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In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as 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.
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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 from now on all mappings are continuous functions from a topological space to another. Given a map
there exists a homotopy
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
If the map
Note that this is the definition of fibration in the sense of Hurewicz, which is more restrictive than fibration in the sense of Serre, for which homotopy lifting only for
Generalization: The Homotopy Lifting Extension Property
There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces
there exists a homotopy
The homotopy lifting property of
The homotopy extension property of