Cofibration

In mathematics, in particular homotopy theory, a continuous mapping

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

where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. 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.

A more general notion of cofibration is developed in the theory of model categories.

Basic theorems

• 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.

Examples

• Cofibrations are preserved under pushouts and composition, as one sees from the definition via diagram-chasing.
• 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.

Discussion

The homotopy colimit generalizes the notion of a cofibration.

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.