Mapping cylinder

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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.