- Homotopy lifting property
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. 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.
Assume from now on all mappings are continuous functions from a topological space to another. Given a map , and a space X, one says that (X,π) has the homotopy lifting property, or that π has the homotopy lifting property with respect to X, if:
- for any homotopy , and
- for any map lifting (i.e. so that ),
there exists a homotopy lifting f (i.e. so that ) with .
The following diagram visualizes 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. Also compare this to the visualization of the homotopy extension property.
If the map π satisfies the homotopy lifting property with respect to all spaces X, then π is called a fibration, or one sometimes simply says that π has the homotopy lifting property.
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 , for simplicity we denote . Given additionally a map , one says that (X,Y,π) has the homotopy lifting extension property if:
- for any homotopy , and
- for any lifting of g = f | T,
there exists a homotopy which extends (i.e. such that ).
The homotopy lifting property of (X,π) is obtained by taking Y = ø, so that T above is simply .
The homotopy extension property of (X,Y) is obtained by taking π to be a constant map, so that π is irrelevant in that every map to E is trivially the lift of a constant map to the image point of π).
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