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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 because 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. ==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'',〔 page 24〕〔 page 7〕 or that has the ''homotopy lifting property'' with respect to , if: *for any homotopy , and *for any map lifting . The following diagram depicts this situation. :File:Homotopy lifting property.png 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''. Note that 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「homotopy lifting property」の詳細全文を読む スポンサード リンク
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