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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.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
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}}} .
Then, by the homotopy lifting property, we can lift the homotopy () to w such that w restricts to . In particular, we have g 1 ∼ g 1 ′ {\displaystyle g_{1}\sim g_{1}'} , establishing the claim. It is clear from the construction that the map is a homomorphism: if γ ( 1 ) = β ( 0 ) {\displaystyle \gamma (1)=\beta (0)} ,
Then any lift of this path to SP(X) is of the form x t α t with α t ∈ A for every t. But this means that its endpoint aα 1 is a multiple of a, hence different from the basepoint, so the Homotopy lifting property fails to be fulfilled. Verifying the fourth axiom can be done quite elementary, in contrast to the previous one.
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.
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