Formal Definition
Assume from now on all mappings are continuous functions from a 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 ) 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 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.
N.B. 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 a CW complex is required.
Read more about this topic: Homotopy Lifting Property
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