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lifting of maps
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(Definition)
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Let $p\co E\to B$ and $f\co X\to B$ be (continuous) maps. Then a lifting of $f$ to $E$ is a (continuous) map $\tilde f\co X\to E$ such that $p\circ \tilde f=f$ . The terminology is justified by the following commutative diagram
which expresses this definition. $\tilde f$ is also said to lift $f$ or to be over $f$ .
This notion is especially useful if $p\co E\to B$ is a fiber bundle. In particular lifting of paths is instrumental in the investigation of covering spaces.
This terminology is used in more general contexts: $X$ , $E$ and $B$ could be objects (and $p$ , $f$ and $\tilde f$ be morphisms) in any category.
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"lifting of maps" is owned by Dr_Absentius.
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lifting, lift |
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Cross-references: category, morphisms, objects, covering spaces, paths, fiber bundle, commutative diagram, maps, continuous
There are 18 references to this entry.
This is version 3 of lifting of maps, born on 2003-02-02, modified 2004-01-24.
Object id is 3961, canonical name is LiftingOfMaps.
Accessed 7545 times total.
Classification:
| AMS MSC: | 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ &{E}\ar[d]^{p}\ {X}\ar@{-->}[ur]^{\tilde f}\ar[r]_{f} & {B} } $](http://images.planetmath.org:8080/cache/objects/3961/js/img1.png "$\displaystyle \xymatrix{ &{E}\ar[d]^{p}\ {X}\ar@{-->}[ur]^{\tilde f}\ar[r]_{f} & {B} } $")