Mapping cylinder

Summary

The mapping cylinder of a continuous map $f \colon X \to Y$ is the space obtained from the cylinder $X \times [0, 1]$ by gluing its end $X \times \set{1}$ to $Y$ along $f$. It contains $Y$ as a deformation retract and contains $X$ as the other end. So every map is, up to homotopy equivalence, the inclusion of a subspace. The construction is used to build mapping cones, cofibrations and long exact sequences.