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 X to another one, Y. It is designed to support the picture of X 'above' Y, by allowing a homotopy taking place in Y to be moved 'upstairs' to X. 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, 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. One says that
- p: A → B
has the homotopy lifting property with respect to a space X if for any homotopy
- g: X × [0,1] → B
and map
- h: X → A
such that
- p o h = g|X × 0
there is a homotopy
- f : X × [0,1] → A
such that
- p o f = g
and f|X × 0 equals h.
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.
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.