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
Famous quotes containing the words formal and/or definition:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (18691950)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on life (based on wording in the First Edition, 1935)