# Mapping cylinder

From Topospaces

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.

View more such constructs

## Contents

## Definition

Let be a function. Then the **mapping cylinder** of is defined as the quotient of the disjoint union of with , modulo the equivalence relation:

Here, is the unit interval.

## Facts

The significance of the mapping cylinder is that it is homotopy-equivalent to , and moreover the inclusion of (say via ) in the mapping cylinder is equivalent to the map .

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

Further, the inclusion of 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 , with continuous maps from to and to , we take and collapse and onto and via the continuous maps | Case where and the map is the identity map. |

### More specific constructions

Name of construction | How it arises as a special case |
---|---|

cone space | Set as a one-point space and as the map sending everything to one point. |