##  [Mapping Cylinder](/mapping-cylinder-0) 

 Definition

Given a continuous map f: X→Y, the mapping cylinder Mf is the space obtained by taking X×[0,1] disjoint union Y and identifying each (x,1)∈X×{1} with f(x)∈Y; equivalently it is X×[0,1] ⊔_f Y. It provides a model for a homotopy between f and a cofibration inclusion of X into Mf.

 

 

 

 

 

 





## Principle

Principle

Attach a cylinder on X to Y along f at one end so that X embeds as a subspace (via x↦(x,0)) and f factors as X → Mf → Y with the second map a deformation retraction onto Y under mild hypotheses; mapping cylinders facilitate factorizations of maps into cofibrations followed by homotopy equivalences.

 

 

 

 

 





## Demonstration

Demonstration

For the inclusion i: A→X, the mapping cylinder Mi is homeomorphic to the union of X with A×[0,1] glued along A×{1}, producing a space in which A is embedded with a cylinder collar; for any f that is a homotopy equivalence, Mf deformation retracts onto Y.

 

 

 

 

## Misapplication

Misapplication

Using the mapping cylinder without checking topological hypotheses (e.g. non-Hausdorff or ill-behaved maps) or confusing Mf with the mapping cone (collapsing the wrong slice) can mislead about cofibration properties and homotopy cofibre sequences.

 

 

 

 

 





## Consequence

Consequence

The mapping cylinder gives a canonical cofibration X→Mf and a projection Mf→Y homotopic to the original map f; it is a basic tool for constructing homotopies, proving factorization theorems, and building relative cell attachments.

 

 

 

 

## Reversal

Reversal

Collapsing X×{0} in Mf yields the mapping cone, which encodes homotopy cofiber information; reversing the cylinder construction by removing the cylinder collar recovers the original map only up to homotopy, not generally by homeomorphism.

 

 

 

 

 





## Boundary

Boundary

Precise homotopical statements require working in convenient categories (compactly generated, Hausdorff, CW complexes) or adding basepoint data for pointed versions; Mf depends on the specific map f and is not functorial in Y alone.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Mapping cylinder is often conflated with product cylinder or mapping cone: the cylinder of f attaches X×[0,1] to Y along {1}, whereas the mapping cone further collapses X×{0}; these differences determine whether one constructs a cofibration or a cofiber.

 

 

 

 

 





## Synthesis

Synthesis

The mapping cylinder of f: X→Y is X×[0,1] attached to Y via f at the top slice; it factors f as a cofibration followed by a map homotopic to f and serves as a concrete device to produce collars, homotopies, and cofibration factorizations.