Mapping cone (topology)

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