Homotopy lifting property

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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) 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 to do with the fact that 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.

[edit] Formal definition

Assume from now on all mappings are continuous functions from a topological space to another. One says that a map

has the homotopy lifting property with respect to a space X if for any homotopy and any map lifting (i.e. so that ) there exists a homotopy lifting f (so that ) with .

If a map satisfies the homotopy lifting property with respect to all spaces X, one sometimes simply says that it satisfies the homotopy lifting property. Such a map is called a fibration. This is the definition of fibration in the sense of Hurewicz, which is more restrictive than the fibration in the sense of Serre, for which homotopy lifting only for X a CW complex is required.

[edit] Generalizations

There is also the more general concept of the homotopy lifting property with respect to a pair (X, Y). Here one requires that given a homotopy

X × [0,1] → B,

a lift of that map on X × 0, and a lift on Y × [0,1] such that the two lifts agree on Y × 0, the lift can be extended to a lift of the homotopy. The homotopy lifting property is obtained by taking Y = ø.