nLab
shrinkable map

shrinkable map \Rightarrow Dold fibration

Contents

Definition

A shrinkable map p:XYp:X \to Y in the category Top is a map with a section s:YXs:Y \to X such that sp:XXs\circ p:X \to X is vertically homotopic to id Xid_X (i.e. is homotopic to id X\id_X in the slice category Top/YTop/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 iYU_i \to Y be a numerable open cover. Then the geometric realization of the Cech nerve NCˇ(U i)N\check{C}(U_i) comes with a canonical map |NCˇ(U i)|Y|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.

Revised on August 16, 2010 07:24:37 by David Roberts (203.24.207.134)