In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of 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.
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particular in homotopy theory within algebraic topology, the homotopyliftingproperty (also known as an instance of the right liftingproperty or the covering...
on a larger space. The homotopy extension property of cofibrations is dual to the homotopyliftingproperty that is used to define fibrations. Let X {\displaystyle...
the homotopyliftingproperty for every CW-complex.: 379 A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopylifting property...
dual to that of a fibration, which is required to satisfy the homotopyliftingproperty with respect to all spaces; this is one instance of the broader...
possessing the homotopyliftingproperty with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups...
the liftingproperty is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of...
since all coverings have the homotopyliftingproperty, covering spaces are an important tool in the calculation of homotopy groups. A standard example...
has the homotopyliftingproperty and it follows that Mp and the fiber F = p − 1 ( b 0 ) {\displaystyle F=p^{-1}(b_{0})} have the same homotopy type. It...
to p−1(Δ). Because fibrations satisfy the homotopyliftingproperty, and Δ is contractible; p−1(Δ) is homotopy equivalent to F. So this partially defined...
an approximate fibration is a sort of fibration such that the homotopyliftingproperty holds only approximately. The notion was introduced by Coram and...
map is a projection. Since p is a fibration, by the homotopyliftingproperty, h lifts to a homotopy g : p − 1 ( b ) × I → E {\displaystyle g:p^{-1}(b)\times...
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other....
and use the homotopyliftingproperty to "follow" paths on the base space X (we assume it path-connected for simplicity) as they are lifted up into the...
has the homotopy liftingproperty or homotopy covering property (see Steenrod (1951, 11.7) for details). This is the defining property of a fibration....
third. Lifting: acyclic cofibrations have the left liftingproperty with respect to fibrations, and cofibrations have the left liftingproperty with respect...
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial...
{\displaystyle p\circ h_{t}=g_{t}} . (The above property is called the homotopyliftingproperty.) A covering map is a basic example of a fibration. fibration...
definition is very similar to that of fibrations in topology (see also homotopyliftingproperty), whence the name "fibration". Using the correspondence between...
Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It...
characterized by having a right liftingproperty with respect to any trivial cofibration in the category. This property makes fibrant objects the "correct"...
mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right liftingproperty with respect to the horn inclusions...
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