in ${\displaystyle {\mathcal {A}}}$. In addition, the cofibrant objects are the complexes whose objects are all projective objects in ${\displaystyle {\mathcal {A}}}$.

Semi-simplicial sets[]

For the category ${\displaystyle ss{\textbf {Set}}}$ of semi-simplicial sets^{[2]pg 1.3} (meaning there are no co-degeneracy maps going up in degree), there is a model category structure with fibrations given by Kan-fibrations, cofibrations injective maps, and weak equivalences given by weak equivalences after geometric realization.

Properties[]

• For Hausdorff spaces, every cofibration is a closed inclusion (injective with closed image); the result also generalizes to weak Hausdorff spaces.
• The pushout of a cofibration is a cofibration. That is, if ${\displaystyle g\colon A\to B}$ is any (continuous) map (between compactly generated spaces), and ${\displaystyle i\colon A\to X}$ is a cofibration, then the induced map ${\displaystyle B\to B\cup _{g}X}$ is a cofibration.
• The mapping cylinder can be understood as the pushout of ${\displaystyle i\colon A\to X}$ and the embedding (at one end of the unit interval) ${\displaystyle i_{0}\colon A\to A\times I}$. That is, the mapping cylinder can be defined as ${\displaystyle Mi=X\cup _{i}(A\times I)}$. By the universal property of the pushout, ${\displaystyle i}$ is a cofibration precisely when a mapping cylinder can be constructed for every space X.
• Every map can be replaced by a cofibration via the mapping cylinder construction. That is, given an arbitrary (continuous) map ${\displaystyle f\colon X\to Y}$ (between compactly generated spaces), one defines the mapping cylinder

${\displaystyle Mf=Y\cup _{f}(X\times I)}$.

One then decomposes ${\displaystyle f}$ into the composite of a cofibration and a homotopy equivalence. That is, ${\displaystyle f}$ can be written as the map

${\displaystyle X{\xrightarrow {j}}Mf{\xrightarrow {r}}Y}$

with ${\displaystyle f=rj}$, when ${\displaystyle j\colon x\mapsto (x,0)}$ is the inclusion, and ${\displaystyle r\colon y\mapsto y}$ on ${\displaystyle Y}$ and ${\displaystyle r\colon (x,s)\mapsto f(x)}$ on ${\displaystyle X\times I}$.

• There is a cofibration (A, X), if and only if there is a retraction from ${\displaystyle X\times I}$ to ${\displaystyle (A\times I)\cup (X\times \{0\})}$, since this is the pushout and thus induces maps to every space sensible in the diagram.
• Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.

Constructions with cofibrations[]

Cofibrant replacement[]

Note that in a model category ${\displaystyle {\mathcal {M}}}$ if ${\displaystyle i:*\to X}$ is not a cofibration, then the mapping cylinder ${\displaystyle Mi}$ forms a cofibrant replacement. In fact, if we work in just the category of topological spaces, the cofibrant replacement for any map from a point to a space forms a cofibrant replacement.

Cofiber[]

For a cofibration ${\displaystyle A\to X}$ we define the cofiber to be the induced quotient space ${\displaystyle X/A}$. In general, for ${\displaystyle f:X\to Y}$, the cofiber^{[1]pg 59} is defined as the quotient space

${\displaystyle C_{f}=M_{f}/(A\times \{0\})}$

which is the mapping cone of ${\displaystyle f}$. Homotopically, the cofiber acts as a homotopy cokernel of the map ${\displaystyle f:X\to Y}$. In fact, for pointed topological spaces, the homotopy colimit of

${\displaystyle {\underset {\to }{\text{hocolim}}}\left({\begin{matrix}X&\xrightarrow {f} &Y\\\downarrow &&\\*\end{matrix}}\right)=C_{f}}$

In fact, the sequence of maps ${\displaystyle X\to Y\to C_{f}}$ comes equipped with the which acts like a distinguished triangle in triangulated categories.

See also[]

• Fibration
• Homotopy colimit
• Homotopy fiber

References[]

• Peter May, "A Concise Course in Algebraic Topology" : chapter 6 defines and discusses cofibrations, and they are used throughout
• Ronald Brown, "Topology and Groupoids" ; Chapter 7 is entitled "Cofibrations", and has many results not found elsewhere.