Homotopy lifting property


In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E.
For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.

Formal definition

Assume all maps are continuous functions between topological spaces. Given a map, and a space, one says that has the homotopy lifting property, or that has the homotopy lifting property with respect to, if:
  • for any homotopy, and
  • for any map lifting ,
there exists a homotopy lifting which also satisfies.
The following diagram depicts this situation:
The outer square commutes if and only if the hypotheses of the lifting property are true. A lifting corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the homotopy extension property; this duality is loosely referred to as Eckmann–Hilton duality.
If the map satisfies the homotopy lifting property with respect to all spaces, then is called a fibration, or one sometimes simply says that has the homotopy lifting property.
A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes.

Generalization: homotopy lifting extension property

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces, for simplicity we denote. Given additionally a map, one says that has the homotopy lifting extension property if:
  • For any homotopy, and
  • For any lifting of, there exists a homotopy which covers and extends .
The homotopy lifting property of is obtained by taking, so that above is simply.
The homotopy extension property of is obtained by taking to be a constant map, so that is irrelevant in that every map to E is trivially the lift of a constant map to the image point of.