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.
Read more about Homotopy Lifting Property: Formal Definition, Generalization: The Homotopy Lifting Extension Property
Famous quotes containing the words lifting and/or property:
“Thales claimed that everything was water. He also put wine into water to sterilize it. Did he really believe he was putting water into water to sterilize it? Parmenides, like most Greeks, knew that wine was not water. But while lifting a glass of wine to his lips, he denied that motion was possible. Did he really believe that the glass was not moving when he lifted it?”
—Avrum Stroll (b. 1921)
“To throw obstacles in the way of a complete education is like putting out the eyes; to deny the rights of property is like cutting off the hands. To refuse political equality is like robbing the ostracized of all self-respect, of credit in the market place, of recompense in the world of work, of a voice in choosing those who make and administer the law, a choice in the jury before whom they are tried, and in the judge who decides their punishment.”
—Elizabeth Cady Stanton (1815–1902)