Homotopy lifting property

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 ${\displaystyle \pi :E\to B}$, and a space ${\displaystyle Y}$, one says that ${\displaystyle (Y,\pi )}$ has the homotopy lifting property, or that ${\displaystyle \pi }$ has the homotopy lifting property with respect to ${\displaystyle Y}$, if:

• for any homotopy ${\displaystyle f_{\bullet }:Y\times I\to B}$, and
• for any map ${\displaystyle {\tilde{f}}_0:Y\to E}$ lifting ${\displaystyle f_0=f_{\bullet }{|}_{Y\times \{0\}}}$ (i.e., so that ${\displaystyle f_{\bullet }\circ {\iota }_0=f_0=\pi \circ {\tilde{f}}_0}$),

there exists a homotopy ${\displaystyle {\tilde{f}}_{\bullet }:Y\times I\to E}$ lifting ${\displaystyle f_{\bullet }}$ (i.e., so that ${\displaystyle f_{\bullet }=\pi \circ {\tilde{f}}_{\bullet }}$) which also satisfies ${\displaystyle {\tilde{f}}_0={\tilde{f}|}_{Y\times \{0\}}}$.

The following diagram depicts this situation:

The outer square (without the dotted arrow) commutes if and only if the hypotheses of the lifting property are true. A lifting ${\displaystyle {\tilde{f}}_{\bullet }}$ 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 ${\displaystyle \pi }$ satisfies the homotopy lifting property with respect to all spaces ${\displaystyle Y}$, then ${\displaystyle \pi }$ is called a fibration, or one sometimes simply says that ${\displaystyle \pi }$ has the homotopy lifting property.

A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes ${\displaystyle Y}$.