From Wikipedia, the free encyclopedia
  (Redirected from Closed inclusion)
Jump to: navigation, search

In mathematics, in particular homotopy theory, a continuous mapping


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[edit]

  • 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 is any (continuous) map (between compactly generated spaces), and is a cofibration, then the induced map is a cofibration.
  • The mapping cylinder can be understood as the pushout of and the embedding (at one end of the unit interval) . That is, the mapping cylinder can be defined as . By the universal property of the pushout, 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 (between compactly generated spaces), one defines the mapping cylinder
One then decomposes into the composite of a cofibration and a homotopy equivalence. That is, can be written as the map
with , when is the inclusion, and on and on .
  • There is a cofibration (A, X), if and only if there is a retraction from to , 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 is a CW pair, then is a cofibration). This follows from the previous fact since is a cofibration for every , and pushouts are the gluing maps to the skeleton.


The homotopy colimit generalizes the notion of a cofibration.