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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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.
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.|