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 .
Read more about this topic: Homotopy Lifting Property
Famous quotes containing the words lifting, extension and/or property:
“The little lifting helplessness, the queer
Whimper-whine; whose unridiculous
Lost softness softly makes a trap for us.
And makes a curse.”
—Gwendolyn Brooks (b. 1917)
“Slavery is founded on the selfishness of man’s nature—opposition to it on his love of justice. These principles are in eternal antagonism; and when brought into collision so fiercely as slavery extension brings them, shocks and throes and convulsions must ceaselessly follow.”
—Abraham Lincoln (1809–1865)
“General education is the best preventive of the evils now most dreaded. In the civilized countries of the world, the question is how to distribute most generally and equally the property of the world. As a rule, where education is most general the distribution of property is most general.... As knowledge spreads, wealth spreads. To diffuse knowledge is to diffuse wealth. To give all an equal chance to acquire knowledge is the best and surest way to give all an equal chance to acquire property.”
—Rutherford Birchard Hayes (1822–1893)