In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) 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 π : E → B {\displaystyle \pi \colon E\to B} , and a space Y {\displaystyle Y\,} , one says that ( Y , π ) {\displaystyle (Y,\pi )} has the homotopy lifting property, or that π {\displaystyle \pi \,} has the homotopy lifting property with respect to Y {\displaystyle Y} , if:
for any homotopy f ∙ : Y × I → B {\displaystyle f_{\bullet }\colon Y\times I\to B} , and for any map f ~ 0 : Y → E {\displaystyle {\tilde {f}}_{0}\colon Y\to E} lifting f 0 = f ∙ | Y × { 0 } {\displaystyle f_{0}=f_{\bullet }|_{Y\times \{0\}}} (i.e., so that f ∙ ∘ ι 0 = f 0 = π ∘ f ~ 0 {\displaystyle f_{\bullet }\circ \iota _{0}=f_{0}=\pi \circ {\tilde {f}}_{0}} ), there exists a homotopy f ~ ∙ : Y × I → E {\displaystyle {\tilde {f}}_{\bullet }\colon Y\times I\to E} lifting f ∙ {\displaystyle f_{\bullet }} (i.e., so that f ∙ = π ∘ f ~ ∙ {\displaystyle f_{\bullet }=\pi \circ {\tilde {f}}_{\bullet }} ) which also satisfies f ~ 0 = f ~ | Y × { 0 } {\displaystyle {\tilde {f}}_{0}=\left.{\tilde {f}}\right|_{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 f ~ ∙ {\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 π {\displaystyle \pi } satisfies the homotopy lifting property with respect to all spaces Y {\displaystyle Y} , then π {\displaystyle \pi } is called a fibration, or one sometimes simply says that π {\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 Y {\displaystyle Y} .
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