# Mapping cylinder

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
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## 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 $(X \times I) \sqcup Y \sqcup Z$ 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.