Paths and cylinders
Given a continuous map of topological spaces, one can define its mapping cylinder as a pushout
in Top, where (the unit interval) and is given by . By tradition, homotopy theorists sometimes use the inverted (upside-down) mapping cylinder where is replaced by . Of course the two mapping cylinders are homeomorphic so it is matter of convention which one to use, of course, compatibly with other constructions depending on the orientation of .
Set-theoretically, the mapping cylinder is usually represented as the quotient space where is the smallest equivalence relation identifying for all .
As any other pushout, the mapping cylinder has a universal property: for any space and mapping , such that for all , there is a unique , such that the composition equals and the composition equals .
For a continuous function, the canonical map is a homotopy equivalence. In fact its homotopy inverse can be chosen a deformation retraction. In particular every continuous function factors as a map into its mapping cylinder followed by a deformation retraction.
We exhibit as a homotopy equivalence by constructing its homotopy inverse given by , where is a class of and for . Clearly this map is well-defined and . On the other hand, . Homotopy is given by
It is easy to see that , hence .
A continuous map is a Hurewicz cofibration iff there is a retraction for the canonical map .
For any , the composition
is a Hurewicz cofibration. Furthermore, the map determined by (for all and ) and (for ) is well defined and a homotopy equivalence.
The composition , hence this is a decomposition of a continuous map into a cofibration followed by a homotopy equivalence.
In homotopy type theory mapping cyclinders can be constructed as higher inductive types. See there.