nLab shrinkable map

shrinkable map $\Rightarrow$ Dold fibration

Definition

A shrinkable map $p:X \to Y$ in the category Top is a map with a section $s:Y \to X$ such that $s\circ p:X \to X$ is vertically homotopic to $id_X$ (i.e. is homotopic to $\id_X$ in the slice category $Top/Y$). In particular, a shrinkable map is a homotopy equivalence.

Properties

Every shrinkable map is a Dold fibration. This result follows from a theorem of Dold about locally homotopy trivial map?s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.

Examples

Example:(Segal) Let $U_i \to Y$ be a numerable open cover. Then the geometric realization of the Cech nerve $N\check{C}(U_i)$ comes with a canonical map $|N\check{C}(U_i)| \to Y$ which is shrinkable.

There are extensions of this to other categories with a notion of homotopy.

References

This definition is due to Dold, in his 1963 Annals paper Partitions of unity in the theory of fibrations.