where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines fibrations. For a more general notion of cofibration see the article about model categories.
- For Hausdorff spaces a cofibration is a closed inclusion (injective with closed image); for suitable spaces, a converse holds
- Every map can be replaced by a cofibration via the mapping cylinder construction
- There is a cofibration (A, X), if and only if there is a retraction from
since this is the pushout and thus induces maps to every space sensible in the diagram.
- 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 .
- Peter May, "A Concise Course in Algebraic Topology" : chapter 6 defines and discusses cofibrations, and they are used throughout