shrinkable map Dold fibration
A shrinkable map in the category Top is a map with a section such that is vertically homotopic to (i.e. is homotopic to in the slice category ). In particular, a shrinkable map is a homotopy equivalence.
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.
There are extensions of this to other categories with a notion of homotopy.
This definition is due to Dold, in his 1963 Annals paper Partitions of unity in the theory of fibrations.