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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.
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category.It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms.
The homotopy lifting property is used to characterize fibrations. Another useful property involving homotopy is the homotopy extension property , which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself.
Then a map : is said to satisfy the right lifting property or the RLP if satisfies the above lifting property for each in . Similarly, a map i : A → X {\displaystyle i:A\to X} is said to satisfy the left lifting property or the LLP if it satisfies the lifting property for each p {\displaystyle p} in c {\displaystyle {\mathfrak {c}}} .
In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups.
Because fibrations satisfy the homotopy lifting property, and Δ is contractible; p −1 (Δ) is homotopy equivalent to F. So this partially defined section assigns an element of π n (F) to every (n + 1)-simplex. This is precisely the data of a π n (F)-valued simplicial cochain of degree n + 1 on B, i.e. an element of C n + 1 (B; π n (F)).
In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations .
In topology, a fibration is a mapping : that has certain homotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or homotopy covering property (see Steenrod (1951, 11.7) for details). This is the defining property of a fibration.