Mapping Cylinder - Interpretation

Interpretation

The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense:

Given a map, the mapping cylinder is a space, together with a cofibration and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f.

Thus the space Y gets replaced with a homotopy equivalent space, and the map f with a lifted map . Equivalently, the diagram

gets replaced with a diagram

together with a homotopy equivalence between them.

The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.

Note that pointwise, a cofibration is a closed inclusion.