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

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

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

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

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

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

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

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

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.