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 $(X \times I) ⊔ Y ⊔ Z$ 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.