# Cofibration

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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*. 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.

## Basic theorems

- 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.

## Examples

- 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 .

## References

- Peter May, "A Concise Course in Algebraic Topology" : chapter 6 defines and discusses cofibrations, and they are used throughout