Editing Homotopy lifting property (section)

==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>.