Cofibration

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, a 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 $g\colon A\to B$ is any (continuous) map (between compactly generated spaces), and $i\colon A\to X$ is a cofibration, then the induced map $B\to B\cup _{g}X$ is a cofibration.

The mapping cylinder can be understood as the pushout of $i\colon A\to X$ and the embedding (at one end of the unit interval) $i_{0}\colon A\to A\times I$ . That is, the mapping cylinder can be defined as $Mi=X\cup _{i}(A\times I)$ . By the universal property of the pushout, $i$ 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 $f\colon X\to Y$ (between compactly generated spaces), one defines the mapping cylinder

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

One then decomposes

f

into the composite of a cofibration and a homotopy equivalence. That is,

f

can be written as the map

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

with

f=rj

, when

j\colon x\mapsto (x,0)

is the inclusion, and

r\colon y\mapsto y

Y

and

r\colon (x,s)\mapsto f(x)

X\times I

There is a cofibration (A, X), if and only if there is a retraction from $X\times I$ to $(A\times I)\cup (X\times \{0\})$ , 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.

Cofibrations are preserved under push-outs 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 $(X,A)$ is a CW pair, then $A\to X$ is a cofibration). This follows from the previous fact since $S^{n-1}\to D^{n}$ is a cofibration for every $n$ , and pushouts are the gluing maps to the $n-1$ skeleton.

The homotopy colimit generalizes the notion of a cofibration.

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.

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