# Homotopy lifting property

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.

## Formal definition[edit]

Assume from now on all maps are continuous functions from one topological space to another. Given a map , and a space , one says that has the homotopy lifting property,^{[1]}^{[2]} or that has the homotopy lifting property with respect to , if:

- for any homotopy , and
- for any map lifting (i.e., so that ),

there exists a homotopy lifting (i.e., so that ) which also satisfies .

The following diagram depicts this situation.

The outer square (without the dotted arrow) commutes if and only if the hypotheses of the lifting property are true. A lifting corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the homotopy extension property; this duality is loosely referred to as Eckmann–Hilton duality.

If the map satisfies the homotopy lifting property with respect to *all* spaces *X*, then is called a fibration, or one sometimes simply says that * has the homotopy lifting property*.

A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes .

## Generalization: homotopy lifting extension property[edit]

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces , for simplicity we denote . Given additionally a map , one says that * has the homotopy lifting extension property* if:

- For any homotopy , and
- For any lifting of , there exists a homotopy which covers (i.e., such that ) and extends (i.e., such that ).

The homotopy lifting property of is obtained by taking , so that above is simply .

The homotopy extension property of is obtained by taking to be a constant map, so that is irrelevant in that every map to *E* is trivially the lift of a constant map to the image point of .

## See also[edit]

## Notes[edit]

**^**Hu, Sze-Tsen (1959).*Homotopy Theory*. page 24**^**Husemoller, Dale (1994).*Fibre Bundles*. page 7

## References[edit]

- Steenrod, Norman (1951).
*The Topology of Fibre Bundles*. Princeton: Princeton University Press. ISBN 0-691-00548-6. - Hu, Sze-Tsen (1959).
*Homotopy Theory*(Third Printing, 1965 ed.). New York: Academic Press Inc. ISBN 0-12-358450-7. - Husemoller, Dale (1994).
*Fibre Bundles*(Third ed.). New York: Springer. ISBN 978-0-387-94087-8. - Hatcher, Allen (2002),
*Algebraic Topology*, Cambridge: Cambridge University Press, ISBN 0-521-79540-0. - Jean-Pierre Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in
*The Architecture of Modern Mathematics*, J. Ferreiros & J.J. Gray, editors, Oxford University Press ISBN 978-0-19-856793-6

## External links[edit]

- A.V. Chernavskii (2001) [1994], "Covering homotopy",
*Encyclopedia of Mathematics*, EMS Press - homotopy lifting property at the
*n*Lab