Mapping cone (topology)

In mathematics, especially homotopy theory, the mapping cone is a construction ${\displaystyle C_{f}}$ of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated ${\displaystyle Cf}$. Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder ${\displaystyle Mf}$, with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces.