lifting of maps
LiftingOfMaps
2003-02-02 17:21:51
2004-01-24 12:52:13
Definition
lifting
lift
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Let $p\co E\to B$ and $f\co X\to B$ be (continuous) maps. Then a
\emph{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
$$\xymatrix{
&{E}\ar[d]^{p}\\
{X}\ar@{-->}[ur]^{\tilde f}\ar[r]_{f} & {B} }
$$
which expresses this definition. $\tilde f$ is also said to \emph{lift}
$f$ or to be \emph{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.