In mathematics, in particular homotopy theory, a continuous mapping
i : A → 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.
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 g : A → B is any (continuous) map (between compactly generated spaces), and i : A → X is a cofibration, then the induced map B → B ∪ g X is a cofibration.The mapping cylinder can be understood as the pushout of i : A → X and the embedding (at one end of the unit interval) i 0 : A → A × I . That is, the mapping cylinder can be defined as M i = X ∪ i ( A × 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 : X → Y (between compactly generated spaces), one defines the mapping cylinderOne then decomposes
f into the composite of a cofibration and a homotopy equivalence. That is,
f can be written as the map
X → j M f → r Y with
f = r j , when
j : x ↦ ( x , 0 ) is the inclusion, and
r : y ↦ y on
Y and
r : ( x , s ) ↦ f ( x ) on
X × I .
There is a cofibration (A, X), if and only if there is a retraction from X × I to ( A × I ) ∪ ( X × { 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 → X is a cofibration). This follows from the previous fact since S n − 1 → 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.
