Homotopy lifting property
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.
Assume from now on all maps are continuous functions from one topological space to another. Given a map , and a space , one says that has the homotopy lifting property, or that has the homotopy lifting property with respect to , if:
- for any homotopy , and
- for any map lifting (i.e., so that ),
there exists a homotopy lifting (i.e., so that ) which also satisfies .
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 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 satisfies the homotopy lifting property with respect to all spaces X, then is called a fibration, or one sometimes simply says that has the homotopy lifting property.
Note that this is the definition of fibration in the sense of Witold Hurewicz, which is more restrictive than fibration in the sense of Jean-Pierre Serre, for which homotopy lifting only for a CW complex is required.
Generalization: homotopy lifting extension property
There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces , for simplicity we denote . Given additionally a map , one says that has the homotopy lifting extension property if:
- For any homotopy , and
- For any lifting of ,
there exists a homotopy which covers (i.e., such that ) and extends (i.e., such that ).
The homotopy lifting property of is obtained by taking , so that above is simply .
The homotopy extension property of is obtained by taking to be a constant map, so that is irrelevant in that every map to E is trivially the lift of a constant map to the image point of .
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