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 has the homotopy lifting extension property if:
- for any homotopy, and
- for any lifting of ,
there exists a homotopy which 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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