Editing Homotopy lifting property

{{Short description|Homotopy theory in algebraic topology}}
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 space]]s. 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==
Assume all maps are continuous functions between topological spaces. Given a map <math>\pi\colon E \to B</math>, and a space <math>Y\,</math>, one says that <math>(Y, \pi)</math> has the homotopy lifting property,<ref>{{cite book | last = Hu | first = Sze-Tsen |authorlink=Sze-Tsen Hu|title = Homotopy Theory | url = https://archive.org/details/homotopytheory0000hust | url-access = registration | year=1959}} page 24</ref><ref>{{cite book | last = Husemoller | first = Dale | authorlink=Dale Husemoller|title = Fibre Bundles| year=1994 }} page 7</ref> or that <math>\pi\,</math> has the homotopy lifting property with respect to <math>Y</math>, if:

*for any [[homotopy]] <math>f_\bullet \colon Y \times I \to B</math>, and 
*for any map <math>\tilde{f}_0 \colon Y \to E</math> lifting <math>f_0 = f_\bullet|_{Y\times\{0\}}</math> (i.e., so that <math>f_\bullet\circ \iota_0 = f_0 = \pi\circ\tilde{f}_0</math>),

there exists a homotopy <math>\tilde{f}_\bullet \colon Y \times I \to E</math> lifting <math>f_\bullet</math> (i.e., so that <math>f_\bullet = \pi\circ\tilde{f}_\bullet</math>) which also satisfies <math>\tilde{f}_0 = \left.\tilde{f}\right|_{Y\times\{0\}}</math>.

The following diagram depicts this situation:

[[File:Homotopy lifting property bulleted.svg|175 px|center]]

The outer square (without the dotted arrow) commutes if and only if the hypotheses of the [[lifting property]] are true. A lifting <math>\tilde{f}_\bullet</math> 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 <math>\pi</math> satisfies the homotopy lifting property with respect to ''all'' spaces <math>Y</math>, then <math>\pi</math> is called a [[fibration]], or one sometimes simply says that ''<math>\pi</math> has the homotopy lifting property''.

A weaker notion of fibration is [[Fibration#Serre fibrations|Serre fibration]], for which homotopy lifting is only required for all [[CW complex|CW complexes]] <math>Y</math>.

==Generalization: homotopy lifting extension property==
There is a common generalization of the homotopy lifting property and the [[homotopy extension property]].  Given a pair of spaces <math>X \supseteq Y</math>, for simplicity we denote <math>T \mathrel{:=} (X \times \{0\}) \cup (Y \times [0, 1]) \subseteq X\times [0, 1]</math>.  Given additionally a map <math>\pi \colon E \to B</math>, one says that ''<math>(X, Y, \pi)</math> has the '''homotopy lifting extension property''''' if:
* For any [[homotopy]] <math>f \colon X \times [0, 1] \to B</math>, and
* For any lifting <math>\tilde g \colon T \to E</math> of <math>g = f|_T</math>, there exists a homotopy <math>\tilde f \colon X \times [0, 1] \to E</math> which covers <math>f</math> (i.e., such that <math>\pi\tilde f = f</math>) and extends <math>\tilde g</math> (i.e., such that <math>\left.\tilde f\right|_T = \tilde g</math>).

The homotopy lifting property of <math>(X, \pi)</math> is obtained by taking <math>Y = \emptyset</math>, so that <math>T</math> above is simply <math>X \times \{0\}</math>.

The homotopy extension property of <math>(X, Y)</math> is obtained by taking <math>\pi</math> to be a constant map, so that <math>\pi</math> is irrelevant in that every map to ''E'' is trivially the lift of a constant map to the image point of <math>\pi</math>.

==See also==
* [[Covering space]]
* [[Fibration]]

==Notes==
{{Reflist}}

==References==
* {{cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | url = https://archive.org/details/topologyoffibreb0000stee | url-access = registration | publisher = Princeton University Press | location = Princeton | year = 1951 | isbn = 0-691-00548-6}}
* {{cite book | last = Hu | first = Sze-Tsen | title = Homotopy Theory | url = https://archive.org/details/homotopytheory0000hust | url-access = registration | publisher = Academic Press Inc. | edition = Third Printing, 1965 |location = New York | year=1959 | isbn= 0-12-358450-7}}
* {{cite book | last = Husemoller | first = Dale | title = Fibre Bundles | publisher = Springer | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}}
* {{citation| last=Hatcher |first= Allen |title=Algebraic Topology |url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |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 & [[Jeremy Gray|J.J. Gray]], editors, [[Oxford University Press]] {{ISBN|978-0-19-856793-6}}

==External links==
* {{springer|author=A.V. Chernavskii|title=Covering homotopy|id=C/c026940}}
* {{nlab|id=homotopy%20lifting%20property|title=homotopy lifting property}}

[[Category:Homotopy theory]]
[[Category:Algebraic topology]]