Mapping cylinder

Definition

Let $f:X\to Y$ be a function. Then the mapping cylinder of $f$ is defined as the quotient of the disjoint union of $X\times I$ with $Y$, modulo the equivalence relation:

$(x,1)\sim f(x)$

Here, $I=[0,1]$ is the unit interval.

Facts

The significance of the mapping cylinder is that it is homotopy-equivalent to $Y$, and moreover the inclusion of $X$ (say via $x\mapsto (x,0)$) in the mapping cylinder is equivalent to the map $f$.

Thus, starting from an arbitrary continuous map, we have got a homotopy-equivalent map which is an inclusion.

Further, the inclusion of $X$ in the mapping cylinder is a cofibration, which makes it even nicer.

Relation with other constructions

More general constructions

Name of construction	Description of construction	How the mapping cylinder is a special case
double mapping cylinder	spaces $X,Y,Z$, with continuous maps from $X$ to $Y$ and $X$ to $Z$, we take $(X\times I)\bigsqcup Y\bigsqcup Z$ and collapse $X\times \{0\}$ and $X\times \{1\}$ onto $Z$ and $Y$ via the continuous maps	Case where $X=Z$ and the map $X\to Z$ is the identity map.

More specific constructions

Name of construction	How it arises as a special case
cone space	Set $Y$ as a one-point space and $f:X\to Y$ as the map sending everything to one point.