Cofibration

Basic theorems

• For Hausdorff spaces, every cofibration is a closed inclusion ; the result also generalizes to weak Hausdorff spaces.
• The pushout of a cofibration is a cofibration. That is, if is any map, and is a cofibration, then the induced map is a cofibration.
• The mapping cylinder can be understood as the pushout of and the embedding . 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 map , one defines the mapping cylinder
• There is a cofibration, 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.

  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. This follows from the previous fact since is a cofibration for every, and pushouts are the gluing maps to the skeleton.

  Discussion