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.

Given a map ${\displaystyle f\colon X\to Y}$, the mapping cone ${\displaystyle C_{f}}$ is defined to be the quotient space of the mapping cylinder ${\displaystyle (X\times I)\sqcup _{f}Y}$ with respect to the equivalence relation ${\displaystyle \forall x,x'\in X,(x,0)\sim \left(x',0\right)\,}$, ${\displaystyle (x,1)\sim f(x)}$. Here ${\displaystyle I}$ denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1.

Visually, one takes the cone on X (the cylinder ${\displaystyle X\times I}$ with one end (the 0 end) collapsed to a point), and glues the other end onto Y via the map f (the 1 end).

The above is the definition for a map of unpointed spaces; for a map of pointed spaces ${\displaystyle f\colon (X,x_{0})\to (Y,y_{0})}$ (so ${\displaystyle f\colon x_{0}\mapsto y_{0}}$), one also identifies all of ${\displaystyle x_{0}\times I}$. Formally, ${\displaystyle (x_{0},t)\sim \left(x_{0},t'\right)}$. Thus one end and the "seam" are all identified with ${\displaystyle y_{0}.}$

If ${\displaystyle X}$ is the circle ${\displaystyle S^{1}}$, the mapping cone ${\displaystyle C_{f}}$ can be considered as the quotient space of the disjoint union of Y with the disk ${\displaystyle D^{2}}$ formed by identifying each point x on the boundary of ${\displaystyle D^{2}}$ to the point ${\displaystyle f(x)}$ in Y.

Consider, for example, the case where Y is the disk ${\displaystyle D^{2}}$, and ${\displaystyle f\colon S^{1}\to Y=D^{2}}$ is the standard inclusion of the circle ${\displaystyle S^{1}}$ as the boundary of ${\displaystyle D^{2}}$. Then the mapping cone ${\displaystyle C_{f}}$ is homeomorphic to two disks joined on their boundary, which is topologically the sphere ${\displaystyle S^{2}}$.

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder ${\displaystyle X\times I}$ joined on one end to a space ${\displaystyle Y_{1}}$ via a map

and joined on the other end to a space ${\displaystyle Y_{2}}$ via a map